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Correlation functions provide a statistical description of the dynamics of molecular variables; however, it remains unclear how they are related to experimental observables. You have probably sensed this from the perspective that correlation functions are complex, and how can observables be complex? Also, correlation functions describe equilibrium dynamics, but from a realistic point of view, exerting external forces should move the system away from equilibrium. What happens as a result? These questions fall into the realm of nonequilibrium statistical mechanics, an area of active research for which formal theories are limited and approximation methods are the primary tool. Linear response theory is the primary approximation method, which describes the evolution away or toward equilibrium under perturbative conditions.
• 11.1: Classical Linear Response Theory
We will use linear response theory as a way of describing a real experimental observable and deal with a nonequilibrium system. We will show that when the changes are small away from equilibrium, the equilibrium fluctuations dictate the nonequilibrium response! Thus knowledge of equilibrium dynamics is useful in predicting the outcome of nonequilibrium processes.
• 11.2: Quantum Linear Response Functions
To develop a quantum description of the linear response function, we start by recognizing that the response of a system to an applied external agent is a problem we can solve in the interaction picture.
• 11.3: The Response Function and Energy Absorption
Let’s investigate the relationship between the linear response function and the absorption of energy from the external agent—in this case an electromagnetic field.
• 11.4: Relaxation of a Prepared State
The impulse response function R(t) describes the behavior of a system initially at equilibrium that is driven by an external field. Alternatively, we may need to describe the relaxation of a prepared state, in which we follow the return to equilibrium of a system initially held in a nonequilibrium state. This behavior is described by step response function.
11: Linear Response Theory
We will use linear response theory as a way of describing a real experimental observable. Specifically this will tell us how an equilibrium system changes in response to an applied potential. The quantity that will describe this is a response function, a real observable quantity. We will go on to show how it is related to correlation functions. Embedded in this discussion is a particularly important observation. We will now deal with a nonequilibrium system, but we will show that when the changes are small away from equilibrium, the equilibrium fluctuations dictate the nonequilibrium response! Thus knowledge of equilibrium dynamics is useful in predicting the outcome of nonequilibrium processes.
So, the question is “How does the system respond if you drive it away from equilibrium?” We will examine the case where an equilibrium system, described by a Hamiltonian $H_0$ interacts weakly with an external agent, $V(t)$. The system is moved away from equilibrium by the external agent, and the system absorbs energy from the external agent. How do we describe the time-dependent properties of the system? We first take the external agent to interact with the system through an internal variable $A$. So the Hamiltonian for this problem is given by
$H = H _ {0} - f (t) A \label{10.1}$
Here $f(t)$ is the time-dependent action of the external agent, and the deviation from equilibrium is linear in the internal variable. We describe the behavior of an ensemble initially at thermal equilibrium by assuming that each member of the ensemble is subject to the same interaction with the external agent, and then ensemble averaging. Initially, the system is described by $H_0$. It is at equilibrium and the internal variable is characterized by an equilibrium ensemble average $\langle A \rangle$. The external agent is then applied at time t0, and the system is moved away from equilibrium, and is characterized through a nonequilibrium ensemble average, $\overline {A}$. $\langle A \rangle \neq \overline {A (t)}$ as a result of the interaction.
For a weak interaction with the external agent, we can describe $\overline {A (t)}$ by performing an expansion in powers of $f(t)$
\begin{align} \overline {A (t)} &= \left( \text {terms} f^{( 0 )} \right) + \left( \text {terms} f^{( 1 )} \right) + \ldots \label{10.2} \[4pt] &= \langle A \rangle + \int d t _ {0} R \left( t , t _ {0} \right) f \left( t _ {0} \right) + \ldots \label{10.3} \end{align}
In this expression the agent is applied at 0 t , and we observe the system att. The leading term in this expansion is independent of f, and is therefore equal to A . The next term in Equation \ref{10.3} describes the deviation from the equilibrium behavior in terms of a linear dependence on the external agent. $R \left( t , t _ {0} \right)$ is the linear response function, the quantity that contains the microscopic information on the system and how it responds to the applied agent. The integration in the last term of Equation \ref{10.3} indicates that the nonequilibrium behavior depends on the full history of the application of the agent $f \left( t _ {0} \right)$ and the response of the system to it. We are seeking a quantum mechanical description of $R$.
Properties of the Response Function
1. Causal: Causality refers to the common sense observation that the system cannot respond before the force has been applied. Therefore $R \left( t , t _ {0} \right) = 0$ for $t < t$, and the time-dependent change in $A$ is
$\overline {\delta A (t)} = \overline {A (t)} - \langle A \rangle = \int _ {- \infty}^{t} d t _ {0} R \left( t , t _ {0} \right) f \left( t _ {0} \right) \label{10.4}$
The lower integration limit is set to $- \infty$ to reflect that the system is initially at equilibrium, and the upper limit is the time of observation. We can also make the statement of causality explicit by writing the linear response function with a step response: $\Theta \left( t - t _ {0} \right) R \left( t , t _ {0} \right)$, where
$\Theta \left( t - t _ {0} \right) \equiv \left\{\begin{array} {l l} {0} & {\left( t < t _ {0} \right)} \ {1} & {\left( t \geq t _ {0} \right)} \end{array} \right. \label{10.5}$
2. Stationary: Similar to our discussion of correlation functions, the time-dependence of the system only depends on the time interval between application of the potential and observation. Therefore we write
$R \left( t , t _ {0} \right) = R \left( t - t _ {0} \right)$
and
$\delta \overline {A (t)} = \int _ {- \infty}^{t} d t _ {0} R \left( t - t _ {0} \right) f \left( t _ {0} \right) \label{10.6}$
This expression says that the observed response of the system to the agent is a convolution of the material response with the time-development of the applied force. Rather than the absolute time points, we can define a time-interval $\tau = t - t _ {0}$, so that we can write
$\delta \overline {A (t)} = \int _ {0}^{\infty} d \tau R ( \tau ) f ( t - \tau ) \label{10.7}$
3. Impulse response: Note that for a delta function perturbation:
$f (t) = \lambda \delta \left( t - t _ {0} \right) \label{10.8}$
We obtain
$\overline {\delta A (t)} = \lambda R \left( t - t _ {0} \right) \label{10.9}$
Thus, $R$ describes how the system behaves when an abrupt perturbation is applied and is often referred to as the impulse response function. An impulse response kicks the system away from the equilibrium established under H0, and therefore the shape of a response function will always rise from zero and ultimately return to zero. In other words, it will be a function that can be expanded in sines. Thus the response to an arbitrary f(t) can be described through a Fourier analysis, suggesting that a spectral representation of the response function would be useful.
The Susceptibility
The observed temporal behavior of the nonequilibrium system can also be cast in the frequency domain as a spectral response function, or susceptibility. We start with Equation \ref{10.7} and Fourier transform both sides:
\left.\begin{aligned} \overline {\delta A ( \omega )} & \equiv \int _ {- \infty}^{+ \infty} d t \delta \overline {A (t)} e^{i \omega t} \ & = \int _ {- \infty}^{+ \infty} d t \left[ \int _ {0}^{\infty} d \tau R ( \tau ) f ( t - \tau ) \right] e^{i \omega t} \end{aligned} \right. \label{10.10}
Now we insert $e^{- i \omega \tau} e^{+ i \omega \tau} = 1$ and collect terms to give
\begin{align} \delta \overline {A ( \omega )} &= \int _ {- \infty}^{+ \infty} d t \int _ {0}^{\infty} d \tau R ( \tau ) f ( t - \tau ) e^{i \omega ( t - \tau )} e^{i \omega \tau} \label{10.11} \[4pt] &= \int _ {- \infty}^{+ \infty} d t^{\prime} e^{i \omega r^{\prime}} f \left( t^{\prime} \right) \int _ {0}^{\infty} d \tau R ( \tau ) e^{i \omega \tau} \label{10.12} \end{align}
or
$\delta \overline {A ( \omega )} = \tilde {f} ( \omega ) \chi ( \omega ) \label{10.13}$
In Equation \ref{10.12} we switched variables, setting $t^{\prime} = t - \tau$. The first term $\tilde {f} ( \omega )$ is a complex frequency domain representation of the driving force, obtained from the Fourier transform of $f \left( t^{\prime} \right)$. The second term $\chi ( \omega )$ is the susceptibility which is defined as the Fourier–Laplace transform (i.e., single-sided Fourier transform) of the impulse response function. It is a frequency domain representation of the linear response function. Switching between time and frequency domains shows that a convolution of the force and response in time leads to the product of the force and response in frequency. This is a manifestation of the convolution theorem:
$A (t) \otimes B (t) \equiv \int _ {- \infty}^{\infty} d \tau A ( t - \tau ) B ( \tau ) = \int _ {- \infty}^{\infty} d \tau A ( \tau ) B ( t - \tau ) = \mathcal {H}^{- 1} [ \tilde {A} ( \omega ) \tilde {B} ( \omega ) ] \label{10.14}$
Here $\otimes$ refers to convolution, $\tilde {A} ( \omega ) = \mathcal {F} [ A (t) ]]$, $\mathcal {F}$ is a Fourier transform, and $\mathcal {F}^{- 1} [ \cdots ]$ is an inverse Fourier transform.
Note that $R(\tau)$ is a real function, since the response of a system is an observable. The susceptibility $\chi ( \omega )$ is complex:
$\chi ( \omega ) = \chi^{\prime} ( \omega ) + i \chi^{\prime \prime} ( \omega ) \label{10.15}$
Since
$\chi ( \omega ) = \int _ {0}^{\infty} d \tau R ( \tau ) e^{i \omega \tau} \label{10.16}$
However, the real and imaginary contributions are not independent. We have
$\chi^{\prime} = \int _ {0}^{\infty} d \tau R ( \tau ) \cos \omega \tau \label{10.17}$
and
$\chi^{\prime \prime} = \int _ {0}^{\infty} d \tau R ( \tau ) \sin \omega \tau \label{10.18}$
$\chi^{\prime}$ and $\chi^{\prime \prime}$ are even and odd functions of frequency:
$\chi^{\prime} ( \omega ) = \chi^{\prime} ( - \omega ) \label{10.19}$
$\chi^{\prime \prime} ( \omega ) = - \chi^{\prime \prime} ( - \omega ) \label{10.20}$
so that $\chi ( - \omega ) = \chi^{*} ( \omega ) \label{10.21}$
Notice also that Equation \ref{10.21} allows us to write
$\chi^{\prime} ( \omega ) = \frac {1} {2} [ \chi ( \omega ) + \chi ( - \omega ) ] \label{10.22}$
$\chi^{\prime \prime} ( \omega ) = \frac {1} {2 i} [ \chi ( \omega ) - \chi ( - \omega ) ] \label{10.23}$
Kramers–Krönig relations
Since they are cosine and sine transforms of the same function, $\chi^{\prime} ( \omega )$ is not independent of $\chi^{\prime \prime} ( \omega )$. The two are related by the Kramers–Krönig relationships:
\begin{align} \chi^{\prime} ( \omega ) &= \frac {1} {\pi} P \int _ {- \infty}^{+ \infty} \frac {\chi^{\prime \prime} \left( \omega^{\prime} \right)} {\omega^{\prime} - \omega} d \omega^{\prime} \label{10.24} \[4pt] \chi^{\prime \prime} ( \omega ) &= - \frac {1} {\pi} P \int _ {- \infty}^{+ \infty} \frac {\chi^{\prime} \left( \omega^{\prime} \right)} {\omega^{\prime} - \omega} d \omega^{\prime} \label{10.25} \end{align}
These are obtained by substituting the inverse sine transform of Equation \ref{10.18} into Equation \ref{10.17}
\begin{align} \chi^{\prime} ( \omega ) &= \frac {1} {\pi} \int _ {0}^{\infty} d t \cos \omega t \int _ {- \infty}^{+ \infty} \chi^{\prime \prime} \left( \omega^{\prime} \right) \sin \omega^{\prime} t d \omega^{\prime} \[4pt] &= \frac {1} {\pi} \lim _ {L \rightarrow \infty} \int _ {- \infty}^{+ \infty} d \omega^{\prime} \chi^{\prime \prime} \left( \omega^{\prime} \right) \int _ {0}^{L} \cos \omega t \sin \omega^{\prime} t \,d t \end{align}
Using $\cos a x \sin b x=\frac{1}{2} \sin (a+b) x+\frac{1}{2} \sin (b-a) x$ this can be written as
$\chi^{\prime} ( \omega ) = \frac {1} {\pi} \lim _ {L \rightarrow \infty} \mathrm {P} \int _ {- \infty}^{+ \infty} d \omega^{\prime} \chi^{\prime \prime} ( \omega ) \frac {1} {2} \left[ \frac {- \cos \left( \omega^{\prime} + \omega \right) L + 1} {\omega^{\prime} + \omega} - \frac {\cos \left( \omega^{\prime} - \omega \right) L + 1} {\omega^{\prime} - \omega} \right] \label{10.27}$
If we choose to evaluate the limit $L \rightarrow \infty$, the cosine terms are hard to deal with, but we expect they will vanish since they oscillate rapidly. This is equivalent to averaging over a monochromatic field. Alternatively, we can average over a single cycle: $L=2 \pi /\left(\omega^{\prime}-\omega\right)$ to obtain eq. (10.24). The other relation can be derived in a similar way. Note that the Kramers– Krönig relationships are a consequence of causality, which dictate the lower limit of $T_{tinitial}=0$ on the first integral evaluated above.
Example $1$: Driven Harmonic Oscillator
One can classically model the absorption of light through a resonant interaction of the electromagnetic field with an oscillating dipole, using Newton’s equations for a forced damped harmonic oscillator:
$\ddot {x} - \gamma \dot {x} + \omega _ {0}^{2} x = F (t) / m \label{10.28}$
Here the $x$ is the coordinate being driven, $\gamma$ is the damping constant, and $\omega_{0}=\sqrt{k / m}$ is the natural frequency of the oscillator. We originally solved this problem is to take the driving force to have the form of a monochromatic oscillating source
$F (t) = F _ {0} \cos \omega t \label{10.29}$
Then, Equation \ref{10.28} has the solution
$x (t) = \frac {F _ {0}} {m} \left( \left( \omega^{2} - \omega _ {0}^{2} \right)^{2} + \gamma^{2} \omega^{2} \right)^{- 1 / 2} \sin ( \omega t + \delta ) \label{10.30}$
with
$\tan \delta = \omega _ {0}^{2} - \omega^{2} / \gamma \omega \label{10.31}$
This shows that the driven oscillator has an oscillation period that is dictated by the driving frequency $\omega$, and whose amplitude and phase shift relative to the driving field is dictated by its detuning from resonance. If we cycle-average to obtain the average absorbed power from the field, the absorption spectrum is
\begin{align*} P _ {a v g} ( \omega ) &= \langle F (t) \cdot \dot {x} (t) \rangle \label{10.32} \[4pt] &= = \frac {\gamma \omega^{2} F _ {0}^{2}} {2 m} \left[ \left( \omega _ {0}^{2} - \omega^{2} \right)^{2} + \gamma^{2} \omega^{2} \right]^{- 1 / 2} \end{align*}
To determine the response function for the damped harmonic oscillator, we seek a solution to Equation \ref{10.28} using an impulsive driving force
$F (t) = F _ {0} \delta \left( t - t _ {0} \right) \nonumber$
The linear response of this oscillator to an arbitrary force is
$x (t) = \int _ {0}^{\infty} d \tau R ( \tau ) F ( t - \tau ) \label{10.33}$
so that time-dependence with an impulsive driving force is directly proportional to the response function, $x(t)=F_{0} R(t)$. For this case, we obtain
$R ( \tau ) = \frac {1} {m \Omega} \exp \left( - \frac {\gamma} {2} \tau \right) \sin \Omega \tau \label{10.34}$
The reduced frequency is defined as
$\Omega = \sqrt {\omega _ {0}^{2} - \gamma^{2} / 4} \label{10.35}$
From this, we evaluate eq. (10.16) and obtain the susceptibility
$\chi ( \omega ) = \frac {1} {m \left( \omega _ {0}^{2} - \omega^{2} - i \gamma \omega \right)} \label{0.36}$
As we will see shortly, the absorption of light by the oscillator is proportional to the imaginary part of the susceptibility
$\chi^{\prime \prime} ( \omega ) = \frac {\gamma \omega} {m \left[ \left( \omega _ {0}^{2} - \omega^{2} \right)^{2} + \gamma^{2} \omega^{2} \right]} \label{10.37}$
The real part is
$\chi^{\prime} ( \omega ) = \frac {\omega _ {0}^{2} - \omega^{2}} {m \left[ \left( \omega _ {0}^{2} - \omega^{2} \right)^{2} + \gamma^{2} \omega^{2} \right]} \label{10.38}$
For the case of weak damping $\gamma < < \omega _ {0}$ commonly encountered in molecular spectroscopy, Equation \ref{10.36} is written as a Lorentzian lineshape by using the near-resonance approximation
$\omega^{2} - \omega _ {0}^{2} = \left( \omega + \omega _ {0} \right) \left( \omega - \omega _ {0} \right) \approx 2 \omega \left( \omega - \omega _ {0} \right) \label{10.39}$
$\chi ( \omega ) \approx \frac {1} {2 m \omega _ {0}} \frac {1} {\omega - \omega _ {0} + i \gamma / 2} \label{10.40}$
Then the imaginary part of the susceptibility shows asymmetric lineshape with a line width of $\gamma$ full width at half maximum.
$\chi^{\prime \prime} ( \omega ) \approx \frac {1} {2 m \omega _ {0}} \frac {\gamma} {\left( \omega - \omega _ {0} \right)^{2} + \gamma^{2} / 4} \label{10.41}$
$\chi^{\prime} ( \omega ) \approx \frac {1} {m \omega _ {0}} \frac {\left( \omega - \omega _ {0} \right)} {\left( \omega - \omega _ {0} \right)^{2} + \gamma^{2} / 4} \label{10.42}$
Nonlinear Response Functions
If the system does not respond in a manner linearly proportional to the applied potential but still perturbative, we can include nonlinear terms, i.e. higher expansion orders of $\overline {A (t)}$ in Equation \ref{10.3}.
Let’s look at second order:
$\delta \overline {A (t)}^{( 2 )} = \int d t _ {1} \int d t _ {2} R^{( 2 )} \left( t ; t _ {1} , t _ {2} \right) f _ {1} \left( t _ {1} \right) f _ {2} \left( t _ {2} \right) \label{10.43}$
Again we are integrating over the entire history of the application of two forces $f_1$ and $f_2$, including any quadratic dependence on $f$. In this case, we will enforce causality through a time ordering that requires
1. that all forces must be applied before a response is observed and
2. that the application of $f_2$ must follow $f_1$. That is $t \geq t _ {2} \geq t _ {1}$ or
$R^{( 2 )} \left( t ; t _ {1} , t _ {2} \right) \Rightarrow R^{( 2 )} \cdot \Theta \left( t - t _ {2} \right) \cdot \Theta \left( t _ {2} - t _ {1} \right) \label{10.44}$
which leads to
$\delta \overline {A (t)}^{( 2 )} = \int _ {- \infty}^{t} d t _ {2} \int _ {- \infty}^{t _ {2}} d t _ {1} R^{( 2 )} \left( t ; t _ {1} , t _ {2} \right) f _ {1} \left( t _ {1} \right) f _ {2} \left( t _ {2} \right) \label{10.45}$
Now we will call the system stationary so that we are only concerned with the time intervals between consecutive interaction times. If we define the intervals between adjacent interactions
$\left. \begin{array} {l} {\tau _ {1} = t _ {2} - t _ {1}} \ {\tau _ {2} = t - t _ {2}} \end{array} \right. \label{10.46}$
Then we have
$\delta \overline {A (t)}^{( 2 )} = \int _ {0}^{\infty} d \tau _ {1} \int _ {0}^{\infty} d \tau _ {2} R^{( 2 )} \left( \tau _ {1} , \tau _ {2} \right) f _ {1} \left( t - \tau _ {1} - \tau _ {2} \right) f _ {2} \left( t - \tau _ {2} \right) \label{10.47}$
Readings
1. Berne, B. J., Time-Dependent Propeties of Condensed Media. In Physical Chemistry: An Advanced Treatise, Vol. VIIIB, Henderson, D., Ed. Academic Press: New York, 1971.
2. Berne, B. J.; Pecora, R., Dynamic Light Scattering. R. E. Krieger Publishing Co.: Malabar, FL, 1990.
3. Chandler, D., Introduction to Modern Statistical Mechanics. Oxford University Press: New York, 1987.
4. Mazenko, G., Nonequilibrium Statistical Mechanics. Wiley-VCH: Weinheim, 2006.
5. Slichter, C. P., Principles of Magnetic Resonance, with Examples from Solid State Physics. Harper & Row: New York, 1963.
6. Wang, C. H., Spectroscopy of Condensed Media: Dynamics of Molecular Interactions. Academic Press: Orlando, 1985.
7. Zwanzig, R., Nonequilibrium Statistical Mechanics. Oxford University Press: New York, 2001.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/11%3A_Linear_Response_Theory/11.01%3A_Classical_Linear_Response_Theory.txt
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To develop a quantum description of the linear response function, we start by recognizing that the response of a system to an applied external agent is a problem we can solve in the interaction picture. Our time-dependent Hamiltonian is
\begin{align} H (t) &= H _ {0} - f (t) \hat {A} \[4pt] &= H _ {0} + V (t) \label{10.48} \end{align}
$H_o$ is the material Hamiltonian for the equilibrium system. The external agent acts on the equilibrium system through $\hat{A}$, an operator in the system states, with a time-dependence $f(t)$. We take $V(t)$ to be a small change, and treat this problem with perturbation theory in the interaction picture.
We want to describe the nonequilibrium response $\overline {A(t)}$, which we will get by ensemble averaging the expectation value of $\hat{A}$, i.e. $\overline {\langle A (t) \rangle}$. Remember the expectation value for a pure state in the interaction picture is
\begin{align} \langle A (t) \rangle & = \left\langle \psi _ {I} (t) \left| A _ {I} (t) \right| \psi _ {I} (t) \right\rangle \[4pt] & = \left\langle \psi _ {0} \left| U _ {I}^{\dagger} A _ {I} U _ {I} \right| \psi _ {0} \right\rangle \label{10.49} \end{align}
The interaction picture Hamiltonian for Equation \ref{10.48} is
\left.\begin{aligned} V _ {I} (t) & = U _ {0}^{\dagger} (t) V (t) U _ {0} (t) \[4pt] & = - f (t) A _ {I} (t) \end{aligned} \right. \label{10.50}
To calculate an ensemble average of the state of the system after applying the external potential, we recognize that the nonequilibrium state of the system characterized by described by $| \psi _ {I} (t) \rangle$ is in fact related to the initial equilibrium state of the system $| \psi _ o\rangle$ through a time-propagator, as seen in Equation \ref{10.49}. So the nonequilibrium expectation value $\overline {A (t)}$ is in fact obtained by an equilibrium average over the expectation value of $U _ {I}^{\dagger} A _ {I} U _ {I}$:
$\overline {A (t)} = \sum _ {n} p _ {n} \left\langle n \left| U _ {I}^{\dagger} A _ {I} U _ {I} \right| n \right\rangle \label{10.51}$
Again $| n \rangle$ are eigenstates of $H_o$. Working with the first order solution to $U_I(t)$
$U _ {I} \left( t - t _ {0} \right) = 1 + \dfrac {i} {\hbar} \int _ {t _ {0}}^{t} d t^{\prime} f \left( t^{\prime} \right) A _ {I} \left( t^{\prime} \right)\label{10.52}$
we can now calculate the value of the operator $\hat{A}$ at time $t$, integrating over the history of the applied interaction $f(t')$:
\left.\begin{aligned} A (t) & = U _ {I}^{\dagger} A _ {I} U _ {I} \ & = \left\{1 - \frac {i} {\hbar} \int _ {t _ {0}}^{t} d t^{\prime} f \left( t^{\prime} \right) A _ {I} \left( t^{\prime} \right) \right\} A _ {I} (t) \left\{1 + \frac {i} {\hbar} \int _ {t _ {0}}^{t} d t^{\prime} f \left( t^{\prime} \right) A _ {I} \left( t^{\prime} \right) \right\} \end{aligned} \right. \label {10.53}
Here note that $f$ is the time-dependence of the external agent. It does not involve operators in $H_o$ and commutes with $A$. Working toward the linear response function, we just retain the terms linear in
\left.\begin{aligned} A (t) & \cong A _ {I} (t) + \dfrac {i} {\hbar} \int _ {t _ {0}}^{t} d t^{\prime}\, f \left( t^{\prime} \right) \left\{A _ {I} (t) A _ {I} \left( t^{\prime} \right) - A _ {I} \left( t^{\prime} \right) A _ {I} (t) \right\} \[4pt] & = A _ {I} (t) + \dfrac {i} {\hbar} \int _ {t _ {0}}^{t} d t^{\prime} \,f \left( t^{\prime} \right) \left[ A _ {I} (t) , A _ {I} \left( t^{\prime} \right) \right] \end{aligned} \right. \label{10.54}
Since our system is initially at equilibrium, we set $t _ {0} = - \infty$ and switch variables to the time interval $\tau = t - t^{\prime}$ and using
$A _ {I} (t) = U _ {0}^{\dagger} (t) A U _ {0} (t)$
to obtain
$A (t) = A _ {I} (t) + \dfrac {i} {\hbar} \int _ {0}^{\infty} d \tau \,f ( t - \tau ) \left[ A _ {I} ( \tau ) , A _ {I} ( 0 ) \right] \label{10.55}$
We can now calculate the expectation value of $A$ by performing the ensemble-average described in Equation \ref{10.51}. Noting that the force is applied equally to each member of ensemble, we have
$\overline {A (t)} = \langle A \rangle + \dfrac {i} {\hbar} \int _ {0}^{\infty} d \tau f ( t - \tau ) \left\langle \left[ A _ {I} ( \tau ) , A _ {I} ( 0 ) \right] \right\rangle \label{10.56}$
The first term is independent of $f$, and so it comes from an equilibrium ensemble average for the value of $A$.
$\langle A (t) \rangle = \sum _ {n} p _ {n} \left\langle n \left| A _ {I} \right| n \right\rangle = \langle A \rangle \label{10.57}$
The second term is just an equilibrium ensemble average over the commutator in $A_I(t)$:
$\left\langle \left[ A _ {I} ( \tau ) , A _ {I} ( 0 ) \right] \right\rangle = \sum _ {n} p _ {n} \left\langle n \left| \left[ A _ {I} ( \tau ) , A _ {I} ( 0 ) \right] \right| n \right\rangle \label{10.58}$
Comparing Equation \ref{10.56} with the expression for the linear response function, we find that the quantum linear response function is
$\left. \begin{array} {r l} {R ( \tau )} & {= - \dfrac {i} {\hbar} \left\langle \left[ A _ {I} ( \tau ) , A _ {I} ( 0 ) \right] \right\rangle} & {\tau \geq 0} \[4pt] {} & {= 0} & {\tau < 0} \end{array} \right. \label{10.59}$
or as it is sometimes written with the unit step function in order to enforce causality:
$R ( \tau ) = - \dfrac {i} {\hbar} \Theta ( \tau ) \left\langle \left[ A _ {I} ( \tau ) , A _ {I} ( 0 ) \right] \right\rangle \label{10.60}$
The important thing to note is that the time-development of the system with the applied external potential is governed by the dynamics of the equilibrium system. All of the time-dependence in the response function is under $H_o$.
The linear response function is therefore the sum of two correlation functions with the order of the operators interchanged, which is the imaginary part of the correlation function $C''(\tau)$
\left.\begin{aligned} R ( \tau ) & = - \dfrac {i} {\hbar} \Theta ( \tau ) \left\{\left\langle A _ {I} ( \tau ) A _ {I} ( 0 ) \right\rangle - \left\langle A _ {I} ( 0 ) A _ {I} ( \tau ) \right\rangle \right\} \[4pt] & = - \dfrac {i} {\hbar} \Theta ( \tau ) \left( C _ {A A} ( \tau ) - C _ {A A}^{*} ( \tau ) \right) \[4pt] & = \dfrac {2} {\hbar} \Theta ( \tau ) C^{\prime \prime} ( \tau ) \end{aligned} \right.\label{10.61}
As we expect for an observable, the response function is real. If we express the correlation function in the eigenstate description:
$C (t) = \sum _ {n , m} p _ {n} \left| A _ {m n} \right|^{2} e^{- i \omega _ {m n} t} \label{10.62}$
then
$R (t) = \dfrac {2} {\hbar} \Theta (t) \sum _ {n , m} p _ {n} \left| A _ {m n} \right|^{2} \sin \omega _ {m n} t \label{10.63}$
$R(t)$ can always be expanded in sines—an odd function of time. This reflects that fact that the impulse response must have a value of 0 (the deviation from equilibrium) at $t = t_o$, and move away from 0 at the point where the external potential is applied.
Readings
1. Mukamel, S., Principles of Nonlinear Optical Spectroscopy. Oxford University Press: New York, 1995; Ch. 5.
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Let’s investigate the relationship between the linear response function and the absorption of energy from the external agent—in this case an electromagnetic field. We will relate this to the absorption coefficient $\alpha = \dot {E} / I$ which we have described previously. For this case,
$H = H _ {0} - f (t) A = H _ {0} - \mu \cdot E (t) \label{10.64}$
This expression gives the energy of the system, so the rate of energy absorption averaged over the nonequilibrium ensemble is described by:
$\dot {E} = \dfrac {\partial \overline {H}} {\partial t} = - \dfrac {\partial f} {\partial t} \overline {A (t)} \label{10.65}$
We will want to cycle-average this over the oscillating field, so the time-averaged rate of energy absorption is
\begin{align} \dot {E} & = \dfrac {1} {T} \int _ {0}^{T} d t \left[ - \dfrac {\partial f} {\partial t} \overline {A (t)} \right] \[4pt] & = \dfrac {1} {T} \int _ {0}^{T} d t \dfrac {\partial f (t)} {\partial t} \left[ \langle A \rangle + \int _ {0}^{\infty} d \tau R ( \tau ) f ( t - \tau ) \right] \label{10.66} \end{align}
Here the response function is
$R ( \tau ) = - i \langle [ \mu ( \tau ) , \mu ( 0 ) ] \rangle / \hbar.$
For a monochromatic electromagnetic field, we can write (and expand)
\begin{align} f (t) &= E _ {0} \cos \omega t \[4pt] &= \dfrac {1} {2} \left[ E _ {0} e^{- i \omega t} + E _ {0}^{*} e^{i \omega t} \right] \label{10.67} \end{align}
which leads to the following for the second term in Equation \ref{10.66}:
$\dfrac {1} {2} \int _ {0}^{\infty} d \tau R ( \tau ) \left[ E _ {0} e^{- i \omega ( t - \tau )} + E _ {0}^{*} e^{i \omega ( t - \tau )} \right] = \dfrac {1} {2} \left[ E _ {0} e^{- i \omega t} \chi ( \omega ) + E _ {0}^{*} e^{i \omega t} \chi ( - \omega ) \right] \label{10.68}$
By differentiating Equation \ref{10.67}, and using it with Equation \ref{10.68} in Equation \ref{10.66}, we have
$\dot {E} = - \dfrac {1} {T} \langle A \rangle [ f ( T ) - f ( 0 ) ] - \dfrac {1} {4 T} \int _ {0}^{T} d t \left[ - i \omega E _ {0} e^{- i \omega t} + i \omega E _ {0}^{*} e^{i \omega t} \right] \left[ E _ {0} e^{- i \omega t} \chi ( \omega ) + E _ {0}^{*} e^{i \omega t} \chi ( - \omega ) \right] \label{10.69}$
We will now cycle-average this expression, setting $T = 2 \pi / \omega$. The first term vanishes and the cross terms in second integral vanish, because
$\dfrac {1} {T} \int _ {0}^{T} d t e^{- i \omega t} e^{+ i \omega t} = 1$
and
$\int _ {0}^{T} d t e^{- i \omega t} e^{- i \omega t} = 0.$
The rate of energy absorption from the field is
\left.\begin{aligned} \dot {E} & = \dfrac {i} {4} \omega \left| E _ {0} \right|^{2} [ \chi ( - \omega ) - \chi ( \omega ) ] \[4pt] & = \dfrac {\omega} {2} \left| E _ {0} \right|^{2} \chi^{\prime \prime} ( \omega ) \end{aligned} \right. \label{10.70}
So, the absorption of energy by the system is related to the imaginary part of the susceptibility. Now, from the intensity of the incident field,
$I = \dfrac{c \left| E _ {0} \right|^{2}}{8 \pi}$
the absorption coefficient is
$\alpha ( \omega ) = \dfrac {\dot {E}} {I} = \dfrac {4 \pi \omega} {c} \chi^{\prime \prime} ( \omega ) \label{10.71}$
Readings
1. McQuarrie, D. A., Statistical Mechanics. Harper & Row: New York, 1976.
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The impulse response function $R(t)$ describes the behavior of a system initially at equilibrium that is driven by an external field. Alternatively, we may need to describe the relaxation of a prepared state, in which we follow the return to equilibrium of a system initially held in a nonequilibrium state. This behavior is described by step response function $S(t)$. The step response comes from holding the system with a constant field $H = H _ {0} - f A$ until a time $t_0$ when the system is released, and it relaxes to the equilibrium state governed by $H=H_o$.
We can anticipate that the forms of these two functions are related. Just as we expect that the impulse response to rise from zero and be expressed as an odd function in time, the step response should decay from a fixed value and look even in time. In fact, we might expect to describe the impulse response by differentiating the step response, as seen in the classical case.
$R (t) = \frac {1} {k T} \frac {d} {d t} S (t) \label{10.72}$
An empirical derivation of the step response begins with a few observations. First, response functions must be real since they are proportional to observables, however quantum correlation functions are complex and follow
$C ( - t ) = C^{*} (t).$
Classical correlation functions are real and even,
$C (t) = C ( - t )$
and have the properties of a step response. To obtain the relaxation of a real observable that is even in time, we can construct a symmetrized function, which is just the real part of the correlation function:
\begin{align} S _ {A A} (t) & = \frac {1} {2} \left\{\left\langle A _ {I} (t) A _ {I} ( 0 ) \right\rangle + \left\langle A _ {I} ( 0 ) A _ {I} (t) \right\rangle \right\} \ & = \frac {1} {2} \left\{C _ {A A} (t) + C _ {A A} ( - t ) \right\} \ & = C _ {A A}^{\prime} (t) \end{align} \label{10.74}
The step response function $S$ defined as follows for $t \ge 0$.
$S ( \tau ) \equiv \frac {1} {\hbar} \Theta ( \tau ) \left\langle \left[ A _ {I} ( \tau ) , A _ {I} ( 0 ) \right] \right\rangle _ {+}$
From the eigenstate representation of the correlation function,
$C (t) = \sum _ {n , m} p _ {n} \left| A _ {m n} \right|^{2} e^{- i \omega _ {m n} t} \label{10.75}$
we see that the step response function can be expressed as an expansion in cosines
$S (t) = \frac {2} {\hbar} \Theta (t) \sum _ {n , m} p _ {n} \left| A _ {m n} \right|^{2} \cos \omega _ {m n} t \label{10.76}$
Further, one can readily show that the real and imaginary parts are related by
\begin{align} \omega \dfrac {d C^{\prime}} {d t} &= C^{\prime \prime} \[4pt] \omega \dfrac {d C^{\prime \prime}} {d t} &= C^{\prime} \end{align} \label{10.77}
Which shows how the impulse response is related to the time-derivative of the step response.
In the frequency domain, the spectral representation of the step response is obtained from the Fourier–Laplace transform
$S _ {A A} ( \omega ) = \int _ {0}^{\infty} d t S _ {A A} (t) e^{i \omega t} \label{10.78}$
\begin{align} S _ {A A} ( \omega ) & = \frac {1} {2} \left[ C _ {A A} ( \omega ) + C _ {A A} ( - \omega ) \right] \ & = \frac {1} {2} \left( 1 + e^{- \beta \hbar \omega} \right) C _ {A A} ( \omega ) \label{10.79} \end{align}
Now, with the expression for the imaginary part of the susceptibility,
$\chi^{\prime \prime} ( \omega ) = \frac {1} {2 \hbar} \left( 1 - e^{- \beta \hbar \omega} \right) C _ {AA} ( \omega ) \label{10.80}$
we obtain the relationship
$\chi^{\prime \prime} ( \omega ) = \frac {1} {\hbar} \tanh \left( \frac {\beta \hbar \omega} {2} \right) S _ {A A} ( \omega ) \label{10.81}$
Equation \ref{10.81} is the formal expression for the fluctuation-dissipation theorem, proven in 1951 by Callen and Welton. It followed an observation made many years earlier (1930) by Lars Onsager for which he was awarded the 1968 Nobel Prize in Chemistry: “The relaxation of macroscopic nonequilibrium disturbance is governed by the same laws as the regression of spontaneous microscopic fluctuations in an equilibrium state.”
Noting that
$\tanh (x) = \dfrac{e^{x} - e^{- x}}{e^{x} + e^{- x}}$
and
$\tanh (x) \rightarrow x$
for $x \gg 1$, we see that in the high temperature (classical) limit
$\chi^{\prime \prime} ( \omega ) \Rightarrow \frac {1} {2 k T} \omega S _ {A A} ( \omega ) \label{10.82}$
Appendix: Derivation of step response function
We can show more directly how the impulse and step response are related. To begin, let’s consider the step response experiment,
$H = \left\{\begin{array} {l l} {H _ {0} - f A} & {t < 0} \ {H _ {0}} & {t \geq 0} \end{array} \right. \label{10.83}$
and write the expectation values of the internal variable A for the system equilibrated under $H$ at time $t = 0$ and $t = ∞$.
$\langle A \rangle _ {0} = \left\langle \frac {e^{- \beta \left( H _ {0} - f A \right)}} {Z _ {0}} A \right\rangle \label{10.84A}$
with
$Z _ {0} = \left\langle e^{- \beta \left( H _ {0} - f A \right)} \right\rangle \label{10.84B}$
and
$\langle A \rangle _ {\infty} = \left\langle \frac {e^{- \beta H _ {0}}} {Z _ {\infty}} A \right\rangle \label{10.85A}$
with
$Z _ {\infty} = \left\langle e^{- \beta H _ {0}} \right\rangle \label{10.85B}$
If we make the classical linear response approximation, which states that when the applied potential $fA$ is very small relative to $0_o$, then
$e^{- \beta \left( H _ {0} - f A \right)} \approx e^{- \beta H _ {0}} ( 1 + \beta f A ) \label{10.86}$
and $Z _ {0} \approx Z _ {\infty}$, that
$\delta A = \langle A \rangle _ {0} - \langle A \rangle _ {\infty} \approx \beta f \left\langle A^{2} \right\rangle \label{10.87}$
and the time dependent relaxation is given by the classical correlation function
$\delta A (t) = \beta f \langle A ( 0 ) A (t) \rangle \label{10.88}$
For a description that works for the quantum case, let’s start with the system under $H_o$ at $t=-∞$, ramp up the external potential at a slow rate $\eta$ until $t=0$, and then abruptly shut off the external potential and watch the system. We will describe the behavior in the limit $\eta → 0$.
$H = \left\{\begin{array} {l l} {H _ {0} + f A e^{\eta t}} & {t < 0} \ {H _ {0}} & {t \geq 0} \end{array} \right. \label{10.89}$
Writing the time-dependence in terms of a convolution over the impulse response function $R$, we have
$\overline {\delta A (t)} = \lim _ {\eta \rightarrow 0} \int _ {- \infty}^{0} d t^{\prime} \Theta \left( t - t^{\prime} \right) R \left( t - t^{\prime} \right) e^{\eta t^{\prime}} f \label{10.90}$
Although the integral over the applied force (t’) is over times t<0, the step response factor ensures that t≥0. Now, expressing R as a Fourier transform over the imaginary part of the susceptibility, we obtain
\left.\begin{aligned} \overline {\delta A (t)} & = \lim _ {\eta \rightarrow 0} \frac {f} {2 \pi} \int _ {- \infty}^{0} d t^{\prime} \int _ {- \infty}^{\infty} d \omega e^{( \eta - i \omega ) t^{\prime}} e^{i \omega t} \chi^{\prime \prime} ( \omega ) \ & = \frac {f} {2 \pi} \int _ {- \infty}^{\infty} d \omega P P \left( \frac {1} {- i \omega} \right) \chi^{\prime \prime} ( \omega ) e^{i \omega t} \ & = \frac {f} {2 \pi i} \int _ {- \infty}^{\infty} d \omega \chi^{\prime} ( \omega ) e^{i \omega t} \ & = f C^{\prime} (t) \end{aligned} \right. \label{10.91}
A more careful derivation of this result that treats the quantum mechanical operators properly is found in the references.
Readings
1. Mazenko, G., Nonequilibrium Statistical Mechanics. Wiley-VCH: Weinheim, 2006.
2. Zwanzig, R., Nonequilibrium Statistical Mechanics. Oxford University Press: New York, 2001.
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• 12.1: A Classical Description of Spectroscopy
The traditional quantum mechanical treatment of spectroscopy is a static representation of a very dynamic process. An oscillating light field acts to drive bound charges in matter, which under resonance conditions leads to efficient exchange of energy between the light and matter. This dynamical picture emerges from a time-domain description, which shares many similarities to a classical description.
• 12.2: Time-Correlation Function Description of Absorption Lineshape
A time-correlation function for the dipole operator can be used to describe the dynamics of an equilibrium ensemble that dictate an absorption spectrum. We will make use of the transition rate expressions from first-order perturbation theory that we derived in the previous section to express the absorption of radiation by dipoles as a correlation function in the dipole operator. L
• 12.3: Different Types of Spectroscopy Emerge from the Dipole Operator
The absorption spectrum in any frequency region is given by the Fourier transform over the dipole correlation function that describes the time-evolving change distributions in molecules, solids, and nanosystems. Let’s consider how this manifests itself in a few different spectroscopies, which have different contributions to the dipole operator.
• 12.4: Ensemble Averaging and Line-Broadening
There are numerous processes that can influence the lineshape. These can be separated by dynamic processes intrinsic to the molecular system, which is termed homogeneous broadening, and static effects known as inhomogeneous broadening, which can be considered an ensemble averaging effect.
12: Time-domain Description of Spectroscopy
The traditional quantum mechanical treatment of spectroscopy is a static representation of a very dynamic process. An oscillating light field acts to drive bound charges in matter, which under resonance conditions leads to efficient exchange of energy between the light and matter. This dynamical picture emerges from a time-domain description, which shares many similarities to a classical description. Since much of the physical intuition that is helpful in understanding spectroscopy naturally emerges from the classical view, we will describe it first.
The classical view begins with the observation that atoms and molecules are composed of charged particles, and these charges are the handle by which an electromagnetic field exerts a force on the atom or molecule. The force exerted on the molecules depends on the form of the potential binding the charges together, the magnitude of the charges, and the strength of the external field.
The simplest elements of a model that captures what happens in absorption spectroscopy require us to consider a charged particle in a bound potential interacting with an oscillating driving force. The matter can be expressed in terms of a particle with charge $z$ in a harmonic potential (the leading term in any expansion of the potential in the coordinate $Q$):
$V _ {r e s} (t) = \dfrac {1} {2} \kappa Q^{2} \label{11.1}$
Here $k$ is the restoring force constant. For the light field, we use the traditional expression
$V _ {e x t} (t) = - \overline {\mu} \cdot \overline {E} (t) \label{11.2}$
for an external electromagnetic field interacting with the dipole moment of the system, $\overline {\mu} = z Q$. We describe the behavior of this system using Newton’s equation of motion F=ma, which we write as
$m \dfrac {\partial^{2} Q} {\partial t^{2}} = F _ {r e s} + F _ {d a m p} + F _ {e x t} \label{11.3}$
On the right hand side of Equation \ref{11.3} there are three forces: the harmonic restoring force, a damping force, and the driving force exerted by the light. Remembering that
$F = - ( \partial V / \partial Q )$
we can write Equation \ref{11.3} as
$m \dfrac {\partial^{2} Q} {\partial t^{2}} = - \kappa Q - b \dfrac {\partial Q} {\partial t} + F _ {0} \cos ( \omega t ) \label{11.4}$
Here, $b$ describes the rate of damping. For the field, we have only considered the time-dependence
$\overline {E} (t) = \overline {E} _ {0} \cos ( \omega t )$
and the amplitude of the driving force
$F _ {0} = \left( \dfrac {\partial \overline {\mu}} {\partial Q} \right) \cdot \overline {E} _ {0} \label{11.5}$
Equation \ref{11.5} indicates that increasing the force on the oscillator is achieved by raising the magnitude of the field, increasing how much the charge is displaced, or improving the alignment between the electric field polarization and the transition dipole moment. We can rewrite Equation \ref{11.4} as the driven harmonic oscillator equation:
$\dfrac {\partial^{2} Q} {\partial t^{2}} + 2 \gamma \dfrac {\partial Q} {\partial t} + \omega _ {0}^{2} Q = \dfrac {F _ {0}} {m} \cos ( \omega t ) \label{11.6}$
Here the damping constant $\gamma = b / 2 m$ and the harmonic resonance frequency $\omega _ {0} = \sqrt {\kappa / m}$.
Let’s look at the solution to Equation \ref{11.6} for a couple of simple cases.
First, for the case (red curve) that there is no damping or driving force ($\gamma = F _ {0} = 0$), we have simple harmonic solutions in which oscillate at a frequency $\omega _ {0}$:
$Q (t) = A \sin \left( \omega _ {0} t \right) + B \cos \left( \omega _ {0} t \right)$
Let’s just keep the $sin$ term for now. Now if you add damping to the equation:
$Q (t) = A e^{- \gamma t} \sin \Omega _ {0} t$
The coordinate oscillates at a reduced frequency
$\Omega _ {0} = \sqrt {\omega _ {0}^{2} - \gamma^{2}}$
As we continue, let’s assume a case with weak damping for which $\Omega _ {0} \approx \omega _ {0}$ (blue curve).
The solution to Equation \ref{11.6} takes the form
$Q (t) = \dfrac {F _ {0} / m} {\sqrt {\left( \omega _ {0}^{2} - \omega^{2} \right)^{2} + 4 \gamma^{2} \omega^{2}}} \sin ( \omega t + \beta ) \label{11.7}$
where the phase factor is
$\tan \beta = \left( \omega _ {0}^{2} - \omega^{2} \right) / 2 \gamma \omega \label{11.8}$
So this solution to the displacement of the particle says that the amplitude certainly depends on the magnitude of the driving force, but more importantly on the resonance condition. The frequency of the driving field should match the natural resonance frequency of the system, $\omega _ {0} \approx \infty$ … like pushing someone on a swing. When you drive the system at the resonance frequency there will be an efficient transfer of power to the oscillator, but if you push with arbitrary frequency, nothing will happen. Indeed, that is what an absorption spectrum is: a measure of the power absorbed by the system from the field.
Notice that the coordinate oscillates at the driving frequency ω and not at the resonance frequency $\omega_0$. Also, the particle oscillates as a sin, that is, 90° out-of-phase with the field when driven on resonance. This reflects the fact that the maximum force can be exerted on the particle when it is stationary at the turning points. The phase shift $\beta$, depends varies with the detuning from resonance. Now we can make some simplifications to Equation \ref{11.7} and calculate the absorption spectrum. For weak damping $\gamma < < \omega _ {0}$ and near resonance $\omega _ {0} \approx \infty$, we can write
$\left( \omega _ {0}^{2} - \omega^{2} \right)^{2} = \left( \omega _ {0} - \omega \right)^{2} \left( \omega _ {0} + \omega \right)^{2} \approx 4 \omega _ {0}^{2} \left( \omega _ {0} - \omega \right)^{2} \label{11.9}$
The absorption spectrum is a measure of the power transferred to the oscillator, so we can calculate it by finding the power absorbed from the force on the oscillator times the velocity, averaged over a cycle of the field.
\begin{align} P _ {a v g} &= \left\langle F (t) \cdot \dfrac {\partial Q} {\partial t} \right\rangle _ {a v g} \[4pt] &= \dfrac {\gamma F _ {0}^{2}} {2 m} \dfrac {1} {\left( \omega - \omega _ {0} \right)^{2} + \gamma^{2}} \label{11.10} \end{align}
This is the Lorentzian lineshape, which is peaked at the resonance frequency and has a line width of $2\gamma$ (full width half-maximum, FWHM). The area under the lineshape is $\pi F _ {0}^{2} / 4 m$.
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The interaction of light and matter as we have described from Fermi’s Golden Rule gives the rates of transitions between discrete eigenstates of the material Hamiltonian $H_0$. The frequency dependence to the transition rate is proportional to an absorption spectrum. We also know that interaction with the light field prepares a superposition of the eigenstates of $H_0$, and this leads to the periodic oscillation of amplitude between the states. Nonetheless, the transition rate expression really seems to hide any time-dependent description of motions in the system. An alternative approach to spectroscopy is to recognize that the features in a spectrum are just a frequency domain representation of the underlying molecular dynamics of molecules. For absorption, the spectrum encodes the time-dependent changes of the molecular dipole moment for the system, which in turn depends on the position of electrons and nuclei.
A time-correlation function for the dipole operator can be used to describe the dynamics of an equilibrium ensemble that dictate an absorption spectrum. We will make use of the transition rate expressions from first-order perturbation theory that we derived in the previous section to express the absorption of radiation by dipoles as a correlation function in the dipole operator. Let’s start with the rate of absorption and stimulated emission between an initial state $| \ell \rangle$ and final state $| k \rangle$ induced by a monochromatic field
$w _ {k \ell} = \dfrac {\pi E _ {0}^{2}} {2 \hbar^{2}} | \langle k | \hat {\varepsilon} \cdot \overline {\mu} | \ell \rangle |^{2} \left[ \delta \left( \omega _ {k \ell} - \omega \right) + \delta \left( \omega _ {k \ell} + \omega \right) \right] \label{11.11}$
For shorthand we have written
$\left| \overline {\mu} _ {k \ell} \right|^{2} = | \langle k | \hat {\mathcal {E}} \cdot \overline {\mu} | \ell \rangle |^{2}.$
We would like to use this to calculate the experimentally observable absorption coefficient (cross-section) which describes the transmission through the sample
$T = \exp [ - \Delta N \alpha ( \omega ) L ] \label{11.12}$
The absorption cross section describes the rate of energy absorption per unit time relative to the intensity of light incident on the sample
$\alpha = \dfrac {- \dot {E} _ {r a d}} {I} \label{11.13}$
The incident intensity is
$I = \frac {c} {8 \pi} E _ {0}^{2} \label{11.14}$
If we have two discrete states $| m \rangle$ and $| n \rangle$ with $E _ {m} > E _ {n}$, the rate of energy absorption is proportional to the absorption rate and the transition energy
$- \dot {E} _ {r a d} = w _ {n n} \cdot \hbar \omega _ {n m} \label{11.15}$
For an ensemble this rate must be scaled by the probability of occupying the initial state.
More generally, we want to consider the rate of energy loss from the field as a result of the difference in rates of absorption and stimulated emission between states populated with a thermal distribution.
So, summing all possible initial and final states $| \ell \rangle$ and $| k \rangle$ over all possible upper and lower states $| m \rangle$ and $| n \rangle$ with
\left.\begin{aligned} - \dot {E} _ {\text {rad}} & = \sum _ {\ell , k \in [ m , n \}} p _ {\ell} w _ {k \ell} \hbar \omega _ {k \ell} \ & = \dfrac {\pi E _ {0}^{2}} {2 \hbar} \sum _ {\ell , k \in [ m , n \}} \omega _ {k \ell} p _ {\ell} \left| \overline {\mu} _ {k \ell} \right|^{2} \left[ \delta \left( \omega _ {k \ell} - \omega \right) + \delta \left( \omega _ {k \ell} + \omega \right) \right] \end{aligned} \right. \label{11.16}
The cross section including the net change in energy as a result of absorption $| n \rangle \rightarrow | m \rangle$ and stimulated emission $| m \rangle \rightarrow | n \rangle$ is:
$\alpha ( \omega ) = \dfrac {4 \pi^{2}} {\hbar c} \sum _ {n , m} \left[ \omega _ {m n} p _ {n} \left| \overline {\mu} _ {m n} \right|^{2} \delta \left( \omega _ {m n} - \omega \right) + \omega _ {n m} p _ {m} \left| \overline {\mu} _ {n m} \right|^{2} \delta \left( \omega _ {n m} + \omega \right) \right] \label{11.17}$
To simplify Equation \ref{11.17}, we note:
1. Since $\delta (x) = \delta ( - x )$, then $\delta \left( \omega _ {n n} + \omega \right) = \delta \left( - \omega _ {m n} + \omega \right) = \delta \left( \omega _ {m n} - \omega \right).$
2. The matrix elements squared in the two terms of Equation \ref{11.17} are the same: $\left| \overline {\mu} _ {m n} \right|^{2} = \left| \overline {\mu} _ {n m} \right|^{2}.$
3. and as a result of the delta function enforcing this equality: $\omega _ {m n} = - \omega _ {n m} = \omega$
So,
$\alpha ( \omega ) = \dfrac {4 \pi^{2} \omega} {\hbar c} \sum _ {n , m} \left( p _ {n} - p _ {m} \right) \left| \overline {\mu} _ {m n} \right|^{2} \delta \left( \omega _ {m n} - \omega \right) \label{11.18}$
Here we see that the absorption coefficient depends on the population difference between the two states. This is expected since absorption will lead to loss of intensity, whereas stimulated emission leads to gain. With equal populations in the upper and lower state, no change to the incident field would be expected. Since
$p _ {n} - p _ {m} = p _ {n} \left( 1 - \exp \left[ - \beta \hbar \omega _ {m n} \right] \right) \label{11.19}$
$\alpha ( \omega ) = \dfrac {4 \pi^{2}} {\hbar c} \omega \left( 1 - e^{- \beta \hbar \omega} \right) \sum _ {n , m} p _ {n} \left| \overline {\mu} _ {m n} \right|^{2} \delta \left( \omega _ {m n} - \omega \right) \label{11.20}$
Again the $\omega _ {m n}$ factor has been replaced with $\omega$. We can now separate $\alpha$ into a product of factors that represent the field, and the matter, where the matter is described by $\sigma ( \omega )$, the absorption lineshape.
$\alpha ( \omega ) = \dfrac {4 \pi^{2}} {\hbar c} \omega \left( 1 - e^{- \beta \hbar \omega} \right) \sigma ( \omega ) \label{11.21}$
$\sigma ( \omega ) = \sum _ {n , m} p _ {n} \left| \overline {\mu} _ {m n} \right|^{2} \delta \left( \omega _ {m n} - \omega \right) \label{11.22}$
To express the lineshape in terms of a correlation function we use one representation of the delta function through a Fourier transform of a complex exponential:
$\delta \left( \omega _ {m n} - \omega \right) = \dfrac {1} {2 \pi} \int _ {- \infty}^{+ \infty} d t \,e^{i \left( \omega _ {m n} - \omega \right) t} \label{11.23}$
to write
$\sigma ( \omega ) = \dfrac {1} {2 \pi} \int _ {- \infty}^{+ \infty} d t \sum _ {n , m} p _ {n} \langle n | \overline {\mu} | m \rangle \langle m | \overline {\mu} | n \rangle e^{i \left( \omega _ {m n} - \omega \right) t} \label{11.24}$
Now equating
$U _ {0} | n \rangle = e^{- i H _ {0} t / \hbar} | n \rangle = e^{- i E _ {n} t / \hbar} | n \rangle$
and recognizing that our expression contains the projection operator
$\sigma ( \omega ) = \dfrac {1} {2 \pi} \int _ {- \infty}^{+ \infty} d t \sum _ {n , m} p _ {n} \langle n | \overline {\mu} | m \rangle \left\langle m \left| U _ {0}^{\dagger} \overline {\mu} U _ {0} \right| n \right\rangle e^{- i \omega t}$
$= \dfrac {1} {2 \pi} \int _ {- \infty}^{+ \infty} d t \sum _ {n , m} p _ {n} \left\langle n \left| \overline {\mu} _ {I} ( 0 ) \overline {\mu} _ {I} (t) \right| n \right\rangle e^{- i \omega t} \label{11.25}$
But this last expression is just a dipole moment correlation function: the equilibrium thermal average over a pair of time-dependent dipole operators:
$\sigma ( \omega ) = \dfrac {1} {2 \pi} \int _ {- \infty}^{+ \infty} d t e^{- i \omega t} \left\langle \overline {\mu} _ {I} ( 0 ) \overline {\mu} _ {I} (t) \right\rangle \label{11.26}$
The absorption lineshape is given by the Fourier transform of the dipole correlation function. The correlation function describes the time-dependent behavior or spontaneous fluctuations in the dipole moment in absence of E field and contains information on states of system and broadening due to relaxation. Additional manipulations can be used to switch the order of operators by taking the complex conjugate of the exponential
$\sigma ( \omega ) = \dfrac {1} {2 \pi} \int _ {- \infty}^{+ \infty} d t e^{i \omega t} \left\langle \overline {\mu} _ {I} (t) \overline {\mu} _ {I} ( 0 ) \right\rangle \label{11.27}$
and we can add back the polarization of the light field to the matrix element
$\sigma ( \omega ) = \dfrac {1} {2 \pi} \int _ {- \infty}^{+ \infty} d t e^{i \omega t} \left\langle \hat {\varepsilon} \cdot \overline {\mu} _ {I} (t) \hat {\varepsilon} \cdot \overline {\mu} _ {I} ( 0 ) \right\rangle \label{11.28}$
to emphasize the orientational component to this correlation function. Here we have written operators emphasizing the interaction picture representation. As we move forward, we will drop this notation, and take it as understood that for the purposes of spectroscopy, the dipole operator is expressed in the interaction picture and evolves under the material Hamiltonian $H_0$.
Readings
1. McHale, J. L., Molecular Spectroscopy. 1st ed.; Prentice Hall: Upper Saddle River, NJ, 1999.
2. McQuarrie, D. A., Statistical Mechanics. Harper & Row: New York, 1976; Ch. 21.
3. Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006.
4. Schatz, G. C.; Ratner, M. A., Quantum Mechanics in Chemistry. Dover Publications: Mineola, NY, 2002; Section 6.2.
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So the absorption spectrum in any frequency region is given by the Fourier transform over the dipole correlation function that describes the time-evolving change distributions in molecules, solids, and nanosystems. Let’s consider how this manifests itself in a few different spectroscopies, which have different contributions to the dipole operator. In general the dipole operator is a relatively simple representation of the charged particles of the system:
$\vec{\mu} = \sum _ {i} q _ {i} \left( \vec{r} _ {i} - \vec{r} _ {0} \right) \label{11.29}$
The complexity arises from the time-dependence of this operator, which evolves under the full Hamiltonian for the system:
$\vec{\mu} (t) = e^{i H _ {0} t / \hbar} \vec{\mu} ( 0 ) e^{- i H _ {0} t / \hbar} \label{11.30}$
where
\begin{align} H _ {0} = H _ {e l e c} + H _ {v i b} + H _ {r o t} + H _ {t r a n s} + H _ {s p i n} + \cdots + H _ {b a t h} + \cdots \[4pt] + \sum _ {i , j \in [ e , v , r , t , s , b , E M \}} H _ {i - j} + \cdots \label{11.31} \end{align}
The full Hamiltonian accounts for the dynamics of all electronic, nuclear, and spin degrees of freedom. It is expressed in Equation \ref{11.31} in terms of separable contributions to all possible degrees of freedom and a bath Hamiltonian that contains all of the dark degrees of freedom not explicitly included in the dipole operator. We could also include an electromagnetic field. The last term describes pairwise couplings between different degrees of freedom, and emphasizes that interactions such as electron-nuclear interactions $He_{e-v}$ and spin-orbit coupling $He_{e-s}$. The wavefunction for the system can be expressed in terms of product states of the wavefunctions for the different degrees of freedom,
$| \psi \rangle = | \psi _ {e l e c} \psi _ {v i b} \psi _ {r o t} \cdots \rangle \label{11.32}$
When the $H_{i-j}$ interaction terms are neglected, the correlation function can be separated into a product of correlation functions from various sources:
$C _ {\mu \mu} (t) = C _ {e l e c} (t) C _ {v i b} (t) C _ {r o t} (t) \cdots \label{11.33}$
which are each expressed in the form shown here for the vibrational states
$C _ {\mu \mu} (t) = C _ {e l e c} (t) C _ {v i b} (t) C _ {r o t} (t) \cdots \label{11.34}$
$\Phi _ {n}$ is the wavefunction for the nth vibrational eigenstate. The net correlation function will have oscillatory components at many frequencies and its Fourier transform will give the full absorption spectrum from the ultraviolet to the microwave regions of the spectrum. Generally speaking the highest frequency contributions (electronic or UV/Vis) will be modulated by contributions from lower frequency motions (… such as vibrations and rotations). However, we can separately analyze each of these contributions to the spectrum.
Atomic Transitions
$H _ {0} = H _ {\text {atom}}$. For hydrogenic orbitals, $| n \rangle \rightarrow | n \ell m _ {\ell} \rangle$.
Rotational Spectroscopy
From a classical perspective, the dipole moment can be written in terms of a permanent dipole moment with amplitude and direction
$\vec{\mu} = \mu _ {0} \hat {u} \label{11.35}$
$\sigma ( \omega ) = \int _ {- \infty}^{+ \infty} d t e^{- i \omega t} \mu _ {0}^{2} \langle \hat {\varepsilon} \cdot \hat {u} ( 0 ) \hat {\varepsilon} \cdot \hat {u} (t) \rangle \label{11.36}$
The lineshape is the Fourier transform of the rotational motion of the permanent dipole vector in the laboratory frame. $\mu _ {0}$ is the magnitude of the permanent dipole moment averaged over the fast electronic and vibrational degrees of freedom. The frequency of the resonance would depend on the rate of rotation—the angular momentum and the moment of inertia. Collisions or other damping would lead to the broadening of the lines.
Quantum mechanically we expect a series of rotational resonances that mirror the thermal occupation and degeneracy of rotational states for the system. Taking the case of a rigid rotor with cylindrical symmetry as an example, the Hamiltonian is
$H _ {m t} = \dfrac {\overline {L}^{2}} {2 I} \label{11.37}$
and the wavefunctions are spherical harmonics, $Y _ {J , M} ( \theta , \phi )$ which are described by
$\overline {L}^{2} | Y _ {J , M} \rangle = \hbar^{2} J ( J + 1 ) | Y _ {J , M} \rangle \label{11.38A}$
with $J = 0,1,2 \ldots \label{11.38B}$ and
$L _ {z} | Y _ {J , M} \rangle = M \hbar | Y _ {J , M} \rangle \label{11.38C}$
with
$M = - J , - J + 1 , \ldots , J \label{11.38D}$
where $J$ is the rotational quantum number and $M$ (or $M_J$) refers to its projection onto an axis (z), and has a degeneracy of $g_M(J)=2J+1$. The energy eigenvalues for Hrot are
$E _ {J , M} = \tilde{B} J ( J + 1 ) \label{11.39}$
where the rotational constant, here in units of joules, is
$\tilde{B} = \frac {\hbar^{2}} {2 I} \label{11.40}$
If we take a dipole operator in the form of Equation \ref{11.35}, then the far-infrared rotational spectrum will be described by the correlation function
$C _ {r o t} (t) = \sum _ {J , M} p _ {J , M} \left| \mu _ {0} \right|^{2} \left\langle Y _ {J , M} \left| e^{i H _ {m t} t h} ( \hat {u} \cdot \hat {\varepsilon} ) e^{- i H _ {m} t / h} ( \hat {u} \cdot \hat {\varepsilon} ) \right| Y _ {J , M} \right\rangle \label{11.41}$
The evaluation of this correlation function involves an orientational average, which is evaluated as follows
$\left\langle Y _ {J , M} | f ( \theta , \phi ) | Y _ {J , M} \right\rangle = \frac {1} {4 \pi} \int _ {0}^{2 \pi} d \varphi \int _ {0}^{\pi} \sin \theta d \theta Y _ {J , M}^{*} f ( \theta , \phi ) Y _ {J , M} \label{11.42}$
Recognizing that
$\left( \hat {u} \cdot \hat {\varepsilon} _ {z} \right) = \cos \theta$, we can evaluate Equation \ref{11.41} using the reduction formula ,
$\cos \theta | Y _ {J , M} \rangle = c _ {J +} | Y _ {J + 1 , M} \rangle + c _ {J -} | Y _ {J - 1 , M} \rangle \label{11.43}$
with
$c _ {J +} = \sqrt {\frac {( J + 1 )^{2} - M^{2}} {4 ( J + 1 )^{2} - 1}}$
and
$c _ {J -} = \sqrt {\frac {J^{2} + M^{2}} {4 J^{2} - 1}} \label{11.44}$
and the orthogonality of spherical harmonics
$\left\langle Y _ {J^{\prime} , M^{\prime}} | Y _ {J , M} \right\rangle = 4 \pi \delta _ {J , J} \delta _ {M^{\prime} , M} \label{11.45}$
The factor $p _ {J , M}$ in Equation \ref{11.41} is the probability of thermally occupying a particular $J,\,M$ level. For this we recognize that
$p _ {J , M} = g _ {M} ( J ) e^{- \beta E _ {J}} / Z _ {r o t}$
so that Equation \ref{11.41} leads to the correlation function
$\left\langle Y _ {J^{\prime} , M^{\prime}} | Y _ {J , M} \right\rangle = 4 \pi \delta _ {J , J} \delta _ {M^{\prime} , M} \label{11.46}$
Fourier transforming Equation \ref{11.46} leads to the lineshape
$\sigma _ {r o t} ( \omega ) = \frac {\left| \mu _ {0} \right|^{2}} {Z _ {r o t}} \hbar \sum _ {J} ( 2 J + 1 ) e^{- \beta \overline {B} J ( J + 1 ) / \hbar} [ \delta ( \hbar \omega - 2 \overline {B} ( J + 1 ) ) + \delta ( \hbar \omega + 2 \overline {B} J ) ] \label{11.47}$
The two terms reflect the fact that each thermally populated level with $J > 0$ contributes both to absorptive and stimulated emission processes, and the observed intensity reflects the difference in populations.
IR Vibrational Spectroscopy
Vibrational spectroscopy can be described by taking the dipole moment to be weakly dependent on the displacement of vibrational coordinates
$\overline {\mu} = \overline {\mu} _ {0} + \left. \frac {\partial \overline {\mu}} {\partial q} \right| _ {q = q _ {0}} q + \cdots \label{11.48}$
Here the first expansion term is the permanent dipole moment and the second term is the transition dipole moment. If we are performing our ensemble average over vibrational states, the lineshape becomes the Fourier transform of a correlation function in the vibrational coordinate
$\sigma ( \omega ) = \left| \frac {\partial \overline {\mu}} {\partial q} \right|^{2} \int _ {- \infty}^{+ \infty} d t \, e^{- i \omega t} \langle q ( 0 ) q (t) \rangle \label{11.49}$
The vector nature of the transition dipole has been dropped here. So the time-dependent dynamics of the vibrational coordinate dictate the IR lineshape.
This approach holds for the classical and quantum mechanical cases. In the case of quantum mechanics, the change in charge distribution in the transition dipole moment is replaced with the equivalent transition dipole matrix element
$| \partial \overline {\mu} / \partial q |^{2} \Rightarrow \left| \overline {\mu} _ {k \ell} \right|^{2}$
If we take the vibrational Hamiltonian to be that of a harmonic oscillator,
$H _ {v i b} = \frac {1} {2 m} p^{2} + \frac {1} {2} m \omega _ {0}^{2} q^{2} = \hbar \omega _ {0} \left( a^{\dagger} a + \frac {1} {2} \right)$
then the time-dependence of the vibrational coordinate, expressed as raising and lowering operators is
$q (t) = \sqrt {\frac {\hbar} {2 m \omega _ {0}}} \left( a^{\dagger} e^{i \omega _ {0} t} + a e^{- i \omega _ {0} t} \right)$
The absorption lineshape is then obtained from Equation \ref{11.49}.
$\sigma _ {v i b} ( \omega ) = \frac {1} {Z _ {v i b}} \sum _ {n} e^{- \beta n \hbar \omega _ {0}} \left[ \left| \overline {\mu} _ {( n + 1 ) n} \right|^{2} ( \overline {n} + 1 ) \delta \left( \omega - \omega _ {0} \right) + \left| \overline {\mu} _ {( n - 1 ) n} \right|^{2} \overline {n} \delta \left( \omega + \omega _ {0} \right) \right]$
where $\overline {n} = \left( e^{\beta \hbar \omega _ {0}} - 1 \right)^{- 1}$ is the thermal occupation number. For the low temperature limit applicable to most vibrations under room temperature conditions $\overline {n} \rightarrow 0$ and
$\sigma _ {v i b} ( \omega ) = \left| \overline {\mu} _ {10} \right|^{2} \delta \left( \omega - \omega _ {0} \right)$
Raman Spectroscopy
Technically, we need second-order perturbation theory to describe Raman scattering, because transitions between two states are induced by the action of two light fields whose frequency difference equals the energy splitting between states. But much the same result is obtained is we replace the dipole operator with an induced dipole moment generated by the incident field: $\overline {\mu} \Rightarrow \overline {\mu} _ {i n d}$. The incident field $E_i$ polarizes the molecule,
$\overline {\mu} _ {i n d} = \overline {\overline {\alpha}} \cdot \overline {E} _ {i} (t) \label{11.54}$
($\overline {\alpha}$ is the polarizability tensor), and the scattered light field results from the interaction with this induced dipole
\begin{align} V (t) & = - \overline {\mu} _ {i n d} \cdot \overline {E} _ {s} (t) \[4pt] & = \overline {E} _ {s} (t) \cdot \overline {\overline {\alpha}} \cdot \overline {E} _ {i} (t) \[4pt] & = E _ {s} (t) E _ {i} (t) \left( \hat {\varepsilon} _ {s} \cdot \overline {\alpha} \cdot \hat {\varepsilon} _ {i} \right) \label{11.55} \end{align}
Here we have written the polarization components of the incident ($i$) and scattered ($s$) light projecting onto the polarizability tensor $\overline{\overline {\alpha}}$. Equation \ref{11.55} leads to an expression for the Raman lineshape as
\begin{align} \sigma ( \omega ) &= \int _ {- \infty}^{+ \infty} d t e^{- i \omega t} \left\langle \hat {\varepsilon} _ {s} \cdot \overline {\alpha} ( 0 ) \cdot \hat {\varepsilon} _ {i} \hat {\varepsilon} _ {s} \cdot \overline {\overline {\alpha}} (t) \cdot \hat {\varepsilon} _ {i} \right\rangle \[4pt] &= \int _ {- \infty}^{+ \infty} d t e^{- i \omega t} \langle \overline {\overline {\alpha}} ( 0 ) \overline {\overline {\alpha}} (t) \rangle \label{11.56} \end{align}
To evaluate this, the polarizability tensor can also be expanded in the nuclear coordinates
$\overline {\overline {\alpha}} = \overline {\overline {\alpha} _ {0}} + \left. \frac {\partial \overline {\overline {\alpha}}} {\partial q} \right| _ {q = q _ {0}} q + \cdots \label{11.57}$
where the leading term would lead to Raleigh scattering and rotational Raman spectra, and the second term would give vibrational Raman scattering. Also remember that the polarizability tensor is a second rank tensor that tells you how well a light field polarized along $i$ can induce a dipole moment (light-field-induced charge displacement) in the s direction. For cylindrically symmetric systems which have a polarizability component $\alpha _ {\|}$ along the principal axis of the molecule and a component $\alpha _ {\perp}$ perpendicular to that axis, this usually takes the form
$\overline {\overline {\alpha}} = \left( \begin{array} {c c} {\alpha _ {\|}} & {} \ {} & {\alpha _ {\perp}} \ {} & {} & {\alpha _ {\perp}} \end{array} \right) = \alpha \mathbf {I} + \frac {1} {3} \beta \left( \begin{array} {c c} {2} & {} \ {} & {- 1} \ {} & {} & {- 1} \end{array} \right) \label{11.58}$
where $\alpha$ is the isotropic component of polarizability tensor and $\beta$ is the anisotropic component.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/12%3A_Time-domain_Description_of_Spectroscopy/12.03%3A_Different_Types_of_Spectroscopy_Emerge_from_the_Dipole_O.txt
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We have seen that an absorption lineshape can represent the dynamics of the dipole or be broadened by energy relaxation, for instance through coupling to a continuum. However, there are numerous processes that can influence the lineshape. These can be separated by dynamic processes intrinsic to the molecular system, which is termed homogeneous broadening, and static effects known as inhomogeneous broadening, which can be considered an ensemble averaging effect. To illustrate, imagine that the dipole correlation function has an oscillatory, damped form
$C _ {\mu \mu} (t) = \sum _ {g , e} p _ {g} \left| \mu _ {e g} \right|^{2} \exp \left[ - i \omega _ {e g} t - \Gamma t \right] \label{11.59}$
Then the Fourier transform would give a lineshape
$\operatorname {Re} \left[ \tilde {C} _ {\mu \mu} ( \omega ) \right] = \sum _ {g , e} p _ {g} \frac {\left| \mu _ {e g} \right|^{2} \Gamma} {\left( \omega - \omega _ {e g} \right)^{2} - \Gamma^{2}} \label{11.60}$
Here the homogeneous effects are reflected in the factor $\Gamma$, the damping rate and linewidth, whereas inhomogeneous effects arise from averaging over the ensemble.
Homogeneous Broadening
Several dynamical mechanisms can potentially contribute to damping and line-broadening. These intrinsically molecular processes, often referred to as homogeneous broadening, are commonly assigned a time scale $T _ {2} = \Gamma^{- 1}$.
Population Relaxation
Population relaxation refers to decay in the coherence created by the light field as a result of the finite lifetime of the coupled states, and is often assigned a time scale $T_1$. This can have contributions from radiative decay, such as spontaneous emission, or non-radiative processes such as relaxation as a result of coupling to a continuum.
$\frac {1} {T _ {1}} = \frac {1} {\tau _ {r a d}} + \frac {1} {\tau _ {N R}} \label{11.61}$
The observed population relaxation time depends on both the relaxation times of the upper and lower states ($m$ and $n$) being coupled by the field:
$1 / T _ {1} = w _ {m n} + w _ {n m}.$
When the energy splitting is high compared to $k_BT$, only the downward rate contributes, which is why the rate is often written $1/2T_1$.
Pure Dephasing
Pure dephasing is characterized by a time constant $T_2^{*}$ that characterizes the randomization of phase within an ensemble as a result of molecular interactions. This is a dynamic effect in which memory of the phase of oscillation of a molecule is lost as a result of intermolecular interactions that randomize the phase. Examples include collisions in a dense gas, or fluctuations induced by a solvent. This process does not change the population of the states involved.
Orientational Relaxation
Orientational relaxation ($\tau_{or}$) also leads to relaxation of the dipole correlation function and to line-broadening. Since the correlation function depends on the projection of the dipole onto a fixed axis in the laboratory frame, randomization of the initial dipole orientations is an ensemble averaged dephasing effect. In solution, this process is commonly treated as an orientational diffusion problem in which $\tau_{or}$ is proportional to the diffusion constant.
If these homogeneous processes are independent, the rates for different processes contribute additively to the damping and line width:
$\frac {1} {T _ {2}} = \frac {1} {T _ {1}} + \frac {1} {T _ {2}^{*}} + \frac {1} {\tau _ {o r}} \label{11.62}$
Inhomogeneous Broadening
Absorption lineshapes can also be broadened by a static distribution of frequencies. If molecules within the ensemble are influenced static environmental variations more than other processes, then the observed lineshape reports on the distribution of environments. This inhomogeneous broadening is a static ensemble averaging effect, which hides the dynamical content in the homogeneous linewidth. The origin of the inhomogeneous broadening can be molecular (for instance a distribution of defects in crystals) or macroscopic (i.e., an inhomogeneous magnetic field in NMR).
The inhomogeneous linewidth is dictated the width of the distribution $\Delta$.
Total Linewidth
The total observed broadening of the absorption lineshape reflects the contribution of all of these effects:
$C _ {\mu \mu} \propto \exp \left[ - i \omega _ {e g} t - \left( \frac {1} {T _ {2}^{*}} + \frac {1} {2 T _ {1}} + \frac {1} {\tau _ {o r}} \right) t - \frac {\Delta^{2}} {2} t^{2} \right] \label{11.63}$
These effects can be wrapped into a lineshape function $g(t)$. The lineshape for the broadening of a given transition can be written as the Fourier transform over the oscillating transition frequency damped and modulated by a complex $g(t)$:
$\sigma ( \omega ) = \int _ {- \infty}^{+ \infty} d t \, e^{i \omega t} e^{- i \omega _ {eg} t - g (t)} \label{11.64}$
All of these effects can be present simultaneously in an absorption spectrum.
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• 13.1: The Displaced Harmonic Oscillator Model
Here we will discuss the displaced harmonic oscillator, a widely used model that describes the coupling of nuclear motions to electronic states. Although it has many applications, we will look at the specific example of electronic absorption experiments, and thereby gain insight into the vibronic structure in absorption spectra.
• 13.2: Coupling to a Harmonic Bath
The displaced harmonic oscillator model is readily generalized to many vibrations or a continuum of nuclear motions. Coupling to a continuum, or a harmonic bath, is the starting point for developing how an electronic system interacts with a continuum of intermolecular motions and phonons typical of condensed phase systems.
• 13.3: Semiclassical Approximation to the Dipole Correlation Function
The semiclassical approximation is a useful representation of the dipole correlation function when one wants to describe the dark degrees of freedom (the bath) using classical molecular dynamics simulations.
13: Coupling of Electronic and Nuclear Motion
Here we will discuss the displaced harmonic oscillator (DHO), a widely used model that describes the coupling of nuclear motions to electronic states. Although it has many applications, we will look at the specific example of electronic absorption experiments, and thereby gain insight into the vibronic structure in absorption spectra. Spectroscopically, it can also be used to describe wavepacket dynamics; coupling of electronic and vibrational states to intramolecular vibrations or solvent; or coupling of electronic states in solids or semiconductors to phonons. As we will see, further extensions of this model can be used to describe fundamental chemical rate processes, interactions of a molecule with a dissipative or fluctuating environment, and Marcus Theory for nonadiabatic electron transfer.
The DHO and Electronic Absorption
Molecular excited states have geometries that are different from the ground state configuration as a result of varying electron configuration. This parametric dependence of electronic energy on nuclear configuration results in a variation of the electronic energy gap between states as one stretches bond vibrations of the molecule. We are interested in describing how this effect influences the electronic absorption spectrum, and thereby gain insight into how one experimentally determines the coupling of between electronic and nuclear degrees of freedom. We consider electronic transitions between bound potential energy surfaces for a ground and excited state as we displace a nuclear coordinate $q$. The simplified model consists of two harmonic oscillators potentials whose 0-0 energy splitting is $E _ {e} - E _ {g}$ and which depends on $q$. We will calculate the absorption spectrum in the interaction picture using the time-correlation function for the dipole operator.
We start by writing a Hamiltonian that contains two terms for the potential energy surfaces of the electronically excited state $| E \rangle$ and ground state $| G \rangle$
$H _ {0} = H _ {G} + H _ {E} \label{12.1}$
These terms describe the dependence of the electronic energy on the displacement of a nuclear coordinate $q$. Since the state of the system depends parametrically on the level of vibrational excitation, we describe it using product states in the electronic and nuclear configuration, $| \Psi \rangle = | \psi _ {\text {elec}} , \Phi _ {n u c} \rangle$, or in the present case
\begin{align} | G \rangle &= | g , n _ {g} \rangle \[4pt] | E \rangle &= | e , n _ {e} \rangle \label{12.2} \end{align}
Implicit in this model is a Born-Oppenheimer approximation in which the product states are the eigenstates of $H_0$, i.e.
$H _ {G} | G \rangle = \left( E _ {g} + E _ {n _ {g}} \right) | G \rangle$
The Hamiltonian for each surface contains an electronic energy in the absence of vibrational excitation, and a vibronic Hamiltonian that describes the change in energy with nuclear displacement.
\begin{align} H _ {G} & = | g \rangle E _ {g} \langle g | + H _ {g} ( q ) \[4pt] H _ {E} & = | e \rangle E _ {e} \langle e | + H _ {e} ( q ) \label{12.3} \end{align}
For our purposes, the vibronic Hamiltonian is harmonic and has the same curvature in the ground and excited states, however, the excited state is displaced by d relative to the ground state along a coordinate $q$.
\begin{align} H _ {g} &= \frac {p^{2}} {2 m} + \frac {1} {2} m \omega _ {0}^{2} q^{2} \label{12.4} \[4pt] H _ {e} &= \frac {p^{2}} {2 m} + \frac {1} {2} m \omega _ {0}^{2} ( q - d )^{2} \label{12.5} \end{align}
The operator $q$ acts only to changes the degree of vibrational excitation on the $| E \rangle$ or $| G \rangle$ surface.
We now wish to evaluate the dipole correlation function
\begin{align} C _ {\mu \mu} (t) & = \langle \overline {\mu} (t) \overline {\mu} ( 0 ) \rangle \[4pt] & = \sum _ {\ell = E , G} p _ {\ell} \left\langle \ell \left| e^{i H _ {0} t / h} \overline {\mu} e^{- i H _ {0} t / h} \overline {\mu} \right| \ell \right\rangle \label{12.6} \end{align}
Here $p_{\ell}$ is the joint probability of occupying a particular electronic and vibrational state, $p _ {\ell} = p _ {\ell , e l e c} p _ {\ell , v i b}$. The time propagator is
$e^{- i H _ {d} t / h} = | G \rangle e^{- i H _ {c} t h} \langle G | + | E \rangle e^{- i H _ {E} t / h} \langle E | \label{12.7}$
We begin by making the Condon Approximation, which states that there is no nuclear dependence for the dipole operator. It is only an operator in the electronic states.
$\overline {\mu} = | g \rangle \mu _ {g e} \langle e | + | e \rangle \mu _ {e g} \langle g | \label{12.8}$
This approximation implies that transitions between electronic surfaces occur without a change in nuclear coordinate, which on a potential energy diagram is a vertical transition.
Under typical conditions, the system will only be on the ground electronic state at equilibrium, and substituting Equations \ref{12.7} and \ref{12.8} into Equation \ref{12.6}, we find:
$C _ {\mu \mu} (t) = \left| \mu _ {e g} \right|^{2} e^{- i \left( E _ {e} - E _ {g} \right) t h} \left\langle e^{i H _ {g} t h} e^{- i H _ {\ell} t / h} \right\rangle \label{12.9}$
Here the oscillations at the electronic energy gap are separated from the nuclear dynamics in the final factor, the dephasing function:
\begin{align} F (t) & = \left\langle e^{i H _ {g} t / \hbar} e^{- i H _ {c} t / h} \right\rangle \[4pt] & = \left\langle U _ {g}^{\dagger} U _ {e} \right\rangle \label{12.10} \end{align}
The average $\langle \ldots \rangle$ in Equations \ref{12.9} and \ref{12.10} is only over the vibrational states $| n _ {g} \rangle$. Note that physically the dephasing function describes the time-dependent overlap of the nuclear wavefunction on the ground state with the time-evolution of the same wavepacket initially projected onto the excited state
$F (t) = \left\langle \varphi _ {g} (t) | \varphi _ {e} (t) \right\rangle \label{12.11}$
This is a perfectly general expression that does not depend on the particular form of the potential. If you have knowledge of the nuclear and electronic eigenstates or the nuclear dynamics on your ground and excited state surfaces, this expression is your route to the absorption spectrum.
For further information on this see:
• Schatz, G. C.; Ratner, M. A., Quantum Mechanics in Chemistry. Dover Publications: Mineola, NY, 2002; Ch. 9.
• Reimers, J. R.; Wilson, K. R.; Heller, E. J., Complex time dependent wave packet technique for thermal equilibrium systems: Electronic spectra. J. Chem. Phys. 1983, 79, 4749-4757. 12-4
To evaluate $F(t)$ for this problem, it helps to realize that we can write the nuclear Hamiltonians as
\begin{align} H _ {g} &= \hbar \omega _ {0} \left( a^{\dagger} a + \ce{1/2} \right) \label{12.12} \[4pt] H _ {e} &= \hat {D} H _ {g} \hat {D}^{\dagger} \label{12.13} \end{align}
Here $\hat {D}$ is the spatial displacement operator
$\hat {D} = \exp ( - i \hat {p} d / \hbar ) \label{12.14}$
which shifts an operator in space as:
$\hat {D} \hat {q} \hat {D}^{\dagger} = \hat {q} - d \label{12.15}$
Note $\hat{p}$ is only an operator in the vibrational degree of freedom. We can now express the excited state Hamiltonian in terms of a shifted ground state Hamiltonian in Equation \ref{12.13}, and also relate the time propagators on the ground and excited states
$e^{- i H _ {c} t / h} = \hat {D} e^{- i H _ {g} t / h} \hat {D}^{\dagger} \label{12.16}$
Substituting Equation \ref{12.16} into Equation \ref{12.10} allows us to write
\begin{align} F (t) & = \left\langle U _ {g}^{\dagger} e^{- i d p / h} U _ {g} e^{i d p / h} \right\rangle \ & = \left\langle \hat {D} (t) \hat {D}^{\dagger} ( 0 ) \right\rangle \label{12.17} \end{align}
Equation \ref{12.17} says that the effect of the nuclear motion in the dipole correlation function can be expressed as a time-correlation function for the displacement of the vibration.
To evaluate Equation \ref{12.17} we write it as
$F (t) = \left\langle e^{- i d \hat {p} (t) / \hbar} e^{i d \hat {p} ( 0 ) / \hbar} \right\rangle \label{12.18}$
since
$\hat {p} (t) = U _ {g}^{\dagger} \hat {p} ( 0 ) U _ {g} \label{12.19}$
The time-evolution of $\hat{p}$ is obtained by expressing it in raising and lowering operator form,
$\hat {p} = i \sqrt {\frac {m \hbar \omega _ {0}} {2}} \left( a^{\dagger} - a \right) \label{12.20}$
and evaluating Equation \ref{12.19} using Equation \ref{12.12}. Remembering $a^{\dagger} a = n$, we find
$\left. \begin{array} {l} {U _ {g}^{\dagger} a U _ {g} = e^{i n \omega _ {0} t} a e^{- i n \omega _ {0} t} = a e^{i ( n - 1 ) \omega _ {0} t} e^{- i n \omega _ {0} t} = a e^{- i \omega _ {0} t}} \ {U _ {g}^{\dagger} a^{\dagger} U _ {g} = a^{\dagger} e^{+ i \omega _ {0} t}} \end{array} \right. \label{12.21}$
which gives
$\hat {p} (t) = i \sqrt {\frac {m \hbar \omega _ {0}} {2}} \left( a^{\dagger} e^{i \omega _ {0} t} - a e^{- i \omega _ {0} t} \right) \label{12.22}$
So for the dephasing function we now have
$F (t) = \left\langle \exp \left[ d \left( a^{\dagger} e^{i \omega _ {0} t} - a e^{- i \omega _ {0} t} \right) \right] \exp \left[ - d \left( a^{\dagger} - a \right) \right] \right\rangle \label{12.23}$
where we have defined a dimensionless displacement variable
$\underset{\sim}{d} = d \sqrt {\frac {m \omega _ {0}} {2 \hbar}} \label{12.24}$
Since $a^{\dagger}$ and $a$ do not commute ($\left[ a^{\dagger} , a \right] = - 1$), we split the exponential operators using the identity
$e^{\hat {A} + \hat {B}} = e^{\hat {A}} e^{\hat {B}} e^{- \frac {1} {2} [ \hat {A} , \hat {B} ]} \label{12.25}$
or specifically for $a^{\dagger}$ and $a$,
$e^{\lambda a^{\dagger} + \mu a} = e^{\lambda a^{\dagger}} e^{\mu a} e^{\frac {1} {2} \lambda \mu} \label{12.26}$
This leads to
$F (t) = \left \langle \exp \left[ \underset{\sim}{d} \,a^{\dagger}\, e^{i \omega _ {0} t} \right] \exp \left[ - \underset{\sim}{d}\, a\, e^{- i \omega _ {0} t} \right] \exp \left[ - \frac {1} {2} \underset{\sim}{d}^{2} \right] \exp \left[ - \underset{\sim}{d}\, a^{\dagger} \right] \exp [ \underset{\sim}{d}\, a ] \exp \left[ - \frac {1} {2} \underset{\sim}{d}^{2} \right] \right \rangle \label{12.27}$
Now to simplify our work further, let’s specifically consider the low temperature case in which we are only in the ground vibrational state at equilibrium $| n _ {s} \rangle = | 0 \rangle$. Since $a | 0 \rangle = 0$ and $\langle 0 | a^{t} = 0$
\begin{align} e^{-\lambda a} | 0 \rangle &= | 0 \rangle \[4pt] \langle 0 | e^{\lambda a^{\dagger}} &= \langle 0 | \label{12.28} \end{align}
and
$F (t) = e^{- \underset{\sim}{d}^{2}} \left\langle 0 \left| \exp \left[ - \underset{\sim}{d} a e^{- i \omega _ {b} t} \right] \exp \left[ - \underset{\sim}{d} a^{\dagger} \right] \right| 0 \right \rangle \label{12.29}$
In principle these are expressions in which we can evaluate by expanding the exponential operators. However, the evaluation becomes much easier if we can exchange the order of operators. Remembering that these operators do not commute, and using
$e^{\hat {A}} e^{\hat {B}} = e^{\hat {B}} e^{\hat {A}} e^{- [ \hat {B} , \hat {A} ]} \label{12.30}$
we can write
\begin{align} F (t) & {= e^{- \underset{\sim}{d}^{2}} \langle 0 \left| \exp \left[ - \underset{\sim}{d} a^{\dagger} \right] \exp \left[ - \underset{\sim}{d} \,a \, e^{- i \omega _ {0} t} \right] \exp \left[ \underset{\sim}{d}^{2} e^{- i \omega _ {0} t} \right] \| _ {0} \right\rangle} \ & = \exp \left[ \underset{\sim}{d}^{2} \left( e^{- i \omega _ {0} t} - 1 \right) \right] \label{12.31} \end{align}
So finally, we have the dipole correlation function:
$C _ {\mu \mu} (t) = \left| \mu _ {e g} \right|^{2} \exp \left[ - i \omega _ {e g} t + D \left( e^{- i \omega _ {v} t} - 1 \right) \right] \label{12.32}$
$D$ is known as the Huang-Rhys parameter (which should be distinguished from the displacement operator $\hat{D}$). It is a dimensionless factor related to the mean square displacement
$D = d^{2} = \underset{\sim}{d}^{2} \frac {m \omega _ {0}} {2 \hbar} \label{12.33}$
and therefore represents the strength of coupling of the electronic states to the nuclear degree of freedom. Note our correlation function has the form
$C _ {\mu \mu} (t) = \sum _ {n} p _ {n} \left| \mu _ {m n} \right|^{2} e^{- i \omega _ {m n} t - g (t)} \label{12.34}$
Here $g(t)$ is our lineshape function
$g (t) = - D \left( e^{- i \omega _ {0} t} - 1 \right) \label{12.35}$
To illustrate the form of these functions, below is plotted the real and imaginary parts of $C _ {\mu \mu} (t)$, $F(t)$, $g(t)$ for $D = 1$, and $\omega_{eg} = 10\omega_0$. $g(t)$ oscillates with the frequency of the single vibrational mode. $F(t)$ quantifies the overlap of vibrational wavepackets on ground and excited state, which peaks once every vibrational period. $C _ {\mu \mu} (t)$ has the same information as $F(t)$, but is also modulated at the electronic energy gap $\omega_{eg}$.
Absorption Lineshape and Franck-Condon Transitions
The absorption lineshape is obtained by Fourier transforming Equation \ref{12.32}
\begin{align} \sigma _ {a b s} ( \omega ) & = \int _ {- \infty}^{+ \infty} d t \,e^{i \omega t} C _ {\mu \mu} (t) \[4pt] & = \left| \mu _ {e g} \right|^{2} e^{- D} \int _ {- \infty}^{+ \infty} d t\, e^{i \omega t} e^{- i \omega _ {e s} t} \exp \left[ D e^{- i \omega _ {0} t} \right] \label{12.36} \end{align}
If we now expand the final term as
$\exp \left[ D \mathrm {e}^{- i \omega _ {0} t} \right] = \sum _ {n = 0}^{\infty} \frac {1} {n !} D^{n} \left( e^{- i \omega _ {0} t} \right)^{n} \label{12.37}$
the lineshape is
$\sigma _ {a b s} ( \omega ) = \left| \mu _ {e g} \right|^{2} \sum _ {n = 0}^{\infty} e^{- D} \frac {D^{n}} {n !} \delta \left( \omega - \omega _ {e g} - n \omega _ {0} \right) \label{12.38}$
The spectrum is a progression of absorption peaks rising from $\omega_{eg}$, separated by $\omega_0$ with a Poisson distribution of intensities. This is a vibrational progression accompanying the electronic transition. The amplitude of each of these peaks are given by the Franck–Condon coefficients for the overlap of vibrational states in the ground and excited states
$\left| \left\langle n _ {g} = 0 | n _ {e} = n \right\rangle \right|^{2} = | \langle 0 | \hat {D} | n \rangle |^{2} = e^{- D} \frac {D^{n}} {n !}$
The intensities of these peaks are dependent on $D$, which is a measure of the coupling strength between nuclear and electronic degrees of freedom. Illustrated below is an example of the normalized absorption lineshape corresponding to the correlation function for $D = 1$ in Figure $3$.
Now let’s investigate how the absorption lineshape depends on $D$.
For $D = 0$, there is no dependence of the electronic energy gap $\omega_{eg}$ on the nuclear coordinate, and only one resonance is observed. For $D < 1$, the dependence of the energy gap on $q$ is weak and the absorption maximum is at $\omega_{eg}$, with the amplitude of the vibronic progression falling off as $D^n$. For $D >1$, the strong coupling regime, the transition with the maximum intensity is found for peak at $n \approx D$. So $D$ corresponds roughly to the mean number of vibrational quanta excited from $q = 0$ in the ground state. This is the Franck-Condon principle, that transition intensities are dictated by the vertical overlap between nuclear wavefunctions in the two electronic surfaces.
To investigate the envelope for these transitions, we can perform a short time expansion of the correlation function applicable for $t < 1/\omega_{0}$ and for $D \gg 1$. If we approximate the oscillatory term in the lineshape function as
$\exp \left( - i \omega _ {0} t \right) \approx 1 - i \omega _ {0} t - \frac {1} {2} \omega _ {0}^{2} t^{2} \label{12.40}$
then the lineshape envelope is
\begin{align} \sigma _ {e n v} ( \omega ) & = \left| \mu _ {e g} \right|^{2} \int _ {- \infty}^{+ \infty} d t e^{i \omega t} e^{- i \omega _ {e g} t} e^{D \left( \exp \left( - i \omega _ {0} t \right) - 1 \right)} \ & \approx \left| \mu _ {e g} \right|^{2} \int _ {- \infty}^{+ \infty} d t e^{i \left( \omega - \omega _ {e g} t \right)} e^{D \left[ - i \omega _ {0} t - \frac {1} {2} \omega _ {0}^{2} t^{2} \right]} \ & = \left| \mu _ {e g} \right|^{2} \int _ {- \infty}^{+ \infty} d t e^{i \left( \omega - \omega _ {e g} - D \omega _ {0} \right) t} e^{- \frac {1} {2} D \omega _ {0}^{2} t^{2}} \label{12.41} \end{align}
This can be solved by completing the square, giving
$\sigma _ {e n v} ( \omega ) = \left| \mu _ {e g} \right|^{2} \sqrt {\frac {2 \pi} {D \omega _ {0}^{2}}} \exp \left[ - \frac {\left( \omega - \omega _ {e g} - D \omega _ {0} \right)^{2}} {2 D \omega _ {0}^{2}} \right] \label{12.42}$
The envelope has a Gaussian profile which is centered at Franck–Condon vertical transition
$\omega = \omega _ {e g} + D \omega _ {0} \label{12.43}$
Thus we can equate $D$ with the mean number of vibrational quanta excited in $| E \rangle$ on absorption from the ground state. Also, we can define the vibrational energy vibrational energy in $| E \rangle$ on excitation at $q=0$
\begin{align} \lambda &= D \hbar \omega _ {0} \[4pt] &= \frac {1} {2} m \omega _ {0}^{2} d^{2} \label{12.44} \end{align}
$\lambda$ is known as the reorganization energy. This is the value of $H_e$ at $q=0$, which reflects the excess vibrational excitation on the excited state that occurs on a vertical transition from the ground state. It is therefore the energy that must be dissipated by vibrational relaxation on the excited state surface as the system re-equilibrates following absorption.
Illustration of how the strength of coupling $D$ influences the absorption lineshape $\sigma$ (Equation \ref{12.38}) and dipole correlation function $C _ {\mu \mu}$ (Equation \ref{12.32}).
Also shown, the Gaussian approximation to the absorption profile (Equation \ref{12.42}), and the dephasing function (Equation \ref{12.31}).
Fluorescence
The DHO model also leads to predictions about the form of the emission spectrum from the electronically excited state. The vibrational excitation on the excited state potential energy surface induced by electronic absorption rapidly dissipates through vibrational relaxation, typically on picosecond time scales. Vibrational relaxation leaves the system in the ground vibrational state of the electronically excited surface, with an average displacement that is larger than that of the ground state. In the absence of other non-radiative processes relaxation processes, the most efficient way of relaxing back to the ground state is by emission of light, i.e., fluorescence. In the Condon approximation this occurs through vertical transitions from the excited state minimum to a vibrationally excited state on the ground electronic surface. The difference between the absorption and emission frequencies reflects the energy of the initial excitation which has been dissipated non-radiatively into vibrational motion both on the excited and ground electronic states, and is referred to as the Stokes shift.
From the DHO model, the emission lineshape can be obtained from the dipole correlation function assuming that the initial state is equilibrated in $| e , 0 \rangle$, centered at a displacement $q= d$, following the rapid dissipation of energy $\lambda$ on the excited state. Based on the energy gap at $q=d$, we see that a vertical emission from this point leaves $\lambda$ as the vibrational energy that needs to be dissipated on the ground state in order to re-equilibrate, and therefore we expect the Stokes shift to be $2\lambda$
Beginning with our original derivation of the dipole correlation function and focusing on emission, we find that fluorescence is described by
\begin{align} C _ {\Omega} & = \langle e , 0 | \mu (t) \mu ( 0 ) | e , 0 \rangle = C _ {\mu \mu}^{*} (t) \ & = \left| \mu _ {e g} \right|^{2} e^{- i \omega _ {\mathrm {g}} t} F^{*} (t) \label{12.45} \[4pt] F^{*} (t) & = \left\langle U _ {e}^{\dagger} U _ {g} \right\rangle \[4pt] & = \exp \left[ D \left( e^{i \omega _ {0} t} - 1 \right) \right] \label{12.46} \end{align}
We note that $C _ {\mu \mu}^{*} (t) = C _ {\mu \mu} ( - t )$ and $F^{*} (t) = F ( - t )$.
Then we can obtain the fluorescence spectrum
\begin{align} \sigma _ {f} ( \omega ) & = \int _ {- \infty}^{+ \infty} d t \,e^{i \omega t} C _ {\mu \mu}^{*} (t) \[4pt] & = \left| \mu _ {e g} \right|^{2} \sum _ {n = 0}^{\infty} e^{- D} \frac {D^{n}} {n !} \delta \left( \omega - \omega _ {e g} + n \omega _ {0} \right) \end{align} \label{12.47}
This is a spectrum with the same features as the absorption spectrum, although with mirror symmetry about $\omega_{eg}$.
A short time expansion confirms that the splitting between the peak of the absorption and emission lineshape envelopes is $2 D \hbar \omega_0$, or $2\lambda$. Further, one can establish that
\left.\begin{aligned} \sigma _ {a b s} ( \omega ) & = \int _ {- \infty}^{+ \infty} dt\, e^{i \left( \omega - \omega _ {e g} \right) t + g (t)} \ \sigma _ {f l} ( \omega ) & = \int _ {- \infty}^{+ \infty} dt\, e^{i \left( \omega - \omega _ {e g} \right) t + g^{*} (t)} \ g (t) & = D \left( e^{- i \omega _ {0} t} - 1 \right) \end{aligned} \right. \label{12.48}
Note that our description of the fluorescence lineshape emerged from our semiclassical treatment of the light–matter interaction, and in practice fluorescence involves spontaneous emission of light into a quantum mechanical light field. However, while the light field must be handled differently, the form of the dipole correlation function and the resulting lineshape remains unchanged. Additionally, we assumed that there was a time scale separation between the vibrational relaxation in the excited state and the time scale of emission, so that the system can be considered equilibrated in $| e , 0 \rangle$. When this assumption is not valid then one must account for the much more complex possibility of emission during the course of the relaxation process.
Readings
1. Mukamel, S., Principles of Nonlinear Optical Spectroscopy. Oxford University Press: New York, 1995; p. 189, p. 217.
2. Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006; Section 12.5.
3. Reimers, J. R.; Wilson, K. R.; Heller, E. J., Complex time dependent wave packet technique for thermal equilibrium systems: Electronic spectra. J. Chem. Phys. 1983, 79, 4749-4757.
4. Schatz, G. C.; Ratner, M. A., Quantum Mechanics in Chemistry. Dover Publications: Mineola, NY, 2002; Ch. 9.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/13%3A_Coupling_of_Electronic_and_Nuclear_Motion/13.01%3A_The_Displaced_Harmonic_Oscillator_Model.txt
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It is worth noting a similarity between the Hamiltonian for this displaced harmonic oscillator problem, and a general form for the interaction of an electronic “system” that is observed in an experiment with a harmonic oscillator “bath” whose degrees of freedom are invisible to the observable, but which influence the behavior of the system. This reasoning will in fact be developed more carefully later for the description of fluctuations. While the Hamiltonians we have written so far describe coupling to a single bath degree of freedom, the DHO model is readily generalized to many vibrations or a continuum of nuclear motions. Coupling to a continuum, or a harmonic bath, is the starting point for developing how an electronic system interacts with a continuum of intermolecular motions and phonons typical of condensed phase systems.
So, what happens if the electronic transition is coupled to many vibrational coordinates, each with its own displacement? The extension is straightforward if we still only consider two electronic states ($e$ and $g$) to which we couple a set of independent modes, i.e., a bath of harmonic normal modes. Then we can write the Hamiltonian for $N$ vibrations as a sum over all the independent harmonic modes
$H _ {e} = \sum _ {\alpha = 1}^{N} H _ {e}^{( \alpha )} = \sum _ {\alpha = 1}^{N} \left( \frac {p _ {\alpha}^{2}} {2 m _ {\alpha}} + \frac {1} {2} m _ {\alpha} \omega _ {a}^{2} \left( q _ {\alpha} - d _ {\alpha} \right)^{2} \right) \label{12.49}$
each with their distinct frequency and displacement. We can specify the state of the system in terms of product states in the electronic and nuclear occupation, i.e.,
\left.\begin{aligned} | G \rangle & = | g ; n _ {1} , n _ {2} , \ldots , n _ {N} \rangle \ & = | g \rangle \prod _ {\alpha = 1}^{N} | n _ {\alpha} \rangle \end{aligned} \right. \label{12.50}
Additionally, we recognize that the time propagator on the electronic excited potential energy surface is
$U _ {e} = \exp \left[ - \frac {i} {\hbar} H _ {e} t \right] = \prod _ {\alpha = 1}^{N} U _ {e}^{( \alpha )} \label{12.51}$
where
$U _ {e}^{( \alpha )} = \exp \left[ - \frac {i} {\hbar} H _ {e}^{( \alpha )} t \right] \label{12.52}$
Defining $D _ {\alpha} = d _ {\alpha}^{2} \left( m \omega _ {\alpha} / 2 \hbar \right)$
\left.\begin{aligned} F^{( \alpha )} & = \left\langle \left[ U _ {g}^{( \alpha )} \right]^{\dagger} U _ {e}^{( \alpha )} \right\rangle \ & = \exp \left[ D _ {\alpha} \left( e^{- i \omega _ {a} t} - 1 \right) \right] \end{aligned} \right. \label{12.53}
the dipole correlation function is then just a product of multiple dephasing functions that characterize the time-evolution of the different vibrations.
$C _ {\mu \mu} (t) = \left| \mu _ {e g} \right|^{2} e^{- i \omega _ {e g} t} \cdot \prod _ {\alpha = 1}^{N} F^{( \alpha )} (t) \label{12.54}$
or
$C _ {\mu \mu} (t) = \left| \mu _ {e g} \right|^{2} e^{- i \omega _ {e g} t + g (t)} \label{12.55}$
with
$g (t) = \sum _ {\alpha} D _ {\alpha} \left( e^{- i \omega _ {a} t} - 1 \right) \label{12.56}$
In the time domain this is a complex beating pattern, which in the frequency domain appears as a spectrum with several superimposed vibronic progressions that follow the rules developed above. Also, the reorganization energy now reflects to total excess nuclear potential energy required to make the electronic transition:
$\lambda = \sum _ {\alpha} D _ {\alpha} \hbar \omega _ {\alpha} \label{12.57}$
Taking this a step further, the generalization to a continuum of nuclear states emerges naturally. Given that we have a continuous frequency distribution of normal modes characterized by a density of states, $W(\omega)$, and a continuously varying frequency-dependent coupling $D ( \omega )$, we can change the sum in Equation \ref{12.56} to an integral over the density of states:
$g (t) = \int d \omega \,W ( \omega ) D ( \omega ) \left( e^{- i \omega t} - 1 \right) \label{12.58}$
Here the product $W ( \omega )D ( \omega )$ is a coupling-weighted density of states, and is commonly referred to as a spectral density.
What this treatment does is provide a way of introducing a bath of states that the spectroscopically interrogated transition couples with. Coupling to a bath or continuum provides a way of introducing relaxation effects or damping of the electronic coherence in the absorption spectrum. You can see that if $g(t)$ is associated with a constant $\Gamma$, we obtain a Lorentzian lineshape with width $\Gamma$. This emerges under certain circumstances, for instance if the distribution of states and coupling is large and constant, and if the integral in Equation \ref{12.58} is over a distribution of low frequencies, such that $e^{- i \omega t} \approx 1 - i \omega t$. More generally the lineshape function is complex, and the real part describes damping and the imaginary part modulates the primary frequency and leads to fine structure. We will discuss these circumstances in more detail later.
An Ensemble at Finite Temperature
As described above, the single mode DHO model above is for a pure state, but the approach can be readily extended to describe a canonical ensemble. In this case, the correlation function is averaged over a thermal distribution of initial states. If we take the initial state of the system to be in the electronic ground state and its vibrational levels ($n_g$) to be occupied as a Boltzmann distribution, which is characteristic of ambient temperature samples, then the dipole correlation function can be written as a thermally averaged dephasing function:
$C _ {\mu \mu} (t) = \left| \mu _ {e g} \right|^{2} e^{- i \omega _ {g} t} F (t) \label{12.59}$
$F (t) = \sum _ {n _ {g}} p \left( n _ {g} \right) \left\langle n _ {g} \left| U _ {g}^{\dagger} U _ {e} \right| n _ {g} \right\rangle \label{12.60}$
$p \left( n _ {g} \right) = \frac {e^{- \beta \ln a _ {b}}} {Z} \label{12.61}$
Evaluating these expressions using the strategies developed above leads to
$C _ {\mu \mu} (t) = \left| \mu _ {e g} \right|^{2} e^{- i \omega _ {e g} t} \exp \left[ D \left[ ( \overline {n} + 1 ) \left( e^{- i \omega _ {0} t} - 1 \right) + \overline {n} \left( e^{+ i \omega _ {0} t} - 1 \right) \right] \right\rceil \label{12.62}$
$\overline {n}$ is the thermally averaged occupation number of the harmonic vibrational mode.
$\overline {n} = \left( e^{\beta \hbar \omega _ {0}} - 1 \right)^{- 1} \label{12.63}$
Note that in the low temperature limit, $\overline {n} \rightarrow 0$, and Equation \ref{12.62} equals our original result Equation \ref{12.32}. The dephasing function has two terms underlined in Equation \ref{12.62}, of which the first describes those electronic absorption events in which the vibrational quantum number increases or is unchanged ($n_e≥n_g$), whereas the second are for those processes where the vibrational quantum number decreases or is unchanged ($n_e≤n_g$). The latter are only allowed at elevated temperature where thermally excited states are populated and are known as “hot bands”.
Now, let’s calculate the lineshape. If we separate the dephasing function into a product of two exponential terms and expand each of these exponentials, we can Fourier transform to give
$\sigma _ {a b s} ( \omega ) = \left| \mu _ {e g} \right|^{2} e^{- D ( 2 \overline {n} + 1 )} \sum _ {j = 0}^{\infty} \sum _ {k = 0}^{\infty} \left( \frac {D^{j + k}} {j ! k !} \right) ( \overline {n} + 1 )^{j} \overline {n}^{k} \delta \left( \omega - \omega _ {e g} - ( j - k ) \omega _ {0} \right) \label{12.64}$
Here the summation over $j$ describes $n_e≥n_g$ transitions, whereas the summation over $k$ describes $n_e≤n_g$. For any one transition frequency, ($\omega \mathrm {eg}^{+} n \omega 0$), the net absorption is a sum over all possible combination of transitions at the energy splitting with $n=(j-k)$. Again, if we set $\overline {n} \rightarrow 0$, we obtain our original result Equation 13.1.47. The contributions where $k<j$ leads to the hot bands.
Examples of temperature dependence to lineshape and dephasing functions for $D = 1$. The real part changes in amplitude, growing with temperature, whereas the imaginary part is unchanged.
We can extend this description to describe coupling to a many independent nuclear modes or coupling to a continuum. We write the state of the system in terms of the electronic state and the nuclear quantum numbers, i.e., $| E \rangle = | e ; n _ {1} , n _ {2} , n _ {3} \dots \rangle$ and from that:
$F (t) = \exp \left[ \sum _ {j} D _ {j} \left[ \left( \overline {n} _ {j} + 1 \right) \left( e^{- i \omega _ {j} t} - 1 \right) + \overline {n} _ {j} \left( e^{i \omega _ {j} t} - 1 \right) \right] \right] \label{12.65}$
or changing to an integral over a continuous frequency distribution of normal modes characterized by a density of states, $W(\omega)$
$F (t) = \exp \left[ \int d \omega \, W ( \omega ) D ( \omega ) \left[ ( \overline {n} ( \omega ) + 1 ) \left( e^{- i \omega t} - 1 \right) + \overline {n} ( \omega ) \left( e^{i \omega t} - 1 \right) \right] \right] \label{12.66}$
$D ( \omega )$ is the frequency dependent coupling. Let’s look at the envelope of the nuclear structure on the transition by doing a short-time expansion on the complex exponential as in Equation 13.1.49
$F (t) = \exp \left[ \int d \omega \,D ( \omega ) W ( \omega ) \left( - i \omega t - ( 2 \overline {n} + 1 ) \frac {\omega^{2} t^{2}} {2} \right) \right] \label{12.67}$
The lineshape is calculated from
$\sigma _ {a b s} ( \omega ) = \int _ {- \infty}^{+ \infty} d t \,e^{i \left( \omega - \omega _ {e g} \right) t} \exp [ - i \langle \omega \rangle t ] \exp \left[ - \frac {1} {2} \left\langle \omega^{2} \right\rangle t^{2} \right] \label{12.68}$
where we have defined the mean vibrational excitation on absorption
\begin{align} \langle \omega \rangle & = \int d \omega \, W ( \omega ) D ( \omega ) \omega \[4pt] & = \lambda / \hbar \label{12.69} \end{align}
and
$\left\langle \omega^{2} \right\rangle = \int d \omega\, W ( \omega ) D ( \omega ) \omega^{2} ( 2 \overline {n} ( \omega ) + 1 ) \label{12.70}$
$\left\langle \omega^{2} \right\rangle$ reflects the thermally averaged distribution of accessible vibrational states. Completing the square of Equation \ref{12.68} gives
$\sigma _ {a b s} ( \omega ) = \left| \mu _ {e g} \right|^{2} \sqrt {\frac {2 \pi} {\left\langle \omega^{2} \right\rangle}} \exp \left[ \frac {- \left( \omega - \omega _ {e g} - \langle \omega \rangle \right)^{2}} {2 \left\langle \omega^{2} \right\rangle} \right] \label{12.71}$
The lineshape is Gaussian, with a transition maximum at the electronic resonance plus reorganization energy. Although the frequency shift $\langle \omega \rangle$ is not temperature dependent, the width of the Gaussian is temperature-dependent as a result of the thermal occupation factor in Equation \ref{12.70}.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/13%3A_Coupling_of_Electronic_and_Nuclear_Motion/13.02%3A_Coupling_to_a_Harmonic_Bath.txt
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In introducing the influence of dark degrees of freedom on the spectroscopy of a bright state, we made some approximations that are not always valid, such as the Condon approximation and the Second Cumulant Approximation. To develop tools that allow us to work outside of these approximations, it is worth revisiting the evaluation of the dipole correlation function and looking at this a bit more carefully. In particular, we will describe the semiclassical approximation, which is a useful representation of the dipole correlation function when one wants to describe the dark degrees of freedom (the bath) using classical molecular dynamics simulations.
For a quantum mechanical material system interacting with a light field, the full Hamiltonian is
$H = H _ {0} + V (t) \label{12.72}$
$V (t) = - \overline {m} \cdot \overline {E} (t) \label{12.73}$
$\overline {m} = \sum _ {i} z _ {i} \overline {r} _ {i}$ is the quantum mechanical dipole operator, where $z_i$ are charges. The absorption lineshape is given by the Fourier transformation of the dipole autocorrelation function $C _ {\mu \mu}$:
$C _ {\mu \mu} ( \tau ) = \langle \overline {m} (t) \overline {m} ( 0 ) \rangle = \operatorname {Tr} \left( \rho _ {e q} \overline {m} (t) \overline {m} ( 0 ) \right) \label{12.74)}$
and the time dependence in $\overline {m}$ is expressed in terms of the usual time-propagator:
$\overline {m} (t) = \hat {U} _ {0}^{\dagger} \overline {m} \hat {U} _ {0} \label{12.75}$
$\hat {U} _ {0} = e _ {+}^{- \frac {i} {\hbar} \int _ {0}^{t} H _ {0} (t) d t} \label{12.76}$
In principle, the time development of the dipole moment for all degrees of freedom can be obtained directly from ab initio molecular dynamics simulations.
For a more practical expression in which we wish to focus on one or a few bright degrees of freedom, we next partition the Hamiltonian into system and bath
$H _ {0} = H _ {S} ( Q ) + H _ {B} ( q ) + H _ {s b} ( Q , q ) \label{12.77}$
For purposes of spectroscopy, the system $H_S$ refers to those degrees of freedom ($Q$) with which the light will interacts, and which will be those in which we calculate matrix elements. The bath $H_B$ refers to all of the other degrees of freedom ($q$), and the interaction between the two is accounted for in $H_{SB}$. Although the interaction of the light depends on how $\overline {m}$ varies with $Q$, the dipole operator remains a function of system and bath coordinates: $\overline {m} ( Q , q )$.
We now use the interaction picture transformation to express the time propagator under the full material Hamiltonian $\hat {U} _ {0}$ in terms of a product of propagators in the individual terms in $H_0$:
$\hat {U} _ {0} = U _ {S} U _ {B} U _ {S B} \label{12.78}$
$\mathbf {H} _ {\mathrm {sB}} (t) = e^{i \left( H _ {S} + H _ {B} \right) t} H _ {S B} e^{- i \left( H _ {S} + H _ {B} \right) t} \label{12.79}$
$\mathbf {H} _ {\mathrm {sB}} (t) = e^{i \left( H _ {S} + H _ {B} \right) t} H _ {S B} e^{- i \left( H _ {S} + H _ {B} \right) t} \label{12.80}$
Then the dipole autocorrelation function becomes
$C _ {\mu \mu} = \sum _ {n} p _ {n} \left\langle n \left| U _ {S B}^{\dagger} U _ {B}^{\dagger} U _ {S}^{\dagger} \overline {m} U _ {S} U _ {B} U _ {S B} \overline {m} \right| n \right\rangle \label{12.81}$
where
$p _ {n} = \left\langle n \left| e^{- \beta H _ {0}} \right| n \right\rangle / \operatorname {Tr} \left( e^{- \beta H _ {0}} \right)$
Further, to make this practical, we make an adiabatic separation between the system and bath coordinates, and say that the interaction between the system and bath is weak. This allows us to write the state of the system as product states in the system ($a$) and bath ($\alpha$); $| n \rangle = | a , \alpha \rangle$:
$\left( H _ {S} + H _ {B} \right) | a , \alpha \rangle = \left( E _ {a} + E _ {\alpha} \right) | a , \alpha \rangle \label{12.82}$
With this we evaluate Equation \ref{12.81} as
\begin{align} C _ {\mu \mu} & = \sum _ {a , \alpha} p _ {a} p _ {\alpha} \left\langle a , \alpha \left| U _ {S B}^{\dagger} U _ {B}^{\dagger} U _ {S}^{\dagger} \overline {m} U _ {B} U _ {B B} \overline {m} \right| a , \alpha \right\rangle \ & = \sum _ {a , b} p _ {a} p _ {\alpha} \left\langle \alpha \left| \left\langle a \left| U _ {S B}^{\dagger} U _ {S}^{\dagger} U _ {B}^{\dagger} \overline {m} U _ {B} U _ {S} U _ {S B} \right| b \right\rangle \overline {m} _ {b a} \right| \alpha \right\rangle \label{12.83} \end{align}
where $\overline {m} _ {b a} = \langle b | \overline {m} | a \rangle$, and we have made use of the fact that $H_S$ and $H_B$ commute. Also,
$p _ {a} = e^{- E _ {j} / k T} / Q _ {s}.$
Now, by recognizing that the time propagators in the system and system-bath Hamiltonians describe time evolution at the system eigenstate energy plus any modulations that the bath introduces to it
$U _ {S} U _ {S B} | b \rangle = e^{- i H _ {s} t} | b \rangle e^{- i \int _ {0}^{t} d^{\prime} \delta E _ {b} \left( t^{\prime} \right)} = | b \rangle e^{- i \int _ {0}^{t} d t^{\prime} E _ {b} \left( t^{\prime} \right)} \label{12.84}$
and we can write our correlation function as
$C _ {\mu \mu} = \sum _ {a , b} p _ {a} p _ {\alpha} \left\langle \alpha \left| e^{i \int _ {0}^{t} d t E _ {a} \left( t^{\prime} \right)} U _ {B}^{\dagger} \overline {m} _ {a b} U _ {B} e^{- i \int _ {0}^{t} d t^{\prime} E _ {b} \left( t^{\prime} \right)} \overline {m} _ {b a} \right| \alpha \right\rangle \label{12.85}$
$C _ {\mu \mu} = \left\langle \overline {m} _ {a b} (t) \overline {m} _ {b a} ( 0 ) e^{- i \int _ {0}^{t} d t^{\prime} \omega _ {b a} \left( t^{\prime} \right)} \right\rangle _ {B} \label{12.86}$
$\overline {m} _ {a b} (t) = e^{- i H _ {B} t} \overline {m} _ {a b} e^{- i H _ {B} t} \label{12.87}$
Equation \ref{12.86} is the first important result. It describes a correlation function in the dipole operator expressed in terms of an average over the time-dependent transition moment, including its orientation, and the fluctuating energy gap. The time dependence is due to the bath and refers to a trace over the bath degrees of freedom.
Let’s consider the matrix elements. These will reflect the strength of interaction of the electromagnetic field with the motion of the system coordinate, which may also be dependent on the bath coordinates. Since we have made an adiabatic approximation, to evaluate the matrix elements we would typically expand the dipole moment in the system degrees of freedom, $Q$. As an example for one system coordinate ($Q$) and many bath coordinates $q$, we can expand:
$\overline {m} ( Q , q ) = \overline {m} _ {0} + \frac {\partial \overline {m}} {\partial Q} Q + \sum _ {\alpha} \frac {\partial^{2} \overline {m}} {\partial Q \partial q _ {\alpha}} Q q _ {\alpha} + \cdots \label{12.88}$
$\overline {m} _ {0}$ is the permanent dipole moment, which we can take as a constant. In the second term, $\partial \overline {m} / \partial Q$ is the magnitude of the transition dipole moment. The third term includes the dependence of the transition dipole moment on the bath degrees of freedom, i.e., non-Condon terms. So now we can evaluate
\left.\begin{aligned} \overline {m} _ {a b} & = \left\langle a \left| \overline {m} _ {0} + \frac {\partial \overline {m}} {\partial Q} Q + \sum _ {\alpha} \frac {\partial^{2} \overline {m}} {\partial Q \partial q _ {\alpha}} Q q _ {\alpha} \right| b \right\rangle \ & = \frac {\partial \overline {m}} {\partial Q} \langle a | Q | b \rangle + \sum _ {\alpha} \frac {\partial} {\partial q _ {\alpha}} \frac {\partial \overline {m}} {\partial Q} \langle a | Q | b \rangle q _ {\alpha} \end{aligned} \right. \label{12.89}
We have set $\left\langle a \left| \overline {m} _ {0} \right| b \right\rangle = 0$. Now defining the transition dipole matrix element,
$\overline {\mu} _ {a b} = \frac {\partial \overline {m}} {\partial Q} \langle a | Q | b \rangle \label{12.90}$
we can write
$\overline {m} _ {a b} = \overline {\mu} _ {a b} \left( 1 + \sum _ {\alpha} \frac {\partial \overline {\mu} _ {a b}} {\partial q _ {\alpha}} q _ {\alpha} \right) \label{12.91}$
Remember that $\overline {\mu} _ {a b}$ is a vector. The bath can also change the orientation of the transition dipole moment. If we want to separate the orientational and remaining dynamics this we could split the matrix element into an orientational component specified by a unit vector along $\partial \overline {m} / \partial Q$ and a scalar that encompasses the amplitude factors: $\overline {\mu} _ {a b} = \hat {u} _ {a b} \mu _ {a b}$. Then Equation \ref{12.86} becomes
$\overline {m} _ {a b} = \overline {\mu} _ {a b} \left( 1 + \sum _ {\alpha} \frac {\partial \overline {\mu} _ {a b}} {\partial q _ {\alpha}} q _ {\alpha} \right) \label{12.92}$
Mixed quantum-classical spectroscopy models apply a semiclassical approximation to Equation \ref{12.86}. Employing the semiclassical approximation says that we will replace the quantum mechanical operator mab (t) with a classicalMab (t), i.e., we replace the time propagator $U_B$ with classical propagation of the dynamics. Also, the trace over the bath in the correlation function becomes an equilibrium ensemble average over phase space.
How do you implement the semiclassical approximation? Replacing the time propagator $U_B$ with classical dynamics amounts to integrating Newton’s equations for all of the bath degrees of freedom. Then you must establish how the bath degrees of freedom influence $\omega _ {b a} (t)$ and $\overline {m} _ {a b} (t)$. For the quantum operator $\overline {m} ( Q , q , t )$, only the system coordinate $Q$ remains quantized, and following Equation \ref{12.91} we can express the orientation and magnitude of the dipole moment and the dynamics depends on the classical degrees of freedom $\tilde {q} _ {\alpha}$.
$\overline {m} _ {a b} = \overline {\mu} _ {a b} \left( 1 + \sum _ {\alpha} a _ {\alpha} \tilde {q} _ {\alpha} \right) \label{12.93}$
$a _ {\alpha}$ is a (linear) mapping coefficient
$a _ {\alpha} = \partial \overline {\mu} _ {a b} / \partial \tilde {q} _ {\alpha}$
between the bath and the transition dipole moment.
In practice, use of this approximation has been handled in different ways, but practical considerations have dictated that $\omega _ {b a} (t)$ and $\overline {m} _ {a b} (t)$ are not separately calculated for each time step, but are obtained from a mapping of these variables to the bath coordinates $q$. This mapping may be to local or collective bath coordinates, and to as many degrees of freedom as are necessary to obtain a highly correlated single valued mapping of $\omega _ {b a} (t)$ and $\overline {m} _ {a b} (t)$. Examples of these mappings include correlating $\omega_{ba}$ with the electric field of the bath acting on the system coordinate.
Appendix
Let’s evaluate the dipole correlation function for an arbitrary HSB and an arbitrary number of system eigenstates. From Equation \ref{12.83} we have
$C _ {\mu \mu} = \sum _ {a b c d \ \alpha} p _ {a} p _ {\alpha} \left\langle \alpha \left| \left\langle a \left| U _ {s B}^{\dagger} \right| c \right\rangle U _ {B}^{\dagger} \left\langle c \left| U _ {s}^{\dagger} \overline {m} U _ {s} \right| d \right\rangle U _ {B} \left\langle d \left| U _ {s B} \right| b \right\rangle \langle b | \overline {m} | a \rangle \right| \alpha \right\rangle \label{12.94}$
$\left\langle c \left| U _ {s}^{\dagger} \overline {m} U _ {S} \right| d \right\rangle = e^{- i \left( E _ {d} - E _ {c} \right) t} \overline {m} _ {c d} \label{12.95}$
$\overline {m} _ {c d} (t) = U _ {B}^{\dagger} \overline {m} _ {c d} U _ {B} \label{12.96}$
$\left\langle a \left| U _ {S B}^{\dagger} \right| c \right\rangle = \left\langle a \left| e^{i \int _ {0}^{t} d t^{\prime} \mathbf {H} _ {\mathrm {sb}} \left( t^{\prime} \right)} \right| c \right\rangle = \exp \left[ i \int _ {0}^{t} d t^{\prime} \left[ \mathbf {H} _ {\mathrm {sB}} \right] _ {a c} \left( t^{\prime} \right) \right] \label{12.97}$
\begin{align} C _ {\mu \mu} &= \sum _ {c \kappa t d} p _ {a} \left \langle e^{- i \omega _ {d c} t} e^{i \int _ {0}^{t} d t^{\prime} \left[ H _ {S B} \right] _ {a c}} \overline {m} _ {c d} e^{- i \int _ {0}^{t} d t^{\prime} \left[ H _ {S B} \right] _ {d b} \left( t^{\prime} \right)} \overline {m} _ {b a} \right \rangle _ {B} \label{12.98} \[4pt] &= \left\langle \overline {m} _ {c d} (t) \overline {m} _ {b a} ( 0 ) \exp \left[ - i \omega _ {d c} t - i \int _ {0}^{t} d t^{\prime} \left[ H _ {S B} \right] _ {d b} \left( t^{\prime} \right) - \left[ H _ {S B} \right] _ {a c} \left( t^{\prime} \right) \right] \right\rangle _ {B} \label{12.99} \end{align}
Readings
1. Auer, B. M.; Skinner, J. L., Dynamical effects in line shapes for coupled chromophores: Time-averaging approximation. J. Chem. Phys. 2007, 127 (10), 104105.
2. Corcelli, S. A.; Skinner, J. L., Infrared and Raman Line Shapes of Dilute HOD in Liquid H2O and D2O from 10 to 90 °C. J. Phys. Chem. A 2005, 109 (28), 6154-6165.
3. Gorbunov, R. D.; Nguyen, P. H.; Kobus, M.; Stock, G., Quantum-classical description of the amide I vibrational spectrum of trialanine. J. Chem. Phys. 2007, 126 (5), 054509.
4. Mukamel, S., Principles of Nonlinear Optical Spectroscopy. Oxford University Press: New York, 1995.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/13%3A_Coupling_of_Electronic_and_Nuclear_Motion/13.03%3A_Semiclassical_Approximation_to_the_Dipole_Correlation_.txt
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Here we will describe how fluctuations are observed in experimental observables, as is common to experiments in molecular condensed phases. As our example, we will focus on absorption spectroscopy and how environmentally induced dephasing influences the absorption lineshape. Our approach will be to calculate a dipole correlation function for transition dipole interacting with a fluctuating environment, and show how the time scale and amplitude of fluctuations are encoded in the lineshape. Although the description here is for the case of a spectroscopic observable, the approach can be applied to any such problems in which the deterministic motions of an internal variable of a quantum system are influenced by a fluctuating environment.
We also aim to establish a connection between this problem and the Displaced Harmonic Oscillator model. Specifically, we will show that a frequency-domain representation of the coupling between a transition and a continuous distribution of harmonic modes is equivalent to a time-domain picture in which the transition energy gap fluctuates about an average frequency with a statistical time scale and amplitude given by the distribution of coupled modes. Thus an absorption spectrum is merely a spectral representation of the dynamics experienced by a experimentally probed transition.
• 14.1: Fluctuations and Randomness - Some Definitions
For chemical problems in the condensed phase we constantly come up against the problem of random fluctuations to dynamical variables as a result of their interactions with their environment. It is unreasonable to think that you will come up with an equation of motion for the internal deterministic variable, but we should be able to understand the behavior statistically and come up with equations of motion for probability distributions. Models of this form are commonly referred to as stochastic.
• 14.2: Line-Broadening and Spectral Diffusion
The interactions of this chromophore with its environment can shift it resonance frequency. In condensed matter, time-dependent interactions with the surroundings can lead to time-dependent frequency shifts, known as spectral diffusion. How these dynamics influence the line width and lineshape of absorption features depends on the distribution of frequencies available to your system and the time scale of sampling varying environments.
• 14.3: Gaussian-Stochastic Model for Spectral Diffusion
We will begin with a classical description of how random fluctuations in frequency influence the absorption lineshape, by calculating the dipole correlation function for the resonant transition. This is a Gaussian stochastic model for fluctuations, meaning that we will describe the time-dependence of the transition energy as random fluctuations about an average value through a Gaussian distribution.
• 14.4: The Energy Gap Hamiltonian
In describing fluctuations in a quantum mechanical system, we describe how an experimental observable is influenced by its interactions with a thermally agitated environment. For this, we work with the specific example of an electronic absorption spectrum and return to the Displaced Harmonic Oscillator model.
• 14.5: Correspondence of Harmonic Bath and Stochastic Equations of Motion
So, why does the mathematical model for coupling of a system to a harmonic bath give the same results as the classical stochastic equations of motion for fluctuations? Why does coupling to a continuum of bath states have the same physical manifestation as perturbation by random fluctuations? The answer is that in both cases, we really have imperfect knowledge of the behavior of all the particles present. Observing a small subset of particles will have dynamics with a random character.
14: Fluctuations in Spectroscopy
“Fluctuations” is my word for the time-evolution of a randomly perturbed system at or near equilibrium. For chemical problems in the condensed phase we constantly come up against the problem of random fluctuations to dynamical variables as a result of their interactions with their environment. It is unreasonable to think that you will come up with an equation of motion for the internal deterministic variable, but we should be able to understand the behavior statistically and come up with equations of motion for probability distributions. Models of this form are commonly referred to as stochastic. A stochastic equation of motion is one which includes a random component to the time-development.
When we introduced correlation functions, we discussed the idea that a statistical description of a system is commonly formulated in terms of probability distribution functions $P$. Observables are commonly described by moments of this distribution, which are obtained by integrating over $P$, for instance
\left.\begin{aligned} \langle x \rangle & = \int d x \,x \mathrm {P} (x) \ \left\langle x^{2} \right\rangle & = \int d x \,x^{2} \mathrm {P} (x) \end{aligned} \right. \label{13.1}
For time-dependent processes, we recognize that it is possible that the probability distribution carries a time-dependence.
\begin{align} \langle x (t) \rangle & = \int d x \,x (t) \mathrm {P} ( x , t ) \ \left\langle x^{2} (t) \right\rangle & = \int d x \,x^{2} (t) \mathrm {P} ( x , t ) \label{13.2} \end{align}
Correlation functions go a step further and depend on joint probability distributions $\mathrm {P} \left( t^{\prime \prime} , A ; t^{\prime} , B \right)$ that give the probability of observing a value of $A$ at time $t''$ and a value of $B$ at time $t'$:
$\left\langle A \left( t^{\prime \prime} \right) B \left( t^{\prime} \right) \right\rangle = \int d A \int d B \, A B \,\mathrm {P} \left( t^{\prime \prime} , A ; t^{\prime} , B \right)\label{13.3}$
The statistical description of random fluctuations are described through these time-dependent probability distributions, and we need a stochastic equation of motion to describe their behavior. A common example of such a process is Brownian motion, the fluctuating position of a particle under the influence of a thermal environment.
It is not practical to describe the absolute position of the particle, but we can formulate an equation of motion for the probability of finding the particle in time and space given that you know its initial position. Working from a random walk model, one can derive an equation of motion that takes the form of the well-known diffusion equation, here written in one dimension:
$\dfrac {\partial \mathrm {P} ( x , t )} {\partial t} = \mathcal {D} \dfrac {\partial^{2}} {\partial x^{2}} \mathrm {P} ( x , t ) \label{13.4}$
Here $\mathcal {D}$ is the diffusion constant which sets the time scale and spatial extent of the random motion. [Note the similarity of this equation to the time-dependent Schrödinger equation for a free particle if $\mathcal {D}$ is taken as imaginary]. Given the initial condition $\mathrm {P} \left( x , t _ {0} \right) = \delta \left( x - x _ {0} \right)$, the solution is a conditional probability density
$\mathrm {P} \left( x , t ; x _ {0} , t _ {0} \right) = \dfrac {1} {\sqrt {2 \pi \mathcal {D} \left( t - t _ {0} \right)}} \exp \left( - \dfrac {\left( x - x _ {0} \right)^{2}} {4 \mathcal {D} \left( t - t _ {0} \right)} \right) \label{13.5}$
The probability distribution describes the statistics for fluctuations in the position of a particle averaged over many trajectories. Analyzing the moments of this probability density using Equation \ref{13.2} we find that
\begin{align} \langle x (t) \rangle & = x _ {0} \[4pt] \left\langle \delta x (t)^{2} \right\rangle & = 2 \mathcal {D} t .\end{align}
where
$\delta x (t) = x (t) - x _ {0}$
So, the distribution maintains a Gaussian shape centered at $x_0$, and broadens with time as $2\mathcal {D}t$.
Brownian motion is an example of a Gaussian-Markovian process. Here Gaussian refers to cases in which we describe the probability distribution for a variable $P(x)$ as a Gaussian normal distribution. Here in one dimension:
\begin{align} \mathrm {P} (x) &= A e^{- \left( x - x _ {0} \right)^{2} / 2 \Delta^{2}} \[4pt] \Delta^{2} &= \left\langle x^{2} \right\rangle - \langle x \rangle^{2} \label{13.6} \end{align}
The Gaussian distribution is important, because the central limit theorem states that the distribution of a continuous random variable with finite variance will follow the Gaussian distribution. Gaussian distributions also are completely defined in terms of their first and second moments, meaning that a time-dependent probability density $P(x,t)$ is uniquely characterized by a mean value in the observable variable $x$ and a correlation function that describes the fluctuations in $x$. Gaussian distributions for systems at thermal equilibrium are also important for the relationship between Gaussian distributions and parabolic free energy surfaces:
$G (x) = - k _ {B} T \ln \mathrm {P} (x) \label{13.7}$
If the probability density is Gaussian along $x$, then the system’s free energy projected onto this coordinate (often referred to as a potential of mean force) has a harmonic shape. Thus Gaussian statistics are effective for describing fluctuations about an equilibrium mean value $x_o$.
Markovian means that the time-dependent behavior of a system does not depend on its earlier history, statistically speaking. Naturally the state of any one molecule depends on its trajectory through phase space, however we are saying that from the perspective of an ensemble there is no memory of the state of the system at an earlier time. This can be stated in terms of joint probability functions as
$\mathrm {P} \left( x _ {2} , t _ {2} ; x _ {1} , t _ {1} ; x _ {0} , t _ {0} \right) = \mathrm {P} \left( x _ {2} , t _ {2} ; x _ {1} , t _ {1} \right) \mathrm {P} \left( x _ {1} , t _ {1} ; x _ {0} , t _ {0} \right)$
or
$\mathrm {P} \left( t _ {2} ; t _ {1} ; t _ {0} \right) = \mathrm {P} \left( t _ {2} ; t _ {1} \right) \mathrm {P} \left( t _ {1} ; t _ {0} \right)$
The probability of observing a trajectory that takes you from state 1 at time 1 to state 2 at time 2 does not depend on where you were at time 0. Further, given the knowledge of the probability of executing changes during a single time interval, you can exactly describe $P$ for any time interval. Markovian therefore refers to time-dependent processes on a time scale long compared to correlation time for the internal variable that you care about. For instance, the diffusion equation only holds after the particle has experienced sufficient collisions with its surroundings that it has no memory of its earlier position and momentum: $t > \tau_c$.
Readings
1. Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006; Ch. 1.5 and Ch. 7.
14.02: Line-Broadening and Spectral Diffusion
We will investigate how a fluctuating environment influences measurements of an experimentally observed internal variable. Specifically we focus on the spectroscopy of a chromophore, and how the chromophore’s interactions with its environment influence its transition frequency and absorption lineshape. In the absence of interactions, the resonance frequency that we observe is $\omega_{eg}$. However, we have seen that interactions of this chromophore with its environment can shift this frequency. In condensed matter, time-dependent interactions with the surroundings can lead to time-dependent frequency shifts, known as spectral diffusion. How these dynamics influence the line width and lineshape of absorption features depends on the distribution of frequencies available to your system ($\Delta$) and the time scale of sampling varying environments ($\tau_c$). Consider the following cases of line broadening:
1. Homogeneous. Here, the absorption lineshape is dynamically broadened by rapid variations in the frequency or phase of dipoles. Rapid sampling of a distribution of frequencies acts to average the experimentally observed resonance frequency. The result in a “motionally narrowed” line width that is narrower than the distribution of frequencies available and proportional to the rate of fluctuation induced dephasing.
2. Inhomogeneous. In this limit, the lineshape reflects a static distribution of resonance frequencies, and the width of the line represents the distribution of frequencies, ', which arise, from different structural environments available to the system.
3. Spectral Diffusion. More generally, every system lies between these limits. Given a distribution of configurations that the system can adopt, for instance an electronic chromophore in a liquid, an equilibrium system will be ergodic, and over a long enough time any molecule will sample all configurations available to it. Under these circumstances, we expect that every molecule will have a different “instantaneous frequency” $\omega_i(t)$ which evolves in time as a result of its interactions with a dynamically evolving system. This process is known as spectral diffusion. The homogeneous and inhomogeneous limits can be described as limiting forms for the fluctuations of a frequency $\omega_i(t)$ through a distribution of frequencies $\Delta$. If $\omega_i(t)$ evolves rapidly relative to $\Delta^{-1}$, the system is homogeneously broadened. If $\omega_i(t)$ evolves slowly the system is inhomogeneous broadened. This behavior can be quantified through the transition frequency time-correlation function $C _ {e g} (t) = \left\langle \omega _ {e g} (t) \omega _ {e g} ( 0 ) \right\rangle \label{13.8}$ Our job will be to relate the transition frequency correlation function $C _ {e g} (t)$ with the dipole correlation function that determines the lineshape, $C _ {\mu \mu} (t)$.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/14%3A_Fluctuations_in_Spectroscopy/14.01%3A_Fluctuations_and_Randomness_-_Some_Definitions.txt
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We will begin with a classical description of how random fluctuations in frequency influence the absorption lineshape, by calculating the dipole correlation function for the resonant transition. This is a Gaussian stochastic model for fluctuations, meaning that we will describe the time-dependence of the transition energy as random fluctuations about an average value through a Gaussian distribution.
\begin{align} \omega (t) &= \langle \omega \rangle + \delta \omega (t) \label{13.9} \[4pt] \langle \delta \omega (t) \rangle &= 0 \label{13.10} \end{align}
The fluctuations in $\omega$ allow the system to explore a Gaussian distribution of transitions frequencies characterized by a variance:
$\Delta = \sqrt {\left\langle \omega^{2} \right\rangle - \langle \omega \rangle^{2}} = \sqrt {\left\langle \delta \omega^{2} \right\rangle} \label{13.11}$
In many figures the width of the Gaussian distribution is labeled with the standard deviation (here $\Delta$). This is meant to symbolize that $\Delta$ is the parameter that determines the width, and not that it is the line width. For Gaussian distributions, the full line width at half maximum amplitude (FWHM) is $2.35 \Delta$.
The time scales for the frequency shifts will be described in terms of a frequency correlation function
$C _ {\delta \omega s \omega} (t) = \langle \delta \omega (t) \delta \omega ( 0 ) \rangle \label{13.12}$
Furthermore, we will describe the time scale of the random fluctuations through a correlation time $\tau_c$.
The absorption lineshape is described with a dipole time-correlation function. Let’s treat the dipole moment as an internal variable to the system, whose value depends on that of $\omega$. Qualitatively, it is possible to write an equation of motion for $\mu$ by associating the dipole moment with the displacement of a bound particle ($x$) times its charge, and using our intuition regarding how the system behaves. For a unperturbed state, we expect that $x$ will oscillate at a frequency $\omega$, but with perturbations, it will vary through the distribution of available frequencies. One function that has this behavior is
$x (t) = x _ {0} e^{- i \omega (t) t} \label{13.13}$
If we differentiate this equation with respect to time and multiply by charge we have
$\frac {\partial \mu} {\partial t} = - i \omega (t) \mu (t) \label{13.14}$
Although it is a classical equation, note the similarity to the quantum Heisenberg equation for the dipole operator:
$\partial \mu / \partial t = i H (t) \mu / \hbar + h . c .$
The correspondence of $\omega (t)$ with $H (t) / \hbar$ offers some insight into how the quantum version of this problem will look.
The solution to Equation \ref{13.14} is
$\mu (t) = \mu ( 0 ) \exp \left[ - i \int _ {0}^{t} d \tau \, \omega ( \tau ) \right] \label{13.15}$
Substituting this expression and Equation \ref{13.9} into the dipole correlation function gives or
$C _ {\mu \mu} (t) = | \mu |^{2} e^{- i \langle \omega \rangle t} F (t) \label{13.16}$
where
$F (t) = \left\langle \exp \left[ - i \int _ {0}^{t} d \tau \, \delta \omega ( \tau ) \right] \right\rangle \label{13.17}$
The dephasing function ($F(t)$) is obtained by performing an equilibrium average of the exponential argument over fluctuating trajectories. For ergodic systems, this is equivalent to averaging long enough over a single trajectory.
The dephasing function is a bit complicated to work with as written. However, for the case of Gaussian statistics for the fluctuations, it is possible to simplify $F(t)$ by expanding it as a cumulant expansion of averages (see appendix below for details).
$F (t) = \exp \left[ - i \int _ {0}^{t} d \tau^{\prime} \, \left\langle \delta \omega \left( \tau^{\prime} \right) \right\rangle + \frac {i^{2}} {2 !} \int _ {0}^{t} d \tau^{\prime} \, \int _ {0}^{t} d \tau^{\prime \prime} \, \left\{\left\langle \delta \omega \left( \tau^{\prime} \right) \delta \omega \left( \tau^{\prime \prime} \right) \right\rangle - \left\langle \delta \omega \left( \tau^{\prime} \right) \right\rangle \left\langle \delta \omega \left( \tau^{\prime \prime} \right) \right\rangle \right\} \right] \label{13.18}$
In this expression, the first term is zero, since $\langle \delta \omega \rangle = 0$. Only the second term survives for a system with Gaussian statistics. Now recognizing that we have a stationary system, we have
$F (t) = \exp \left[ - \frac {1} {2} \int _ {0}^{t} d \tau^{\prime} \, \int _ {0}^{t} d \tau^{\prime \prime} \, \left\langle \delta \omega \left( \tau^{\prime} - \tau^{\prime \prime} \right) \delta \omega ( 0 ) \right\rangle \right] \label{13.19}$
We have rewritten the dephasing function n terms of a correlation function that describes the fluctuating energy gap. Note that this is a classical exception, so there is no time-ordering to the exponential. $F(t)$ can be rewritten through a change of variables ($\tau = \tau^{\prime} - \tau^{\prime \prime}$):
$F (t) = \exp \left[ - \int _ {0}^{t} d \tau ( t - \tau ) \langle \delta \omega ( \tau ) \delta \omega ( 0 ) \rangle \right] \label{13.20}$
So the Gaussian stochastic model allows the influence of the frequency fluctuations on the lineshape to be described by $C _ {\delta \omega \delta v} (t)$ a frequency correlation function that follows Gaussian statistics. Note, we are now dealing with two different correlation functions $C _ {\delta \omega \delta \omega}$ and $C _ {\mu \mu}$. The frequency correlation function encodes the dynamics that result from molecules interacting with the surroundings, whereas the dipole correlation function describes how the system interacts with a light field and thereby the absorption spectrum.
Now, we will calculate the lineshape assuming that $C _ {\delta \omega \delta \omega}$ decays with a correlation time $\tau_c$ and takes on an exponential form
$C _ {\delta \omega \delta \omega} (t) = \Delta^{2} \exp \left[ - t / \tau _ {c} \right] \label{13.21}$
Then Equation \ref{13.20} gives
$F (t) = \exp \left[ - \Delta^{2} \tau _ {c}^{2} \left( \exp \left( - t / \tau _ {c} \right) + t / \tau _ {c} - 1 \right) \right] \label{13.22}$
which is in the form we have seen earlier $F (t) = \exp ( - g (t) )$
$g (t) = \Delta^{2} \tau _ {c}^{2} \left( \exp \left( - t / \tau _ {c} \right) + t / \tau _ {c} - 1 \right) \label{13.23}$
to interpret this lineshape function, let’s look at its limiting forms.
Long correlation times ($t \ll \tau_c$)
This corresponds to the inhomogeneous case where $C _ {\delta \omega \delta \omega} (t) = \Delta^{2}$, a constant. For $t \ll \tau _ {c}$, we can perform a short time expansion of exponential
$e^{- t / \tau _ {c}} \approx 1 - \frac {t} {\tau _ {c}} + \frac {t^{2}} {2 \tau _ {c}^{2}} + \ldots \label{13.24}$
and from Equation \ref{13.23} we obtain
$g (t) = \Delta^{2} t^{2} / 2 \label{13.25}$
At short times, the dipole correlation function will have a Gaussian decay with a rate given by $\Delta^{2}$:
$F (t) = \exp \left( - \Delta^{2} t^{2} / 2 \right)$
This has the proper behavior for a classical correlation function, i.e., even in time
$C _ {\mu \mu} (t) = C _ {\mu \mu} ( - t ).$
In this limit, the absorption lineshape is:
\begin{align} \sigma ( \omega ) & = | \mu |^{2} \int _ {- \infty}^{+ \infty} d t \, e^{i \omega t} e^{- i ( \omega ) t - g (t)} \ & = | \mu |^{2} \int _ {- \infty}^{+ \infty} d t \, e^{i ( \omega - ( \omega ) ) t} e^{- \Delta^{2} t^{2} / 2} \ & = \sqrt {\frac {2 \pi} {\Delta^{2}}} | \mu |^{2} \exp \left( - \frac {( \omega - \langle \omega \rangle )^{2}} {2 \Delta^{2}} \right) \label{13.26} \end{align}
We obtain a Gaussian inhomogeneous lineshape centered at the mean frequency with a width dictated by the frequency distribution.
Short Correlation Times ($t \gg \tau_c$)
This corresponds to the homogeneous limit in which you can approximate
$C _ {\delta \omega \delta \omega} (t) = \Delta^{2} \delta (t).$
For $t \gg \tau _ {c}$ we set $e^{- t / \tau _ {c}} \approx 0$, $t / \tau _ {c} > > 1$, and Equation \ref{13.23} gives
$g (t) = - \Delta^{2} \tau _ {c} t \label{13.27}$
If we define the constant
$\Delta^{2} \tau _ {c} \equiv \Gamma \label{13.28}$
we see that the dephasing function has an exponential decay:
$F (t) = \exp [ - \Gamma t ] \label{13.29}$
The lineshape for short correlation times (or fast fluctuations) takes on a Lorentzian shape
$\begin{array} {c} {\sigma ( \omega ) = | \mu |^{2} \int _ {- \infty}^{+ \infty} d t \, e^{i ( \omega - \langle \omega ) t} e^{- \Gamma t}} \ {\operatorname {Re} \sigma ( \omega ) = | \mu |^{2} \frac {\Gamma} {( \omega - \langle \omega \rangle )^{2} + \Gamma^{2}}} \end{array} \label{13.30}$
This represents the homogeneous limit. Even with a broad distribution of accessible frequencies, if the system explores all of these frequencies on a time scale fast compared to the inverse of the distribution ($\Delta \tau _ {\mathrm {c}} > 1$, then the resonance will be “motionally narrowed” into a Lorentzian line.
More generally, the envelope of the dipole correlation function will look Gaussian at short times and exponential at long times.
The correlation time is the separation between these regimes. The behavior for varying time scales of the dynamics ($\tau_c$) are best characterized with respect to the distribution of accessible frequencies ($\Delta$). So we can define a factor
$\kappa = \Delta \cdot \tau _ {c} \label{13.31}$
$\kappa \ll 1$ is the fast modulation limit and $\kappa \gg 1$ is the slow modulation limit. Let’s look at how $C _ {\delta o \delta \omega}$, $F (t)$, and $\sigma _ {a b s} ( \omega )$ change as a function of $\kappa$.
We see that for a fixed distribution of frequencies $\Delta$ the effect of increasing the time scale of fluctuations through this distribution (decreasing $\tau_c$) is to gradually narrow the observed lineshape from a Gaussian distribution of static frequencies with width (FWHM) of $2.35\Delta$ to a motionally narrowed Lorentzian lineshape with width (FWHM) of
$\Delta^{2} \tau _ {c} / \pi = \Delta \cdot \kappa / \pi.$
This is analogous to the motional narrowing effect first described in the case of temperature dependent NMR spectra of two exchanging species. Assume we have two resonances at $\omega_A$ and $\omega_B$ associated with two chemical species that are exchanging at a rate $k_{AB}$
$\ce{A <=> B}$
If the rate of exchange is slow relative to the frequency splitting, $k _ {A B} < < \omega _ {A} - \omega B$ then we expect two resonances, each with a linewidth dictated by the molecular relaxation processes ($T_2$) and transfer rate of each species. On the other hand, when the rate of exchange between the two species becomes faster than the energy splitting, then the two resonances narrow together to form one resonance at the mean frequency.
Appendix: The Cumulant Expansion
For a statistical description of the random variable $x$, we wish to characterize the moments of $x$: $\langle x \rangle$, $\langle x^2 \rangle$, .... Then the average of an exponential of $x$ can be expressed as an expansion in moments
$\underbrace{\left\langle e^{i k x} \right\rangle = \sum _ {n = 0}^{\infty} \frac {( i k )^n} {n !} \left\langle x^n \right\rangle}_{\text{expansion in moments}} \label{13.31A}$
An alternate way of expressing this expansion is in terms of cumulants
$\underbrace{\left\langle e^{i k x} \right\rangle = \exp \left( \sum _ {n = 1}^{\infty} \frac {( i k )^{n}} {n !} c _ {n} (x) \right)}_{\text{expansion in cumulants}} \label{13.32}$
where the first few cumulants are:
\begin{align} c _ {1} (x) &= \langle x \rangle \tag{mean} \label{13.33} \[4pt] c _ {2} (x) &= \left\langle x^{2} \right\rangle - \langle x \rangle^{2} \label{13.34} \tag{variance} \[4pt] c _ {3} (x) &= \left\langle x^{3} \right\rangle - 3 \langle x \rangle \left\langle x^{2} \right\rangle + 2 \langle x \rangle^{3} \tag{skewness} \label{13.35} \end{align}
An expansion in cumulants converges much more rapidly than an expansion in moments, particularly when you consider that $x$ may be a time-dependent variable. Particularly useful is the observation that all cumulants with $n > 2$ vanish for a system that obeys Gaussian statistics.
We obtain the cumulants above by expanding Equation \ref{13.31} and \ref{13.32}, and comparing terms in powers of $x$. We start by postulating that, instead of expanding the exponential directly, we can instead expand the exponential argument in powers of an operator or variable $H$
$F = \exp [ c ] = 1 + c + \frac {1} {2} c^{2} + \cdots \label{13.36}$
$c = c _ {1} H + \frac {1} {2} c _ {2} H^{2} + \cdots \label{13.37}$
Inserting Equation \ref{13.37} into Equation \ref{13.36} and collecting terms in orders of $H$ gives
\begin{aligned} F & = 1 + \left( c _ {1} H + \frac {1} {2} c _ {2} H^{2} + \cdots \right) + \frac {1} {2} \left( c _ {1} H + \frac {1} {2} c _ {2} H^{2} + \cdots \right)^{2} + \cdots \ & = 1 + \left( c _ {1} \right) H + \frac {1} {2} \left( c _ {2} + c _ {1}^{2} \right) H^{2} + \cdots \end{aligned} \label{13.38}
Now comparing this with the expansion of the exponential operator (of $H$)
\begin{align} F & = \exp [ f H ] \ & = 1 + f _ {1} H + \frac {1} {2} f _ {2} H^{2} + \cdots \label{13.39} \end{align}
allows one to see that
$\begin{array} {l} {c _ {1} = f _ {1}} \ {c _ {2} = f _ {2} - f _ {1}^{2}} \end{array} \label{13.40}$
The cumulant expansion can also be applied to time-correlations. Applying this to the time-ordered exponential operator we obtain:
\begin{align} F (t) & = \left\langle \exp _ {+} \left[ - i \int _ {0}^{t} d t \omega (t) \right] \right\rangle \ & \approx \exp \left[ c _ {1} (t) + c _ {2} (t) \right] \label{13.42} \end{align}
\begin{aligned} c _ {1} & = - i \int _ {0}^{t} d \tau \langle \omega ( \tau ) \rangle \ c _ {2} & = - \int _ {0}^{t} d \tau _ {2} \int _ {0}^{\tau _ {2}} d \tau _ {1} \left\{\left\langle \omega \left( \tau _ {2} \right) \omega \left( \tau _ {1} \right) \right\rangle - \left\langle \omega \left( \tau _ {2} \right) \right\rangle \left\langle \omega \left( \tau _ {1} \right) \right\rangle \right\} \ & = - \int _ {0}^{t} d \tau _ {2} \int _ {0}^{\tau _ {2}} d \tau _ {1} \left\langle \delta \omega \left( \tau _ {2} \right) \delta \omega \left( \tau _ {1} \right) \right\rangle \end{aligned} \label{13.43}
For Gaussian statistics, all higher cumulants vanish.
Readings
1. Kubo, R., A Stochastic Theory of Line-Shape and Relaxation. In Fluctuation, Relaxation and Resonance in Magnetic Systems, Ter Haar, D., Ed. Oliver and Boyd: Edinburgh, 1962; pp 23- 68.
2. McHale, J. L., Molecular Spectroscopy. 1st ed.; Prentice Hall: Upper Saddle River, NJ, 1999.
3. Mukamel, S., Principles of Nonlinear Optical Spectroscopy. Oxford University Press: New York, 1995.
4. Schatz, G. C.; Ratner, M. A., Quantum Mechanics in Chemistry. Dover Publications: Mineola, NY, 2002; Sections 7.4 and 7.5.
5. W. Anderson, P., A Mathematical Model for the Narrowing of Spectral Lines by Exchange or Motion. Journal of the Physical Society of Japan 1954, 9, 316-339.
6. Wang, C. H., Spectroscopy of Condensed Media: Dynamics of Molecular Interactions. Academic Press: Orlando, 1985.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/14%3A_Fluctuations_in_Spectroscopy/14.03%3A_Gaussian-Stochastic_Model_for_Spectral_Diffusion.txt
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In describing fluctuations in a quantum mechanical system, we describe how an experimental observable is influenced by its interactions with a thermally agitated environment. For this, we work with the specific example of an electronic absorption spectrum and return to the Displaced Harmonic Oscillator (DHO) model. We previously described this model in terms of the eigenstates of the material Hamiltonian $H_0$, and interpreted the dipole correlation function and resulting lineshape in terms of the overlap between two wave packets evolving on the ground and excited surfaces $| E \rangle$ and $| G \rangle$.
$C _ {\mu \mu} (t) = \left| \mu _ {e g} \right|^{2} e^{- i \left( E _ {e} - E _ {g} \right) t / \hbar} \left\langle \varphi _ {g} (t) | \varphi _ {e} (t) \right\rangle \label{13.43}$
It is worth noting a similarity between the DHO Hamiltonian, and a general form for the interaction of an electronic two-level “system” with a harmonic oscillator “bath” whose degrees of freedom are dark to the observation, but which influence the behavior of the system.
Expressed in a slightly different physical picture, we can also conceive of this process as nuclear motions that act to modulate the electronic energy gap $\omega _ {e g}$. We can imagine rewriting the same Hamiltonian in a form with a new physical picture that desscribes the electronic energy gap’s dependence on $q$, i.e., its variation relative to $\omega _ {e g}$. If we define an Energy Gap Hamiltonian:
$H _ {e g} = H _ {e} - H _ {g}$
we can rewrite the DHO Hamiltonian
$H _ {0} = | e \rangle E _ {e} \langle e | + | g \rangle E _ {g} \langle g | + H _ {e} + H _ {g} \label{13.44}$
as an electronic transition linearly coupled to a harmonic oscillator:
$H _ {0} = | e \rangle E _ {e} \langle e | + | g \rangle E _ {g} \langle g | + H _ {e g} + 2 H _ {g} \label{13.44B}$
Noting that
$H _ {g} = \frac {p^{2}} {2 m} + \frac {1} {2} m \omega _ {0}^{2} q^{2} \label{13.44C}$
we can write this as a system-bath Hamiltonian:
$H _ {0} = H _ {S} + H _ {B} + H _ {S B} \label{13.44D}$
where $H_{SB}$ describes the interaction of the electronic system ($H_S$) with the vibrational bath ($H_B$). Here
$H _ {S} = | e \rangle E _ {e} \langle e | + | g \rangle E _ {g} \langle g |$
$H _ {B} = 2 H _ {g}$
and
\begin{align} H _ {S B} = H _ {e g} &= \dfrac {1} {2} m \omega _ {0}^{2} ( q - d )^{2} - \frac {1} {2} m \omega _ {0}^{2} q^{2} \ &= - m \omega _ {0}^{2} d q + \frac {1} {2} m \omega _ {0}^{2} d^{2} \ &= - c q + \lambda \end{align}
The Energy Gap Hamiltonian describes a linear coupling between the electronic transition and a harmonic oscillator. The strength of the coupling is $c$ and the Hamiltonian has a constant energy offset value given by the reorganization energy ($\lambda$). Any motion in the bath coordinate $q$ introduces a proportional change in the electronic energy gap.
In an alternate form, the Energy Gap Hamiltonian can also be written to incorporate the reorganization energy into the system:
\begin{align*} H _ {0} &= | e \rangle E _ {e} \langle e | + | g \rangle E _ {g} \langle g | + H _ {e g} + 2 H _ {g} \label{13.44E} \[4pt] H _ {S}^{\prime} &= | e \rangle \left( E _ {e} + \lambda \right) \langle e | + | g \rangle E _ {g} \langle g | \[4pt] H _ {B}^{\prime} &= \frac {p^{2}} {2 m} + \frac {1} {2} m \omega _ {0}^{2} q^{2} \[4pt] H _ {S B}^{\prime} &= - m \omega _ {0}^{2} d q \end{align*}
This formulation describes fluctuations about the average value of the energy gap $\hbar \omega _ {e g} + \lambda$, however, the observables calculated are the same.
From the picture of a modulated energy gap one can begin to see how random fluctuations can be treated by coupling to a harmonic bath. If each oscillator modulates the energy gap at a given frequency, and the phase between oscillators is random as a result of their independence, then time-domain fluctuations and dephasing can be cast in terms of a Fourier spectrum of couplings to oscillators with continuously varying frequency.
Energy Gap Hamiltonian
Now let’s work through the description of electronic spectroscopy with the Energy Gap Hamiltonian more carefully. Working from Equations \ref{13.43} and \ref{13.44} we express the energy gap Hamiltonian through reduced coordinates for the momentum, coordinate, and displacement of the oscillator.
$p = \hat {p} \left( 2 \hbar \omega _ {0} m \right)^{- 1 / 2}$
$q = \hat {q} \left( m \omega _ {0} / 2 \hbar \right)^{1 / 2}$
$d = d \left( m \omega _ {0} / 2 \hbar \right)^{1 / 2}$
with
\begin{align} H _ {e} &= \hbar \omega _ {0} \left( p^{2} + ( q - d )^{2} \right) \[4pt] H_{g} &= \hbar \omega _ {0} \left( p^{2} + q^{2} \right) \end{align} \label{13.48}
From Equation \ref{13.43} we have
\left.\begin{aligned} H _ {e g} & = - 2 \hbar \omega _ {0} d q + \hbar \omega _ {0} d^{2} \ & = - m \omega _ {0}^{2} d q + \lambda \end{aligned} \right. \label{13.49}
The energy gap Hamiltonian describes a linear coupling of the electronic system to the coordinate q. The slope of $H_{eg}$ versus $q$ is the coupling strength, and the average value of $H_{eg}$ in the ground state, $H _ {e g} ( q = 0 )$, is offset by the reorganization energy $\lambda$. We note that the average value of the energy gap Hamiltonian is $\left\langle H _ {e g} \right\rangle = \lambda$.
To obtain the absorption lineshape from the dipole correlation function
$C _ {\mu \mu} (t) = \left| \mu _ {e g} \right|^{2} e^{- i \omega _ {e g} t} F (t) \label{13.50}$
we must evaluate the dephasing function.
$F (t) = \left\langle e^{i H _ {g} t} e^{- i H _ {e} t} \right\rangle = \left\langle U _ {g}^{\dagger} U _ {e} \right\rangle \label{13.51}$
We want to rewrite the dephasing function in terms of the time dependence to the energy gap $H_{eg}$; that is, if $F (t) = \left\langle U _ {c g} \right\rangle$, then what is $U _ {e g}$? This involves a unitary transformation of the dynamics to a new frame of reference. The transformation from the DHO Hamiltonian to the EG Hamiltonian is similar to our derivation of the interaction picture.
Transformation of time-propagators
If we have a time dependent quantity of the form
$e^{i H _ {A} t} A e^{- i H _ {B} t} \label{13.52}$
we can also express the dynamics through the difference Hamiltonian $H _ {B A} = H _ {B} - H _ {A}$
$A e^{- i \left( H _ {B} - H _ {A} \right) t} = A e^{- i H _ {B A} t} \label{13.53}$
using a commonly performed unitary transformation. If we write
$H _ {B} = H _ {A} + H _ {B A} \label{13.54}$
we can use the same procedure for partitioning the dynamics in the interaction picture to write
$e^{- i H _ {B t} t} = e^{- i H _ {A} t} \exp _ {+} \left[ - \frac {i} {\hbar} \int _ {0}^{t} d \tau H _ {B A} ( \tau ) \right] \label{13.55}$
where
$H _ {B A} ( \tau ) = e^{i H _ {A} t} H _ {B A} e^{- i H _ {A} t} \label{13.56}$
Then, we can also write:
$e^{i H _ {A} t} e^{- i H _ {B} t} = \exp _ {+} \left[ - \frac {i} {\hbar} \int _ {0}^{t} d \tau H _ {B A} ( \tau ) \right] \label{13.57}$
Noting the mapping to the interaction picture
$H _ {e} = H _ {g} + H _ {e g} \quad \Leftrightarrow \quad H = H _ {0} + V \label{13.58}$
we see that we can represent the time dependence of the electronic energy gap $H_{eg}$ using
$e^{- i H _ {c} t / h} = e^{- i H _ {g} t / h} \exp _ {+} \left[ \frac {- i} {\hbar} \int _ {0}^{t} d \tau H _ {e g} ( \tau ) \right] \label{13.59}$
$U _ {e} = U _ {g} U _ {e g}$
where
\begin{align} H _ {e g} (t) & = e^{i H _ {g} t / \hbar} H _ {e g} e^{- i H _ {g} t / \hbar} \ & = U _ {g}^{\dagger} H _ {e g} U _ {g} \label{13.60} \end{align}
Remembering the equivalence between the harmonic mode $H_g$ and the bath mode(s) $H_B$ indicates that the time dependence of the EG Hamiltonian reflects how the electronic energy gap is modulated as a result of the interactions with the bath. That is $U _ {g} \Leftrightarrow U _ {B}$.
Equation \ref{13.59} immediately implies that
$F (t) = \left\langle e^{i H _ {g} t / \hbar} e^{- i H _ {e} t / \hbar} \right\rangle = \left\langle \exp _ {+} \left[ \frac {- i} {\hbar} \int _ {0}^{t} d \tau H _ {e g} ( \tau ) \right] \right\rangle \label{13.61}$
Now the quantum dephasing function is in the same form as we saw in our earlier classical derivation. Using the second-order cumulant expansion allows the dephasing function to be written as
$F (t) = \left\langle e^{i H _ {g} t / \hbar} e^{- i H _ {e} t / \hbar} \right\rangle = \left\langle \exp _ {+} \left[ \frac {- i} {\hbar} \int _ {0}^{t} d \tau H _ {e g} ( \tau ) \right] \right\rangle \label{13.62}$
Note that the cumulant expansion is here written as a time-ordered expansion. The first exponential term depends on the mean value of $H_{eg}$
$\left\langle H _ {e g} \right\rangle = \hbar \omega _ {0} d^{2} = \lambda \label{13.63}$
This is a result of how we defined $H_{eg}$. Alternatively, the EG Hamiltonian could have been defined relative to the energy gap at $Q=0$: $H _ {e g} = H _ {e} - H _ {g} + \lambda$. In this case the leading term in Equation \ref{13.62} would be zero, and the mean energy gap that describes the high frequency (system) oscillation in the dipole correlation function is $\omega _ {e g} + \lambda$.
The second exponential term in Equation \ref{13.62} is a correlation function that describes the time dependence of the energy gap
$\left. \begin{array} {c} {\left\langle H _ {e g} \left( \tau _ {2} \right) H _ {e g} \left( \tau _ {1} \right) \right\rangle - \left\langle H _ {e g} \left( \tau _ {2} \right) \right\rangle \left\langle H _ {e g} \left( \tau _ {1} \right) \right\rangle} \ {= \left\langle \delta H _ {e g} \left( \tau _ {2} \right) \delta H _ {e g} \left( \tau _ {1} \right) \right\rangle} \end{array} \right. \label{13.64}$
where
\left.\begin{aligned} \delta H _ {e g} & = H _ {e g} - \left\langle H _ {e g} \right\rangle \ & = - m \omega _ {0}^{2} d q \end{aligned} \right. \label{13.65}
Defining the time-dependent energy gap transition frequency in terms of the EG Hamiltonian as
$\delta \hat {\omega} _ {e g} \equiv \frac {\delta H _ {e g}} {\hbar} \label{13.66}$
we can write the energy gap correlation function
$C _ {e g} \left( \tau _ {2} , \tau _ {1} \right) = \left\langle \delta \hat {\omega} _ {e g} \left( \tau _ {2} - \tau _ {1} \right) \delta \hat {\omega} _ {e g} ( 0 ) \right\rangle \label{13.68}$
It follows that
$F (t) = e^{- i \lambda t / \hbar} e^{- g (t)}$
and
$g (t) = \int _ {0}^{t} d \tau _ {2} \int _ {0}^{\tau _ {2}} d \tau _ {1} C _ {e g} \left( \tau _ {2} , \tau _ {1} \right) \label{13.69}$
and the dipole correlation function can be expressed as
$C _ {\mu \mu} (t) = \left| \mu _ {e g} \right|^{2} e^{- i \left( E _ {e} - E _ {g} + \lambda \right) t / \hbar} e^{- g (t)} \label{13.70}$
This is the correlation function expression that determines the absorption lineshape for a timedependent energy gap. It is a general expression at this point, for all forms of the energy gap correlation function. The only approximation made for the bath is the second cumulant expansion.
Now, let’s look specifically at the case where the bath we are coupled to is a single harmonic mode. The energy gap correlation function is evaluated from
\left.\begin{aligned} C _ {e g} (t) & = \sum _ {n} p _ {n} \left\langle n \left| \delta \hat {\omega} _ {e g} (t) \delta \hat {\omega} _ {e g} ( 0 ) \right| n \right\rangle \ & = \frac {1} {\hbar^{2}} \sum _ {n} p _ {n} \left\langle n \left| e^{i H _ {g} t / \hbar} \delta H _ {e g} e^{- i H _ {g} t / \hbar} \delta H _ {e g} \right| n \right\rangle \end{aligned} \right. \label{13.71}
Noting that the bath oscillator correlation function
$C _ {q q} (t) = \langle q (t) q ( 0 ) \rangle = \frac {\hbar} {2 m \omega _ {0}} \left[ ( \overline {n} + 1 ) e^{- i \omega _ {0} t} + \overline {n} e^{i \omega _ {0} t} \right] \label{13.72}$
we find
$C _ {e g} (t) = \omega _ {0}^{2} D \left[ ( \overline {n} + 1 ) e^{- i \omega _ {0} t} + \overline {n} e^{i \omega _ {0} t} \right] \label{13.73}$
Here, as before, $\beta = 1 / k _ {B} T$, $\overline {n}$ is the thermally averaged occupation number for the oscillator
$\overline {n} = \sum _ {n} p _ {n} \left\langle n \left| a^{\dagger} a \right| n \right\rangle = \left( e^{\beta \hbar \omega _ {b}} - 1 \right)^{- 1} \label{13.74}$
and $\beta = 1 / \mathrm {kB} \mathrm {T}$. Note that the energy gap correlation function is a complex function. We can separate the real and imaginary parts of $C_{eg}$ as
$C _ {e g} (t) = C _ {e g}^{\prime} + i C _ {e g}^{\prime \prime} \label{13.75}$
with
\begin{align} C _ {e g}^{\prime} (t) &= \omega _ {0}^{2} D \operatorname {coth} \left( \beta \hbar \omega _ {0} / 2 \right) \cos \left( \omega _ {0} t \right) \[4pt] C _ {e g}^{\prime \prime} (t) &= \omega _ {0}^{2} D \sin \left( \omega _ {0} t \right) \end{align} \label{13.76}
where we have made use of the relation
$2 \overline {n} ( \omega ) + 1 = \operatorname {coth} ( \beta \hbar \omega / 2 ) \label{13.77}$
and
$\operatorname{coth}(x)=\left(e^{x}+e^{-x}\right) /\left(e^{x}-e^{-x}\right)$
We see that the imaginary part of the energy gap correlation function is temperature independent. The real part has the same amplitude at $T=0$, and rises with temperature. We can analyze the high and low temperature limits of this expression from
\begin{align} \lim_{x \rightarrow \infty} \operatorname {coth} (x) = 1 \[4pt] \lim_{x \rightarrow 0} \operatorname {coth} (x) \approx \frac {1} {x} \end{align} \label{13.78}
Looking at the low temperature limit $\operatorname{coth}\left(\beta \hbar \omega_{0} / 2\right) \rightarrow 1$ and $\overline {n} \rightarrow 0$ we see that Equation \ref{13.82} reduces to Equation \ref{13.84}.
In the high temperature limit $k T > \star \omega _ {0}$, $\operatorname {coth} \left( \hbar \omega _ {0} / 2 k T \right) \rightarrow 2 k T / \hbar \omega _ {0}$ and we recover the expected classical result. The magnitude of the real component dominates the imaginary part $\left| C _ {e g}^{\prime} \right| > > \left| C _ {e g}^{\prime \prime} \right|$ and the energy gap correlation function ($C_{eq}(t)$ becomes real and even in time.
Similarly, we can evaluate Equation \ref{13.69}, the lineshape function
$g (t) = - D \left[ ( \overline {n} + 1 ) \left( e^{- i \omega _ {0} t} - 1 \right) + \overline {n} \left( e^{i \omega _ {0} t} - 1 \right) \right] - i D \omega _ {0} t \label{13.79}$
The leading term in Equation \ref{13.79} gives us a vibrational progression, the second term leads to hot bands, and the final term is the reorganization energy ($- i D \omega _ {0} t = - i \lambda t / \hbar$). The lineshape function can be written in terms of its real and imaginary parts
$g(t)=g^{\prime}+i g^{\prime \prime}$
with
\begin{align} g^{\prime} (t) &= D \operatorname {coth} \left( \beta \hbar \omega _ {0} / 2 \right) \left( 1 - \cos \omega _ {0} t \right) \[4pt] g^{\prime \prime} (t) &= D \left( \sin \omega _ {0} t - \omega _ {0} t \right) \label{13.81} \end{align}
Because these enter into the dipole correlation function as exponential arguments, the imaginary part of $g(t)$ will reflect the bath-induced energy shift of the electronic transition gap and vibronic structure, and the real part will reflect damping, and therefore the broadening of the lineshape. Similarly to $C_{eg}(t)$, in the high temperature limit $g' \gg g''$. Now, using Equation \ref{13.68}, we see that the dephasing function is given by
\begin{align} F(t) &=\exp \left[D\left((\bar{n}+1)\left(e^{-i \omega_{0} t}-1\right)+\bar{n}\left(e^{i \omega_{0} t}-1\right)\right)\right] \[4pt] &=\exp \left[D\left(\operatorname{coth}\left(\frac{\beta \hbar \omega}{2}\right)(1-\cos \omega t)+i \sin \omega t\right)\right] \end{align} \label{13.82}
Let’s confirm that we get the same result as with our original DHO model, when we take the low temperature limit. Setting $\overline {n} \rightarrow 0$ in Equation \ref{13.82}, we have our original result
$F_{k T=0}(t)=\exp \left[D\left(e^{-i \omega_{0} t}-1\right)\right]\label{13.84}$
In the high temperature limit $g' \gg g''$, and from Equation \ref{13.78} we obtain
\left.\begin{aligned} F (t) & = \exp \left[ \frac {2 D k T} {\hbar \omega _ {0}} \cos \left( \omega _ {0} t \right) \right] \ & = \sum _ {j = 0}^{\infty} \frac {1} {j !} \left( \frac {2 D k T} {\hbar \omega _ {0}} \right)^{j} \cos^{j} \left( \omega _ {0} t \right) \end{aligned} \right. \label{13.85}
which leads to an absorption spectrum which is a series of sidebands equally spaced on either side of $\text {oleg}$.
Spectral representation of energy gap correlation function
Since time- and frequency-domain representations are complementary, and one form may be preferable over another, it is possible to express the frequency correlation function in terms of its spectrum. For a complex spectrum of vibrational motions composed of many modes, representing the nuclear motions in terms of a spectrum rather than a beat pattern is often easier. It turns out that calculation are often easier performed in the frequency domain. To start we define a Fourier transform pair that relates the time and frequency domain representations:
$\tilde {C} _ {e g} ( \omega ) = \int _ {- \infty}^{+ \infty} e^{i \omega t} C _ {e g} (t) d t \label{13.86}$
$C _ {e g} (t) = \frac {1} {2 \pi} \int _ {- \infty}^{+ \infty} e^{- i \omega t} \tilde {C} _ {e g} ( \omega ) d \omega \label{13.87}$
Since the energy gap correlation function has the property
$C _ {e g} ( - t ) = C _ {e g}^{*} (t)$
it also follows from Equation \ref{13.86} that the energy gap correlation spectrum is entirely real:
$\tilde {C} _ {e g} ( \omega ) = 2 \operatorname {Re} \int _ {0}^{\infty} e^{i \omega t} C _ {e g} (t) d t \label{13.88}$
or
$\tilde {C} _ {e g} ( \omega ) = \tilde {C} _ {e g}^{\prime} ( \omega ) + \tilde {C} _ {e g}^{\prime \prime} ( \omega ) \label{13.89}$
Here $\tilde {C} _ {e s}^{\prime} ( \omega )$ and $\tilde {C} _ {e g}^{\prime \prime} ( \omega )$ are the Fourier transforms of the real and imaginary components of $C _ {e s} (t)$, respectively. $\tilde {C} _ {e s}^{\prime} ( \omega )$ and $\tilde {C} _ {e g}^{\prime \prime} ( \omega )$ are even and odd in frequency. Thus while $\tilde {C} _ {e s} ( \omega )$ is entirely real valued, it is asymmetric about $\omega = 0$.
With these definitions in hand, we can write the spectrum of the energy gap correlation function for coupling to a single harmonic mode spectrum (Equation \ref{13.71}):
$\tilde {C} _ {e g} \left( \omega _ {\alpha} \right) = \omega _ {\alpha}^{2} D \left( \omega _ {\alpha} \right) \left[ \left( \overline {n} _ {\alpha} + 1 \right) \delta \left( \omega - \omega _ {\alpha} \right) + \overline {n} _ {\alpha} \delta \left( \omega + \omega _ {\alpha} \right) \right] \label{13.90}$
This is a spectrum that characterizes how bath vibrational modes of a certain frequency and thermal occupation act to modify the observed energy of the system. The first and second terms in Equation \ref{13.90} describe upward and downward energy shifts of the system, respectively. Coupling to a vibration typically leads to an upshift of the energy gap transition energy since energy must be put into the system and bath. However, as with hot bands, when there is thermal energy available in the bath, it also allows for down-shifts in the energy gap. The net balance of upward and downward shifts averaged over the bath follows the detailed balance expression
$\tilde {C} ( - \omega ) = e^{- \beta \hbar \omega} \tilde {C} ( \omega ) \label{13.91}$
The balance of rates tends toward equal with increasing temperature. Fourier transforms of Equation \ref13.76} gives two other representations of the energy gap spectrum
$\tilde {C} _ {e g}^{\prime} \left( \omega _ {\alpha} \right) = \omega _ {\alpha}^{2} D \left( \omega _ {\alpha} \right) \operatorname {coth} \left( \beta \hbar \omega _ {\alpha} / 2 \right) \left[ \delta \left( \omega - \omega _ {\alpha} \right) + \delta \left( \omega + \omega _ {\alpha} \right) \right] \label{13.92}$
$\tilde {C} _ {e g}^{\prime \prime} \left( \omega _ {\alpha} \right) = \omega _ {\alpha}^{2} D \left( \omega _ {\alpha} \right) \left[ \delta \left( \omega - \omega _ {\alpha} \right) + \delta \left( \omega + \omega _ {\alpha} \right) \right]. \label{13.93}$
The representations in Equation \ref{13.90}, \ref{13.92}, and \ref{13.93} are not independent, but can be related to one another through
$\tilde{C}_{e g}^{\prime}\left(\omega_{\alpha}\right)=\operatorname{coth}\left(\beta \hbar \omega_{\alpha} / 2\right) \tilde{C}_{e g}^{\prime \prime}\left(\omega_{\alpha}\right)$
$\tilde {C} _ {e g} \left( \omega _ {\alpha} \right) = \left( 1 + \operatorname {coth} \left( \beta \hbar \omega _ {\alpha} / 2 \right) \right) \tilde {C} _ {e g}^{\prime \prime} \left( \omega _ {\alpha} \right) \label{13.95}$
That is, given either the real or imaginary part of the energy gap correlation spectrum, we can predict the other part. As we will see, this relationship is one manifestation of the fluctuationdissipation theorem that we address later. Due to its independence on temperature, the spectral density $\tilde {C} _ {e g}^{\prime \prime} \left( \omega _ {\alpha} \right)$ is the commonly used representation.
Also from Equations.\ref{13.69} and \ref{13.87} we obtain the lineshape function as
\left.\begin{aligned} g (t) & = \int _ {- \infty}^{+ \infty} d \omega \frac {1} {2 \pi} \frac {\tilde {C} _ {e g} ( \omega )} {\omega^{2}} [ \exp ( - i \omega t ) + i \omega t - 1 ] \ & = \int _ {0}^{\infty} d \omega \frac {\tilde {C} _ {e g}^{\prime \prime} ( \omega )} {\pi \omega^{2}} \left[ \operatorname {coth} \left( \frac {\beta \hbar \omega} {2} \right) ( 1 - \cos \omega t ) + i ( \sin \omega t - \omega t ) \right] \end{aligned} \right. \label{13.96}
The first expression relates g(t) to the complex energy gap correlation function, whereas the second separates the real and the imaginary parts and relates them to the imaginary part of the energy gap correlation function.
Coupling to a Harmonic Bath
More generally for condensed phase problems, the system coordinates that we observe in an experiment will interact with a continuum of nuclear motions that may reflect molecular vibrations, phonons, or intermolecular interactions. We describe this continuum as continuous distribution of harmonic oscillators of varying mode frequency and coupling strength. The Energy Gap Hamiltonian is readily generalized to the case of a continuous distribution of motions if we statistically characterize the density of states and the strength of interaction between the system and this bath. This method is also referred to as the Spin-Boson Model used for treating a two level spin-½ system interacting with a quantum harmonic bath.
Following our earlier discussion of the DHO model, the generalization of the EG Hamiltonian to the multimode case is
$H _ {0} = \hbar \omega _ {e g} + H _ {e g} + H _ {B} \label{13.97}$
$H _ {B} = \sum _ {\alpha} \hbar \omega _ {\alpha} \left( p _ {\sim}^{2} + q _ {\alpha}^{2} \right) \label{13.98}$
$H _ {e g} = \sum _ {\alpha} 2 \hbar \omega _ {\alpha} d _ {\alpha} q _ {\alpha} + \lambda \label{13.99}$
$\lambda = \sum _ {\alpha} \hbar \omega _ {\alpha} d _ {\alpha}^{2} \label{13.100}$
Note that the time-dependence to $H_{eg}$ results from the interaction with the bath:
$H _ {e g} (t) = e^{i H _ {B} t / \hbar} H _ {e g} e^{- i H _ {B} t / \hbar} \label{13.101}$
Also, since the harmonic modes are normal to one another, the dephasing function and lineshape function are obtained from Equation \ref{13.102}
$F(t)=\prod_{\alpha} F_{\alpha}(t) \quad g(t)=\sum_{\alpha} g_{\alpha}(t)\label{13.102}$
For a continuum, we assume that the number of modes are so numerous as to be continuous, and that the sums in the equations above can be replaced by integrals over a continuous distribution of states characterized by a density of states W Z . Also the interaction with modes of a particular frequency are equal so that we can simply average over a frequency dependent coupling constant 2 D d Z Z . For instance, Equation \ref{13.102} becomes
$g (t) = \int d \omega _ {\alpha} W \left( \omega _ {\alpha} \right) g \left( t , \omega _ {\alpha} \right) \label{13.103}$
Coupling to a continuum leads to dephasing resulting from interaction to a continuum of modes of varying frequency. This will be characterized by damping of the energy gap frequency correlation function
$C _ {e g} (t) = \int d \omega _ {\alpha} C _ {e g} \left( \omega _ {\alpha} , t \right) W \left( \omega _ {\alpha} \right) \label{13.104}$
Here $C _ {e g} \left( \omega _ {\alpha} , t \right) = \left\langle \delta \omega _ {e g} \left( \omega _ {\alpha} , t \right) \delta \omega _ {e g} \left( \omega _ {\alpha} , 0 \right) \right\rangle$ refers to the energy gap frequency correlation function for a single harmonic mode given in Equation \ref{13.71}. While Equation \ref{13.104} expresses the modulation of the energy gap in the time domain, we can alternatively express the continuous distribution of coupled bath modes in the frequency domain:
$\tilde {C} _ {e g} ( \omega ) = \int d \omega _ {\alpha} W \left( \omega _ {\alpha} \right) \tilde {C} _ {e g} \left( \omega _ {\alpha} \right) \label{13.105}$
An integral of a single harmonic mode spectrum over a continuous density of states provides a coupling weighted density of states that reflects the action spectrum for the system-bath interaction. We evaluate this with the single harmonic mode spectrum, Equation \ref{13.90}. We see that the spectrum of the correlation function for positive frequencies is related to the product of the density of states and the frequency dependent coupling
$\tilde{C}_{e g}(\omega)=\omega^{2} D(\omega) W(\omega)(\bar{n}+1) \quad(\omega>0) \label{13.106}$
$\tilde{C}_{e g}(\omega)=\omega^{2} D(\omega) W(\omega) \bar{n} \quad(\omega<0) \label{13.107}$
This is an action spectrum that reflects the coupling weighted density of states of the bath that contributes to the spectrum.
In practice, the unusually symmetry of $\tilde {C} _ {e g} ( \omega )$ and its growth as $\omega^{2}$ make it difficult to work with. Therefore we choose to express the frequency domain representation of the coupling-weighted density of states in Equation \ref{13.106} as a spectral density, defined as
\left.\begin{aligned} \rho ( \omega ) & \equiv \frac {\tilde {C} _ {e g}^{\prime \prime} ( \omega )} {\pi \omega^{2}} \ & = \frac {1} {\pi} \int d \omega _ {\alpha} W \left( \omega _ {\alpha} \right) D \left( \omega _ {\alpha} \right) \delta \left( \omega - \omega _ {\alpha} \right) \ & = \frac {1} {\pi} W ( \omega ) D ( \omega ) \end{aligned} \right. \label{13.108}
This expression is real and defined only for positive frequencies. Note $\tilde {C} _ {e g}^{\prime \prime} ( \omega )$ is an odd function in $\infty$, and therefore $\rho(\infty)$ is also.
The reorganization energy can be obtained from the first moment of the spectral density
$\lambda = \hbar \int _ {0}^{\infty} d \omega \omega \rho ( \omega ) \label{13.109}$
Furthermore, from Equation \ref{13.69} and \ref{13.105} we obtain the lineshape function in two forms
\left.\begin{aligned} g (t) & = \int _ {- \infty}^{+ \infty} d \omega \frac {1} {2 \pi} \frac {\tilde {C} _ {e g} ( \omega )} {\omega^{2}} [ \exp ( - i \omega t ) + i \omega t - 1 ] \[4pt] & = - \frac {i \lambda t} {\hbar} + \int _ {0}^{\infty} d \omega \rho ( \omega ) \left[ \operatorname {coth} \left( \frac {\beta \hbar \omega} {2} \right) ( 1 - \cos \omega t ) + i \sin \omega t \right] \end{aligned} \right. \label{13.110}
In this expression the temperature dependence implies that in the high temperature limit, the real part of $g(t)$ will dominate, as expected for a classical system. This is a perfectly general expression for the lineshape function in terms of an arbitrary spectral distribution describing the time scale and amplitude of energy gap fluctuations. Given a spectral density $\rho(\infty)$, you can calculate various spectroscopic observables and other time-dependent processes in a fluctuating environment.
Now, let’s evaluate the behavior of the lineshape function and absorption lineshape for different forms of the spectral density. To keep things simple, we will consider the high temperature limit, $k _ {B} T \ll \hbar \omega$. Here
$\operatorname {coth} ( \beta \hbar \omega / 2 ) \rightarrow 2 / \beta \hbar \omega$
and we can neglect the imaginary part of the frequency correlation function and lineshape function. These examples are motivated by the spectral densities observed for random or noisy processes. Depending on the frequency range and process of interest, noise tends to scale as $U \approx Z^{-n}$, where $n = 0$, $1$ or $2$. This behavior is often described in terms of a spectral density of the form
$\rho ( \omega ) \propto \omega _ {c}^{1 - s} \omega^{s - 2} e^{- \omega / \omega _ {c}} \label{13.111}$
where $Z_c$ is a cut-off frequency, and the units are inverse frequency. These spectral densities have the desired property of being an odd function in $Z$, and can be integrated to a finite value. The case $s = 1$ is known as the Ohmic spectral density, whereas $s > 1$ is super-ohmic and $s < 1$ is sub-ohmic.
Step 1
Let’s first consider the example when $U$ drops as $1/Z$ with frequency, which refers to the Ohmic spectral density with a high cut-off frequency. This is the spectral density that corresponds to an energy gap correlation function that decays infinitely fast: $C_{e g}(t) \sim \delta(t)$. To choose a definition consistent with Equation \ref{13.109}, we set
$\rho ( \omega ) = \lambda / \Lambda \hbar \omega \label{13.112}$
where $\Lambda$ is a finite high frequency integration limit that we enforce to keep $U$ well behaved. $\Lambda$ has units of frequency, it is equated with the inverse correlation time for the fast decay of $C_{eg}(t)$.
Now we evaluate
\begin{aligned} g (t) & = \int _ {0}^{\infty} d \omega \frac {2 k _ {B} T} {\Lambda \hbar \omega} \rho ( \omega ) ( 1 - \cos \omega t ) - \frac {i \lambda t} {\hbar} \ & = \int _ {0}^{\infty} d \omega \frac {2 \lambda k _ {B} T ( 1 - \cos \omega t )} {\omega^{2}} - \frac {i \lambda t} {\hbar} \ & = \lambda \frac {\pi k _ {B} T} {\Lambda \hbar^{2}} t - \frac {i \lambda t} {\hbar} \end{aligned} \label{13.113}
Then we obtain the dephasing function
$F (t) = e^{- \Gamma t} \label{13.114}$
where we have defined the exponential damping constant as
$\Gamma = \lambda \frac {\pi k T} {\Lambda \hbar^{2}} \label{13.115}$
From this we obtain the absorption lineshape
$\sigma _ {a b s} \propto \frac {\left| \mu _ {e g} \right|^{2}} {\left( \omega - \omega _ {e g} \right) + i \Gamma} \label{13.116}$
Thus, a spectral density that scales as $1 / \omega$ has a rapidly fluctuating bath and leads to a homogeneous Lorentzian lineshape with a half-width $\Gamma$.
Step 2
Now take the case that we choose a Lorentzian spectral density centered at $Z= 0$. To keep the proper odd function of $Z$ and definition of $O$ we write:
$\rho ( \omega ) = \frac {\lambda} {\hbar \omega} \frac {\Lambda} {\omega^{2} + \Lambda^{2}} \label{13.117}$
Note that for frequencies $\omega \ll \Lambda$ this has the ohmic form of Equation \ref{13.112}. This is a spectral density that corresponds to an energy gap correlation function that drops exponentially as $C_{e g}(t) \sim \exp (-\Lambda t)$. Here, in the high temperature (classical) limit $k T>>\hbar \Lambda$, neglecting the imaginary part, we find
$g (t) \approx \frac {\pi \lambda k T} {\hbar^{2} \Lambda^{2}} [ \exp ( - \Lambda t ) + \Lambda t - 1 ] \label{13.118}$
This expression looks familiar. If we equate
$\Delta^{2} = \lambda \frac {\pi k T} {\hbar^{2}} \label{13.119}$
and
$\tau _ {c} = \frac {1} {\Lambda} \label{13.120}$
we obtain the same lineshape function as the classical Gaussian-stochastic model:
$g (t) = \Delta^{2} \tau _ {c}^{2} \left[ \exp \left( - t / \tau _ {c} \right) + t / \tau _ {c} - 1 \right] \label{13.121}$
So, the interaction of an electronic transition with a harmonic bath leads to line broadening that is equivalent to random fluctuations of the energy gap. As we noted earlier, for the homogeneous limit, we find $\Gamma = \Delta^{2} \tau _ {c}$.
Readings
1. Mukamel, S., Principles of Nonlinear Optical Spectroscopy. Oxford University Press: New York, 1995; Ch. 7 and Ch. 8.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/14%3A_Fluctuations_in_Spectroscopy/14.04%3A_The_Energy_Gap_Hamiltonian.txt
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So, why does the mathematical model for coupling of a system to a harmonic bath give the same results as the classical stochastic equations of motion for fluctuations? Why does coupling to a continuum of bath states have the same physical manifestation as perturbation by random fluctuations? The answer is that in both cases, we really have imperfect knowledge of the behavior of all the particles present. Observing a small subset of particles will have dynamics with a random character. These dynamics can be quantified through a correlation function or a spectral density for the time-scales of motion of the bath. In this section, we will demonstrate a more formal relationship that illustrates the equivalence of these pictures.
To take our discussion further, let’s again consider the electronic absorption spectrum from a classical perspective. It’s quite common to think that the electronic transition of interest is coupled to a particular nuclear coordinate $Q$ which we will call a local coordinate. This local coordinate could be an intramolecular normal vibrational mode, an intermolecular rattling in a solvent shell, a lattice vibration, or another motion that influences the electronic transition. The idea is that we take the observed electronic transition to be linearly dependent on one or more local coordinates. Therefore describing $Q$ allows us to describe the spectroscopy. However, since this local mode has further degrees of freedom that it may be interacting with, we are extracting a particular coordinate out or a continuum of other motions, the local mode will appear to feel a fluctuating environment—a friction.
Classically, we describe fluctuations in $Q$ as Brownian motion, typically through a Langevin equation. In the simplest sense, this is an equation that restates Newton’s equation of motion $F=ma$ for a fluctuating force acting on a particle with position $Q$. For the case that this particle is confined in a harmonic potential,
$m \ddot{Q}(t)+m \omega_{0}^{2} Q^{2}+m \gamma \dot{Q}=f_{R}(t) \label{13.122}$
Here the terms on the left side represent a damped harmonic oscillator. The first term is the force due to acceleration of the particle of mass $m\left(F_{a c c}=m a\right)$. The second term is the restoring force of the potential, $F_{r e s}=-\partial V / \partial Q=m \omega_{0}^{2}$. The third term allows friction to damp the motion of the coordinate at a rate $\gamma$. The motion of $Q$ is under the influence of $f_{R}(t)$, a random fluctuating force exerted on $Q$ by its surroundings.
Under steady-state conditions, it stands to reason that the random force acting on $Q$ is the origin of the damping. The environment acts on $Q$ with stochastic perturbations that add and remove kinetic energy, which ultimately leads to dissipation of any excess energy. Therefore, the Langevin equation is modelled as a Gaussian stationary process. We take $f_{R}(t)$ to have a timeaveraged value of zero,
$\left\langle f _ {R} (t) \right\rangle = 0 \label{13.123}$
and obey the classical fluctuation-dissipation theorem:
$\gamma = \frac {1} {2 m k _ {B} T} \int _ {- \infty}^{\infty} \left\langle f _ {R} (t) f _ {R} ( 0 ) \right\rangle \label{13.124}$
This shows explicitly how the damping is related to the correlation time for the random force. We will pay particular attention to the Markovian case
$\left\langle f _ {R} (t) f _ {R} ( 0 ) \right\rangle = 2 m \gamma k _ {B} T \delta (t) \label{13.125}$
which indicate that the fluctuations immediately lose all correlation on the time scale of the evolution of Q. The Langevin equation can be used to describe the correlation function for the time dependence of Q. For the Markovian case, Equation \ref{13.122} leads to
$C _ {Q Q} (t) = \frac {k _ {B} T} {m \omega _ {0}^{2}} \left( \cos \zeta t + \frac {\gamma} {2 \zeta} \sin \zeta t \right) e^{- \gamma t / 2} \label{13.126}$
where the reduced frequency $\zeta=\sqrt{\omega_{0}^{2}-\gamma^{2} / 4}$. The frequency domain expression, obtained by Fourier transformation, is
$\tilde {C} _ {Q Q} ( \omega ) = \frac {\gamma k T} {m \pi} \frac {1} {\left( \omega _ {0}^{2} - \omega^{2} \right)^{2} + \omega^{2} \gamma^{2}} \label{13.127}$
Remembering that the absorption lineshape was determined by the quantum mechanical energy gap correlation function $\langle q(t) q(0)\rangle$, one can imagine an analogous classical description of the spectroscopy of a molecule that experiences interactions with a fluctuating environment. In essence this is what we did when discussing the Gaussian stochastic model of the lineshape. A more general description of the position of a particle subject to a fluctuating force is the Generalized Langevin Equation. The GLE accounts for the possibility that the damping may be time-dependent and carry memory of earlier configurations of the system:
$m \ddot {Q} (t) + m \omega _ {0}^{2} Q^{2} + m \int _ {0}^{t} d \tau \gamma ( t - \tau ) \dot {Q} ( \tau ) = f (t) \label{13.128}$
The memory kernel, $\gamma ( t - \tau )$, is a correlation function that describes the time-scales over which the fluctuating force retains memory of its previous state. The force due to friction on $Q$ depends on the history of the system through $\tau$, the time preceding $t$, and the relaxation of $\gamma ( t - \tau )$. The classical fluctuation-dissipation relationship relates the magnitude of the fluctuating forces on the system coordinate to the damping
$\left\langle f_{R}(t) f_{R}(\tau)\right\rangle=2 m k_{B} T \gamma(t-\tau) \label{13.129}$
As expected, for the case that $\gamma ( t - \tau ) = \gamma \delta ( t - \tau )$, the GLE reduces to the Markovian case, Equation \ref{13.122}.
To demonstrate that the classical dynamics of the particle described under the GLE is related to the quantum mechanical dynamics for a particle interacting with a harmonic bath, we will outline the derivation of a quantum mechanical analog of the classical GLE. To do this we will derive an expression for the time-evolution of the system under the influence of the harmonic bath. We work with a Hamiltonian with a linear coupling between the system and the bath
$H _ {H B} = H _ {S} ( P , Q ) + H _ {B} \left( p _ {\alpha} , q _ {\alpha} \right) + H _ {S B} ( Q , q ) \label{13.130}$
We take the system to be a particle of mass M, described through variables P and Q, whereas $m_{\alpha}$, $p_{\alpha}$, and $q_{\alpha}$ are bath variables. For the present case, we will take the system to be a quantum harmonic oscillator,
$H _ {s} = \frac {P^{2}} {2 M} + \frac {1} {2} M \Omega^{2} Q^{2} \label{13.131}$
and the Hamiltonian for the bath and its interaction with the system is written as
$H _ {B} + H _ {S B} = \sum _ {\alpha} \left( \frac {p _ {\alpha}^{2}} {2 m _ {\alpha}} + \frac {m _ {\alpha} \omega _ {\alpha}^{2}} {2} \left( q _ {\alpha} - \frac {c _ {\alpha}} {m _ {\alpha} \omega _ {\alpha}^{2}} Q \right)^{2} \right) \label{13.132}$
This expression explicitly shows that each of the bath oscillators is displaced with respect to the system by an amount dependent on their mutual coupling. In analogy to our work with the Displaced Harmonic Oscillator, if we define a displacement operator
$\hat {D} = \exp \left( - \frac {i} {\hbar} \sum _ {\alpha} \hat {p} _ {\alpha} \xi _ {\alpha} \right) \label{13.133}$
where
$\xi _ {\alpha} = \frac {c _ {\alpha}} {m _ {\alpha} \omega _ {\alpha}^{2}} Q \label{13.134}$
then
$H _ {B} + H _ {S B} = \hat {D}^{\dagger} H _ {B} \hat {D} \label{13.135}$
Equation \ref{13.132} is merely a different representation of our earlier harmonic bath model. To see this we write Equation \ref{13.132} as
$H _ {B} + H _ {S B} = \sum _ {\alpha} \hbar \omega _ {\alpha} \left( p _ {\alpha}^{2} + \left( q _ {\alpha} - c _ {\alpha} Q \right)^{2} \right) \label{13.136}$
where the coordinates and momenta are written in reduced form
$\begin{array}{l} \underline{Q}=Q \sqrt{m \omega_{0} / 2 \hbar} \ q_{\alpha}=q_{\alpha} \sqrt{m_{\alpha} \omega_{\alpha} / 2 \hbar} \ p_{\alpha}=p_{\alpha} / \sqrt{2 \hbar m_{\alpha} \omega_{\alpha}} \end{array} \label{13.137}$
Also, the reduced coupling is of the system to the $\alpha^{\text {th }}$ oscillator is
$\mathcal {C} _ {\alpha} = c _ {\alpha} / \omega _ {\alpha} \sqrt {m _ {\alpha} \omega _ {\alpha} m \omega _ {0}} \label{13.138}$
Expanding Equation \ref{13.136} and collecting terms, we find that we can separate terms as in the harmonic bath model
$H _ {B} = \sum _ {\alpha} \hbar \omega _ {\alpha} \left( p _ {\alpha}^{2} + q _ {\alpha}^{2} \right) \label{13.139}$
$H _ {S B} = - 2 \sum _ {\alpha} \hbar \omega _ {\alpha} d _ {\alpha} q _ {\alpha} + \lambda _ {B} \label{13.140}$
The reorganization energy due to the bath oscillators is
$\lambda _ {B} = \sum _ {\alpha} \hbar \omega _ {\alpha} d _ {\alpha}^{2} \label{13.141}$
and the unit less bath oscillator displacement is
$d _ {\alpha} = \underset {\mathcal {Q}} {\approx} \mathcal {C} _ {\alpha} \label{13.142}$
For our current work we regroup the total Hamiltonian (Equation \ref{13.130}) as
$H _ {H B} = \left[ \frac {P^{2}} {2 M} + \frac {1} {2} M \overline {\Omega}^{2} Q^{2} \right] + \sum _ {\alpha} \hbar \omega _ {\alpha} \left( p _ {\alpha}^{2} + q _ {\alpha}^{2} \right) - 2 \sum _ {\alpha} \hbar \omega _ {\alpha} c _ {\alpha} Q q _ {\alpha} \label{13.143}$
where the renormalized frequency is
$\overline {\Omega}^{2} = \Omega^{2} + \Omega \sum _ {\alpha} \omega _ {\alpha} c _ {\alpha}^{2} \label{13.144}$
To demonstrate the equivalence of the dynamics under this Hamiltonian and the GLE, we can derive an equation of motion for the system coordinate $Q$. We approach this by first expressing these variables in terms of ladder operators
$\hat{P}=i\left(\hat{a}^{\dagger}-\hat{a}\right) \quad \hat{p}_{\alpha}=i\left(\hat{b}_{\alpha}^{\dagger}-\hat{b}_{\alpha}\right) \label{13.145}$
$\hat{Q}=\left(\hat{a}^{\dagger}+\hat{a}\right) \quad \hat{q}_{\alpha}=\left(\hat{b}_{\alpha}^{\dagger}+\hat{b}_{\alpha}\right) \label{13.146}$
Here $\hat {a}$, $\hat {a}^{\dagger}$ are system operators, $\hat {b}$ and $\hat {b}^{\dagger}$ are bath operators. If the observed particle is taken to be bound in a harmonic potential, then the Hamiltonian in Equation \ref{13.130} can be written as
$H _ {H B} = \hbar \overline {\Omega} \left( \hat {a}^{\dagger} \hat {a} + \frac {1} {2} \right) + \sum _ {\alpha} \hbar \omega _ {\alpha} \left( \hat {b} _ {\alpha}^{\dagger} \hat {b} _ {\alpha} + \frac {1} {2} \right) - \left( \hat {a}^{\dagger} + \hat {a} \right) \sum _ {\alpha} \hbar \omega _ {\alpha} c _ {\alpha} \left( \hat {b} _ {\alpha}^{\dagger} + \hat {b} _ {\alpha} \right) \label{13.147}$
The equations of motion for the operators in Equations \ref{13.145} and \ref{13.146} can be obtained from the Heisenberg equation of motion.
$\dot {\hat {a}} = \frac {i} {\hbar} \left[ H _ {H B} , \hat {a} \right] \label{13.148}$
from which we find
$\dot {\hat {a}} = - i \overline {\Omega} \hat {a} + i \sum _ {\alpha} \omega _ {\alpha} c _ {\alpha} \left( \hat {b} _ {\alpha}^{\dagger} + \hat {b} _ {\alpha} \right) \label{13.149}$
$\dot {\hat {b}} _ {\alpha} = - i \omega _ {\alpha} \hat {b} _ {\alpha} + i \omega _ {\alpha} \mathcal {C} _ {\alpha} \left( \hat {a}^{\dagger} + \hat {a} \right)\label{13.150}$
To derive an equation of motion for the system coordinate, we begin by solving for the time evolution of the bath coordinates by directly integrating Equation \ref{13.150},
$\hat {b} _ {\alpha} (t) = e^{- i \omega _ {a} t} \int _ {0}^{t} e^{i \omega _ {a} t^{\prime}} \left( i \omega _ {\alpha} \mathcal {C} _ {\alpha} \left( \hat {a}^{\dagger} + \hat {a} \right) \right) d t^{\prime} + \hat {b} _ {\alpha} ( 0 ) e^{- i \omega _ {a} t} \label{13.151}$
and insert the result into Equation \ref{13.149}. This leads to
$\dot {\hat {a}} + i \overline {\Omega} \hat {a} - i \sum _ {\alpha} \omega _ {\alpha} c _ {\alpha}^{2} \left( \hat {a}^{\dagger} + \hat {a} \right) + i \int _ {0}^{t} d t^{\prime} \kappa \left( t - t^{\prime} \right) \left( \dot {\hat {a}}^{\dagger} \left( t^{\prime} \right) + \dot {\hat {a}} \left( t^{\prime} \right) \right) = i F (t) \label{13.152}$
where
$\kappa (t) = \sum _ {\alpha} \omega _ {\alpha} c _ {\alpha}^{2} \cos \left( \omega _ {\alpha} t \right) \label{13.153}$
and
$F (t) = \sum _ {\alpha} c _ {\alpha} \left[ \hat {b} _ {\alpha} ( 0 ) - \omega _ {\alpha} c _ {\alpha} \left( \hat {a}^{\dagger} ( 0 ) + \hat {a} ( 0 ) \right) \right] e^{- i \omega _ {a} t} + h . c . \label{13.154}$
Now, recognizing that the time-derivative of the system variables is given by
$\dot {\hat {P}} = i \left( \dot {\hat {a}}^{\dagger} - \dot {\hat {a}} \right) \label{13.155}$
$\hat {\hat {Q}} \left( \dot {\hat {a}}^{\dagger} + \dot {\hat {a}} \right) \label{13.156}$
and substituting Equation \ref{13.152} into \ref{13.155}, we can write an equation of motion
$\dot {P} (t) + \left( \overline {\Omega} - 2 \sum _ {\alpha} \frac {2 \mathcal {c} _ {\alpha}^{2}} {\omega _ {\alpha}} \right) Q + \int _ {0}^{t} d t^{\prime} 2 \kappa \left( t - t^{\prime} \right) \hat {Q} \left( t^{\prime} \right) = F (t) + F^{\dagger} (t) \label{13.157}$
Equation \ref{13.157} bears a striking resemblance to the classical GLE, Equation \ref{13.128}. In fact, if we define
$\gamma(t)=2 \bar{\Omega} \kappa(t)$
$=\frac{1}{M} \sum_{\alpha} \frac{c_{\alpha}^{2}}{m_{\alpha} \omega_{\alpha}^{2}} \cos \omega_{\alpha} t \label{13.158}$
$f_{R}(t)=\sqrt{2 \hbar M \Omega}\left[F(t)+F^{\dagger}(t)\right]$
$=\sum_{\alpha} c_{\alpha}\left[q_{\alpha}(0) \cos \omega_{\alpha} t+\frac{p_{\alpha}(0)}{m_{\alpha} \omega_{\alpha}} \sin \omega_{\alpha} t\right] \label{13.159}$
then the resulting equation is isomorphic to the classical GLE
$\dot{P}(t)+M \Omega^{2} Q(t)+M \int_{0}^{t} d t^{\prime} \gamma\left(t-t^{\prime}\right) \dot{Q}\left(t^{\prime}\right)=f_{R}(t)\label{13.160}$
This demonstrates that the quantum harmonic bath acts a dissipative environment, whose friction on the system coordinate is given by Equation \ref{13.158}. What we have shown here is an outline of the proof, but detailed discussion of these relationships can be found elsewhere.
Readings
1. Calderia, A. O.; Legget, A. J., H.O.-bath model;theory. Ann. Phys 1983, 149, 372-456.
2. Fleming, G. R.; Cho, M., Chromophore-Solvent Dynamics. Annual Review of Physical Chemistry 1996, 47 (1), 109-134.
3. Leggett, A.; Chakravarty, S.; Dorsey, A.; Fisher, M.; Garg, A.; Zwerger, W., Dynamics of the dissipative two-state system. Reviews of Modern Physics 1987, 59 (1), 1-85.
4. Mukamel, S., Principles of Nonlinear Optical Spectroscopy. Oxford University Press: New York, 1995; Ch. 8
5. Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006; Ch. 8.
6. Schatz, G. C.; Ratner, M. A., Quantum Mechanics in Chemistry. Dover Publications: Mineola, NY, 2002; Sections 6.5, 12.2, 12.5.
7. Weiss, U., Quantum Dissipative Systems. 3rd ed.; World Scientific: Hackensack, N.J., 2008.
8. Yan, Y. J.; Xu, R. X., Quantum mechanics of dissipative systems. Annual Review of Physical Chemistry 2005, 56, 187-219.
1 Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006.
2 Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006; Ch. 8.
3 Calderia, A. O.; Legget, A. J., Ann. Phys 1983, 149, 372-456
4 4 Weiss, U. Quantum Dissipative Systems. 3rd ed.; World Scientific: Hackensack, N.J. , 2008; Leggett, A. J.; Chakravarty, S.; Dorsey, A. T.; Fisher, M. P. A.; Garg, A.; Zwerger, W. Dynamics of the dissipative two-state system. Reviews of Modern Physics 1987, 59 (1), 1-85; Yan, Y.; Xu, R. Quantum Mechanics of Dissipative Systems. Annual Review of Physical Chemistry 2005, 56 (1), 187-219. 13-3
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/14%3A_Fluctuations_in_Spectroscopy/14.05%3A_Correspondence_of_Harmonic_Bath_and_Stochastic_Equations_of_Motion.txt
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• 15.1: Electronic Interactions
In this section we will describe processes that result from the interaction between two or more molecular electronic states, such as the transport of electrons or electronic excitation.
• 15.2: Förster Resonance Energy Transfer (FRET)
Förster resonance energy transfer (FRET) refers to the nonradiative transfer of an electronic excitation from a donor molecule to an acceptor molecule. This electronic excitation transfer, whose practical description was first given by Förster, arises from a dipole–dipole interaction between the electronic states of the donor and the acceptor, and does not involve the emission and reabsorption of a light field.
• 15.3: Excitons in Molecular Aggregates
The absorption spectra of periodic arrays of interacting molecular chromophores show unique spectral features that depend on the size of the system and disorder of the environment. We will investigate some of these features, focusing on the delocalized eigenstates of these coupled chromophores, known as excitons. These principles apply to the study of molecular crystals, Jaggregates, photosensitizers, and light-harvesting complexes in photosynthesis.
• 15.4: Multiple Particles and Second Quantization
In the case of a large number of nuclear or electronic degrees of freedom (or for photons in a quantum light field), it becomes tedious to write out the explicit product-state form of the state vector. Under these circumstances it becomes useful to define creation and annihilation operators. This representation is sometimes referred to as “second quantization”
• 15.5: Marcus Theory for Electron Transfer
Here we describe the rates of electron transfer between weakly coupled donor and acceptor states when the potential energy depends on a nuclear coordinate, i.e., nonadiabatic electron transfer. These results reflect the findings of Marcus’ theory of electron transfer.
15: Energy and Charge Transfer
In this section we will describe processes that result from the interaction between two or more molecular electronic states, such as the transport of electrons or electronic excitation. This problem can be formulated in terms of a familiar Hamiltonian
$H = H _ {0} + V$
in which $H_o$ describes the electronic states (including any coupling to nuclear motion), and $V$ is the interaction between the electronic states. In formulating such a problem we will need to consider some basic questions: Is V strong or weak? Are the electronic states described in a diabatic or adiabatic basis? How do nuclear degrees of freedom influence the electronic couplings? For weak couplings, we can describe the transport of electrons and electronic excitation with perturbation theory drawing on Fermi’s Golden Rule:
\begin{align} \overline {w} &= \frac {2 \pi} {\hbar} \sum _ {k , \ell} p _ {\ell} \left| V _ {k \ell} \right|^{2} \delta \left( E _ {k} - E _ {\ell} \right) \[4pt] &= \frac {1} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t \left\langle V _ {I} (t) V _ {I} ( 0 ) \right\rangle \end{align}
This approach underlies common descriptions of electronic energy transport and non-adiabatic electron transfer. We will discuss this regime concentrating on the influence of vibrational motions they are coupled to. However, the electronic couplings can also be strong, in which case the resulting states become delocalized. We will discuss this limit in the context of excitons that arise in molecular aggregates.
To begin, it is useful to catalog a number of electronic interactions of interest. We can use some schematic diagrams to illustrate them, emphasizing the close relationship between the various transport processes. However, we need to be careful, since these are not meant to imply a mechanism or meaningful information on dynamics. Here are a few commonly described processes involving transfer from a donor molecule D to an acceptor molecule A:
a) Resonance Energy Transfer
Applies to the transfer of energy from the electronic excited state of a donor to an acceptor molecule. Arises from a Coulomb interaction that is operative at long range, i.e., distances large compared to molecular dimensions. Requires electronic resonance. Named for the first practical derivations of expressions describing this effect: Förster Resonance Energy Transfer (FRET)
b) Electron Transfer Marcus theory.
Nonadiabatic electron transfer. Requires wavefunction overlap.
Ground state Electron Transfer
Excised state Electron Transfer
Hole Transfer
c) Electron-exchange energy transfer
Dexter Transfer. Requires wavefunction overlap. Singlet or triplet
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/15%3A_Energy_and_Charge_Transfer/15.01%3A_Electronic_Interactions.txt
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Förster resonance energy transfer (FRET) refers to the nonradiative transfer of an electronic excitation from a donor molecule to an acceptor molecule:
$\ce{D}^{*} + \ce{A} \rightarrow \ce{D} + \ce{A}^{*} \label{14.1}$
This electronic excitation transfer, whose practical description was first given by Förster, arises from a dipole–dipole interaction between the electronic states of the donor and the acceptor, and does not involve the emission and reabsorption of a light field. Transfer occurs when the oscillations of an optically induced electronic coherence on the donor are resonant with the electronic energy gap of the acceptor. The strength of the interaction depends on the magnitude of a transition dipole interaction, which depends on the magnitude of the donor and acceptor transition matrix elements, and the alignment and separation of the dipoles. The sharp $1/r^6$ dependence on distance is often used in spectroscopic characterization of the proximity of donor and acceptor.
The electronic ground and excited states of the donor and acceptor molecules all play a role in FRET. We consider the case in which we have excited the donor electronic transition, and the acceptor is in the ground state. Absorption of light by the donor at the equilibrium energy gap is followed by rapid vibrational relaxation that dissipates the reorganization energy of the donor $\lambda _ {D}$ over the course of picoseconds. This leaves the donor in a coherence that oscillates at the energy gap in the donor excited state $\omega _ {e g}^{D} \left( q _ {D} = d _ {D} \right)$. The time scale for FRET is typically nanoseconds, so this preparation step is typically much faster than the transfer phase. For resonance energy transfer we require a resonance condition, so that the oscillation of the excited donor coherence is resonant with the ground state electronic energy gap of the acceptor $\omega _ {e g}^{A} \left( q _ {A} = 0 \right)$. Transfer of energy to the acceptor leads to vibrational relaxation and subsequent acceptor fluorescence that is spectrally shifted from the donor fluorescence. In practice, the efficiency of energy transfer is obtained by comparing the fluorescence emitted from donor and acceptor.
This description of the problem lends itself naturally to treating with a DHO Hamiltonian, However, an alternate picture is also applicable, which can be described through the EG Hamiltonian. FRET arises from the resonance that occurs when the fluctuating electronic energy gap of a donor in its excited state matches the energy gap of an acceptor in its ground state. In other words
$\underbrace {\hbar \omega _ {e g}^{D} - \lambda _ {D}} _ {\Omega _ {D} (t)} = \underbrace {\hbar \omega _ {e g}^{A} - \lambda _ {A}} _ {\Omega _ {A} (t)} \label{14.2}$
These energy gaps are time-dependent with occasion crossings that allow transfer of energy.
Our system includes the ground and excited potentials of the donor and acceptor molecules. The four possible electronic configurations of the system are
$| G _ {D} G _ {A} \rangle , | E _ {D} G _ {A} \rangle , | G _ {D} E _ {A} \rangle , | E _ {D} E _ {A} \rangle$
Here the notation refers to the ground ($G$) or excited ($E$) vibronic states of either donor ($D$) or acceptor ($A$). More explicitly, the states also include the vibrational excitation:
$| E _ {D} G _ {A} \rangle = | e _ {D} n _ {D} ; g _ {A} n _ {A} \rangle$
Thus the system can have no excitation, one excitation on the donor, one excitation on the acceptor, or one excitation on both donor and acceptor. For our purposes, let’s only consider the two electronic configurations that are close in energy, and are likely to play a role in the resonance transfer in Equation \ref{14.2} and
$| E _ {D} G _ {A} \rangle$ and $| G _ {D} E _ {A} \rangle$
Since the donor and acceptor are weakly coupled, we can write our Hamiltonian for this problem in a form that can be solved by perturbation theory ($H = H _ {0} + V$). Working with the DHO. approach, our material Hamiltonian has four electronic manifolds to consider:
$\underbrace {\hbar \omega _ {e g}^{D} - \lambda _ {D}} _ {\Omega _ {D} (t)} = \underbrace {\hbar \omega _ {e g}^{A} - \lambda _ {A}} _ {\Omega _ {A} (t)} \label{14.3}$
Each of these is defined as we did previously, with an electronic energy and a dependence on a displaced nuclear coordinate. For instance
\begin{align} H _ {D}^{E} &= | e _ {D} \rangle E _ {e}^{D} \langle e _ {D} | + H _ {e}^{D} \label{14.4} \[4pt] H _ {e}^{D} &= \hbar \omega _ {0}^{D} \left( \tilde {p} _ {D}^{2} + \left( \tilde {q} _ {D} - \tilde {d} _ {D} \right)^{2} \right) \label{14.5} \end{align}
$E _ {e}^{D}$ is the electronic energy of donor excited state.
Then, what is $V$? Classically it is a Coulomb interaction of the form ,
$V = \sum _ {i j} \frac {q _ {i}^{D} q _ {j}^{A}} {\left| r _ {i}^{D} - r _ {j}^{A} \right|} \label{14.6}$
Here the sum is over all electrons and nuclei of the donor ($i$) and acceptor ($j$).
As is, this is challenging to work with, but at large separation between molecules, we can recast this as a dipole–dipole interaction. We define a frame of reference for the donor and acceptor molecule, and assume that the distance between molecules is large. Then the dipole moments for the molecules are
\begin{aligned} \overline {\mu}^{D} & = \sum _ {i} q _ {i}^{D} \left( r _ {i}^{D} - r _ {0}^{D} \right) \ \overline {\mu}^{A} & = \sum _ {j} q _ {j}^{A} \left( r _ {J}^{A} - r _ {0}^{A} \right) \end{aligned} \label{14.7}
The interaction between donor and acceptor takes the form of a dipole–dipole interaction:
$V = \dfrac {3 \left( \overline {\mu} _ {A} \cdot \hat {r} \right) \left( \overline {\mu} _ {D} \cdot \hat {r} \right) - \overline {\mu} _ {A} \cdot \overline {\mu} _ {D}} {\overline {r}^{3}} \label{14.8}$
where $r$ is the distance between donor and acceptor dipoles and $\hat{r}$ is a unit vector that marks the direction between them. The dipole operators here are taken to only act on the electronic states and be independent of nuclear configuration, i.e., the Condon approximation. We write the transition dipole matrix elements that couple the ground and excited electronic states for the donor and acceptor as
\begin{align} \overline {\mu} _ {A} &= | A \rangle \overline {\mu}_{AA^{*}} \left\langle A^{*} | + | A^{*} \right\rangle \overline {\mu} _ {A^{*} A} \langle A | \label{14.9} \[4pt] \overline {\mu} _ {D} &= | D \rangle \overline {\mu} _ {D D^{*}} \left\langle D^{*} | + | D^{*} \right\rangle \overline {\mu} _ {D^{*} D} \langle D | \label{14.10} \end{align}
For the dipole operator, we can separate the scalar and orientational contributions as
$\overline {\mu} _ {A} = \hat {u} _ {A} \mu _ {A} \label{14.11}$
This allows the transition dipole interaction in Equation \ref{14.8} to be written as
$V = \mu _ {A} \mu _ {B} \frac {\kappa} {r^{3}} [ | D^{*} A \rangle \left\langle A^{*} D | + | A^{*} D \right\rangle \left\langle D^{*} A | \right] \label{14.12}$
All of the orientational factors are now in the term $\kappa$
$\kappa = 3 \left( \hat {u} _ {A} \cdot \hat {r} \right) \left( \hat {u} _ {D} \cdot \hat {r} \right) - \hat {u} _ {A} \cdot \hat {u} _ {D} \label{14.13}$
We can now obtain the rates of energy transfer using Fermi’s Golden Rule expressed as a correlation function in the interaction Hamiltonian:
$w _ {k \ell} = \frac {2 \pi} {\hbar^{2}} \sum _ {\ell} p _ {\ell} \left| V _ {k \ell} \right|^{2} \delta \left( \omega _ {k} - \omega _ {\ell} \right) = \frac {1} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t \left\langle V _ {I} (t) V _ {I} ( 0 ) \right\rangle \label{14.14}$
Note that this is not a Fourier transform! Since we are using a correlation function there is an assumption that we have an equilibrium system, even though we are initially in the excited donor state. This is reasonable for the case that there is a clear time scale separation between the ps vibrational relaxation and thermalization in the donor excited state and the time scale (or inverse rate) of the energy transfer process.
Now substituting the initial state $\ell = | D^{*} A \rangle$ and the final state $k = | A^{*} D \rangle$, we find
$w _ {E T} = \frac {1} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t \frac {\left\langle \kappa^{2} \right\rangle} {r^{6}} \left\langle D^{*} A \left| \mu _ {D} (t) \mu _ {A} (t) \mu _ {D} ( 0 ) \mu _ {A} ( 0 ) \right| D^{*} A \right\rangle \label{14.15}$
where
$\mu _ {D} (t) = e^{i H _ {D} t / \hbar} \mu _ {D} e^{- i H _ {D} t / \hbar}.$
Here, we have neglected the rotational motion of the dipoles. Most generally, the orientational average is
$\left\langle \kappa^{2} \right\rangle = \langle \kappa (t) \kappa ( 0 ) \rangle \label{14.16}$
However, this factor is easier to evaluate if the dipoles are static, or if they rapidly rotate to become isotropically distributed. For the static case $\left\langle \kappa^{2} \right\rangle = 0.475$. For the case of fast loss of orientation:
$\left\langle \kappa^{2} \right\rangle \rightarrow \langle \kappa (t) \rangle \langle \kappa ( 0 ) \rangle = \langle \kappa \rangle^{2} = \dfrac{2}{3}$
Since the dipole operators act only on $A$ or $D^{*}$, and the $D$ and $A$ nuclear coordinates are orthogonal, we can separate terms in the donor and acceptor states.
\begin{aligned} w _ {E T} & = \frac {1} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t \frac {\left\langle \kappa^{2} \right\rangle} {r^{6}} \left\langle D^{*} \left| \mu _ {D} (t) \mu _ {D} ( 0 ) \right| D^{*} \right\rangle \left\langle A \left| \mu _ {A} (t) \mu _ {A} ( 0 ) \right| A \right\rangle \ & = \frac {1} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t \frac {\left\langle \kappa^{2} \right\rangle} {r^{6}} C _ {D^{*} D^{*}} (t) C _ {\mathrm {AA}} (t) \end{aligned} \label{14.17}
The terms in this equation represent the dipole correlation function for the donor initiating in the excited state and the acceptor correlation function initiating in the ground state. That is, these are correlation functions for the donor emission (fluorescence) and acceptor absorption. Remembering that $| D^{*} \rangle$ represents the electronic and nuclear configuration $| d^{*} n _ {D^{*}} \rangle$, we can use the displaced harmonic oscillator Hamiltonian or energy gap Hamiltonian to evaluate the correlation functions. For the case of Gaussian statistics we can write
$C _ {D _ {D}^{*}} \cdot (t) = \left| \mu _ {D D^{*}} \right|^{2} e^{- i \left( \omega _ {D D^{*}} - 2 \lambda _ {D} \right) t^{*} - g _ {D}^{*} (t)} \label{14.18}$
$C _ {A A} (t) = \left| \mu _ {A A} \right|^{2} e^{- i \omega _ {A A} t - g _ {A} (t)} \label{14.19}$
Here we made use of
$\omega _ {D^{*} D} = \omega _ {D D^{*}} - 2 \lambda _ {D}\label{14.20}$
which expresses the emission frequency as a frequency shift of $2 \lambda _ {D}$ relative to the donor absorption frequency. The dipole correlation functions can be expressed in terms of the inverse Fourier transforms of a fluorescence or absorption lineshape:
$C _ {D^{*} D^{\cdot}} (t) = \frac {1} {2 \pi} \int _ {- \infty}^{+ \infty} d \omega e^{- i \omega t} \sigma _ {f l u o r}^{D} ( \omega ) \label{14.21}$
$C _ {A A} (t) = \frac {1} {2 \pi} \int _ {- \infty}^{+ \infty} d \omega e^{- i \omega t} \sigma _ {a b s}^{A} ( \omega ) \label{14.22}$
To express the rate of energy transfer in terms of its common practical form, we make use of Parsival’s Theorem, which states that if a Fourier transform pair is defined for two functions, the integral over a product of those functions is equal whether evaluated in the time or frequency domain:
$\int _ {- \infty}^{\infty} f _ {1} (t) f _ {2}^{*} (t) d t = \int _ {- \infty}^{\infty} \tilde {f} _ {1} ( \omega ) \tilde {f} _ {2}^{*} ( \omega ) d \omega \label{14.23}$
This allows us to express the energy transfer rate as an overlap integral $J_{DA}$ between the donor fluorescence and acceptor absorption spectra:
$w _ {E T} = \frac {1} {\hbar^{2}} \frac {\left\langle \kappa^{2} \right\rangle} {r^{6}} \left| \mu _ {D D^{*}} \right|^{2} \left| \mu _ {A A^{\prime}} \right|^{2} \int _ {- \infty}^{+ \infty} d \omega \sigma _ {a b s}^{A} ( \omega ) \sigma _ {f u o r}^{D} ( \omega ) \label{14.24}$
Here is the lineshape normalized to the transition matrix element squared: $\sigma = \sigma / | \mu |^{2}$. The overlap integral is a measure of resonance between donor and acceptor transitions.
So, the energy transfer rate scales as $r^{-6}$, depends on the strengths of the electronic transitions for donor and acceptor molecules, and requires resonance between donor fluorescence and acceptor absorption. One of the things we have neglected is that the rate of energy transfer will also depend on the rate of excited donor state population relaxation. Since this relaxation is typically dominated by the donor fluorescence rate, the rate of energy transfer is commonly written in terms of an effective distance $R_0$, and the fluorescence lifetime of the donor $\tau_D$:
$w _ {E T} = \frac {1} {\tau _ {D}} \left( \frac {R _ {0}} {r} \right)^{6} \label{14.25}$
At the critical transfer distance $R_0$ the rate (or probability) of energy transfer is equal to the rate of fluorescence. $R_0$ is defined in terms of the sixth-root of the terms in Equation \ref{14.24}, and is commonly written as
$R _ {0}^{6} = \frac {9000 \ln ( 10 ) \phi _ {D} \left\langle \kappa^{2} \right\rangle} {128 \pi^{5} n^{4} N _ {A}} \int _ {0}^{\infty} d \overline {\nu} \frac {\sigma _ {\text {fluur}}^{D} ( \overline {V} ) \varepsilon _ {A} ( \overline {v} )} {\overline {V}^{4}} \label{14.26}$
This is the practical definition that accounts for the frequency dependence of the transitiondipole interaction and non-radiative donor relaxation in addition to being expressed in common units. $\overline {V}$ represents units of frequency in cm-1. The fluorescence spectrum $\sigma_{\text {fluor}}^{D}$ must be normalized to unit area, so that at $\sigma_{\text {fluor }}^{D}(\bar{v})$ is expressed in cm (inverse wavenumbers). The absorption spectrum $\varepsilon_{A}(\bar{v})$ must be expressed in molar decadic extinction coefficient units (liter/mol*cm). $n$ is the index of refraction of the solvent, $N_A$ is Avagadro’s number, and $\phi_D$ is the donor fluorescence quantum yield.
Transition Dipole Interaction
FRET is one example of a quantum mechanical transition dipole interaction. The interaction between two dipoles, $A$ and $D$, in Equation \ref{14.12} is
$V = \frac {\kappa} {r^{3}} \left\langle e \left| \mu _ {A} \right| g \right\rangle \left\langle g \left| \mu _ {D} \right| e \right\rangle \label{14.27}$
Here, $\left\langle g \left| \mu _ {D} \right| e \right\rangle$ is the transition dipole moment in Debye for the ground-to-excited state transition of molecule $A$. $r$ is the distance between the centers of the point dipoles, and $\kappa$ is the unitless orientational factor
$\kappa = 3 \cos \theta _ {1} \cos \theta _ {2} - \cos \theta _ {12}$
The figure below illustrates this function for the case of two parallel dipoles, as a function of the angle between the dipole and the vector defining their separation.
In the case of vibrational coupling, the dipole operator is expanded in the vibrational normal coordinate: $\mu = \mu _ {0} + \left( \partial \mu / \partial Q _ {A} \right) Q _ {A}$ and harmonic transition dipole matrix elements are
$\left\langle 1\left|\mu_{A}\right| 0\right\rangle=\sqrt{\frac{\hbar}{2 c \omega_{A}}} \frac{\partial \mu}{\partial Q_{A}} \label{14.28}$
where $\omega _ {A}$ is the vibrational frequency. If the frequency $V _ {A}$ is given in cm-1, and the transition dipole moment $\partial \mu / \partial Q _ {A}$ is given in units of $$\text { D } Å^{-1} \text {amu }^{-1 / 2}$$, then the matrix element in units of $D$ is
$\left| \left\langle 1 \left| \mu _ {A} \right| 0 \right\rangle \right| = 4.1058 v _ {A}^{- 1 / 2} \left( \partial \mu / \partial Q _ {A} \right)$
If the distance between dipoles is specified in Ångstroms, then the transition dipole coupling from Equation \ref{14.27} in cm-1 is
$V \left( c m^{- 1} \right) = 5034 \kappa r^{- 3}.$
Experimentally, one can determine the transition dipole moment from the absorbance $A$ as
$A = \left( \frac {\pi N _ {A}} {3 c^{2}} \right) \left( \frac {\partial \mu} {\partial Q _ {A}} \right)^{2} \label{14.29}$
Readings
1. Cheam, T. C.; Krimm, S., Transition dipole interaction in polypeptides: Ab initio calculation of transition dipole parameters. Chemical Physics Letters 1984, 107, 613-616.
2. Forster, T., Transfer mechanisms of electronic excitation. Discussions of the Faraday Society 1959, 27, 7-17.
3. Förster, T., Zwischenmolekulare Energiewanderung und Fluoreszenz. Annalen der Physik 1948, 437, 55-75.
4. Förster, T., Experimentelle und theoretische Untersuchung des zwischenmolecularen Uebergangs von Electronenanregungsenergie. Z. Naturforsch 1949, 4A, 321–327
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/15%3A_Energy_and_Charge_Transfer/15.02%3A_Forster_Resonance_Energy_Transfer_%28FRET%29.txt
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The absorption spectra of periodic arrays of interacting molecular chromophores show unique spectral features that depend on the size of the system and disorder of the environment. We will investigate some of these features, focusing on the delocalized eigenstates of these coupled chromophores, known as excitons. These principles apply to the study of molecular crystals, J-aggregates, photosensitizers, and light-harvesting complexes in photosynthesis. Similar topics are used in the description of properties of conjugated polymers and organic photovoltaics, and for extended vibrational states in IR and Raman spectroscopy.
Energy Transfer in the Strong Coupling Limit
Strong coupling between molecules leads to the delocalization of electronic or vibrational eigenstates, under which weak coupling models like FRET do not apply. From our studies of the coupled two-state system, we know that when the coupling between states is much larger that the energy splitting between the states ($\varepsilon _ {1} - \varepsilon _ {2} \ll 2 \mathrm {V}$) then the resulting eigenstates $| \pm \rangle$ are equally weighted symmetric and antisymmetric combinations of the two, whose energy eigenvalues are split by $2V$. Setting $\varepsilon _ {1} = \varepsilon _ {2} = \varepsilon$
$E _ {\pm} = \varepsilon \pm V$
$| \pm \rangle = \frac {1} {\sqrt {2}} ( | 1 \rangle \pm | 2 \rangle )$
If we excite one of these molecules, we expect that the excitation will flow back and forth at the Rabi frequency. So, what happens with multiple coupled chromophores, focusing particular interest on the placement of coupled chromophores into periodic arrays in space? In the strong coupling regime, the variation in the uncoupled energies is small, making this a problem of coupled quasi-degenerate states. With a spatially period structure, the resulting states bear close similarity to simple descriptions of electronic band structure using the tight-binding model.
Excitons
Excitons refer to electronic excited states that are not localized to a particular molecule. But beyond that there are many flavors. We will concentrate on Frenkel excitons, which refer to excited states in which the excited electron and the corresponding hole (or electron vacancy) reside on the same molecule. All molecules remain electrically neutral in the ground and excited states. This corresponds to what one would expect when one has resonant dipole–dipole interactions between molecules. When there is charge transfer character, the electron and hole can reside on different molecules of the coupled complex. These are referred to as Mott–Wannier excitons.
Absorption Spectrum of Molecular Dimer
To describe the spectroscopy of an array of many coupled chromophores, it is first instructive to work through a pair of coupled molecules. This is in essence the two-level problem from earlier. We consider a pair of molecules ($1$ and $2$), which each have a ground and electronically excited state $| e \rangle$ and $| g \rangle$) split by an energy gap $\varepsilon 0$, and a transition dipole moment $\overline {\mu}$. In the absence of coupling, the state of the system can be specified by specifying the electronic state of both molecules, leading to four possible states: $|g g\rangle,|e g\rangle,|g e\rangle,|e e\rangle$ whose energies are $0, \varepsilon_{0,} \varepsilon_{0}$, and $2 \varepsilon 0$, respectively.
For shorthand we define the ground state as $|G\rangle$ and the excited states as $|1\rangle$ and $|2\rangle$ to signify the the electronic excitation is on either molecule $1$ or $2$. In addition, the molecules are spaced by a separation $r_{12}$, and there is a transition dipole interaction that couples the molecules.
$V = J ( | 2 \rangle \langle 1 | + | 1 \rangle \langle 2 | )$
Following our description of transition dipole coupling the coupling strength $J$ is given by
$J=\frac{\left(\bar{\mu}_{1} \cdot \bar{\mu}_{2}\right)\left|\bar{r}_{12}\right|^{2}-3\left(\bar{\mu}_{1} \cdot \bar{r}_{12}\right)\left(\bar{\mu}_{2} \cdot \bar{r}_{12}\right)}{\left|\bar{r}_{12}\right|^{5}}=\frac{\mu_{1} \mu_{2}}{r_{12}^{3}} \kappa$
where the orientational factor is
$\kappa=\left(\hat{\mu}_{1} \cdot \hat{\mu}_{2}\right)-3\left(\hat{\mu}_{1} \cdot \hat{r}_{12}\right)\left(\hat{\mu}_{2} \cdot \hat{r}_{12}\right)$
We assume that the coupling is not too strong, so that we can just concentrate on how it influences $|1 \rangle$ and $|2 \rangle$ but not $|G \rangle$. Then we only need to describe the coupling induced shifts to the singly excited states, which are described by the Hamiltonian
$H = \left( \begin{array} {l l} {\varepsilon _ {0}} & {J} \ {J} & {\varepsilon _ {0}} \end{array} \right)$
As stated above, we find that the eigenvalues are
$E _ {\pm} = \varepsilon _ {0} \pm J$
and that the eigenstates are:
$|\pm\rangle=\frac{1}{\sqrt{2}}(|1\rangle \pm|2\rangle)$
These symmetric and antisymmetric states are delocalized across the two molecules, and in the language of Frenkel excitons are referred to as the one-exciton states. Furthermore, the dipole operator for the dimer is
$\overline {M} = \overline {\mu} _ {1} + \overline {\mu} _ {2}$
and so the transition dipole matrix elements are:
$M _ {\pm} = \langle \pm | \overline {M} | G \rangle = \frac {1} {\sqrt {2}} \left( \overline {\mu} _ {1} \pm \overline {\mu} _ {2} \right)$
$M_+$ and $M_-$ are oriented perpendicular to each other in the molecular frame. If we confine the molecular dipoles to be within a plane, with an angle $2 \theta$ between them, then the amplitude of M+ and M- is given by
\begin{array}{l}
M_{+}=2 \mu \cos \theta \
M_{-}=2 \mu \sin \theta
\end{array}
We can now predict the absorption spectrum for the dimer. We have two transitions from the ground state and the $| \pm \rangle$ states which are resonant at $\hbar \omega = \varepsilon _ {0} \pm J$ and which have an amplitude $\left| M _ {*} \right|^{2}$. The splitting between the peaks is referred to as the Davydov splitting. Note that the relative amplitude of the peaks allows one to infer the angle between the molecular transition dipoles. Also, note for $θ = 0°$ or $90°$, all amplitude appears in one transition with magnitude $2 | \mu |^{2}$, which is referred to as superradiant.
Frenkel Excitons with Periodic Boundary Conditions
Now let’s consider linear aggregate of $N$ periodically arranged molecules. We will assume that each molecule is a two-level electronic system with a ground state and an excited state. We will assume that electronic excitation moves an electron from the ground state to an unoccupied orbital of the same molecule. We will label the molecules with integer values ($n$) between $0$ and $N-1$:
If the molecules are separated along the chain by a lattice spacing $a$, then the size of the chain is $L = \alpha N$. Each molecule has a transition dipole moment $\mu$, which makes an angle $\beta$ with the axis of the chain.
In the absence of interactions, we can specify the state of the system exactly by identifying whether each molecule is in the electronically excited or ground state. If the state of molecule n within the chain is $\varphi _ {n}$, which can take on values of $g$ or $e$, then
$| \psi \rangle = | \varphi _ {0} , \varphi _ {1} , \varphi _ {2} \cdots \varphi _ {n} \cdots \varphi _ {N - 1} \rangle$
This representation of the state of the system is referred to as the site basis, since it is expressed in terms of each molecular site in the chain. For simplicity we write the ground state of the system as
$| G \rangle = | g , g , g \ldots , g \rangle$
If we excite one of the molecules within the aggregate, we have a singly excited state in which the nth molecule is excited, so that
$| \psi \rangle = | g , g , g , \ldots , e , \dots , g \rangle \equiv | n \rangle$
For shorthand, we identify this product state as $| n \rangle$ which is to be distinguished from the molecular eigenfunction at site $n$, $\varphi _ {n}$.
The singly excited state is assigned an energy $\mathcal {E} _ {0}$ corresponding to the electronic energy gap. In the absence of coupling, the singly excited states are $N$-fold degenerate, corresponding to a single excitation at any of the $N$ sites. If two excitations are placed on the chain we can see that there are $N(N-1)$ possible states with energy $2 \varepsilon _ {0}$, recognizing that the Pauli principle does not allow two excitations on the same site. When coupling is introduced, the mixing of these degenerate states leads to the one-exciton and two-exciton bands. For this discussion, we will concentrate on the one-exciton states.
The coupling between molecule $n$ and molecule $n′$ is given by the matrix element $V _ {n n^{\prime}}$. We will assume that a molecule interacts only with its neighbors, and that each pairwise interaction has a magnitude $J$
$V _ {n n^{\prime}} = J \delta _ {n , n^{\prime} \pm 1}$
If $V$ is a dipole–dipole interaction, the orientational factor $\kappa$ dictates that when the transition dipole angle $\beta < 54.7^{\circ}$ then the sign of the coupling $J < 0$, which is the case known as J-aggregates (after Edwin Jelley), and implies an offset stack of chromophores or head-to-tail arrangement. If $\beta > 54.7^{\circ}$ then $J > 0$, and the system is known as an H-aggregate.
To begin, we also apply periodic boundary conditions to this problem, which implies that we are describing the states of an N-molecule chain within an infinite linear chain. In terms of the Hamiltonian, the molecules at the beginning and end of our chain feel the same symmetric interactions to two neighbors as the other molecules. To write this in terms of a finite $N \times N$ matrix, one couples the first and last member of the chain: $J _ {0 , N - 1} = J _ {N - 1,0} = J$
$J _ {0 , N - 1} = J _ {N - 1,0} = J.$
With these observations in mind, we can write the Frenkel Exciton Hamiltonian for the linear aggregate in terms of a system Hamiltonian that reflects the individual sites and their couplings
\begin{align} H _ {0} &= H _ {S} + V \[4pt] H _ {S} &= \sum _ {n = 1}^{N} \varepsilon _ {0} | n \rangle \langle n | \[4pt] V &= \sum _ {n = 1}^{N} J \{| n^{\prime} \rangle \langle n | + | n \rangle \left\langle n^{\prime} | \right\} \delta _ {n , n^{\prime} \pm 1} \label{14.30} \end{align}
Here periodic boundary conditions imply that we replace $| N \rangle \Rightarrow | 0 \rangle$ and $| - 1 \rangle \Rightarrow | N - 1 \rangle$ where they appear.
The optical properties of the aggregate will be obtained by determining the eigenstates of the Hamiltonian. We look for solutions that describe one-exciton eigenstates as an expansion in the site basis.
$| \psi (x) \rangle = \sum _ {n = 0}^{N - 1} c _ {n} ( \mathrm {x} ) | \varphi _ {n} \left( x - x _ {n} \right) \rangle \label{14.31}$
which is written in order to point out the dependence of these wavefunctions on the lattice spacing x, and the position of a particular molecule at xn. Such an expansion should work well when the electronic interactions between sites is weak enough to treat perturbatively. For the electronic structure of solids, this is known as the tight binding model, which describes band structure as a linear combinations of atomic orbitals.
Rather than diagonalizing the Hamiltonian, we can take advantage of its translational symmetry to obtain the eigenstates. The symmetry of the Hamiltonian is such that it is unchanged by any integral number of translations along the chain. That is the results are unchanged for any summation in Equation \ref{14.30} and \ref{14.31} over $N$ consecutive integers. Similarly, the molecular wavefunction at any site is unchanged by such a translation. Written in terms of a displacement operator $D = e^{i p _ {x} \alpha / \hbar}$ that shifts the molecular wavefunction by one lattice constant
$| \varphi ( x + n \alpha ) \rangle = D^{n} | \varphi (x) \rangle \label{14.32}$
These observations underlie Bloch’s theorem, which states that the eigenstates of a periodic system will vary only by a phase shift when displaced by a lattice constant.
$| \psi ( x + \alpha ) \rangle = e^{i k \alpha} | \psi (x) \rangle \label{14.33}$
Here $k$ is the wavevector, or reciprocal lattice vector, a real quantity. Thus the expansion coefficients in Equation \ref{14.31} will have an amplitude that reflects an excitation spread equally among the N sites, and only vary between sites by a spatially varying phase factor. Equivalently, the eigenstates are expected to have a form that is a product of a spatially varying phase factor and a periodic function:
$| \psi (x) \rangle = e^{i k x} u (x) \label{14.34}$
These phase factors are closely related to the lattice displacement operators. If the linear chain has $N$ molecules, the eigenstates must remain unchanged with a translation by the length of the chain $L = \alpha N$:
$| \psi \left( x _ {n} + L \right) \rangle = | \psi \left( x _ {n} \right) \rangle$
Therefore, we see that our wavefunctions must satisfy
$N k \alpha = 2 \pi m\label{14.35}$
where $m$ is an integer. Furthermore, since there are $N$ sites on the chain, unique solutions to Equation \ref{14.35} require that $m$ can only take on $N$ consecutive integer values. Like the site index $n$, there is no unique choice of $m$. Rewriting Equation \ref{14.35}, the wavevector is
$k _ {m} = \frac {2 \pi} {\alpha} \frac {m} {N} \label{14.36}$
We see that for an $N$ site lattice, $m$ can take on the $N$ consecutive integer values, so that $k_m\alpha$ varies over a $2\pi$ range of angles. The wavevector index m labels the $N$ one-exciton eigenstates of an $N$ molecule chain. By convention, $k_m$ is chosen such that
$- \pi / \alpha < k _ {m} \leq \pi / \alpha.$
Then the corresponding values of $m$ are integers from $-N-1)/2$ to $N-1/2$ if there are an odd number of lattice sites or $-N-2)/2$ to $N/2$ for an even number of sites. For example, a 20 molecule chain would have $m = -9,\, -8,\, … 9,\,10$.
These findings lead to the general form for the m one-exciton eigenstates
$| k _ {m} \rangle = \frac {1} {\sqrt {N}} \sum _ {n = 0}^{N - 1} e^{i n k _ {m} \alpha} | n \rangle \label{14.37}$
The factor of $\sqrt{N}$ assures proper normalization of the wavefunction, $\langle \psi | \psi \rangle = 1$. Comparing Equation \ref{14.37} and \ref{14.31} we see that the expansion coefficients for the nth site of the mth eigenstate is
$c _ {m , n} = \frac {1} {\sqrt {N}} e^{i n k _ {m} \alpha} = \frac {1} {\sqrt {N}} e^{i 2 \pi n m / N} \label{14.38}$
We see that for state $| k _ {0} \rangle$, with $m = 0$, the phase factor is the same for all sites. In other words, the transition dipoles of the chain will oscillate in-phase, constructively adding for all sites. For the case that $k _ {m} = \pi / \alpha$, we see that each site is out-of-phase with its nearest neighbors. Looking at the case of the dimer, $N = 2$, we see that $m = 0$ or $1$, $k_m = 0$ or $\pi/2$, and we recover the expected symmetric and antisymmetric eigenstates:
$| k _ {0} \rangle = \frac {1} {\sqrt {2}} \sum _ {n = 0}^{1} e^{i n 0} | n \rangle = \frac {1} {\sqrt {2}} ( | 0 \rangle + | 1 \rangle )$
for $k=0$ and
$| k _ {1} \rangle = \frac {1} {\sqrt {2}} \sum _ {n = 0}^{1} e^{i n \pi} | n \rangle = \frac {1} {\sqrt {2}} ( | 0 \rangle - | 1 \rangle )$
for $k = \pi / \alpha$.
Schematically for $N = 20$, we see how the dipole phase varies with $k_m$, plotting the real and imaginary components of the expansion coefficients.
Also, we can evaluate the one-exciton transition dipole matrix elements, $M(k_m)$, which are expressed as superpositions of the dipole moments at each site, $\overline {\mu} _ {n}$:
$\overline {M} = \sum _ {n = 0}^{N - 1} \overline {\mu} _ {n} \label{14.39}$
\begin{align} M _ {m} = \left\langle k _ {m} | \overline {M} | G \right\rangle \[4pt] = \frac {1} {\sqrt {N}} \sum _ {n = 0}^{N - 1} e^{i n k _ {n} \alpha} \left\langle n \left| \overline {\mu} _ {n} \right| G \right\rangle \label{14.40} \end{align}
The phase of the transition dipoles of the chain matches their phase within each k state. Thus for our problem, in which all of the dipoles are parallel, transitions from the ground state to the $k_m=0$ state will carry all of the oscillator strength. Plotted below is an illustration of the phase relationships between dipoles in a chain with $N = 20$.
Finally, let’s solve for the one-exciton energy eigenvalues by calculating the expectation value of the Hamiltonian operator, Equation \ref{14.30}
\begin{align} E \left( k _ {m} \right) &= \left\langle k \left| H _ {0} \right| k \right\rangle \[4pt] &= \frac {1} {N} \sum _ {n , m = 0}^{N - 1} e^{i ( n - m ) k \alpha} \left\langle m \left| H _ {0} \right| n \right\rangle \label{14.41} \end{align}
$\left\langle k _ {m} \left| H _ {S} \right| k _ {m} \right\rangle = \frac {1} {N} \sum _ {n = 0}^{N - 1} \varepsilon _ {0} = \varepsilon _ {0}$
\begin{align} \left\langle k _ {m} | V | k _ {m} \right\rangle &= \frac {1} {N} \sum _ {n = 0}^{N - 1} \left\{e^{i k _ {m} \alpha} \langle n - 1 | V | n \rangle + e^{- i k _ {m} \alpha} \langle n + 1 | V | n \rangle \right\}\[4pt] &= 2 J \cos \left( k _ {m} \alpha \right) \label{14.42} \end{align}
You predict that the one-exciton band of states varies in energy between $\varepsilon _ {0} - 2 J$ and $\varepsilon _ {0} + 2 J$. If we take J as negative, as expected for the case of J-aggregates (negative couplings), then $k = 0$ is at the bottom of the band. Examples are illustrated below for the $N=20$ aggregate.
Note that the result in Equation \ref{14.42} gives you a splitting of $4J$ between the two states of the dimer, unlike the expected $2J$ splitting from earlier. This is a result of the periodic boundary conditions that we enforce here. We are now in a position to plot the absorption spectrum for aggregate, summing over eigenstates and assuming a Lorentzian lineshape for the system:
$\sigma ( \omega ) = \sum _ {m} \left| M _ {m} \right|^{2} \frac {\Gamma^{2}} {\left( \hbar \omega - E \left( k _ {m} \right) \right) + \Gamma^{2}}$
For a 20 oscillator chain with negative coupling, the spectrum is plotted below. We have one peak corresponding to the k0 mode that is peaked at $\hbar \omega = \varepsilon _ {0} - 2 J$ and carries the oscillator strength of all 20 dipoles.
Absorption spectrum for $N=20$ aggregate with periodic boundary conditions and $J<0$.
Open Boundary Conditions
Similar types of solutions appear without using periodic boundary conditions. For the case of open boundary conditions, in the molecules at the end of the chain are only coupled to the one nearest neighbor in the chain. In this case, it is helpful to label on the sites from $n = 1,2,...,N$. Furthermore, $m = 1,2,...,N$. Under those conditions, one can solve for the eigenstates using use the boundary condition that $\psi = 0$ at sites $0$ and $N+1$. The change in boundary condition gives sine solutions:
$| k _ {m} \rangle = \sqrt {\frac {2} {N + 1}} \sum _ {n = 1}^{N} \sin \left( \frac {\pi m n} {N + 1} \right) | n \rangle$
The energy eigenvalues are
$E _ {m} = \omega _ {0} + 2 J \cos \left( \frac {\pi m} {N + 1} \right)$
Absorption spectrum for N = 20 aggregate with periodic boundary conditions and $J < 0$. Returning to the case of the dimer ($N=2$), we can now confirm that we recover the symmetric and anti-symmetric eigenstates, with an energy splitting of $2J$.
If you calculate the oscillator strength for these transitions using the dipole operator in Equation \ref{14.39}, one finds:
$M _ {m}^{2} = \left| \left\langle k _ {m} | \overline {M} | G \right\rangle \right|^{2} = \left( \frac {1 - ( - 1 )^{m}} {2} \right)^{2} \frac {2 \mu^{2}} {N + 1} \cot^{2} \left( \frac {\pi m} {2 ( N + 1 )} \right)$
This result shows that most of the oscillator strength lies in the $m =1$ state, for which all oscillators are in phase. For large $N$, $M_1^2$ carries 81% of the oscillator strength, with approximately 9% in the transition to the $m = 3$ state.
Absorption spectra for $N=3,7,11$ for negative coupling.
The shift in the peak of the absorption relative to the monomer gives the coupling $J$. Including long-range interactions has the effect of shifting the exciton band asymmetrically about $\omega _ {0}$.
• $\Omega _ {1} = \omega _ {0} + 2.4 J$ ($m=1$, bottom of the band with $J$ negative)
• $\Omega _ {N} = \omega _ {0} - 1.8 J$ (Top of band)
Exchange Narrowing
If the chain is not homogeneous, i.e., all molecules do not have same site energy $\varepsilon _ {0}$, then we can model this effect as Gaussian random disorder. The energy of a given site is
$\varepsilon _ {n} = \varepsilon _ {0} + \delta \omega _ {n}$
We add as an extra term to our earlier Hamiltonian, Equation \ref{14.30}, to account for this variation.
$H _ {0} = H _ {s} + H _ {d i s} + V$
$H _ {d i s} = \sum _ {n} \delta \omega _ {n} | n \rangle \langle n |$
The effect is to shift and mix the homogeneous exciton states. Absorption spectra for N = 3,7,11 for negative coupling
$\delta \Omega _ {k} = \left\langle k \left| H _ {d i s} \right| k \right\rangle = \frac {2} {N + 1} \sum _ {n} \sin^{2} \left( \frac {\pi k n} {N + 1} \right) \delta \omega _ {n}$
We find that these shifts are also Gaussian random variables, with a standard deviation of $\Delta \sqrt {3 / 2 ( N + 1 )}$, where $\Delta$ is the standard deviation for site energies. So, the delocalization of the eigenstate averages the disorder over $N$ sites, which reduces the distribution of energies by a factor scaling as $N$. The narrowing of the absorption lineshape with delocalization is called exchange narrowing. This depends on the distribution of site energies being relatively small: $\Delta \ll 3 \pi | J | / N^{3 / 2}$.
Absorption spectra for $N =2,6,30$ normalized to the number of oscillators. $3\Delta = J$ and $J<0$.
Readings
1. Knoester, J., Optical Properties of Molecular Aggregates. In Proceedings of the International School of Physics "Enrico Fermi" Course CXLIX, Agranovich, M.; La Rocca, G. C., Eds. IOS Press: Amsterdam, 2002; pp 149-186.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/15%3A_Energy_and_Charge_Transfer/15.03%3A_Excitons_in_Molecular_Aggregates.txt
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In the case of a large number of nuclear or electronic degrees of freedom (or for photons in a quantum light field), it becomes tedious to write out the explicit product-state form of the state vector, i.e.,
$| \psi \rangle = | \varphi _ {1} , \varphi _ {2} , \varphi _ {3} \cdots \rangle$
Under these circumstances it becomes useful to define creation and annihilation operators. If $| \psi \rangle$ refers to the state of multiple harmonic oscillators, then the Hamiltonian has the form
$H = \sum _ {\alpha} \left( \frac {p _ {\alpha}^{2}} {2 m _ {\alpha}} + \frac {1} {2} m _ {\alpha} \omega _ {\alpha}^{2} q _ {\alpha}^{2} \right) \label{14.43}$
which can also be expressed as
$H = \sum _ {\alpha} \hbar \omega _ {\alpha} \left( a _ {\alpha}^{\dagger} a _ {\alpha} + \frac {1} {2} \right) \label{14.44}$
and the eigenstates represented in through the occupation of each oscillator
$| \psi \rangle = | n _ {1} , n _ {2} , n _ {3} \dots ).$
This representation is sometimes referred to as “second quantization”, because the classical Hamiltonian was initially quantized by replacing the position and momentum variables by operators, and then these quantum operators were again replaced by raising and lowering operators.
The operator $a _ {\alpha}^{\dagger}$ raises the occupation in mode $| n _ {\alpha} \rangle$, and $a _ {\alpha}$ lowers the excitation in mode $| n _ {\alpha} \rangle$. The eigenvalues of these operators, $n _ {\alpha} \rightarrow n _ {\alpha} \pm 1$, are captured by the commutator relationships:
$\left[ a _ {\alpha} , a _ {\beta}^{\dagger} \right] = \delta _ {\alpha \beta} \label{14.45}$
$\left[ a _ {\alpha} , a _ {\beta} \right] = 0 \label{14.46}$
Equation \ref{14.45} indicates that the raising and lower operators do not commute if they are operators in the same degree of freedom ($\alpha = \beta$), but they do otherwise. Written another way, these expression indicate that the order of operations for the raising and lowering operators in different degrees of freedom commute.
$a _ {\alpha} a _ {\beta}^{\dagger} = a _ {\beta}^{\dagger} a _ {\alpha} \label{14.47}$
$a _ {\alpha} a _ {\beta} = a _ {\beta} a _ {\alpha} \label{14.48A}]$
$a _ {\alpha}^{\dagger} a _ {\beta}^{\dagger} = a _ {\beta}^{\dagger} a _ {\alpha}^{\dagger} \label{14.48B}$
These expressions also imply that the eigenfunctions operations of the forms in Equations \ref{14.47}-\ref{14.48B} are the same, so that these eigenfunctions should be symmetric to interchange of the coordinates. That is, these particles are bosons.
This observations proves an avenue to defining raising and lowering operators for electrons. Electrons are fermions, and therefore antisymmetric to exchange of particles. This suggests that electrons will have raising and lowering operators that change the excitation of an electronic state up or down following the relationship
$b _ {\alpha} b _ {\beta}^{\dagger} = - b _ {\beta}^{\dagger} b _ {\alpha} \label{14.49}$
or
$\left[ b _ {\alpha} , b _ {\beta}^{\dagger} \right] _ {+} = \delta _ {\alpha \beta} \label{14.50}$
where $[ \ldots ] _+$ refers to the anti-commutator. Further, we write
$\left[ b _ {\alpha} , b _ {\beta} \right] _ {+} = 0 \label{14.51}$
This comes from considering the action of these operators for the case where $\alpha = \beta$. In that case, taking the Hermetian conjugate, we see that Equation \ref{14.51} gives
$2 b _ {\alpha}^{\dagger} b _ {\alpha}^{\dagger} = 0 \label{14.52A}$
or
$b _ {\alpha}^{\dagger} b _ {\alpha}^{\dagger} = 0 \label{14.52B}$
This relationship says that we cannot put two excitations into the same state, as expected for Fermions. This relationship indicates that there are only two eigenfunctions for the operators $b _ {\alpha}^{\dagger}$ and $b _ {\alpha}$, namely $| n _ {\alpha} = 0 \rangle$ and $| n _ {\alpha} = 1 \rangle$. This is also seen with Equation \ref{14.50}, which indicates that
$b _ {\alpha}^{\dagger} b _ {\alpha} | n _ {\alpha} \rangle + b _ {\alpha} b _ {\alpha}^{\dagger} | n _ {\alpha} \rangle = | n _ {\alpha} \rangle$
or
$b _ {\alpha} b _ {\alpha}^{\dagger} | n _ {\alpha} \rangle = \left( 1 - b _ {\alpha}^{\dagger} b _ {\alpha} \right) | n _ {\alpha} \rangle \label{14.53}$
If we now set $| n _ {\alpha} \rangle = | 0 \rangle$, we find that Equation \ref{14.53} implies
$\left. \begin{array} {l} {b _ {\alpha} b _ {\alpha}^{\dagger} | 0 \rangle = | 0 \rangle} \ {b _ {\alpha}^{\dagger} b _ {\alpha} | 0 \rangle = 0} \ {b _ {\alpha} b _ {\alpha}^{\dagger} | 1 \rangle = 0} \ {b _ {\alpha}^{\dagger} b _ {\alpha} | 1 \rangle = | 1 \rangle} \end{array} \right. \label{14.54}$
Again, this reinforces that only two states, $| 0 \rangle$ and $| 1 \rangle$, are allowed for electron raising and lowering operators. These are known as Pauli operators, since they implicitly enforce the Pauli exclusion principle. Note, in Equation \ref{14.54}, that $| 0 \rangle$ refers to the eigenvector with an eigenvalue of zero $| \varphi _ {0} \rangle$, whereas “0” refers to the null vector.
Frenkel Excitons
For electronic chromophores, we use the notation $| g \rangle$ and $| e \rangle$ for the states of an electron in its ground or excited state. The state of the system for one excitation in an aggregate
$| n \rangle = | g , g , g , g \dots e \ldots g \rangle$
can then be written as $a _ {n}^{\dagger} | G \rangle$, or simply $a _ {n}^{\dagger}$, and the Frenkel exciton Hamiltonian is
$H _ {0} = \sum _ {n = 0}^{N - 1} \varepsilon _ {0} | n \rangle \langle n | + \sum _ {n , m} J _ {n , m} | n \rangle \langle m | \label{14.55}$
or
$H _ {0} = \sum _ {n} \varepsilon _ {0} b _ {n}^{\dagger} b _ {n} + \sum _ {n , m} J _ {n , m} b _ {n}^{\dagger} b _ {m} \label{14.56}$
Readings
1. Schatz, G. C.; Ratner, M. A., Quantum Mechanics in Chemistry. Dover Publications: Mineola, NY, 2002; p. 119.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/15%3A_Energy_and_Charge_Transfer/15.04%3A_Multiple_Particles_and_Second_Quantization.txt
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The displaced harmonic oscillator (DHO) formalism and the Energy Gap Hamiltonian have been used extensively in describing charge transport reactions, such as electron and proton transfer. Here we describe the rates of electron transfer between weakly coupled donor and acceptor states when the potential energy depends on a nuclear coordinate, i.e., nonadiabatic electron transfer. These results reflect the findings of Marcus’ theory of electron transfer.
We can represent the problem as calculating the transfer or reaction rate for the transfer of an electron from a donor to an acceptor
$\ce{D + A \rightarrow D^{+} + A^{-}}\label{4.57}$
This reaction is mediated by a nuclear coordinate $q$. This need not be, and generally isn’t, a simple vibrational coordinate. For electron transfer in solution, we most commonly consider electron transfer to progress along a solvent rearrangement coordinate in which solvent reorganizes its configuration so that dipoles or charges help to stabilize the extra negative charge at the acceptor site. This type of collective coordinate is illustrated below.
The external response of the medium along the electron transfer coordinate is referred to as “outer shell” electron transfer, whereas the influence of internal vibrational modes that promote ET is called “inner shell”. The influence of collective solvent rearrangements or intramolecular vibrations can be captured with the use of an electronic transition coupled to a harmonic bath.
Normally we associate the rates of electron transfer with the free-energy along the electron transfer coordinate $q$. Pictures such as the ones above that illustrate states of the system with electron localized on the donor or acceptor electrons hopping from donor to acceptor are conceptually represented through diabatic energy surfaces. The electronic coupling $J$ that results in transfer mixes these diabatic states in the crossing region. From this adiabatic surface, the rate of transfer for the forward reaction is related to the flux across the barrier. From classical transition state theory we can associate the rate with the free energy barrier using
$k _ {f} = A \exp \left( - \Delta G^{\dagger} / k _ {B} T \right)$
If the coupling is weak, we can describe the rates of transfer between donor and acceptor in the diabatic basis with perturbation theory. This accounts for nonadiabatic effects and tunneling through the barrier.
To begin we consider a simple classical derivation for the free-energy barrier and the rate of electron transfer from donor to acceptor states for the case of weakly coupled diabatic states. First we assume that the free energy or potential of mean force for the initial and final state,
$\mathrm {G} ( \mathrm {q} ) = - \mathrm {k} _ {\mathrm {B}} \mathrm {T} \ln \mathrm {P} ( \mathrm {q} )$
is well represented by two parabolas.
\begin{align} G _ {D} ( q ) &= \frac {1} {2} m \omega _ {0}^{2} \left( q - d _ {D} \right)^{2} \label{14.58a} \[4pt] G _ {A} ( q ) &= \frac {1} {2} m \omega _ {0}^{2} \left( q - d _ {A} \right)^{2} + \Delta G^{0} \label{14.58b} \end{align}
To find the barrier height $\Delta G^{\dagger}$, we first find the crossing point $dC$ where
$G_D(d_C) = G_A(d_C). \label{14.58c}$
Substituting Equations \ref{14.58a} and \ref{14.58b} into Equation \ref{14.58c}
$\frac {1} {2} m \omega _ {0}^{2} \left( d _ {c} - d _ {D} \right)^{2} = \Delta G^{\circ} + \frac {1} {2} m \omega _ {0}^{2} \left( d _ {C} - d _ {A} \right)^{2}$
and solving for $d_C$ gives
\begin{align} d _ {C} &= \frac {\Delta G^{\circ}} {m \omega _ {0}^{2}} \left( \frac {1} {d _ {A} - d _ {D}} \right) + \frac {d _ {A} + d _ {D}} {2} \[4pt] & = \frac {\Delta G^{\circ}} {2 \lambda} \left( d _ {A} - d _ {D} \right) + \frac {d _ {A} + d _ {D}} {2} \end{align} .
The last expression comes from the definition of the reorganization energy ($\lambda$), which is the energy to be dissipated on the acceptor surface if the electron is transferred at $d_D$,
\begin{align} \lambda & = G _ {A} \left( d _ {D} \right) - G _ {A} \left( d _ {A} \right) \ & = \frac {1} {2} m \omega _ {0}^{2} \left( d _ {D} - d _ {A} \right)^{2} \label{14.59} \end{align}
Then, the free energy barrier to the transfer $\Delta G^{\dagger}$ is
\begin{aligned} \Delta G^{\dagger} & = G _ {D} \left( d _ {C} \right) - G _ {D} \left( d _ {D} \right) \ & = \frac {1} {2} m \omega _ {0}^{2} \left( d _ {C} - d _ {D} \right)^{2} \ & = \frac {1} {4 \lambda} \left[ \Delta G^{\circ} + \lambda \right]^{2} \end{aligned}.
So the Arrhenius rate constant is for electron transfer via activated barrier crossing is
$k _ {E T} = A \exp \left[ \frac {- \left( \Delta G^{\circ} + \lambda \right)^{2}} {4 \lambda k T} \right] \label{14.60}$
This curve qualitatively reproduced observations of a maximum electron transfer rate under the conditions $- \Delta G^{\circ} = \lambda$, which occurs in the barrierless case when the acceptor parabola crosses the donor state energy minimum.
We expect that we can more accurately describe nonadiabatic electron transfer using the DHO or Energy Gap Hamiltonian, which will include the possibility of tunneling through the barrier when donor and acceptor wavefunctions overlap. We start by writing the transfer rates in terms of the potential energy as before. We recognize that when we calculate thermally averaged transfer rates that this is equivalent to describing the diabatic free energy surfaces. The Hamiltonian is
$H = H _ {0} + V \label{14.61}$
with
$H _ {0} = | D \rangle H _ {D} \langle D | + | A \rangle H _ {A} \langle A | \label{14.62}$
Here $| D \rangle$ and $| A \rangle$ refer to the potential where the electron is either on the donor or acceptor, respectively. Also remember that $| D \rangle$ refers to the vibronic states
$| D \rangle = | d , n \rangle.$
These are represented through the same harmonic potential, displaced from one another vertically in energy by
$\Delta E = E _ {A} - E _ {D}$
and horizontally along the reaction coordinate $q$:
\begin{align} H _ {D} &= | d \rangle E _ {D} \langle d | + H _ {d} \[4pt] H _ {A} &= | a \rangle E _ {A} \langle a | + H _ {a} \label{14.63} \end{align}
\left.\begin{aligned} H _ {d} & = \hbar \omega _ {0} \left( p^{2} + \left( q - d _ {D} \right)^{2} \right) \ H _ {a} & = \hbar \omega _ {0} \left( p^{2} + \left( q - d _ {A} \right)^{2} \right) \end{aligned} \right. \label{14.64}
Here we are using reduced variables for the momenta, coordinates, and displacements of the harmonic oscillator. The diabatic surfaces can be expressed as product states in the electronic and nuclear configurations: $| D \rangle = | d , n \rangle$. The interaction between the surfaces is assigned a coupling $J$
$V = J [ | d \rangle \langle a | + | a \rangle \langle d | ] \label{14.65}$
We have made the Condon approximation, implying that the transfer matrix element that describes the electronic interaction has no dependence on nuclear coordinate. Typically this electronic coupling is expected to drop off exponentially with the separation between donor and acceptor orbitals;
$J = J _ {0} \exp \left( - \beta _ {E} \left( R - R _ {0} \right) \right) \label{14.66}$
Here $\beta_E$ is the parameter governing the distance dependence of the overlap integral. For our purposes, even though this is a function of donor-acceptor separation (R), we take this to vary slowly over the displacements investigated here, and therefore be independent of the nuclear coordinate ($Q$).
Marcus evaluated the perturbation theory expression for the transfer rate by calculating Franck-Condon factors for the overlap of donor and acceptor surfaces, in a manner similar to our treatment of the DHO electronic absorption spectrum. Similarly, we can proceed to calculate the rates of electron transfer using the Golden Rule expression for the transfer of amplitude between two states
$w _ {k \ell} = \frac {1} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t \left\langle V _ {I} (t) V _ {I} ( 0 ) \right\rangle \label{14.67}$
Using
$V _ {I} (t) = e^{i H _ {0} t / \hbar} V e^{- i H _ {0} t / \hbar},$
we write the electron transfer rate in the DHO eigenstate form as
$w _ {E T} = \frac {| J |^{2}} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t e^{- i \Delta E t / \hbar} F (t) \label{14.68}$
where
$F (t) = \left\langle e^{i H _ {d} t / h} e^{- i H _ {a} t / h} \right\rangle \label{14.69}$
This form emphasizes that the electron transfer rate is governed by the overlap of vibrational wavepackets on the donor and acceptor potential energy surfaces.
Alternatively, we can cast this in the form of the Energy Gap Hamiltonian. This carries with is a dynamical picture of the electron transfer event. The energy of the two states have time-dependent (fluctuating) energies as a result of their interaction with the environment. Occasionally the energy of the donor and acceptor states coincide that is the energy gap between them is zero. At this point transfer becomes efficient. By integrating over the correlation function for these energy gap fluctuations, we characterize the statistics for barrier crossing, and therefore forward electron transfer.
Similar to before, we define a donor-acceptor energy gap Hamiltonian
$H _ {A D} = H _ {A} - H _ {D} \label{14.70}$
which allows us to write
$F (t) = \left\langle \exp _ {+} \left[ - \frac {i} {\hbar} \int _ {0}^{t} d t^{\prime} H _ {A D} \left( t^{\prime} \right) \right] \right\rangle \label{14.71}$
and
$H _ {A D} (t) = e^{i H _ {d} t / \hbar} H _ {A D} e^{- i H _ {d} t / \hbar} \label{14.72}$
These expressions and application of the cumulant expansion to equation allows us to express the transfer rate in terms of the lineshape function and correlation function
$F (t) = \exp \left[ \frac {- i} {\hbar} \left\langle H _ {A D} \right\rangle t - g (t) \right] \label{14.73}$
$g (t) = \int _ {0}^{t} d \tau _ {2} \int _ {0}^{\tau _ {2}} d \tau _ {1} C _ {A D} \left( \tau _ {2} - \tau _ {1} \right) \label{14.74}$
$C _ {A D} (t) = \frac {1} {\hbar^{2}} \left\langle \delta H _ {A D} (t) \delta H _ {A D} ( 0 ) \right\rangle \label{14.75}$
$\left\langle H _ {A D} \right\rangle = \lambda \label{14.76}$
The lineshape function can also be written as a sum of many coupled nuclear coordinates, $q_{\alpha}$. This expression is commonly applied to the vibronic (inner shell) contributions to the transfer rate:
\begin{align} g (t) &= - \sum _ {\alpha} \left( d _ {\alpha}^{A} - d _ {\alpha}^{D} \right)^{2} \left[ \left( \overline {n} _ {\alpha} + 1 \right) \left( e^{- i \omega _ {\alpha} t} - 1 + i \omega _ {0} t \right) + \overline {n} _ {\alpha} \left( e^{i \omega _ {a} t} - 1 - i \omega _ {0} t \right) \right] \[4pt] &= - \sum _ {\alpha} \left( d _ {\alpha}^{A} - d _ {\alpha}^{D} \right)^{2} \left[ \operatorname {coth} \left( \beta \hbar \omega _ {\alpha} / 2 \right) \left( \cos \omega _ {\alpha} t - 1 \right) - i \left( \sin \omega _ {\alpha} t - \omega _ {\alpha} t \right) \right] \label{14.77} \end{align}
Substituting the expression for a single harmonic mode into the Golden Rule rate expression gives
\begin{align} w _ {E T} &= \frac {| J |^{2}} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t e^{- i \Delta E t / \hbar - g (t)} \label{4.78} \[4pt] &= \frac {| J |^{2}} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t e^{- i ( \Delta E + \lambda ) t / \hbar} \exp \left[ D \left( \operatorname {coth} \left( \beta \hbar \omega _ {0} / 2 \right) \left( \cos \omega _ {0} t - 1 \right) - i \sin \omega _ {0} t \right) \right] \label{14.78} \end{align}
where
$D = \left( d _ {A} - d _ {D} \right)^{2} \label{14.79}$
This expression is very similar to the one that we evaluated for the absorption lineshape of the Displaced Harmonic Oscillator model. A detailed evaluation of this vibronically mediated transfer rate is given in Jortner.
To get a feeling for the dependence of $k$ on $q$, we can look at the classical limit $\hbar \omega \ll k T$. This corresponds to the case where one is describing the case of a low frequency “solvent mode” or “outer sphere” effect on the electron transfer. Now, we neglect the imaginary part of $g(t)$ and take the limit
$\operatorname {coth} ( \beta \hbar \omega / 2 ) \rightarrow 2 / \beta \hbar \omega$
so
$w _ {E T} = \frac {| J |^{2}} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t e^{- i ( \Delta E + \lambda ) t} \exp \left( - \left( \frac {2 D k _ {B} T} {\hbar \omega _ {0}} \right) \left( 1 - \cos \omega _ {0} t \right) \right) \label{14.80}$
Note that the high temperature limit also means the low frequency limit for $\omega _ {0}$. This means that we can expand
$\cos \omega _ {0} t \approx 1 - \left( \omega _ {0} t \right)^{2} / 2,$
and find
$w _ {E T} = \frac {| J |^{2}} {\hbar} \sqrt {\frac {\pi} {\lambda k T}} \exp \left[ \frac {- ( \Delta E + \lambda )^{2}} {4 \lambda k T} \right] \label{14.81}$
where $\lambda = D \hbar \omega _ {0}$. Note that the activation barrier $\Delta E^{\dagger}$ for displaced harmonic oscillators is $\Delta E^{\dagger} = \Delta E + \lambda$. For a thermally averaged rate it is proper to associate the average energy gap with the standard free energy of reaction,
$\left\langle H _ {A} - H _ {D} \right\rangle - \lambda = \Delta G^{0}.$
Therefore, this expression is equivalent to the classical Marcus’ result for the electron transfer rate
$k _ {E T} = A \exp \left[ \frac {- \left( \Delta G^{o} + \lambda \right)^{2}} {4 \lambda k T} \right] \label{14.82}$
where the pre-exponential is
$A = 2 \pi | J |^{2} / \hbar \sqrt {4 \pi \lambda k T} \label{14.83}$
This expression shows the nonlinear behavior expected for the dependence of the electron transfer rate on the driving force for the forward transfer, i.e., the reaction free energy. This is unusual because we generally think in terms of a linear free energy relationship between the rate of a reaction and the equilibrium constant:
$\ln k \propto \ln K _ {e q}.$
This leads to the thinking that the rate should increase as we increase the driving free energy for the reaction $-\Delta G^{0}$. This behavior only hold for a small region in $\Delta G^{0}$. Instead, eq. shows that the ET rate will increase with $-\Delta G^{0}$, until a maximum rate is observed for $-\Delta G^{0}=\lambda$ and the rate then decreases. This decrease of k with increased $-\Delta G^{0}$ is known as the “inverted regime”. The inverted behavior means that extra vibrational excitation is needed to reach the curve crossing as the acceptor well is lowered. The high temperature behavior for coupling to a low frequency mode $\left(100 \mathrm{~cm}^{-1} \text {at } 300 \mathrm{~K}\right)$ is shown at right, in addition to a cartoon that indicates the shift of the curve crossing at $\Delta G^{0}$ in increased.
Particularly in intramolecular ET, it is common that one wants to separately account for the influence of a high frequency intramolecular vibration (inner sphere ET) that is not in the classical limit that applies to the low frequency classical solvent response. If an additional mode of frequency $\omega _ {0}$ and a rate in the form of Equation \ref{14.81} is added to the low frequency mode, Jortner has given an expression for the rate as:
$w _ {E T} = \frac {| J |^{2}} {\hbar} \sqrt {\frac {\pi} {\lambda _ {0} k T}} \sum _ {j = 0}^{\infty} \left( \frac {e^{- D}} {j !} D^{j} \right) \exp \left[ \frac {- \left( \Delta G^{o} + \lambda _ {0} + j \hbar \omega _ {0} \right)^{2}} {4 \lambda _ {0} k T} \right] \label{14.84}$
Here $\lambda _ {0}$ is the solvation reorganization energy. For this case, the same inverted regime exists; although the simple Gaussian dependence of $k$ on $\Delta G^{0}$ no longer exists. The asymmetry here exists because tunneling sees a narrower barrier in the inverted regime than in the normal regime. Examples of the rates obtained with eq. are plotted in the figure below (T= 300 K).
As with electronic spectroscopy, a more general and effective way of accounting for the nuclear motions that mediate the electron transfer process is to describe the coupling weighted density of states as a spectral density. Then we can use coupling to a harmonic bath to describe solvent and/or vibrational contributions of arbitrary form to the transfer event using
$g (t) = \int _ {0}^{\infty} d \omega\, \rho ( \omega ) \left[ \operatorname {coth} \left( \frac {\beta \hbar \omega} {2} \right) ( 1 - \cos \omega t ) + i ( \sin \omega t - \omega t ) \right] \label{14.85}$
Readings
1. Barbara, P. F.; Meyer, T. J.; Ratner, M. A., Contemporary issues in electron transfer research. J. Phys. Chem. 1996, 100, 13148-13168, and references within.
2. Georgievskii, Y.; Hsu, C.-P.; Marcus, R. A., Linear response in theory of electron transfer reactions as an alternative to the molecular harmonic oscillator model. The Journal of Chemical Physics 1999, 110, 5307-5317.
3. Jortner, J., The temperature dependent activation energy for electron transfer between biological molecules. Journal of Chemical Physics 1976, 64, 4860-4867.
4. Marcus, R. A.; Sutin, N., Electron transfers in chemistry and biology. Biochimica et Biophysica Acta (BBA) - Reviews on Bioenergetics 1985, 811, 265-322.
5. Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006; Ch. 10
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/15%3A_Energy_and_Charge_Transfer/15.05%3A_Marcus_Theory_for_Electron_Transfer.txt
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• 16.1: Vibrational Relaxation
We want to address how excess vibrational energy undergoes irreversible energy relaxation as a result of interactions with other intra- and intermolecular degrees of freedom. Why is this process important? It is the fundamental process by which nonequilibrium states thermalize. This plays a particularly important role in chemical reactions, where efficient vibrational relaxation of an activated species is important to stabilizing the product and not allowing re-crossing to the reactant well.
• 16.2: A Density Matrix Description of Quantum Relaxation
Here we will more generally formulate a quantum mechanical picture of coherent and incoherent relaxation processes that occur as the result of interaction between a prepared system and its environment. This description will apply to the case where we separate the degrees of freedom in our problem into a system and a bath that interact. We have limited information about the bath degrees of freedom.
16: Quantum Relaxation Processes
Here we want to address how excess vibrational energy undergoes irreversible energy relaxation as a result of interactions with other intra- and intermolecular degrees of freedom. Why is this process important? It is the fundamental process by which nonequilibrium states thermalize. As chemists, this plays a particularly important role in chemical reactions, where efficient vibrational relaxation of an activated species is important to stabilizing the product and not allowing it to re-cross to the reactant well. Further, the rare activation event for chemical reactions is similar to the reverse of this process. Although we will be looking specifically at vibrational couplings and relaxation, the principles are the same for electronic population relaxation through electron–phonon coupling and spin–lattice relaxation.
For an isolated molecule with few vibrational coordinates, an excited vibrational state must relax by interacting with the remaining internal vibrations or the rotational and translational degrees of freedom. If a lot of energy must be dissipated, radiative relaxation may be more likely. In the condensed phase, relaxation is usually mediated by the interactions with the environment, for instance, the solvent or lattice. The solvent or lattice forms a continuum of intermolecular motions that can absorb the energy of of the vibrational relaxation. Quantum mechanically this means that vibrational relaxation (the annihilation of a vibrational quantum) leads to excitation of solvent or lattice motion (creation of an intermolecular vibration that increases the occupation of higher lying states).
For polyatomic molecules it is common to think of energy relaxation from high lying vibrational states ($k T \ll \hbar \omega _ {0}$) in terms of cascaded redistribution of energy through coupled modes of the molecule and its surroundings leading finally to thermal equilibrium. We seek ways of describing these highly non-equilibrium relaxation processes in quantum systems.
Classically vibrational relaxation reflects the surroundings exerting a friction on the vibrational coordinate, which damps its amplitude and heats the sample. We have seen that a Langevin equation for an oscillator experiencing a fluctuating force $f(t)$ describes such a process:
$\ddot {Q} (t) + \omega _ {0}^{2} Q^{2} - \gamma \dot {Q} = f (t) / m \label{15.1}$
This equation assigns a phenomenological damping rate $\gamma$ to the vibrational relaxation we wish to describe. However, we know in the long time limit, the system must thermalize and the dissipation of energy is related to the fluctuations of the environment through the classical fluctuation-dissipation relationship. Specifically,
$\langle f (t) f ( 0 ) \rangle = 2 m \gamma k _ {B} T \delta (t) \label{15.2}$
More general classical descriptions relate the vibrational relaxation rates to the correlation function for the fluctuating forces acting on the excited coordinate.
In these classical pictures, efficient relaxation requires a matching of frequencies between the vibrational period of the excited oscillator and the spectrum of fluctuation of the environment. Since these fluctuations are dominated by motions are of the energy scale of $k_BT$, such models do not work effectively for high frequency vibrations whose frequency $\omega \gg k_BT/\hbar$. We would like to develop a quantum model that allows for these processes and understand the correspondence between these classical pictures and quantum relaxation.
Let’s treat the problem of a vibrational system $H_S$ that relaxes through weak coupling $V$ to a continuum of bath states $H_B$ using perturbation theory. The eigenstates of $H_S$ are $| a \rangle$ and those of $H_B$ are $| \alpha \rangle$. Although our earlier perturbative treatment did not satisfy energy conservation, here we can take care of it by explicitly treating the bath states.
\begin{align} H &= H _ {0} + V \label{15.3} \[4pt] H _ {0} &= H _ {S} + H _ {B} \label{15.4} \end{align}
with
\begin{align} H _ {S} &= | a \rangle E _ {a} \langle a | + | b \rangle E _ {b} \langle b | \label{15.5} \[4pt] H _ {B} &= \sum _ {\alpha} | \alpha \rangle E _ {\alpha} \langle \alpha | \label{15.6} \[4pt] H _ {0} | a \alpha \rangle &= \left( E _ {a} + E _ {\alpha} \right) | a \alpha \rangle \label{15.7} \end{align}
We will describe transitions from an initial state $| i \rangle = | a \alpha \rangle$ with energy $E _ {a} + E _ {\alpha}$ to a final state $| f \rangle = | b \beta \rangle$ with energy $E _ {b} + E _ {\beta}$. Since we expect energy conservation to hold, this undoubtedly requires that a change in the system states will require an equal and opposite change of energy in the bath.
Initially, we take $p_a=1$ and $p_b=0$. If the interaction potential is $V$, Fermi’s Golden Rule says the transition from $| i \rangle$ to $| f \rangle$ is given by
\begin{align} k _ {f i} &= \frac {2 \pi} {\hbar} \sum _ {i , f} p _ {i} | \langle i | V | f \rangle |^{2} \delta \left( E _ {f} - E _ {i} \right) \label{15.8} \[4pt] &= \frac {2 \pi} {\hbar} \sum _ {a , \alpha , b , \beta} p _ {a , \alpha} | \langle a \alpha | V | b \beta \rangle |^{2} \delta \left( \left( E _ {b} + E _ {\beta} \right) - \left( E _ {a} + E _ {\alpha} \right) \right) \[4pt] &= \frac {1} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t \sum _ {a , \alpha \atop b , \beta} p _ {a , \alpha} \langle a \alpha | V | b \beta \rangle \langle b \beta | V | a \alpha \rangle e^{- i \left( E _ {b} - E _ {a} \right) + \left( E _ {\beta} - E _ {\alpha} \right) ) t / \hbar} \label{15.10} \end{align}
Equation \ref{15.10} is just a restatement of the time domain version of Equation \ref{15.8}
$k _ {f _ {f}} = \frac {1} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t \langle V (t) V ( 0 ) \rangle \label{15.11}$
with
$V (t) = e^{i H _ {0} t} V e^{- i H _ {0} t} \label{15.12}$
Now, the matrix element involves both evaluation in both the system and bath states, but if we write this in terms of a matrix element in the system coordinate $V _ {a b} = \langle a | V | b \rangle$:
$\langle a \alpha | V | b \beta \rangle = \left\langle \alpha \left| V _ {a b} \right| \beta \right\rangle \label{15.13}$
Then we can write the rate as
\begin{align} k_{b a} &=\frac{1}{\hbar^{2}} \int_{-\infty}^{+\infty} d t \sum_{\alpha, \beta} p_{\alpha}\left\langle\alpha\left|e^{+i E_{\alpha} t} V_{a b} e^{-i E_{\beta} t}\right| \beta\right\rangle\left\langle\beta\left|V_{b a}\right| \alpha\right\rangle e^{-i \omega_{b a} t} \label{15.14} \[4pt] &=\frac{1}{\hbar^{2}} \int_{-\infty}^{+\infty} d t\left\langle V_{a b}(t) V_{b a}(0)\right\rangle_{B} e^{-i \omega_{b a} t} \label{15.15} \end{align}
$V _ {a b} (t) = e^{i H _ {B} t} V _ {a b} e^{- i H _ {B} t} \label{15.16}$
Equation \ref{15.15} says that the relaxation rate is determined by a correlation function
$C _ {b a} (t) = \left\langle V _ {a b} (t) V _ {b a} ( 0 ) \right\rangle \label{15.17}$
which describes the time-dependent changes to the coupling between $b$ and $a$. The time dependence of the interaction arises from the interaction with the bath; hence its time evolution under $H_B$. The subscript $\langle \cdots \rangle _ {B}$ means an equilibrium thermal average over the bath states
$\langle \cdots \rangle _ {B} = \sum _ {\alpha} p _ {\alpha} \langle \alpha | \cdots | \alpha \rangle \label{15.18}$
Note also that Equation \ref{15.15} is similar but not quite a Fourier transform. This expression says that the relaxation rate is given by the Fourier transform of the correlation function for the fluctuating coupling evaluated at the energy gap between the initial and final state states.
Alternatively we could think of the rate in terms of a vibrational coupling spectral density, and the rate is given by its magnitude at the system energy gap $\omega _ {b a}$.
$k _ {b a} = \frac {1} {\hbar^{2}} \tilde {C} _ {b a} \left( \omega _ {a b} \right) \label{15.19}$
where the spectral representation $\tilde {C} _ {b a} \left( \omega _ {\omega b} \right)$ is defined as the Fourier transform of $C _ {b a} (t)$.
Vibration Coupled to a Harmonic Bath
To evaluate these expressions, let’s begin by consider the specific case of a system vibration coupled to a harmonic bath, which we will describe by a spectral density. Imagine that we prepare the system in an excited vibrational state in $v=|1\rangle$ and we want to describe relaxation $v=|0\rangle$.
$H _ {S} = \hbar \omega _ {0} \left( P^{2} + Q^{2} \right) \label{15.20}$
$H _ {B} = \sum _ {\alpha} \hbar \omega _ {\alpha} \left( p _ {\alpha}^{2} + q _ {\alpha}^{2} \right) = \sum _ {\alpha} \hbar \omega _ {\alpha} \left( a _ {\alpha}^{\dagger} a _ {\alpha} + \frac {1} {2} \right) \label{15.21}$
We will take the system–bath interaction to be linear in the bath coordinates:
$V = H _ {S B} = \sum _ {\alpha} c _ {\alpha} Q q _ {\alpha} \label{15.22}$
Here $\mathcal {C} \alpha$ is a coupling constant which describes the strength of the interaction between the system and bath mode $\alpha$. Note, that this form suggests that the system vibration is a local mode interacting with a set of normal vibrations of the bath.
For the case of single quantum relaxation from $| a \rangle = | 1 \rangle$ to $b = | 0 \rangle$, we can write the coupling matrix element as
$V _ {b a} = \sum _ {\alpha} \xi _ {a b , \alpha} \left( a _ {\alpha}^{\dagger} + a _ {\alpha} \right) \label{15.23}$
where
$\xi _ {a b , \alpha} = c _ {\alpha} \frac {\sqrt {m _ {\varrho} m _ {q} \omega _ {0} \omega _ {\alpha}}} {2 \hbar} \langle b | Q | a \rangle \label{15.24}$
Note
Note that we are using an equilibrium property, the coupling correlation function, to describe a nonequilibrium process, the relaxation of an excited state. Underlying the validity of the expressions are the principles of linear response. In practice this also implies a time scale separation between the equilibration of the bath and the relaxation of the system state. The bath correlation function should work fine if it has rapidly equilibrated, even though the system may not have. An instance where this would work well is electronic spectroscopy, where relaxation and thermalization in the excited state occurs on picosecond time scales, whereas the electronic population relaxation is on nanosecond time scales.
Here the matrix element $\langle b | Q | a \rangle$ is taken in evaluating $\xi _ {a b , \alpha}$. Evaluating Equation \ref{15.17} is now much the same as problems we’ve had previously:
\begin{align} \left\langle V _ {a b} (t) V _ {b a} ( 0 ) \right\rangle _ {B} &= \left\langle e^{i H _ {B} t} V _ {a b} e^{- i H _ {B} t} V _ {b a} \right\rangle _ {B} \[4pt] &= \sum _ {\alpha} \xi _ {\alpha}^{2} \left[ \left( \overline {n} _ {\alpha} + 1 \right) e^{- i \omega _ {\alpha} t} + \overline {n} _ {\alpha} e^{+ i \omega _ {\alpha} t} \right] \label{15.25} \end{align}
here $\overline {n} _ {\alpha} = \left( e^{\beta \hbar \omega _ {\alpha}} - 1 \right)^{- 1}$ is the thermally averaged occupation number of the bath mode at $\omega_{\alpha}$. In evaluating this we take advantage of relationships we have used before
$\overline {n} _ {\alpha} = \left( e^{\beta \hbar \omega _ {\alpha}} - 1 \right)^{- 1} \label{15.26}$
$\left. \begin{array} {l} {\left\langle a _ {\alpha} a _ {\alpha}^{\dagger} \right\rangle = \overline {n} _ {\alpha} + 1} \ {\left\langle a _ {\alpha}^{\dagger} a _ {\alpha} \right\rangle = \overline {n} _ {\alpha}} \end{array} \right. \label{15.27}$
So, now by Fourier transforming (Equation \ref{15.25}) we have the rate as
$k _ {b a} = \frac {1} {\hbar^{2}} \sum _ {\alpha} \left[ \xi _ {\alpha} \right] _ {a b}^{2} \left[ \left( \overline {n} _ {\alpha} + 1 \right) \delta \left( \omega _ {b a} + \omega _ {\alpha} \right) + \overline {n} _ {\alpha} \delta \left( \omega _ {b a} - \omega _ {\alpha} \right) \right] \label{15.28}$
This expression describes two relaxation processes which depend on temperature. The first is allowed at $T = 0\, K$ and is obeys $- \omega _ {b a} = \omega _ {\alpha}$. This implies that $E _ {a} > E _ {b}$, and that a loss of energy in the system is balanced by an equal rises in energy of the bath. That is $| \beta \rangle = | \alpha + 1 \rangle$. The second term is only allowed for elevated temperatures. It describes relaxation of the system by transfer to a higher energy state $E _ {b} > E _ {a}$, with a concerted decrease of the energy of the bath ($| \beta \rangle = | \alpha - 1 \rangle$). Naturally, this process vanishes if there is no thermal energy in the bath.
Note
There is an exact analogy between this problem and the interaction of matter with a quantum radiation field. The interaction potential is instead a quantum vector potential and the bath is the photon field of different electromagnetic modes. Equation \ref{15.28} describes has two terms that describe emission and absorption processes. The leading term describes the possibility of spontaneous emission, where a material system can relax in the absence of light by emitting a photon at the same frequency.
To more accurately model the relaxation due to a continuum of modes, we can replace the explicit sum over bath states with an integral over a density of bath states $W$
$k _ {b a} = \frac {1} {\hbar^{2}} \int d \omega _ {\alpha} W \left( \omega _ {\alpha} \right) \xi _ {b a}^{2} \left( \omega _ {\alpha} \right) \left[ \left( \overline {n} \left( \omega _ {\alpha} \right) + 1 \right) \delta \left( \omega _ {b a} + \omega _ {\alpha} \right) + \overline {n} \left( \omega _ {\alpha} \right) \delta \left( \omega _ {b a} - \omega _ {\alpha} \right) \right] \label{15.29}$
We can also define a spectral density, which is the vibrational coupling-weighted density of states:
$\rho \left( \omega _ {\alpha} \right) \equiv W \left( \omega _ {\alpha} \right) \xi _ {b a}^{\mathcal {E}} \left( \omega _ {\alpha} \right) \label{15.30}$
Then the relaxation rate is:
\left.\begin{aligned} k _ {b a} & = \frac {1} {\hbar^{2}} \int d \omega _ {\alpha} W \left( \omega _ {\alpha} \right) \xi _ {b a}^{2} \left( \omega _ {\alpha} \right) \left[ \left( \overline {n} \left( \omega _ {\alpha} \right) + 1 \right) \delta \left( \omega _ {b a} + \omega _ {\alpha} \right) + \overline {n} \left( \omega _ {\alpha} \right) \delta \left( \omega _ {b a} - \omega _ {\alpha} \right) \right] \ & = \frac {1} {\hbar^{2}} \left[ \left( \overline {n} \left( \omega _ {b a} \right) + 1 \right) \rho _ {b a} \left( \omega _ {a b} \right) + \overline {n} \left( \omega _ {b a} \right) \rho _ {b a} \left( - \omega _ {a b} \right) \right] \end{aligned} \right. \label{15.31}
We see that the Fourier transform of the fluctuating coupling correlation function, is equivalent to the coupling-weighted density of states, which we evaluate at $\omega _ {b a}$ or $-\omega _ {b a}$ depending on whether we are looking at upward or downward transitions. Note that $\overline {n}$ still refers to the occupation number for the bath, although it is evaluated at the energy splitting between the initial and final system states. Equation \ref{15.31} is a full quantum expression, and obeys detailed balance between the upward and downward rates of transition between two states:
$k _ {b a} = \exp \left( - \beta \hbar \omega _ {a b} \right) k _ {a b} \label{15.32}$
From our description of the two level system in a harmonic bath, we see that high frequency relaxation ($k T < < \hbar \omega _ {0}$) only proceeds with energy from the system going into a mode of the bath at the same frequency, but at lower frequencies ($k T \approx \hbar \omega _ {0}$) that energy can flow both into the bath and from the bath back into the system. When the vibration has energies that are thermally populated in the bath, we return to the classical picture of a vibration in a fluctuating environment that can dissipate energy from the vibration as well as giving kicks that increase the energy of the vibration. Note that in a cascaded relaxation scheme, as one approaches kT, the fraction of transitions that increase the system energy increase. Also, note that the bi-linear coupling in Equation \ref{15.22} and used in our treatment of quantum fluctuations can be associated with fluctuations of the bath that induce changes in energy (relaxation) and shifts of frequency (dephasing).
Multiquantum Relaxation of Polyatomic Molecules
3 Vibrational relaxation of polyatomic molecules in solids or in solution involves anharmonic coupling of energy between internal vibrations of the molecule, also called IVR (internal vibrational energy redistribution). Mechanical interactions between multiple modes of vibrationof the molecule act to rapidly scramble energy deposited into one vibrational coordinate and lead to cascaded energy flow toward equilibrium.
For this problem the bilinear coupling above doesn’t capture the proper relaxation process. Instead we can express the molecular potential energy in terms of well-defined normal modes of vibration for the system and the bath, and these interact weakly through small anharmonic terms in the potential. Then we can extend the perturbative approach above to include the effect of multiple accepting vibrations of the system or bath. For a set of system and bath coordinates, the potential energy for the system and system–bath interaction can be expanded as
$V _ {S} + V _ {S B} = \frac {1} {2} \sum _ {a} \frac {\partial^{2} V} {\partial Q _ {a}^{2}} Q _ {a}^{2} + \frac {1} {6} \sum _ {a , \alpha , \beta} \frac {\partial^{3} V} {\partial Q _ {a} \partial q _ {\alpha} \partial q _ {\beta}} Q _ {a , b , \alpha} q _ {\beta} + \frac {1} {6} \sum _ {a , b , \alpha} \frac {\partial^{3} V} {\partial Q _ {a} \partial Q _ {b} \partial q _ {\alpha}} Q _ {a} Q _ {b} q _ {\alpha} \cdots \label{15.33}$
Focusing explicitly on the first cubic expansion term, for one system oscillator:
$V _ {S} + V _ {S B} = \frac {1} {2} m \Omega^{2} Q^{2} + V^{( 3 )} Q q _ {\alpha} q _ {\beta} \label{15.34}$
Here, the system–bath interaction potential describes the case for a cubic anharmonic coupling that involves one vibration of the system $Q$ interacting weakly with two vibrations of the bath $\frac {9} {2} \alpha$ and $9 _ {\beta}$, so that $\hbar \Omega \gg V^{( 3 )}$. Energy deposited in the system vibration will dissipate to the two vibrations of the bath, a three quantum process. Higher-order expansion terms would describe interactions involving four or more quanta.
Working specifically with the cubic example, we can use the harmonic bath model to calculate the rate of energy relaxation. This picture is applicable if a vibrational mode of frequency $\Omega$ relaxes by transferring its energy to another vibration nearby in energy ($\infty _ {\alpha}$), and the energy difference $\omega _ {\beta}$ being accounted for by a continuum of intermolecular motions. For this case one can show
$k _ {b a} = \frac {1} {\hbar^{2}} \left[ \left( \overline {n} \left( \omega _ {\alpha} \right) + 1 \right) \left( \overline {n} \left( \omega _ {\beta} \right) + 1 \right) \rho _ {b a} \left( \omega _ {a b} \right) + \left( \overline {n} \left( \omega _ {\alpha} \right) + 1 \right) \overline {n} \left( \omega _ {\beta} \right) \rho _ {b a} \left( \omega _ {a b} \right) \right] \label{15.35}$
where $\rho ( \omega ) \equiv W ( \omega ) \left( V^{( 3 )} ( \omega ) \right)^{2}.$. Here we have taken , $\Omega , \omega _ {\alpha} \gg \omega _ {\beta}$. These two terms describe two possible relaxation pathways, the first in which annihilation of a quantum of $\Omega$ leads to a creation of one quantum each of $\omega_{\alpha} \text { and } \omega_{\beta}$. The second term describes the dissipation of energy by coupling to a higher energy vibration, with the excess energy being absorbed from the bath. Annihilation of a quantum of $\Omega$ leads to a creation of one quantum of $\omega_{\alpha}$ and the annihilation of one quantum of $\omega_{\beta}$. Naturally this latter term is only allowed when there is adequate thermal energy present in the bath.
Rate Calculations using Classical Vibrational Relaxation
In general, we would like a practical way to calculate relaxation rates, and calculating quantum correlation functions is not practical. How do we use classical calculations for the bath, for instance drawing on a classical molecular dynamics simulation? Is there a way to get a quantum mechanical rate?
The first problem is that the quantum correlation function is complex $C _ {a b}^{*} (t) = C _ {a b} ( - t )$ and the classical correlation function is real and even $C _ {C l} (t) = C _ {C l} ( - t )$. In order to connect these two correlation functions, one can derive a quantum correction factor that allows one to predict the quantum correlation function on the basis of the classical one. This is based on the assumption that at high temperature it should be possible to substitute the classical correlation function with the real part of the quantum correlation function
$C _ {c} (t) \Rightarrow C _ {b n}^{\prime} (t) \label{15.36}$
To make this adjustment we start with the frequency domain expression derived from the detailed balance expression $\tilde {C} ( - \omega ) = e^{- \beta \hbar \omega} \tilde {C} ( \omega )$
$\tilde {C} ( \omega ) = \frac {2} {1 + \exp ( - \beta \hbar \omega )} \tilde {C}^{\prime} ( \omega ) \label{15.37}$
Here $\tilde {C}^{\prime} ( \omega )$ is defined as the Fourier transform of the real part of the quantum correlation function. So the vibrational relaxation rate is
$k _ {b a} = \frac {4} {\hbar^{2} \left( 1 + \exp \left( - \hbar \omega _ {b a} / k T \right) \right)} \int _ {0}^{\infty} d t e^{- i \omega _ {\omega a} t} \operatorname {Re} \left[ \left\langle V _ {a b} (t) V _ {b a} ( 0 ) \right\rangle \right] \label{15.38}$
Now we will assume that one can replace a classical calculation of the correlation function here as in Equation \ref{15.36}. The leading term out front can be considered a “quantum correction factor” that accounts for the detailed balance of rates encoded in the quantum spectral density.
In practice such a calculation might be done with molecular dynamics simulations. Here one has an explicit characterization of the intermolecular forces that would act to damp the excited vibrational mode. One can calculate the system–bath interactions by expanding the vibrational potential of the system in the bath coordinates
\left.\begin{aligned} V _ {S} + V _ {s B} & = V _ {0} + \sum _ {\alpha} \frac {\partial V^{\alpha}} {\partial Q} Q + \sum _ {\alpha} \frac {\partial^{2} V^{\alpha}} {\partial Q^{2}} Q^{2} + \cdots \ & = V _ {0} + F Q + G Q^{2} + \cdots \end{aligned} \right. \label{15.39}
Here $V^{\alpha}$ represents the potential of an interaction of one solvent coordinate acting on the excited vibrational system coordinate $Q$. The second term in this expansion $FQ$ depends linearly on the system $Q$ and bath $\alpha$ coordinates, and we can use variation in this parameter to calculate the correlation function for the fluctuating interaction potential. Note that $F$ is the force that molecules exert on $Q$! Thus the relevant classical correlation function for vibrational relaxation is a force correlation function
$C _ {C l} (t) = \langle F (t) F ( 0 ) \rangle \label{15.40}$
$k _ {C l} = \frac {1} {k T} \int _ {0}^{\infty} d t \cos \omega _ {b a} t \langle F (t) F ( 0 ) \rangle \label{15.41}$
Readings
1. Egorov, S. A.; Rabani, E.; Berne, B. J., Nonradiative relaxation processes in condensed phases: Quantum versus classical baths. J. Chem. Phys. 1999, 110, 5238-5248.
2. Kenkre, V. M.; Tokmakoff, A.; Fayer, M. D., Theory of vibrational relaxation of polyatomic molecules in liquids. The Journal of Chemical Physics 1994, 101, 10618.
3. Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006; Ch. 11.
4. Oxtoby, D. W., Vibrational population relaxion in liquids. Adv. Chem. Phys. 1981, 47, 487- 519.
5. Skinner, J. L., Semiclassical approximations to golden rule rate constants. The Journal of Chemical Physics 1997, 107, 8717-8718.
6. Slichter, C. P., Principles of Magnetic Resonance, with Examples from Solid State Physics. Harper & Row: New York, 1963.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/16%3A_Quantum_Relaxation_Processes/16.01%3A_Vibrational_Relaxation.txt
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Here we will more generally formulate a quantum mechanical picture of coherent and incoherent relaxation processes that occur as the result of interaction between a prepared system and its environment. This description will apply to the case where we separate the degrees of freedom in our problem into a system and a bath that interact. We have limited information about the bath degrees of freedom. As a statistical mixture, we only have knowledge of the probability of occupying states of the bath and not of the phase relationships required to describe a deterministic quantum system. For such problems, the density matrix is the natural tool.
Mixed States
How does a system get into a mixed state? Generally, if you have two systems and you put these in contact with each other, interaction between the two will lead to a new system that is inseparable. Imagine that I have two systems $H_S$ and $H_B$ for which the eigenstates of $H_S$ are $| a \rangle$ and those of $H_B$ are $| \alpha \rangle$.
$H _ {0} = H _ {S} + H _ {B} \label{15.42}$
with
\begin{align} H _ {S} | a \rangle &= E _ {a} | a \rangle \label{15.43} \[4pt] H _ {B} | \alpha \rangle &= E _ {\alpha} | \alpha \rangle \end{align}
In general, before these systems interact, they can be described in terms of product states in the eigenstates of $H_S$ and $H_B$:
$| \psi \left( t _ {0} \right) \rangle = | \psi _ {S}^{0} \rangle | \psi _ {B}^{0} \rangle \label{15.44}$
with
\begin{align} | \psi _ {S}^{0} \rangle &= \sum _ {a} s _ {a} | a \rangle \label{15.45A} \[4pt] | \psi _ {B}^{0} \rangle &= \sum _ {\alpha} b _ {\alpha} | \alpha \rangle \end{align}
$| \psi _ {0} \rangle = \sum _ {a , \alpha} s _ {a} b _ {\alpha} | a \rangle | \alpha \rangle \label{15.46}$
After these states are allowed to interact, we have a new state vector $| \psi (t) \rangle$. The new state can still be expressed in the zero-order basis, although this does not represent the eigenstates of the new Hamiltonian
$H = H _ {0} + V \label{15.47}$
$| \psi (t) \rangle = \sum _ {a , \alpha} c _ {a \alpha} | a \alpha \rangle \label{15.48}$
For any point in time, $\mathcal {C} _ {a \alpha}$ is the joint probability amplitude for finding particle of $| \psi _ {S} \rangle$ in $| a \rangle$ and simultaneously finding particle of $| \psi _ {B} \rangle$ in $| \alpha \rangle$. At $t=t_o$, $c _ {a \alpha} = S _ {a} b _ {\alpha}$.
Now suppose that you have an operator $A$ that is only an operator in the $| \psi _ {S} \rangle$ coordinates. This might represent an observable for the system that you wish to measure. Let’s calculate the expectation value of $A$
$\langle A (t) \rangle = \langle \psi (t) | A | \psi (t) \rangle = \left\langle \psi _ {S} | A | \psi _ {S} \right\rangle \label{15.49}$
\begin{aligned} \langle A (t) \rangle & = \sum _ {a , \alpha} c _ {a \alpha}^{*} c _ {b \beta} \langle a \alpha | A | b , \beta \rangle \ & = \sum _ {a , \alpha} c _ {a \alpha}^{*} c _ {b \beta} \langle a | A | b \rangle \delta _ {\alpha \beta} \ & = \sum _ {a , \alpha} c _ {a \alpha}^{*} c _ {b \beta} \langle a | A | b \rangle \delta _ {\alpha \beta} \ & = \sum _ {a , \beta} \left( \sum _ {\alpha} c _ {a \alpha}^{*} c _ {b \alpha} \right) A _ {a b} \ & \equiv \sum _ {a , b} \left( \rho _ {S} \right) _ {b a} A _ {a b} \ & = \operatorname {Tr} \left[ \rho _ {S} A \right] \end{aligned}
Here we have defined a density matrix for the degrees of freedom in $| \psi _ {s} \rangle$
$\rho _ {S} = | \psi _ {S} \rangle \langle \psi _ {S} | \label{15.51}$
with density matrix elements that traced over the $| \psi _ {B} \rangle$ states, that is, that are averaged over the probability of occupying the $| \psi _ {B} \rangle$ states.
$| b \rangle \rho _ {S} \langle a | = \sum _ {\alpha} c _ {a \alpha}^{*} c _ {b \alpha} \label{15.52}$
Here the matrix elements in direct product states involve elements of a four-dimensional matrix, which are specified by the tetradic notation.
We have defined a trace of the density matrix over the unobserved degrees of freedom in $| \psi _ {B} \rangle$, i.e. a sum over diagonal elements in $\alpha$. To relate this to our similar prior expression: $\langle A (t) \rangle = \operatorname {Tr} [ \rho A ]$, the following definitions are useful:
\begin{aligned} \rho _ {S} & = T r _ {B} ( \rho ) \ & = \sum _ {a , b} \left( \rho _ {S} \right) _ {b a} A _ {a b} \ & = \operatorname {Tr} \left( \rho _ {S} A \right) \end{aligned} \label{15.53}
Also,
$\operatorname {Tr} ( A \times B ) = \operatorname {Tr} ( A ) \operatorname {Tr} ( B ) \label{15.54}$
Since $\rho _ {S}$ is Hermitian, it can be diagonalized by a unitary transformation $T$, where the new eigenbasis $| m \rangle$ represents the mixed states of the $| \psi _ {S} \rangle$ system.
$\rho _ {S} = \sum _ {m} | m \rangle \langle m | \rho _ {m m} \label{15.55}$
$\sum _ {m} \rho _ {m n} = 1 \label{15.56}$
The density matrix elements represent the probability of occupying state m averaged over the bath degrees of freedom
\begin{aligned} \rho _ {m n} & = \sum _ {a , b} T _ {m b} \rho _ {b a} T _ {a m}^{\dagger} \ & = \sum _ {a , b} a _ {b \alpha} T _ {m b} a _ {a \alpha}^{*} T _ {m a}^{*} \ & = \sum _ {\alpha} f _ {m \alpha} f _ {m \alpha}^{*} \ & = \sum _ {\alpha} \left| f _ {m \alpha} \right|^{2} = p _ {m} \geq 0 \end{aligned} \label{15.57}
The quantum mechanical interaction of one system with another causes the system to be in a mixed state after the interaction. The mixed states are generally not separable into the original states. The mixed state is described by
$| \psi _ {S} \rangle = \sum _ {m} d _ {m} | m \rangle \label{15.58}$
$d _ {m} = \sum _ {\alpha} f _ {m \alpha} \label{15.59}$
If we only observe a few degrees of freedom, we can calculate observables by tracing over unobserved degrees of freedom. This forms the basis for treating relaxation phenomena. A few degrees of freedom that we observe, coupled to many other degrees of freedom, which lend to irreversible relaxation.
Equation of Motion for the Reduced Density Matrix
So now to describe irreversible processes in quantum systems, let’s look at the case where we have partitioned the problem so that we have a few degrees of freedom that we are most interested in (the system), which is governed by $H_S$ and which we observe with a system operator $A$. The remaining degrees of freedom are a bath, which interact with the system. The Hamiltonian is given by Equation \ref{15.42} and \ref{15.47}. In our observations, we will be interested in expectation values in $A$ which we have seen are written
\begin{aligned} \left\langle A _ {S} \right\rangle & = \operatorname {Tr} [ \rho (t) A ] \[4pt] & = \operatorname {Tr} _ {S} [ \sigma (t) A ] \[4pt] & = \sum _ {a , b} \sigma _ {a b} (t) A _ {b a} \[4pt] & = T r _ {S} T r _ {B} [ \rho (t) A ] \end{aligned} \label{15.60}
Here $\sigma$ is the reduced density operator for the system degrees of freedom. This is the more commonly variable used for $\rho _ {S}$.
$\sigma_{ab} = \sum_{\alpha} \langle a \alpha | \rho | b \alpha \rangle = \operatorname {Tr} _ {B} \rho _ {a b} \label{15.61}$
$T r _ {B}$ and $T r _ {S}$ are partial traces over the bath and system respectively. Note, that since
$\operatorname {Tr} ( A \times B ) = \operatorname {Tr} A T r B$
for direct product states, all we need to do is describe time evolution of $\sigma$ to understand the time dependence to $A$.
We obtain the equation of motion for the reduced density matrix beginning with
$\rho (t) = U (t) \rho ( 0 ) U^{\dagger} (t) \label{15.62}$
and tracing over bath:
$\sigma (t) = T r _ {B} \left[ U \rho U^{\dagger} \right] \label{15.63}$
We can treat the time evolution of the reduced density matrix in the interaction picture. From our earlier discussion of the density matrix, we integrate the equation of motion
$\dot {\rho} _ {I} = - \frac {i} {\hbar} \left[ V _ {I} (t) , \rho _ {I} (t) \right] \label{15.64}$
to obtain
$\rho _ {I} (t) = \rho _ {I} ( 0 ) - \frac {i} {\hbar} \int _ {0}^{t} d \tau \left[ V _ {I} ( \tau ) , \rho _ {I} ( \tau ) \right] \label{15.65}$
Remember that the density matrix in the interaction picture is
$\rho _ {I} (t) = U _ {0}^{\dagger} \rho (t) U _ {0} = e^{i \left( H _ {s} + H _ {B} \right) t / \hbar} \rho (t) e^{- i \left( H _ {s} + H _ {B} \right) t / \hbar} \label{15.66}$
and similarly
$V _ {I} (t) = U _ {0}^{\dagger} V U _ {0} = e^{i \left( H _ {S} + H _ {B} \right) t / \hbar} V (t) e^{- i \left( H _ {s} + H _ {B} \right) t / \hbar} \label{15.67}$
Substituting Equation \ref{15.65} into Equation \ref{15.64} we have
$\dot {\rho} _ {I} (t) = - \frac {i} {\hbar} \left[ V _ {I} (t) , \rho _ {I} \left( t _ {0} \right) \right] - \frac {1} {\hbar^{2}} \int _ {0}^{t} d t^{\prime} \left[ V _ {I} (t) , \left[ V _ {I} \left( t^{\prime} \right) , \rho _ {I} \left( t^{\prime} \right) \right] \right] \label{15.68}$
Now taking a trace over the bath states
$\dot {\sigma} _ {I} (t) = - \frac {i} {\hbar} T r _ {B} \left[ V _ {I} (t) , \rho _ {I} \left( t _ {0} \right) \right] - \frac {1} {\hbar^{2}} \int _ {0}^{t} d t^{\prime} T r _ {B} \left[ V _ {I} (t) , \left[ V _ {I} \left( t^{\prime} \right) , \rho _ {I} \left( t^{\prime} \right) \right] \right] \label{15.69}$
If we assume that the interaction of the system and bath is small enough that the system cannot change the bath
$\rho _ {I} (t) \approx \sigma _ {I} (t) \rho _ {B} ( 0 ) = \sigma _ {I} (t) \rho _ {e q}^{B} \label{15.70}$
$\rho _ {e q}^{B} = \frac {e^{- \beta H _ {B}}} {Z} \label{15.71}$
Then we obtain an equation of motion for $\sigma$ to second order:
$\dot {\sigma} _ {I} (t) = - \frac {i} {\hbar} T r _ {B} \left[ V _ {I} (t) , \sigma _ {I} ( 0 ) \rho _ {e q}^{B} \right] - \frac {1} {h^{2}} \int _ {0}^{t} d t^{\prime} T r _ {B} \left[ V _ {I} (t) , \left[ V _ {I} \left( t^{\prime} \right) , \sigma _ {I} \left( t^{\prime} \right) \rho _ {e q}^{B} \right] \right] \label{15.72}$
The last term involves an integral over a correlation function for a fluctuating interaction potential. This looks similar to a linear response function, and also the same form as the relaxation rates from Fermi’s Golden Rule that we just discussed. The first term in Equation \ref{15.72} involves a thermal average over the interaction potential,
$\langle V \rangle _ {B} = T r _ {B} \left[ V \rho _ {e q}^{B} \right].$
If this average value is zero, which would be the case for an off-diagonal form of $V$, we can drop the first term in the equation of motion for $\sigma_I$. If it were not zero, it is possible to redefine the Hamiltonian such that $H_{0} \rightarrow H_{0}+\langle V\rangle_{B} \text { and } V(t) \rightarrow V(t)-\langle V\rangle_{B}$
which recasts it in a form where $\langle V \rangle _ {B} \rightarrow 0$ and the first term can be neglected. Now let’s evaluate the equation of motion for the case where the system–bath interaction can be written as a product of operators in the system $\hat{A}$ and bath $\hat {\beta}$
$H _ {s B} = V = \hat {A} \hat {\beta}\label{15.73}$
This is equivalent to the bilinear coupling form that was used in our prior description of dephasing and population relaxation. There we took the interaction to be linearly proportional to the system and bath coordinate(s): $V = c \varrho q$. The time evolution in the two variables is separable and given by
$\begin{array} {l} {\hat {A} (t) = U _ {S}^{\dagger} \hat {A} \left( t _ {0} \right) U _ {S}} \[4pt] {\hat {\beta} (t) = U _ {B}^{\dagger} \hat {\beta} \left( t _ {0} \right) U _ {B}} \end{array} \label{15.74}$
The equation of motion for $\sigma _ {I}$ becomes
$\dot {\sigma} _ {I} (t) = \frac {1} {\hbar^{2}} \int _ {0}^{t} d t^{\prime} \left[ \hat {A} (t) \hat {A} \left( t^{\prime} \right) \sigma \left( t^{\prime} \right) - \hat {A} \left( t^{\prime} \right) \sigma \left( t^{\prime} \right) \hat {A} (t) \right] \operatorname {Tr} _ {B} \left( \hat {\beta} (t) \hat {\beta} \left( t^{\prime} \right) \rho _ {e q}^{B} \right) - \left[ \hat {A} (t) \sigma \left( t^{\prime} \right) \hat {A} \left( t^{\prime} \right) - \sigma \left( t^{\prime} \right) \hat {A} \left( t^{\prime} \right) \hat {A} (t) \right] \operatorname {Tr} _ {B} \left( \hat {\beta} \left( t^{\prime} \right) \hat {\beta} (t) \rho _ {e q}^{B} \right) \label{15.75}$
Here the history of the evolution of $\hat{A}$ depends on the time dependence of the bath variables coupled to the system. The time dependence of the bath enters as a bath correlation function
\begin{aligned} C _ {\beta \beta} \left( t - t^{\prime} \right) & = \operatorname {Tr} _ {B} \left( \hat {\beta} (t) \hat {\beta} \left( t^{\prime} \right) \rho _ {e q}^{B} \right) \[4pt] & = \left\langle \hat {\beta} (t) \hat {\beta} \left( t^{\prime} \right) \right\rangle _ {B} = \left\langle \hat {\beta} \left( t - t^{\prime} \right) \hat {\beta} ( 0 ) \right\rangle _ {B} \end{aligned} \label{15.76}
The bath correlation function can be evaluated using the methods that we have used in the Energy Gap Hamiltonian and Brownian Oscillator Models. Switching integration variables to the time interval prior to observation
$\tau = t - t^{\prime} \label{15.77}$
we obtain
$\dot {\sigma} _ {I} (t) = - \frac {1} {\hbar^{2}} \int _ {0}^{t} d \tau \left[ \hat {A} (t) , \hat {A} ( t - \tau ) \sigma _ {I} ( t - \tau ) \right] C _ {\beta \beta} ( \tau ) - \left[ \hat {A} (t) , \sigma _ {I} ( t - \tau ) \hat {A} ( t - \tau ) \right] C _ {\beta \beta}^{*} ( \tau ) \label{15.78}$
Here we have made use of $C _ {\beta \beta}^{*} ( \tau ) = C _ {\beta \beta} ( - \tau )$. For the case that the system–bath interaction is a result of interactions with many bath coordinates
$V = \sum _ {\alpha} \hat {A} \hat {\beta} _ {\alpha} \label{15.79}$
then Equation \ref{15.78} becomes
$\dot {\sigma} _ {I} (t) = - \frac {1} {\hbar^{2}} \sum _ {\alpha , \beta} \int _ {0}^{t} d \tau \left[ \hat {A} (t) , \hat {A} ( t - \tau ) \sigma _ {I} ( t - \tau ) \right] C _ {\alpha \beta} ( \tau ) - \left[ \hat {A} (t) , \sigma _ {I} ( t - \tau ) \hat {A} ( t - \tau ) \right] C _ {\alpha \beta}^{*} ( \tau ) \label{15.80}$
with the bath correlation function
$C _ {\alpha \beta} ( \tau ) = \left\langle \hat {\beta} _ {\alpha} ( \tau ) \hat {\beta} _ {\beta} ( 0 ) \right\rangle _ {B} \label{15.81}$
Equation \ref{15.78} or \ref{15.80} indicates that the rates of exchange of amplitude between the system states carries memory of the bath’s influence on the system, that is, $\sigma _ {I} (t)$ is dependent on $\sigma _ {I} ( t - \tau )$. If we make the Markov approximation, for which the dynamics of the bath are much faster than the evolution of the system and where the system has no memory of its past, we would replace
$\sigma \left( t^{\prime} \right) = \sigma \left( t^{\prime} \right) \delta \left( t - t^{\prime} \right) \Rightarrow \sigma (t) \label{15.82}$
in Equation \ref{15.75}, or equivalently in Equation \ref{15.78} set
$\sigma _ {I} ( t - \tau ) \Rightarrow \sigma _ {I} (t) \label{15.83}$
For the subsequent work, we use this approximation. Similarly, the presence of a time scale separation between a slow system and a fast bath allows us to change the upper integration limit in Equation \ref{15.78} from $t$ to $∞$.
$\dot {\sigma} _ {a b} (t) = - i \omega _ {a b} \sigma _ {a b} (t) - \frac {1} {\hbar^{2}} \sum _ {c , d} [ \hat {A} _ {a c} \hat {A} _ {c d} \sigma _ {d b} (t) \tilde {C} _ {\beta \beta} \left( \omega _ {d c} \right) - \hat {A} _ {a c} \hat {A} _ {d b} \sigma _ {c d} (t) \tilde {C} _ {\beta \beta} \left( \omega _ {c a} \right) - \hat {A}^{2} \hat {A} _ {d b} \sigma _ {c d} (t) \tilde {C} _ {\beta \beta}^{*} \left( - \omega _ {d b} \right) + \hat {A} _ {c d} \hat {A} _ {d b} \sigma _ {a c} (t) \tilde {C} _ {\beta \beta}^{*} \left( - \omega _ {c d} \right) ] \label{15.90}$
$\dot {\sigma} _ {a b} (t) = - i \omega _ {a b} \sigma _ {a b} (t) - \frac {1} {\hbar^{2}} \sum _ {c , d} [ \hat {A} _ {a c} \hat {A} _ {c d} \sigma _ {d b} (t) \tilde {C} _ {\beta \beta} \left( \omega _ {d c} \right) - \hat {A} _ {a c} \hat {A} _ {d b} \sigma _ {c d} (t) \tilde {C} _ {\beta \beta} \left( \omega _ {c a} \right) - \hat {A}^{2} \hat {A} _ {d b} \sigma _ {c d} (t) \tilde {C} _ {\beta \beta}^{*} \left( - \omega _ {d b} \right) + \hat {A} _ {c d} \hat {A} _ {d b} \sigma _ {a c} (t) \tilde {C} _ {\beta \beta}^{*} \left( - \omega _ {c d} \right) ] \label{15.91}$
The rate constants are defined through:
$\Gamma _ {a b , c d}^{+} = \frac {1} {\hbar^{2}} A _ {a b} A _ {c d} \tilde {C} _ {\beta \beta} \left( \omega _ {d c} \right) \label{15.92}$
$\Gamma _ {a b , c d}^{-} = \frac {1} {\hbar^{2}} A _ {a b} A _ {c d} \tilde {C} _ {\beta \beta} \left( \omega _ {b a} \right) \label{15.93}$
Here we made use of
$\tilde {C} _ {\beta \beta}^{*} ( \omega ) = \tilde {C} _ {\beta \beta} ( - \omega ).$
Also, it is helpful to note that
$\Gamma _ {a b , c d}^{+} = \left[ \Gamma _ {d c , b a}^{-} \right]^{*} \label{15.94}$
The coupled differential equations in Equation \ref{15.91} express the relaxation dynamics of the system states almost entirely in terms of the system Hamiltonian. The influence of the bath only enters through the bath correlation function.
Evaluating the equation of motion: Redfield Equations
Do describe the exchange of amplitude between system states induced by the bath, we will want to evaluate the matrix elements of the reduced density matrix in the system eigenstates. To begin, we use Equation \ref{15.78} to write the time-dependent matrix elements as
$\dot {\sigma} _ {I} (t) = - \frac {1} {\hbar^{2}} \sum _ {\alpha , \beta} \int _ {0}^{t} d \tau \left[ \hat {A} (t) , \hat {A} ( t - \tau ) \sigma _ {I} ( t - \tau ) \right] C _ {\alpha \beta} ( \tau ) - \left[ \hat {A} (t) , \sigma _ {I} ( t - \tau ) \hat {A} ( t - \tau ) \right] C _ {\alpha \beta}^{*} ( \tau ) \label{15.84}$
Now, let’s convert the time dependence expressed in terms of the interaction picture into a Schrödinger representation using
$\left\langle a | A (t) | b \right\rangle = e^{i \omega _ {a b} t} A _ {a b}$
$\left\langle a \left| \sigma^{I} \right| b \right\rangle = e^{i \omega _ {a b} t} \sigma _ {a b}$
with
$\sigma _ {a b} = \frac {\partial} {\partial t} \left\langle a \left| \sigma^{I} \right| b \right\rangle$
To see how this turns out, consider the first term in Equation \ref{15.84}:
$\dot {\sigma} _ {a b}^{I} (t) = - \sum _ {c , d} \frac {1} {\hbar^{2}} \int _ {0}^{\infty} d \tau \hat {A} _ {a c} (t) \hat {A} _ {c d} ( t - \tau ) \sigma _ {d b}^{I} (t) C _ {\beta \beta} ( \tau ) \label{15.86}$
$\dot {\sigma} _ {a b} (t) e^{i \omega _ {b b} \tau} + i \omega _ {a b} e^{i \omega _ {b b} \tau} \sigma _ {a b} = - \sum _ {c , d} \frac {1} {\hbar^{2}} \hat {A} _ {a c} \hat {A} _ {c d} \sigma _ {d b} (t) e^{i \omega _ {a} t + i \omega _ {c d} t + i \omega _ {b b} t} \int _ {0}^{\infty} d \tau e^{- i \omega _ {c t} \tau} C _ {\beta \beta} ( \tau ) \label{15.87}$
Defining the Fourier-Laplace transform of the bath correlation function:
$\tilde {C} _ {\beta \beta} ( \omega ) = \int _ {0}^{\infty} d \tau e^{i \omega \pi} C _ {\beta \beta} ( \tau )\label{15.88}$
We have
$\dot {\sigma} _ {a b} (t) = - i \omega _ {a b} \sigma _ {a b} - \sum _ {c , d} \frac {1} {\hbar^{2}} \hat {A} _ {a c} \hat {A} _ {c d} \sigma _ {d b} (t) \tilde {C} _ {\beta \beta} \left( \omega _ {d c} \right) \label{15.89}$
Here the spectral representation of the bath correlation function is being evaluated at the energy gap between system states $\omega _ {d c}$. So the evolution of coherences and populations in the system states is governed by their interactions with other system states, governed by the matrix elements, and this is modified depending on the fluctuations of the bath at different system state energy gaps. In this manner, Equation \ref{15.84} becomes
The common alternate way of writing these expressions is in terms of the relaxation superoperator $\mathbf {R}$
$\dot {\sigma} _ {a b} (t) = - i \omega _ {a b} \sigma _ {a b} - \sum _ {c , d} R _ {a b , c d} \sigma _ {c d} (t) \label{15.95}$
or in the interaction picture
$\dot {\sigma} _ {a b}^{I} (t) = \sum _ {c , d} \sigma _ {c d}^{I} (t) R _ {a b , c d} e^{i \left( E _ {a} - E _ {b} - E _ {c} + E _ {d} \right) t / h} \label{15.96}$
Equation \ref{15.95}, the reduced density matrix equation of motion for a Markovian bath, is known as the Redfield equation. It describes the irreversible and oscillatory components of the amplitude in the $| a \rangle \langle b |$ coherence as a result of dissipation to the bath and feeding from other states. $\mathbf {R}$ describes the rates of change of the diagonal and off-diagonal elements of $\sigma _ {I}$ and is expressed as:
$R _ {a b , c d} = \delta _ {d b} \sum _ {k} \Gamma _ {a k , k c}^{+} - \Gamma _ {d b , a d}^{+} - \Gamma _ {d b , a d}^{-} + \delta _ {a c} \sum _ {k} \Gamma _ {d k , k b}^{-} \label{15.97}$
where $k$ refers to a system state. The derivation described above can be performed without assuming a form to the system– bath interaction potential as we did in Equation \ref{15.73}. If so, one can write the relaxation operator in terms of a correlation function for the system–bath interaction ,
$\Gamma _ {a b , c d}^{+} = \frac {1} {\hbar^{2}} \int _ {0}^{\infty} d \tau \left\langle V _ {a b} ( \tau ) V _ {c d} ( 0 ) \right\rangle _ {B} e^{- i \omega _ {c d} \tau} \label{15.98}$
$\Gamma _ {a b , c d}^{-} = \frac {1} {\hbar^{2}} \int _ {0}^{\infty} d \tau \left\langle V _ {a b} ( 0 ) V _ {c d} ( \tau ) \right\rangle _ {B} e^{- i \omega _ {a b} \tau} \label{15.99}$
The tetradic notation for the Redfield relaxation operator allows us to identify four classes of relaxation processes, depending on the number of states involved:
• $aa, aa$: Population relaxation (rate of loss of the population in $a$)
• $ab,ab$: Coherence relaxation or dephasing (damping of the coherence $ab$)
• $aa,bb$: Population transfer (rate of transfer of population from state b to state $a$)
• $ab,cd$: Coherence transfer (rate at which amplitude in an oscillating superposition between two states ($c$ and $d$) couples to form oscillating amplitude between two other states ($a$ and $b$)
The origin and meaning of these terms will be discussed below.
Secular Approximation
From Equation \ref{15.96} we note that the largest changes in matrix elements of $\sigma_I$ result from a resonance condition:
\begin{aligned} \exp \left[ i \left( E _ {a} - E _ {b} - E _ {c} + E _ {d} \right) t / \hbar \right] & \approx 1 \ E _ {a} - E _ {b} - E _ {c} + E _ {d} & \approx 0 \end{aligned} \label{15.100}
which is satisfied when:
$\begin{array} {l l} {a = c ; b = d} & {\Rightarrow R _ {a b , a b}} \ {a = b ; c = d} & {\Rightarrow R _ {a a , c c}} \ {a = b = c = d} & {\Rightarrow R _ {a a , a a}} \end{array} \label{15.101}$
In evaluating relaxation rates, often only these secular terms are retained. Whether this approximation is valid must be considered on a case by case basis and depends on the nature of the system eigenvalues and the bath correlation function.
Population Relaxation and the Master Equation
To understand the information in the relaxation operator and the classification of relaxation processes, let’s first consider the relaxation of the diagonal elements of the reduced density matrix. Using the secular approximation,
$\dot{\sigma}_{a a}(t)=-\sum_{b} R_{a a, b b} \sigma_{b b}(t)$
Considering first the case that $a ≠ b$, Equation \ref{15.97} gives the relaxation operator as
$R_{a a, b b}=-\Gamma_{b a, a b}^{+}-\Gamma_{b a, a b}^{-} \label{15.103}$
Recognizing that $\Gamma^{+}$ and $\Gamma^{-}$ are Hermitian conjugates,
\begin{aligned} R _ {a a , b b} & = - \frac {1} {\hbar^{2}} \left| A _ {a b} \right|^{2} \int _ {0}^{\infty} d \tau C _ {\beta \beta} ( \tau ) e^{- i \omega _ {b a} \tau} + c . c . \ & = - \frac {1} {\hbar^{2}} \int _ {0}^{\infty} d \tau \left\langle V _ {b a} ( \tau ) V _ {a b} ( 0 ) \right\rangle _ {B} e^{- i \omega _ {a b} \tau} + c . c . \end{aligned} \label{15.104}
So $R _ {a a , b b}$, is a real valued quantity. However, since $\left\langle V _ {b a} ( \tau ) V _ {a b} ( 0 ) \right\rangle = \left\langle V _ {b a} ( 0 ) V _ {a b} ( - \tau ) \right\rangle$
$R _ {a a , b b} = - \frac {1} {\hbar^{2}} \int _ {- \infty}^{+ \infty} d t \left\langle V _ {b a} ( \tau ) V _ {a b} ( 0 ) \right\rangle _ {B} e^{i \omega _ {b b} \tau} \label{15.105}$
So we see that the relaxation tensor gives the population relaxation rate between states $a$ and $b$ that we derived from Fermi’s Golden rule:
$R _ {a a , b b} = - w _ {a b} \label{15.106}$
if $a \neq b$.
For the case that $a = b$, Equation \ref{15.97} gives the relaxation operator as
\begin{aligned} R _ {a a , a a} & = - \left( \Gamma _ {a a , a a}^{+} + \Gamma _ {a a , a a}^{-} \right) + \sum _ {k} \left( \Gamma _ {a k , k a}^{+} + \Gamma _ {a k , k a}^{-} \right) \ & = \sum _ {k \neq a} \left( \Gamma _ {a k , k a}^{+} + \Gamma _ {a k , k a}^{-} \right) \end{aligned} \label{15.107}
The relaxation accounts for the bath-induced dissipation for interactions with all states of the system (last term), but with the influence of self-relaxation (first term) removed. The net result is that we are left with the net rate of relaxation from a to all other system states ($a ≠ k$)
$R _ {a a , a a} = \sum _ {k \neq a} w _ {k a} \label{15.108}$
This term $R _ {a a , a a}$, is also referred to as the inverse of $T_1$, the population lifetime of the $a$ state.
The combination of these observations shows that the diagonal elements of the reduced density matrix follow a master equation that describes the net gain and loss of population in a particular state
$\dot {\sigma} _ {a a} (t) = \sum _ {b \neq a} w _ {a b} \sigma _ {b b} (t) - \sum _ {k \neq a} w _ {k a} \sigma _ {a a} (t) \label{15.109}$
Coherence Relaxation
Now let’s consider the relaxation of the off-diagonal elements of the reduced density matrix. It is instructive to limit ourselves at first to one term in the relaxation operator, so that we write the equation of motion as
$\dot {\sigma} _ {a b} (t) = - i \omega _ {a b} \sigma _ {a b} (t) - R _ {a b , a b} \sigma _ {a b} (t) + \cdots \label{15.110}$
The relaxation operator gives
\begin{align} R_{a b, a b} &=-\left(\Gamma_{a a, b b}^{+}+\Gamma_{a a, b b}^{-}\right)+\sum_{k}\left(\Gamma_{a k, k a}^{+}+\Gamma_{b k, k b}^{-}\right) \label{15.111} \[4pt] &=-\left(\Gamma_{a a, b b}^{+}+\Gamma_{a a, b b}^{-}-\Gamma_{a a, a a}^{+}-\Gamma_{b b, b b}^{-}\right)+\left(\sum_{k \neq a} \Gamma_{a k, k a}^{+}+\sum_{k \neq b} \Gamma_{b k, k b}^{-}\right) \label{15.112} \end{align}
In the second step, we have separated the sum into two terms, one involving relaxation constants for the two coherent states, and the second involving all other states. The latter term looks very similar to the relaxation rates. In fact, if we factor out the imaginary parts of these terms and add them as a correction to the frequency in Equation \ref{15.110}, $\omega _ {a b} \rightarrow \omega _ {a b} + \operatorname {Im} [ t e r m 2 ]$, then the remaining expression is directly related to the population lifetimes of the $a$ and $b$ states:
\begin{aligned} \operatorname {Re} \left( \sum _ {k \neq a} \Gamma _ {a k , k a}^{+} + \sum _ {k \neq b} \Gamma _ {b k , k b}^{-} \right) & = \frac {1} {2} \sum _ {k \neq b} w _ {k b} - \frac {1} {2} \sum _ {k \neq a} w _ {k a} \ & = \frac {1} {2} \left( \frac {1} {T _ {1 , a}} + \frac {1} {T _ {1 , b}} \right) \end{aligned} \label{15.113}
This term accounts for the decay of the coherence as a sum of the rates of relaxation of the $a$ and $b$ states.
The meaning of the first term on the right hand side of Equation \ref{15.112} is a little less obvious. If we write out the four contributing relaxation factors explicitly using the system–bath correlation functions in Equation \ref{15.98} and \ref{15.99}, the real part can be written as
\begin{align} \operatorname {Re} \left( \Gamma _ {a a , b b}^{+} + \Gamma _ {a a , b b}^{-} - \Gamma _ {b b , a b}^{+} - \Gamma _ {b b , b b}^{-} \right) &= \int _ {0}^{\infty} d \tau \left\langle \left[ V _ {b b} ( \tau ) - V _ {a a} ( \tau ) \right] \left[ V _ {b b} ( 0 ) - V _ {a a} ( 0 ) \right] \right\rangle _ {B} \[4pt] &\equiv \int _ {0}^{\infty} d \tau \langle \Delta V ( \tau ) \Delta V ( 0 ) \rangle _ {B} \label{15.114B} \end{align}
In essence, this term involves an integral over a correlation function that describes variations in the a-b energy gap that varies as a result of its interactions with the bath. So this term, in essence, accounts for the fluctuations of the energy gap that we previously treated with stochastic models. Of course in the current case, we have made a Markovian bath assumption, so the fluctuations are treated as rapid and only assigned an interaction strength $\Gamma$ which is related to the linewidth. In an identical manner to the fast modulation limit of the stochastic model we see that the relaxation rate is related to the square of the amplitude of modulation times the correlation time for the bath:
\begin{align} \int _ {0}^{\infty} d \tau \langle \Delta V ( \tau ) \Delta V ( 0 ) \rangle _ {B} &= \left\langle \Delta V^{2} \right\rangle \tau _ {c} \label{15.115} \[4pt] &\equiv \Gamma \[4pt] & = \frac {1} {T _ {2}^{*}} \end{align}
As earlier this is how the pure dephasing contribution to the Lorentzian lineshape is defined. It is also assigned a time scale $T_2^*$.
So to summarize, we see that the relaxation of coherences has a contribution from pure dephasing and from the lifetime of the states involved. Explicitly, the equation of motion in Equation \ref{15.110} can be re-written
$\dot {\sigma} _ {a b} (t) = - i \omega _ {a b} \sigma _ {a b} (t) - \frac {1} {T _ {2}} \sigma _ {a b} (t) \label{15.116}$
where the dephasing time is
$\frac {1} {T _ {2}} = \frac {1} {T _ {2}^{*}} + \frac {1} {2} \left( \frac {1} {T _ {1 , a}} + \frac {1} {T _ {1 , b}} \right) \label{15.117}$
and the frequency has been corrected as a result interactions with the bath with the (small) imaginary contributions to $R _ {a b , a b}$:
$\omega _ {a b} = \omega _ {a b} + \operatorname {Im} \left[ R _ {a b , a b} \right] \label{15.118}$
Readings
1. Blum, K., Density Matrix Theory and Applications. Plenum Press: New York, 1981.
2. Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006; Ch. 10.
3. Pollard, W. T.; Friesner, R. A., Solution of the Redfield equation for the dissipative quantum dynamics of multilevel systems. The Journal of Chemical Physics 1994, 100, 5054-5065.
4. Slichter, C. P., Principles of Magnetic Resonance, with Examples from Solid State Physics. Harper & Row: New York, 1963.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/16%3A_Quantum_Relaxation_Processes/16.02%3A_A_Density_Matrix_Description_of_Quantum_Relaxation.txt
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For a weak acid HA in aqueous solution at temperature T and pressure p (which is ambient pressure and so close to the standard pressure) the following chemical equilibrium is established.
$\mathrm{HA}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq})$
The r.h.s. of equation (1.1.3) describes a 1:1 ‘salt’ in aqueous solution. At equilibrium (i.e. at a minimum in Gibbs energy), the thermodynamic description of the solution takes the following form.
$\mu^{e q}(\mathrm{HA} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mu^{\mathrm{eq}}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\mu^{\mathrm{eq}}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)$
We express µeq(H2O;aq;T;p) in terms of the practical osmotic coefficient φ for the solution.
$\mu^{e q}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda ; \mathrm{T} ; \mathrm{p}\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left[\mathrm{m}(\mathrm{HA})+2 \, \mathrm{m}_{\mathrm{j}}\right]^{e q}$
Here mj is the molality of the ‘salt’ H3O+A-. The latter yields 2 moles of ions for each mole of H3O+A-. A full description of the solution takes the following form.
\begin{aligned} \mu^{0}(\mathrm{HA} ; \mathrm{aq})+& \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}(\mathrm{HA}) \, \gamma(\mathrm{HA}) / \mathrm{m}^{0}\right]^{e q} \ &+\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left[\mathrm{m}(\mathrm{HA})+2 \, \mathrm{m}_{\mathrm{j}}\right]^{\mathrm{eq}} \ &=\mu^{0}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm}\right]^{\mathrm{eq}} \quad \end{aligned}
The practical osmotic coefficient φ describes the properties of solvent, water in the aqueous solution; γ± is the mean ionic activity coefficient for the ‘salt’ H3O+A-. By definition, if ambient pressure p is close to the standard pressure p0, the standard Gibbs energy of acid dissociation,
$\begin{gathered} \Delta_{\mathrm{d}} \mathrm{G}^{0}=\mu^{0}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)-\mu^{0}(\mathrm{HA} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu^{0}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda ; \mathrm{T} ; \mathrm{p}\right) \ =-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{A}}^{0}\right) \end{gathered}$
$\mathrm{K}_{\mathrm{A}}^{0}$ is the acid dissociation constant. Combination of equations (1.1.4) and (1.1.5) yields equation (1.1.6).
$\mathrm{K}_{\mathrm{A}}^{0} = \frac{\left[\mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \, \mathrm{A}^{-}\right)^{\mathrm{cq}} \, \gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]^{2} \, \exp \left[\phi \, \mathrm{M}_{1} \,\left(\mathrm{m}(\mathrm{HA})+2 \, \mathrm{m}_{\mathrm{j}}\right)\right]^{\mathrm{cq}}}{\left[\mathrm{m}(\mathrm{HA}) \, \gamma(\mathrm{HA}) / \mathrm{m}^{0}\right]^{\mathrm{eq}}}$
For dilute aqueous solutions, several approximations are valid. The exponential term and γ(HA)eq are close to unity. There are advantages in defining a quantity $\mathrm{K}_{\mathrm{A}}^{0}$ (app) . Further, γ±(H3O+A-) is obtained using the Debye - Huckel Limiting Law, DHLL.
By definition,
$\mathrm{K}_{\mathrm{A}}(\mathrm{app})=\left[\mathrm{m}\left(\mathrm{H}^{+} \mathrm{A}^{-}\right)^{\mathrm{eq}}\right]^{2} / \mathrm{m}(\mathrm{HA})^{\mathrm{eq}} \, \mathrm{m}^{0}$
Then
$\ln \mathrm{K}_{\mathrm{A}}(\operatorname{app})=\ln \mathrm{K}_{\mathrm{A}}^{0}+2 \, \mathrm{S}_{\gamma}\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}$
In other words, with increase in ionic strength $I$, $\mathrm{K}_{\mathrm{A}}(\operatorname{app})$ increases as a consequence of ion - ion interactions which stabilize the dissociated form of the acid.
1.1.02: Acquisitive Convention
This convention refers to the sign associated with communication between a system and its surroundings. The convention describes changes as seen from the standpoint of the system. The convention guides chemists concerning the ‘sign’ of changes in thermodynamic variables
00for a given system. For example, if heat q flows from the surroundings into a system, q is positive. If thermodynamic energy U is lost by a system to the surroundings, ∆U is negative. In fact this convention is intuitively attractive to chemists. For example, when told that the volume of a system increases during a given process, then chemists conclude that the volume of the surroundings (i.e. the rest of the universe!) decreases.
1.1.03: Activity- Solutions and Liquid Mixtures
The concept of activity was introduced by Lewis[1] in descriptions of the properties of liquid mixtures and solutions. By way of illustration we consider the chemical potential of chemical substance $j$, $\mu_{j}(\text { system })$ present in a solution at fixed pressure $p$ and temperature $T$. By definition,
$\mu_{j}(\text { system })=\mu_{j}(\text { ref })+R \, T \, \ln \left(a_{j}\right)$
While we can never know either $\mu_{j}(\text { system })$ or $\mu_{j}(\text { ref })$, the difference is related to the activity $a_{j}$, a dimensionless function of the composition of the system. We as observers of the system are required to define the reference state where the chemical potential of chemical substance $j$ can be clearly defined. Nevertheless the terms $\mu_{j}(\text { system })$, $\mu_{j}(\text { ref })$ and $a_{j}$ in equation (a) are based on somewhat abstract concepts. The link with practical chemistry is made through the differential of equation (a) with respect to pressure at constant temperature.
$\text { Then, } \mathrm{V}_{\mathrm{j}}(\text { system })=\mathrm{V}_{\mathrm{j}}(\text { ref })+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{a}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}$
$\mathrm{V}_{\mathrm{j}}(\text { system })$ and $\mathrm{V}_{\mathrm{j}}(\text { ref })$ are, respectively, the partial molar volumes of chemical substance $j$ in the system and in a convenient reference state. The term $\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{a}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}$ contrasts the role of intermolecular interactions in the two states. Four applications of the concept of activity make the point.
For the binary liquid mixture, ethanol + water at defined temperature and pressure, the activity of, for example, water (substance 1) $a_{1}$ is given by the product, $x_{1} \, f_{1}$ where $f_{1}$ is the (rational) activity coefficient and $x_{1}$ is the mole fraction of water.
$a_{1}=x_{1} \, f_{1}$
The activity of urea (chemical substance $j$) in an aqueous solution is related to the product of activity coefficient $\gamma_{i}$ and molality $m_{j}$ using the reference molality $m^{0}$, namely $1 \mathrm{ mol kg}^{–1}$.
$a_{j}=\left(m_{j} / m^{0}\right) \, \gamma_{j}$
If the concentration of urea in the solution equals $c_{j} \mathrm{ mol dm}^{-3}$, then the activity $a_{j}$ is given by equation (e) where $c_{r}$ is the reference concentration $c_{r}$, $1 \mathrm{ mol dm}^{–3}$, and $y_{j}$ is the solute activity coefficient.
$a_{j}=\left(c_{j} / c_{r}\right) \, y_{j}$
If $x_{j}$ is the mole fraction of urea and $\mathrm{f}_{\mathrm{j}}^{*}$ is the asymmetric activity coefficient, the activity of urea is given by equation (d).
$a_{j}=x_{j} \, f_{j}^{*}$
Equations (d) to (f) describe the same property, namely activity $a_{j}$ of solute $j$ in a given solution.
Footnotes
[1] G. N. Lewis, Proc. Am. Acad. Arts Sci.,1907,43,259.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.01%3A_Activity/1.1.01%3A_Acid_Dissociation_Constants-_Weak_Acids-_Debye-Huckel_Limiting_Law.txt
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A given aqueous solution is prepared using 1 kg of water($\lambda$, molar mass $\mathrm{M}_{1}$, and $m_{j}$ moles of solute at temperature $\mathrm{T}$ and pressure $p$ (which is close to the standard pressure $p_{0}$) At fixed $\mathrm{T}$ and $p$, the activity of water $a_{1}(\mathrm{aq})$ is related to the chemical potential of water in the aqueous solution using equation (a) where $\mu_{1}^{*}(\lambda)$ is the chemical potential of water($\lambda$) at the same $\mathrm{T}$ and $p$.
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{a}_{1}\right)$
Further equation (b) relates $\mu_{1}(\mathrm{aq})$ to the molality of a simple solute, $\mathrm{m}_{j}$ (e.g. urea) where $\mathrm{R}$ is the gas constant ($=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$).
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
Here $\mathrm{m}_{j}$ is the molality of solute $j$ and $\phi$ is the practical osmotic coefficient. If the thermodynamic properties of the aqueous solution are ideal (i.e. no solute-solute interactions) the practical osmotic coefficient is unity.
$\text { At fixed } \mathrm{T} \text { and } \mathrm{p}, \operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0$
$\text { Hence, } \mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
Therefore in the case of an ideal solution, addition of a solute, molality $\mathrm{m}_{j}$, stabilises the solvent; i.e. lowers the chemical potential of the solvent. In the event that solute $j$ is a salt which forms with complete dissociation $ν$ ions for each mole of salt in solution, $\mu_{1}(\mathrm{aq})$ is given by equation (e).
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
We consider an aqueous solution containing a simple neutral solute $j$. In order to understand the properties of this solution, we need to consider water-solute interactions and solute $j$ – solute $j$ interactions. Solute-solute interactions determine the extent to which the properties of a given solution differ from those of the corresponding solution having thermodynamic properties which are ideal [1].
The extent to which the thermodynamic properties of solutions are not ideal also reflects in part the role of water-solute interactions. For example the extent to which urea-urea interactions differ from ethanol-ethanol interactions in aqueous solutions reflects the different hydration characteristics of urea and ethanol.
Comparison of equations (a) and (b) yields the following important equation relating activity of solvent, water, and the molality of simple neutral solute $j$.
$\text { Thus, } \quad \ln \left(\mathrm{a}_{1}\right)=-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
The minus signs in equations (b) and (f) are extremely significant. If the thermodynamic properties of the solutions are ideal, $\phi$ is unity.
$\text { Then, } \ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}=-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=-\mathrm{M}_{1} \, \mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \, \mathrm{M}_{1}=-\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0}$
Here $n_{1}{}^{0}$ is the amount of solvent, water, molar mass $\mathrm{M}_{1}$; $n_{j}$ is the amount of solute $j$. Therefore a plot of $\ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}$ against molality $\mathrm{m}_{j}$ is linear with slope ‘$-\mathrm{M}){1}$’. Furthermore the plots for a range of neutral solutes will be super-imposable. In other words$\ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}$ is related to the ratio of amounts of solute to solvent. By adding a solute to a fixed amount of (solvent) water($\lambda$) we lower the activity of water($\lambda$) , (i.e. the chemical potential of water, µ(aq) in an aqueous solution) and stabilise the solvent.
The chemical potential of solute $j$ in an aqueous solution $\mu_{\mathrm{j}}(\mathrm{aq})$ is related to the molality of solute mj using equation (h) where $\mu_{j}^{0}(\mathrm{aq})$ is the chemical potential of solute $j$ in an aqueous solution, molality $\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}$ and $\gamma_{j}=1$ at all $\mathrm{T}$ and $p$, (taken as close to the standard pressure $p^{0}$ ).
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
$\text { By definition, at all T and } p \operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}} \rightarrow 1$
The role of water activity in determining enzyme activity is an important consideration [2]. .
Footnotes
[1] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London 2nd edn. Revised, 1965.
[2] G. Bell, A. E. M. Janssen and P. J. Halling, Enzyme and Microbial Technology, 1997,20,471.
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Classically, the colligative properties of non-ionic solutions were used to determine the molar mass of solutes. For example, the depression $\Delta \mathrm{T}_{f}$ of the freezing point of water $\mathrm{T}_{f}$ at a given molalilty $\mathrm{m}_{j}$ of solute-$j$ yields an estimate of the relative molar mass of the solute $\mathrm{M}_{j}$. Key thermodynamic assumptions require that (a) on cooling only pure solvent separates out as the solid phase and (b) the thermodynamic properties of the solution are ideal. The key relationship emerges from the Schroder- van Laar equation [1]. The common assumption is that the thermodynamic properties of the solution are ideal. If the properties of a given aqueous solutions are determined to a significant extent by solute-solute interactions, a measured relative molar mass will be in error. Indeed McGlashan[2] was dismissive of the procedures based on Beckmann’s apparatus for the determination of the relative molar mass of solute using freezing point measurements.
The chemical potential of water in an aqueous solution, $\mu_{1}(\mathrm{aq})$ at temperature $\mathrm{T}$ and pressure $p$ (assumed to be close to the standard pressure, $p^{0}$) is related to the molality of solute $j$, $\mathrm{m}_{j}$ using equation (a) where $\mathrm{R}$ is the gas constant, $\phi$ is the practical osmotic coefficient and $\mathrm{M}_{1}$ is the molar mass of water, $0.018015 \mathrm{ kg mol}^{-1}$ where $\mu_{1}^{*}(\lambda)$ is the chemical potential of water($\lambda$) at the same $\mathrm{T}$ and $p$.
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
Chemical potential $\mu_{1}(\mathrm{aq})$ at temperature $\mathrm{T}$ is also related to $\mu_{1}^{*}(\lambda)$ using equation (b) where $a_{1}$ is the activity of water in the aqueous solution.
$\mu_{1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mu_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{a}_{1}\right)$
Comparison of equations (a) and (b) shows that $\ln \left(a_{1}\right)$ is related to the molality of solute $\mathrm{m}_{j}$ using equation (c).
$\ln \left(a_{1}\right)=-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
For a solution having thermodynamic properties which are ideal, the practical osmotic coefficient is unity.
$\text { Then, } \ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}=-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
Hence for a solution having thermodynamic properties which are ideal, $\ln \left(a_{1}\right)$ is a linear function of molality $m_j$, the plot having slope, $-(\mathrm{M}_{1})$. Equation (d) forms a reference for a consideration of the properties of real solutions. For a solution having thermodynamic properties which are ideal, the solvent, water in an aqueous solution is at a lower chemical potential than the pure liquid. This observation is at the heart of the terms ‘depression of freezing point’ and ‘elevation of boiling point’. In the event that the thermodynamic properties of a given solution are not ideal then the form of the plot showing $\ln \left(a_{1}\right)$ as a function of molality $\mathrm{m}_{j}$ is determined by $\phi$ which is, in turn, a function of $\mathrm{m}_{j}$. The dependence of $\phi$ on $\mathrm{m}_{j}$ for a given solute in aqueous solutions (at fixed $\mathrm{T}$ and $p$) is not defined ‘a priori’.
Bower and Robinson[3] report the dependence of osmotic coefficients for urea (aq) at 298 K over the range $0 \leq \mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1} \leq 20.0$; $\phi$ decreases with increase in $\mathrm{m}_{j}$. Similarly Stokes and Robinson [4] report the dependence of $\phi$ on solute molality for sucrose(aq), glucose(aq) and glycerol(aq) over the range $0 \leq \mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1} \leq 7.5$.
For $\mathrm{m}(\text { urea })=8 \mathrm{~mol} \mathrm{~kg}^{-1}$, $\ln \left(a_{1}\right) \text { equals }-12 \times 10^{-2}$ whereas $\ln \left(a_{1}\right)^{i d}$ equals approx. $-15 \times 10^{-2}$. At this molality for urea(aq), $\mu_{1}(\mathrm{aq})>\mu_{1}(\mathrm{aq})^{\mathrm{id}}$ indicating that water in urea(aq) at this molality is at a higher chemical potential than would be the case for a solution where the thermodynamic properties are ideal. On the other hand for the hydrophilic solute sucrose where $\mathrm{m}(\text { sucrose })=6 \mathrm{~mol} \mathrm{~kg}^{-1}, \ln \left(\mathrm{a}_{1}\right) \text { is }-15 \times 10^{-2}$ whereas $\ln \left(a_{1}\right)^{i d}$ equals approx. $-11 \times 10^{-2}$ indicating that adding sucrose at this molality to water lowers the chemical potential of water relative to that for a solution having ideal properties.
For a dilute solution of simple neutral solutes the difference between ideal and real properties can be understood [5,6] in terms of the dependence of pairwise Gibbs energy interaction parameters $g_{jj}$ on molality using equation (e) where $\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{~kg}^{-1}$; the units of $g_{jj}$ are $\mathrm{ J kg}^{-1}$.
$1-\phi=-(1 / R \, T) \, g_{i j} \,\left(1 / m^{0}\right)^{2} \, m_{j}$
Using equation (c),
$\ln \left(a_{1}\right)=-M_{1} \, m_{j} \,\left[1+(R \, T)^{-1} \, g_{i j} \,\left(m^{0}\right)^{-2} \, m_{j}\right]$
or
$\ln \left(a_{1}\right)+M_{1} \, m_{j}=-M_{1} \,(R \, T)^{-1} \, g_{i j} \,\left(m^{0}\right)^{-2} \,\left(m_{j}\right)^{2}$
Hence for dilute solutions $\left[\ln \left(a_{1}\right)+M_{1} \, m_{j}\right]$ is a linear function of $\left(\mathrm{m}_{\mathrm{j}}\right)^{2}$, the gradient of the plot yielding the pairwise Gibbs energy interaction parameter $g_{jj}$. If, for example, $g_{jj}$ is positive indicating solute-solute repulsion, $\left[\ln \left(\mathrm{a}_{1}\right)+\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]$ decreases with increase in $\mathrm{m}_{j}$ such that $\mu_{1}(\mathrm{aq})>\mu_{1}(\mathrm{aq} ; \mathrm{id})$. In the event that solute-solute interactions are attractive, $g_{jj}$ is negative. Hence the difference between the properties of real and ideal solutions is highlighted by the contrast between equations (d) and (f).
The analysis described above is readily extended to aqueous solutions containing two solutes; e.g. urea(aq) + sucrose(aq), [7] and glucose(aq) + sucrose(aq).[8]
Salt Solutions
A given aqueous salt solution contains a single salt j; µ1(aq) and µj(aq) are the chemical potentials of water and salt respectively in the closed system. For water,
$\text { For water, } \mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{v} \, \mathrm{m}_{\mathrm{j}}$
$\text { And } \mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{a}_{1}\right)$
Here $ν$ is the stoichiometric parameter, the number of moles of ions produced by complete dissociation of one mole of salt $j$; for a 1:1 salt, $ν$ equals 2. According to equations (h) and (i),
$\ln \left(a_{1}\right)=-\phi \, M_{1} \, v \, m_{j}$
$\ln \left(a_{1}\right)^{\text {id }}=-M_{1} \, v \, m_{j}$
$\text { If we confine attention to } 1: 1 \text { salts, } \ln \left(a_{1}\right)^{\text {id }}=-2 \, M_{1} \, m_{j}$
With increase in $\mathrm{m}_{j}$, $\ln \left(a_{1}\right)^{i d}$ decreases linearly. With reference to equation (j), a ) $\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0$. With dilution of a salt solution, a plot of ln(a1) against mj approaches a linear dependence.
$\text { For the salt } \mathrm{j}, \mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)$
Here $ν_{+}$ and $ν_{-}$ are the number of moles of cations and anions respectively produced by one mole of salt $j$; $v=v_{+}+v_{-} ; \gamma_{\pm}$ is the mean ionic activity γ coefficient of salt $j$. By definition, $\mathrm{Q}^{\mathrm{v}}=\mathrm{v}_{+}^{\mathrm{v}(+)} \, \mathrm{V}_{-}^{\mathrm{v}(-)}$. Also $\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1.0$ at all $\mathrm{T}$ and $p$; $\mu_{\mathrm{j}}^{0}(\mathrm{aq})$ is the chemical potential of salt $j$ in a solution where $\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}$ and the thermodynamic properties of the solute are ideal; i.e. no ion-ion interactions.
For a 1:1 salt (e.g. $\mathrm{KBr}$),
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)$
For a 1:1 salt where the thermodynamic properties of the solution are ideal,
$\mu_{j}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
According to the Debye-Huckel Limiting Law, for very dilute solutions,
$\ln \left(\gamma_{\pm}\right)=-S_{\gamma} \,\left(m_{j} / m^{0}\right)^{1 / 2}$
where $S_{\gamma}=f\left(T, p, \varepsilon_{r}\right)$ and $\varepsilon_{\mathrm{r}}$ is the relative permittivity of the solvent at the same $\mathrm{T}$ and $p$.
$\text { Further }[8,9], 1-\phi=\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}$
At 298.15 K and ambient pressure [8], $\mathrm{S}_{\gamma}=0.5115$. In other words,
$\phi^{\mathrm{dh} l l}=1-\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}$
Hence using equation (j), for a 1:1 salt,
$\ln \left(\mathrm{a}_{1}\right)^{\mathrm{dhll}}+2 \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=\left[2 \,\left(\mathrm{S}_{\gamma} / 3\right) \, \mathrm{M}_{1} \,\left(\mathrm{m}^{0}\right)^{-1 / 2}\right] \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3 / 2}$
Then $\ln \left(a_{1}\right)^{\text {dhll }}$ indicates that for a salt solution, molality $\mathrm{m}_{j}$, $\ln \left(a_{1}\right)$ exceeds that in the corresponding salt solution having ideal thermodynamic properties . In other words the activity of the solvent, water, is enhanced above that for water having ideal thermodynamic properties. For very dilute solutions $\left[\ln \left(\mathrm{a}_{1}\right)^{\mathrm{dhll}}+2 \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]$ is a linear function of $\left(\mathrm{m}_{\mathrm{j}}\right)^{3 / 2}$. However other than for very dilute solutions equation (q) is inadequate and so a more sophisticated equation is required relating $\phi$ and $\mathrm{m}_{j}$. Extensive compilations of $\phi$ for salt solutions are given in references [8] and [10].
Footnotes
[1] I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London, 1954, equation 22.5.
[2] M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, page 307.
[3] V. E. Bower and R. A. Robinson, J. Phys. Chem.,1963,67,1524.
[4] R. H. Stokes and R. A. Robinson, J. Phys. Chem.,1963,67,2126.
[5] J. J. Savage and R. H. Wood, J. Solution Chem., 1976,5,733.
[6] M. J. Blandamer, J. Burgess, J. B. F. N. Engberts and W. Blokzijl, Annu. Rep. Prog. Chem., Sect C., Phys.Chem.,1990,87,45.
[7] H. Ellerton and P. J. Dunlop. J. Phys.Chem.,1966,70,1831.
[8] R. A.Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn., 1965, chapter 8.
[9] K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 3rd. edition, 1995.
[10] S. Lindenbaum and G. E. Boyd, J. Phys.Chem.,1964,68,911.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.01%3A_Activity/1.1.05%3A_Activity_of_Solvents.txt
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Thermodynamics underpins a classic topic in physical chemistry concerning the depression of freezing point, $\Delta \mathrm{T}_{\mathrm{f}}$ of a liquid by added solute [1,2]. We note the superscript ‘id’ in equation (a) relating the activity of a solvent in an ideal solution, molality $\mathrm{m}_{j}$.
$\ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}=-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=-\mathrm{M}_{1} \, \mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \, \mathrm{M}_{1}=-\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0}$
If the properties of a given aqueous solution are determined to a significant extent by solute-solute interactions, a determined molar mass for a given solute will be in error. Otherwise an observed depression is not a function of solute-solute interactions. Glasstone [2] comments that the ratio $\Delta \mathrm{T}_{\mathrm{f}} / \mathrm{m}_{\mathrm{j}}$ decreases with increasing concentration of solute, emphasising that a simple analysis is only valid for dilute solutions. Nevertheless the general idea is that the depression for a given mj is not a function of the hydration of a solute. Barrow [3] noted that the ratio $\Delta \mathrm{T}_{\mathrm{f}} / \mathrm{m}_{\mathrm{j}}$ for mannitol(aq) in very dilute solutions [4] is effectively constant. A similar opinion is advanced by Adam [5] who nevertheless comments on the importance of the condition ’dilute solution’ in a determination of the molar mass of a given solute.
In summary, classic physical chemistry emphasises the importance of the superscript ‘id’ in equation (a). For very dilute solutions in a given solvent $\ln \left(a_{1}\right)$ is linear function of $\mathrm{m}_{j}$, leading to description of such properties as ‘depression of freezing point ‘ and ‘elevation of boiling point ’ under the heading ‘colligative properties. Only the molality of solute mj is important; solute-solute interactions and hydration characteristics of solutes apparently play no part in determining these colligative properties.
The key to these statements is provided by the Gibbs-Duhem equation. For a solution prepared using 1 kg of water($\lambda$) and $\mathrm{m}_{j}$ moles of a simple solute $j$, the Gibbs energy is given by equation (b).
$\mathrm{G}(\mathrm{aq})=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})$
Then, $\mathrm{G}(\mathrm{aq})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]$
$+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \text { (c) }$
According to the Gibbs - Duhem Equation, the chemical potentials of solvent and solute are linked. At fixed $\mathrm{T}$ and $p$,
$\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0$
Or, $\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]$
$+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0$
$\text { Or, } \quad-\mathrm{d}\left(\phi \, \mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)=0$
$\text { Or, } \quad-\mathrm{d}\left(\phi \, \mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)=0$
$\text { Hence, } \quad(\phi-1) \, \frac{\mathrm{dm}_{\mathrm{j}}}{\mathrm{m}_{\mathrm{j}}}+\mathrm{d} \phi=\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)$
The latter equation forms the basis of the oft-quoted statement that if the thermodynamic properties of a solute are ideal then so are the properties of the solvent. Similarly if the thermodynamic properties of the solute are ideal then so are the properties of the solvent.
$\text { From equation }(h), \phi=1+\frac{1}{m_{j}} \, \int_{0}^{m(j)} m_{j} \, d \ln \left(\gamma_{j}\right)$
The importance of equation (i) emerges from the idea that $\gamma_{j}$ describes the impact of solute-solute interactions on the properties of a given solution. If we can formulate an equation for $\ln \left(\gamma_{j}\right)$ in terms of the properties of the solution, we obtain $\phi$ from equation (i). If the properties of a real solution containing a neutral solute are not ideal, both $\gamma_{j}$ and $\phi$ are linked functions of the solute molality. Pitzer [6] suggests equations (j) and (k) for $\ln \left(\gamma_{j}\right)$ and $\phi$ in terms of solute molalities using two parameters, $\lambda$ and $\mu$.
$\ln \left(\gamma_{\mathrm{j}}\right)=2 \, \lambda \, \mathrm{m}_{\mathrm{j}}+3 \, \mu \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}$
$\phi-1=\lambda \, \mathrm{m}_{\mathrm{j}}+2 \, \mu \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}$
For example in the case of mannitol(aq) and butan-1-ol(aq), Pitzer [6] suggests the following equations for $\ln \left(\gamma_{j}\right)$.
$\text { For mannitol(aq) } \quad \ln \left(\gamma_{\mathrm{j}}\right)=-0.040 \, \mathrm{m}_{\mathrm{j}}$
$\text { For butan-1-ol(aq) } \ln \left(\gamma_{\mathrm{j}}\right)=-0.38 \, \mathrm{m}_{\mathrm{j}}+0.51 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}$
Guggenheim[7] using the mole fraction scale suggests equation (n) where $\mathrm{A}$ and $\mathrm{B}$ are characteristic of the solute.
$1-\phi=A \, x_{j}+B \,\left(x_{j}\right)^{2}$
Prigogine and Defay [1] comment that the non-ideal properties of solutions can be understood in terms of the different molecular sizes of solute and solvent. A similar comment is made by Robinson and Stokes [8] who use a parameter describing the ratio of molar volumes of solute and solvent. The extent to which the properties of a solution differ from ideal can often be traced to a variety of causes including solvation, molecular size and shape.
Footnotes
[1] I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green , London 1953.
[2] S. Glasstone, Physical Chemistry, McMillan, London, 2nd. edn., 1948,page 645.
[3] G. M. Barrow, Physical Chemistry, McGraw-Hill, New York, 4th edn.,1979, page 298.
[4] L. H. Adams, J. Am. Chem. Soc.,1915,37,481.
[5] Neil Kensington Adam, Physical Chemistry, Oxford,1956, page 284.
[6] K. S. Pitzer, Thermodynamics , McGraw-Hill, New York, 3rd. edn., 1995.
[7] E. A. Guggenheim, Thermodynamics, North-Holland Publishing Co., Amsterdam, 1950, pages 252-3
[8] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London 2nd edn. Revised, 1965.
1.1.07: Activity of Water - Foods
An important scientific literature comments on the activity of water in the context of biochemistry and of the very important industry concerned with foods.[1-4]
Scott [5] identified the importance of water activity and microbial growth on foodstuffs; e.g. chilled beef. Hartel reviews the problem of freezing of water in, for example, ice cream; [6] see also comments on water activity in sucrose+ water systems.[1] The importance of water activity in sensory crispness and mechanical deformation of snack products is discussed by Katz and Labula. [7] Water activity is an important variable in fungal spoilage of food.[8,9]
Berg and Bruin review[10] the role of activity, not necessarily the water content, in the context of the deterioration of food, a matter of concern for humans from earliest times. In fact water activity, a1 is a major control variable in food preservation although the chemistry of food deterioration is complicated.
Crucially important in this context are publications produced by the National Institute of Standards and Technology .[11]
Footnotes
[1] P. Walstra, Physical Chemistry of Foods, M. Dekker, New York,2003.
[2] Water Activity: Influences on Food Quality, ed. L. B. Rockland and G. F. Stewart, Academic Press, New York,1981.
[3] A. J. Fontana Jr., Cereal Foods World,2000,45,7.
[4] A. J. Fontana and C. S. Campbell, Handbook of Food Analysis, volume 1, Physical Characterization and Nutrient Analysis , 2nd edn (revised and expanded) e.d. L. M.L. Nollet, 2004, M. Dekker, <New York
[5] W. J. Scott, Adv. Food Res.,1957,7,83.
[6] R.W. Hartel, Crystallizaiton in Foods, Aspen, Maryland, 2001.
[7] E. Katz and T. P. Labuza, J. Food Science,1981,46,403.
[8] R. Beauchat, J. Food Protection, 1983.46,135.
[9] W. H. Sperber, J. Food Protection, 1983.46,142.
[10] C. van den Berg and S. Bruin, Water Activity : Influence on Food Quality , ed. L. B. Rockford and G. F. Stewart, Academic Press, New York1981, pages 1-62.
[11] U. S. Food and Drug Administration, Title 21, Code of Federal Regulations, Parts 108, 110, 113 and 114, U. S. Government Printing Office, Washington, DC,1998.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.01%3A_Activity/1.1.06%3A_Activity_of_Solvents-_Classic_Analysis.txt
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Classical accounts of the physical chemical properties of solutions concentrate attention on the properties of solutes. Many experiments set out to determine activity coefficients $\gamma_{j}$ for solutes in solution; (see also mean activity coefficients $\gamma_{\pm}$ for salts). Information concerning solvent activity is obtained by exploiting the Gibbs-Duhem equation which (at fixed $\mathrm{T}$ and $p$) links the properties of solute and solvent. However recent technological developments allow the activity of water in an aqueous solution to be measured [1].
The classic analysis of colligative properties of solutions by van’t Hoff and others in the 19th Century is successful for dilute solutions. The importance of solute-solute interactions was generally recognised very early on in the 20th Century. However the role of solute-solvent interactions was perhaps underplayed.
We develop a model for a given solution prepared by dissolving $n_{j}$ moles of neutral solute $j$, molar mass $\mathrm{M}_{j}$, in $n_{1}{}^{0}$ moles of water($\lambda$), molar mass $\mathrm{M}_{1}$. The molality of solute as prepared is given by equation (a).
$\mathrm{m}_{\mathrm{j}}(\text { prepared })=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \, \mathrm{M}_{1}$
Then $\ln \left(a_{1} ; \text { prepared }\right)^{\text {id }}$ is given by equation (b).
$\ln \left(a_{1}\right)^{i d}=-$\mathrm{M}_{1} \, \(\mathrm{m}_{j}=-n_{j} / n_{1}^{0}] However in another description of the solution under investigation each mole of solute \(j$ is strongly hydrated by $h$ moles of water. The mass of solvent water, $\mathrm{w}_{1}$ is given by equation (c). \[\mathrm{w}_{1}=\left[\mathrm{n}_{1}^{0}-\mathrm{h} \, \mathrm{n}_{\mathrm{j}}\right] \, \mathrm{M}_{1}$
Hence the molality of hydrated solute is given by equation (d).
$\mathrm{m}_{\mathrm{j}}(\text { hydrated solute })=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1}^{0}-\mathrm{h} \, \mathrm{n}_{\mathrm{j}}\right] \, \mathrm{M}_{1}] In effect the molality of the solute increases because there is less ‘solvent water’ \[\text { By analogy with equation }(b), \ln \left(a_{1} ; \text { hyd }\right)=-n_{j} /\left(n_{1}^{0}-h \, n_{j}\right)$
Therefore for a range of solutions containing different solutes but prepared using the same amount of solute, the activities are a function of the different extents of hydration of the solutes. With increase in $h$, $\ln \left(a_{1} ; \text { hyd }\right)$ decreases (i.e. becomes more negative) indicative of increasing stabilisation of the water in the system by virtue of solute-water hydration interaction.
In a real solution, the properties of a solute are not ideal because there exist solute-solute communication by virtue of the fact that each solute molecule is ‘aware’ that some of the solvent has been ‘removed’ by solute hydration. The amount of solvent ‘available ‘ to each molecule has been depleted by hydration of all solutes in solution. In other words a Gibbs-Duhem communication operates in the solution.
Hydration of Hydrophilic Solutes; Scatchard Model
The model of an aqueous solution described in conjunction with equation (a) is used to obtain the ratio, $n_{1}^{0} / n_{j}$; equation (f).
$\text { Then, } \frac{n_{1}^{0}}{n_{j}}=\frac{1 \mathrm{~kg}}{M_{1}} \, \frac{M_{j}}{w_{j}}$
Thus $n_{1}^{0} / n_{j}$ describes the solution as prepared using $n_{j}$ and $n_{1}{}^{0}$ moles of solutes and solvent respectively. We assert that by virtue of solute hydration an amount of water is removed from the ‘solvent’. In solution the mole fraction of (solvent) water is $x_{1}$, the mole fraction of hydrated solute is $x_{j}$.
$\text { Then, } \quad x_{w}+x_{j}=1$
The mole fraction ratio $\mathbf{X}_{w} / \mathbf{X}_{j}$ is given by equation (h).
$\frac{\mathbf{X}_{w}}{X_{j}}=\frac{X_{w}}{1-X_{w}}] Equation (h) forms the basis of a treatment described by Scatchard [2] in 1921, nearly a century ago. The model proposed by Scatchard [2] was based on a model for water($\lambda$), described as a mixture of hydrols; monohydrols and polymerised water. Scatchard discusses hydration of solutes although not all solutes in a given solution are seen as hydrated; i.e. the solution contains various hydrates. In fact a given solute may be hydrated to varying degrees; i.e. a given solution contains various hydrates. However Scatchard envisaged [2] that one hydrate may be dominant. We note the date when the model was proposed by Scatchard [2]. The concept of hydrogen bonding in aqueous solutions has its origin in a paper published by Latimer and Roedbush in 1920 [3]; see also [4]; i.e. the previous year to publications by Scatchard [2]. Scatchard [2] invoked an assumption called the ‘semi-ideal’ assumption in which mole fraction $x_{\mathrm{w}$ in equation (h) is replaced by the activity of the solvent, water $\mathrm{a}_{\mathrm{w}$. Hence, from equation (h), \[\text { Hence, from equation (h), } \frac{\mathrm{x}_{\mathrm{w}}}{\mathrm{x}_{\mathrm{j}}}=\frac{\mathrm{a}_{\mathrm{w}}}{1-\mathrm{a}_{\mathrm{w}}}$
The difference between amounts of water defined by equations (f) and (i) yields the ‘average degree of hydration’, $h$ of solute $j$.
$\text { Then, } \quad \mathrm{h}=\frac{(1.0 / 0.0180)}{\mathrm{m}_{\mathrm{j}}}-\frac{\mathrm{a}_{\mathrm{w}}}{1-\mathrm{a}_{\mathrm{w}}}$
Equation (j) is Scatchard’s equation. If one can measure $\mathrm{a}_{\mathrm{w}$ for an aqueous solution molality $\mathrm{m}_{j}$ we obtain the ‘average degree of hydration’ for solute $j$. Scatchard using vapour pressure data obtained an estimate of $h$ for sucrose at 30 Celsius. In the case of a solution containing $34 \mathrm{~g}$ of sucrose in $100 \mathrm{~g}$ water($\lambda$), $h$ equals $4.46$, decreasing with increase in the ‘strength ‘ of the solution. The term ‘semi-ideal’ proposed by Scatchard emerges from the identification of solvent activity with mole fraction of depleted solvent. In summary Scatchard [2] obtained a property $h$ but there is no indication of the stability of the hydrate.
Stokes and Robinson [5] extended the Scatchard analysis using a chemical equilibrium involving solute hydrates. The hydration of a given solute is described by a set of equilibrium constants describing i-hydration steps. For solute $\mathrm{S}$,
$\mathrm{S}_{\mathrm{i}-1}+\mathrm{H}_{2} \mathrm{O} \Leftrightarrow \Rightarrow \mathrm{S}_{\mathrm{i}} \quad(\mathrm{i}=1,2 \ldots \ldots \mathrm{n})$
Each step is described by an equilibrium constant, $\mathrm{K}_{i}$. So for a solute hydrated by 3 water molecules there are 3 equilibrium constants. Stokes and Robinson [5] set $n$ equal to $11$ for sucrose. However in this case Stokes and Robinson [5] simplify the analysis by assuming that the equilibrium constants for all hydration steps are equal. The resulting equations are as follows.
$\frac{\left(1 / \mathrm{M}_{1}\right)}{\mathrm{m}_{\mathrm{j}}}=\frac{\mathrm{a}_{\mathrm{w}}}{1-\mathrm{a}_{\mathrm{w}}}+\frac{\sigma}{\Sigma}$
$\text { where } \quad \sigma=\mathrm{K} \, \mathrm{a}_{\mathrm{w}}+\ldots \ldots \ldots+\mathrm{K} \,\left(\mathrm{a}_{\mathrm{w}}\right)^{\mathrm{n}}$
$\text { and } \quad \Sigma=1+K \, a_{w}+\ldots \ldots . .+\left(K \, a_{w}\right)^{n}$
Equation ($\lambda$) is interesting because for a given solute, the dependence of $\mathrm{a}_{\mathrm{w}$ on $\mathrm{m}_{j}$ yields two interesting parameters, $n$ and $\mathrm{K}$, describing hydration of a given solute $j$. The equilibrium constants defined above are dimensionless.
Stokes and Robinson [5] describe a method of data analysis but modern computer-based methods should lighten the arithmetic drudgery. For sucrose(aq) at $298.15 \mathrm{~K}$ Stokes and Robinson estimate that $n = 11$ and $\mathrm{K} = 0.994$. For glucose(aq) $n = 6$ with $\mathrm{K} = 0.786$. Finally we note that the composition of the solutions should be expressed in molalities; i.e. each solution made up by weight. If this is not done, conversion of concentrations to molalities is required. Possibly the literature will yield the required densities of the solutions. The worst approximation sets the density of the solutions at the density of water($\lambda$) at the same temperature.
Footnotes
[1] Water Activity Meter, Decagon Devices Inc. WA 99163, USA.
[2] G. Scatchard, J.Am.Chem.Soc.,1921,43,2387.,2408.
[3] W. M. Latimer and W. H. Rodebush, J. Am. Chem.Soc.,1920,42,1419.
[4] L. Pauling, The Nature of the Chemical Bond, Cornell Univ. Press, Ithaca, New York, 3rd edn.,1960, chapter 12.
[5] R. H. Stokes and R.A Robinson, J. Phys.Chem.,1966, 70,2126.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.01%3A_Activity/1.1.08%3A_Activity_of_Water_-_One_Solute.txt
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A given solution contains two neutral (i.e. non-ionic) solutes, solute-$i$ and solute-$j$. We anticipate, for example, activity coefficient $\gamma_{i}$ for solute –$i$ is a function of the molalities of both solutes, $\mathrm{m}_{i}$ and $\mathrm{m}_{j}$. The thermodynamic properties of this class of solutions are discussed by Bower and Robinson [1] and by Ellerton and Dunlop [2]. Because the analysis discussed by these authors concerns the properties of solvent, water in aqueous solutions, the starting point is isopiestic vapour pressure measurements [1-3]. Analysis of the thermodynamic properties of these mixed aqueous solutions has four themes which we develop separately, drawing the analysis together in a final section.
Theme A
Solution I is prepared by dissolving ni moles of solute-$i$ in water, mass $\mathrm{w}_{1}(\mathrm{I})$ at temperature $\mathrm{T}$ and pressure $p$( which is close to the standard pressure $p^{0}$); $\mathrm{M}_{1}$ is the molar mass of water and $\phi(I)$ is the practical osmotic coefficient of the solvent, water, in solution(I). The contribution $\mathrm{G}_{1}(\mathrm{I})$ of the solvent to the Gibbs energy of the solution is given by equation(a).
$\mathrm{G}_{1}(\mathrm{I})=\left[\mathrm{w}_{1}(\mathrm{I}) / \mathrm{M}_{1}\right] \,\left\{\mu_{1}^{*}(\lambda)-\left[\phi(\mathrm{I}) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})\right]\right\}$
Solution (II) is similarly prepared using $n_{j}$ moles of solute-$j$ in water, mass $\mathrm{w}_{1}(\mathrm{II})$.
$\mathrm{G}_{1}(\mathrm{II})=\left[\mathrm{w}_{1}(\mathrm{II}) / \mathrm{M}_{1}\right] \,\left\{\mu_{1}^{*}(\lambda)-\left[\phi(\mathrm{II}) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]\right\}$
We add a sample of solution (I) containing $1 \mathrm{~kg}$ of water to a sample of solution (II) also prepared using $1 \mathrm{~kg}$ of water. The resulting solution contains $2 \mathrm{~kg}$ of water and the initial molalities $\mathrm{m}_{i}(\mathrm{I})$ and $\mathrm{m}_{j}(\mathrm{II})$ will be reduced by a half. Then we imagine that $1 \mathrm{~kg}$ of water is withdrawn from the solution. This concentration process restores the original molalities of solutes $i$ and $j$. The letter ‘F’ identifies the new solution.
$\mathrm{G}_{1}\left(\text { total } ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]$
$\phi(\mathrm{F}$) is the practical osmotic coefficient of the solution prepared using solutions I and II from which $1 \mathrm{~kg}$ of solvent has been removed. The results of the analysis given above can be summarised in three equations describing the activities of water in the three solutions.
$\ln \left[\mathrm{a}_{1}(\mathrm{I})\right]=-\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I}) \, \mathrm{M}_{1}$
$\ln \left[\mathrm{a}_{1}(\mathrm{II})\right]=-\phi(\text { II }) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{M}_{1}$
$\ln \left[\mathrm{a}_{1}(\mathrm{~F})\right]=-\phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right] \, \mathrm{M}_{1}$
The molalities remain the same as in the original solutions; i.e. $\mathrm{m}_{\mathrm{i}}(\mathrm{F})=\mathrm{m}_{\mathrm{i}}(\mathrm{I})$ and $\mathrm{m}_{\mathrm{j}}(\mathrm{F})=\mathrm{m}_{\mathrm{j}}(\mathrm{II})$.
$\text { By definition, } \Delta \equiv \phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]-\left[\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})+\phi(\mathrm{II}) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]$
Experiments based on isopiestic measurements using the equilibrium between reference and mixed solutions and independently determined $\phi(\mathrm{I})$ and $\phi(\mathrm{II})$ yield the quantity $\Delta$.
Theme B
The starting points are general equations for the activity coefficients $\gamma_{i}$ and $\gamma_{j}$ for solutes $i$ and $j$ respectively as a function of the molalities $\mathrm{m}_{i}$ and $\mathrm{m}_{j}$ in the mixed solutions. Two equations based on Taylor series are used.
$\ln \left(\gamma_{i}\right)=\sum_{k=0}^{k=\infty} \sum_{\lambda=0}^{\lambda=\infty} A_{k \lambda} \,\left(m_{i} / m^{0}\right)^{k} \,\left(m_{j} / m^{0}\right)^{\lambda}$
$\ln \left(\gamma_{k}\right)=\sum_{k=0}^{k=\infty} \sum_{\lambda=0}^{\lambda=\infty} B_{k \lambda} \,\left(m_{i} / m^{0}\right)^{k} \,\left(m_{j} / m^{0}\right)^{\lambda}$
With reference to equations (h) and (i), both $\mathrm{A}_{00}$ and $\mathrm{B}_{00}$ are zero. The dimensionless coefficients $\mathrm{A}_{k} \lambda$ and $\mathrm{B}_{k} \lambda$ are interlinked by the Gibbs-Duhem equation. It also turns out that the series up to and including ‘$k = 4$’ and ‘$\lambda = 4$’ are sufficient in the analysis of experimental results.
According to equation (h), a description of the properties of solute-$i$ is given by equation (j).
\begin{aligned} \ln \left(\gamma_{\mathrm{i}}\right)=& \mathrm{A}_{10} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{A}_{01} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{20} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \ &+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \ &+\mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{A}_{30} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \ &+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{40} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{4} \ &+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \ &+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{4} \end{aligned}
In the event that $\mathrm{m}_{j}$ is zero,
\begin{aligned} \ln \left[\gamma_{\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}=0\right)\right]=\mathrm{A}_{10} \, &\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{A}_{20} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \ &+\mathrm{A}_{30} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{40} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{4} \end{aligned}
Moreover $\ln \left[\gamma_{i}\left(m_{j}=0\right)\right]$ can be calculated from the measured properties of aqueous solutions containing only solute-$i$. Therefore the dependence of $\ln \left[\gamma_{i}\left(m_{j}=0\right)\right]$ on $\mathrm{m}_{i}$ can be analysed using a linear least squares procedure to yield the coefficients $\mathrm{A}_{k 0}$ for $k=1- 4$. Hence $\ln \left(\gamma_{i}\right)$ for the mixed solute system is given by a combination of equations (j) and (k) to yield equation ($\lambda$).
\begin{aligned} &\ln \left(\gamma_{\mathrm{i}}\right)=\ln \left[\gamma_{\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}=0\right)\right] \ &+\mathrm{A}_{01} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \ &\quad+\mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \ &\quad+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \ &\quad+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{4} \end{aligned}
According to equation ($\lambda$), the dependence of $\gamma_{i}$ on $\mathrm{m}_{j}$ at fixed $\mathrm{m}_{i}$ is given by equation (m).
\begin{aligned} {\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})} } &=\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{02} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \ &+\mathrm{A}_{03} \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-1} \ &+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{2}+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \ &+\mathrm{A}_{04} \, 4 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \end{aligned}
A cross-differential link yields the following interesting equation.
$\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}=\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{i}}}\right]_{\mathrm{m}(\mathrm{j})}$
We combine equations (m) and (n).
\begin{aligned} {\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{i}}}\right]_{\mathrm{m}(\mathrm{j})} } &=\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{02} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \ &+\mathrm{A}_{03} \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-1} \ &+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2}+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \ &+\mathrm{A}_{04} \, 4 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \end{aligned}
We integrate the latter equation to yield an equation for $\gamma_{j}\left(m_{i}=0\right)$ where at ‘$\mathrm{m}_{i} = 0$’ , $\gamma_{j}$ is represented as $\gamma_{j}\left(\mathrm{~m}_{\mathrm{i}}=0\right)$. The outcome is an equation for $\ln \left(\gamma_{j}\right)$ in terms of the $\mathrm{A}_{i}$-variables, making the $\mathrm{B}_{i}$ variables somewhat redundant.
\begin{aligned} \ln \left(\gamma_{\mathrm{j}}\right)=& \ln \left[\gamma_{\mathrm{j}}\left(\mathrm{m}_{\mathrm{i}}=0\right)\right] \ +& \mathrm{A}_{01} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\left(\mathrm{A}_{11} / 2\right) \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2}+2 \, \mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}}\right) \,\left(\mathrm{m}^{0}\right)^{-2} \ &+\left(\mathrm{A}_{21} / 3\right) \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1} \ &+3 \, \mathrm{A}_{03} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \ &+(1 / 4) \, \mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{4} \ &+\left(2 \, \mathrm{A}_{22} / 3\right) \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4}+\left(3 \, \mathrm{A}_{13} / 2\right) \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \ &+4 \, \mathrm{A}_{04} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \end{aligned}
Theme C Gibbs-Duhem Equations
A Single Solute
For an aqueous solution at fixed $\mathrm{T}$ and $\mathrm{p}$ containing the single solute-$i$, the Gibbs-Duhem equation yields the following relationship.
$\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{i}} \, \mathrm{d} \mu_{\mathrm{i}}(\mathrm{aq})=0$
\text { Then, } \begin{aligned} \frac{1}{\mathrm{M}_{1}} \, \mathrm{d}\left[\mu_{1}^{*}(\lambda)-\phi_{\mathrm{i}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{i}}\right] \ &+\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}
The symbol $\phi_{i}$ identifies the practical osmotic coefficient in a solution containing solute-$i$.
$\text { Hence }[4], \quad \mathrm{d}\left[\phi_{i} \, \mathrm{m}_{\mathrm{i}}\right]=\mathrm{dm} \mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)$
Similarly for an aqueous solutions containing solute-$j$,
$\mathrm{d}\left[\phi_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)$
Two Solutes
From the Gibbs-Duhem equation (at fixed $\mathrm{T}$ and $p$)
$n_{1} \, d \mu_{1}(a q)+n_{i} \, d \mu_{i}(a q)+n_{j} \, d \mu_{j}(a q)=0$
Then,
\begin{aligned} &\frac{1}{\mathrm{M}_{1}} \, \mathrm{d}\left[\mu_{1}^{*}(\lambda)-\phi_{\mathrm{ij}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \ &\quad+\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right] \ &\quad+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}
The practical osmotic coefficient $\phi_{ij}$ identifies a solution containing two solutes, $i$ and $j$.
$\text { Hence, } \mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right]=\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)$
$\text { Therefore[5] } \mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right]=\mathrm{d}\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)$
According to equation (g)
$\Delta \equiv \phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]-\left[\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})+\phi(\mathrm{II}) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]$
$\text { Then } \mathrm{d} \Delta=\mathrm{d}\left\{\phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]-\mathrm{d}\left[\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})\right]-\mathrm{d}\left[\phi(\mathrm{II}) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]\right.$
Labels (I) and (II) can be dropped when applied to solute molalities. Then using equations (s), (t) and (x),
$\begin{array}{r} \mathrm{d} \Delta=\mathrm{d}\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}(\mathrm{F})\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left[\gamma_{\mathrm{j}}(\mathrm{F})\right] \ \quad-\mathrm{dm}_{\mathrm{i}}-\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left[\gamma_{\mathrm{i}}(\mathrm{I})\right]-\mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left[\gamma_{\mathrm{j}}[\mathrm{II}]\right. \end{array}$
$\text { Or, } \mathrm{d} \Delta=\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left\{\ln \left(\gamma_{\mathrm{i}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{j}}(\mathrm{I})\right]\right\}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left\{\ln \left[\gamma_{\mathrm{j}}(\mathrm{F})\right]-\mathrm{d} \ln \left[\gamma_{\mathrm{j}}[\mathrm{II}]\right\}\right.\right.$
In equation ($\lambda$) we identify $\ln \left[\gamma_{i}\left(m_{j}=0\right)\right]$ with $\ln \left[\gamma_{i}(\mathrm{I})\right]$. Similarly $\ln \left[\gamma_{\mathrm{j}}\left(\mathrm{m}_{\mathrm{i}}=0\right)\right]$ in equation (p) equals $\ln \left[\gamma_{\mathrm{j}}(\mathrm{II})\right]$ in equation (za). Therefore [6]
\begin{aligned} \Delta / & \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1} \ =& \mathrm{A}_{01}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-1}+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2} \ &+(3 / 2) \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2}+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \ &+(4 / 3) \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3}+2 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \ &+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-3} \end{aligned}
Footnotes
[1] V. E. Bower and R. A. Robinson , J.Phys.Chem.,1963,67,1524.
[2] H. D. Ellerton and P. J. Dunlop, J. Phys Chem.1966,70,1831.
[3] H. D. Ellerton, G. Reinfelds, D. E. Mulcahy and P. J. Dunlop, J. Phys. Chem., 1964, 68,398.
[4] Hence, $-\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)=0$
Then, $-\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]+\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{i}}\right)+\ln \left(\gamma_{\mathrm{i}}\right)-\ln \left(\mathrm{m}^{0}\right)\right]=0$
Or, $\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]=\mathrm{m}_{\mathrm{i}} \, \frac{1}{\mathrm{~m}_{\mathrm{i}}} \mathrm{dm} \mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)$
Then, $\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]=\mathrm{dm}_{\mathrm{i}}+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)$
[5] Or,
\begin{aligned} &\mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \ &\quad=\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \end{aligned}
Or
\begin{aligned} &\mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \ &\quad=\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)+\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}_{\mathrm{i}}\right) \, \mathrm{d}\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \end{aligned}
Or,
\begin{aligned} &\mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \ &\quad=\mathrm{d}\left(\mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{i}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \end{aligned}
[6] Differentiation of equation ($\lambda$) yields
\begin{aligned} &\mathrm{d}\left\{\ln \left[\gamma_{\mathrm{i}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{i}}(\mathrm{I})\right]\right\}=\right. \ &\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{j}}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \ &+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}}+2 \, \mathrm{A}_{21} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \ &+2 \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{j}_{\mathrm{j}}+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{m}_{\mathrm{j}} \ &+3 \, \mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \ &+2 \, \mathrm{A}_{22} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+2 \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \ &+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+3 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \ &+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \end{aligned}
Similarly from equation (p),
\begin{aligned} &\mathrm{d}\left\{\ln \left[\gamma_{\mathrm{j}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{j}}(\mathrm{I})\right]\right\}=\right. \ &\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}} \ &+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+2 \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \ &+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{m}_{\mathrm{j}}+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \ &+6 \, \mathrm{A}_{03} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}} \ &+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm} \mathrm{m}_{\mathrm{i}}+(2 / 3) \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm} \ &\mathrm{j} \ &+3 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}} \ &+3 \, \mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}}+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}} \ &+12 \, \mathrm{A}_{04} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \end{aligned}
But according to equation (za),
$\mathrm{d} \Delta=\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left\{\ln \left(\gamma_{\mathrm{i}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{j}}(\mathrm{I})\right]\right\}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left\{\ln \left[\gamma_{\mathrm{j}}(\mathrm{F})\right]-\mathrm{d} \ln \left[\gamma_{\mathrm{j}}[\mathrm{II}]\right\}\right.\right.$
After rearranging one obtains the following equation.
\begin{aligned} &\mathrm{d} \Delta= \ &\mathrm{A}_{01} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{01} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{j}} \ &+2 \, \mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \ &+2 \, \mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+4 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \ &+3 \, \mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}} \ &+3 \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+3 \, \mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}} \ &+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+9 \, \mathrm{A}_{03} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}} \ &+4 \, \mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{4} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \ &+4 \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+(8 / 3) \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \ &+4 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+6 \, \mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \ &+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{4} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+16 \, \mathrm{A}_{04} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \end{aligned}
Term by term integration of the latter equation yields
$\Delta=\int \mathrm{d} \Delta$
As an example we cite the terms containing the coefficient $\mathrm{A}_{11}$.
\begin{aligned} &\int 2 \, A_{11} \, m_{i} \, m_{j} \,\left(m^{0}\right)^{-2} \, d m_{i}+\int A_{11} \,\left(m_{i}\right)^{2} \,\left(m^{0}\right)^{-2} \, d m_{j} \ &=A_{11} \,\left(m^{0}\right)^{-2} \, d m_{i} \, \int\left[2 \, m_{i} \, m_{j} \, d m_{i}+\left(m_{i}\right)^{2} \, d m_{j}\right] \ &=A_{11} \,\left(m^{0}\right)^{-2} \, \int d\left[\left(m_{i}\right)^{2} \, m_{j}\right] \ &=A_{11} \,\left(m^{0}\right)^{-2} \,\left(m_{i}\right)^{2} \, m_{j} \end{aligned}
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.01%3A_Activity/1.1.09%3A_Activity_of_Water_-_Two_Solutes.txt
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A given solution is prepared using n1 moles of water($\lambda$) together with $n_{i}$ and $n_{j}$ moles of neutral solutes $i$ and $j$ respectively at temperature $\mathrm{T}$ and pressure $p$ (which is close to the standard pressure $p^{0}$). The mass of water is $\mathrm{n}_{1} \, \mathrm{M}_{1}$ where $\mathrm{M}_{1}$ is the molar mass of water. Hence the molalities of solutes $i$ and $j$ are $\mathrm{m}_{\mathrm{i}} \left(=\mathrm{n}_{\mathrm{i}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right)$ and $\mathrm{m}_{\mathrm{j}}\left(=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right)$ respectively. The chemical potential of water in the aqueous solution $\mu_{1}(\mathrm{aq})$ is related to the molality of solutes using equation (a) where $\phi$ is the practical osmotic coefficient and $\mu_{1}^{*}(\lambda)$ is the chemical potential water($\lambda$) at the same $\mathrm{T}$ and $p$.
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)$
The chemical potentials of the two solutes, $\mu_{\mathrm{i}}(\mathrm{aq})$ and $\mu_{\mathrm{j}}(\mathrm{aq})$, are related to $\mathrm{m}_{i}$ and $\mathrm{m}_{j}$ together with corresponding activity coefficients, $\gamma_{i}$ and $\gamma_{j}$ using equations (b) and (c).
$\mu_{\mathrm{i}}(\mathrm{aq})=\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)$
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Here $\mu_{i}^{0}(\mathrm{aq})$ is the reference chemical potential for solute $i$ in a solution where $\mathrm{m}_{\mathrm{j}}=0 \mathrm{~mol} \mathrm{~kg}{ }^{-1}$, $\mathrm{m}_{\mathrm{i}}=1 \mathrm{~mol} \mathrm{~kg}$ and $\gamma_{i} = 1$. A similar definition operates for solute $j$. For the mixed solution at all $\mathrm{T}$ and $p$,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{i}}=1$
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1$
In these terms, the thermodynamic properties of solute $i$ are not ideal as a consequence of $i-i$, $j-j$ and $i-j$ solute-solute interactions. A similar comment concerns the thermodynamic properties of solute $j$. With increase in molalities $\mathrm{m}_{i}$ and $\mathrm{m}_{j}$, so the extent to which the thermodynamic properties deviate from ideal increases; i.e. for solute $j$, $\gamma_{j} \neq 1$ and for solute $i$, $\gamma{i} \neq 1$. Such deviations can be understood in terms of $i-i$, $j-j$, and $i-j$ solute-solute interactions.
In some applications of this analysis, solute $i$ is present in trace amounts and so $\gamma_{i}$ in the absence of solute $j$ would be close to unity. However as the molality of solute $j$ is increased, the thermodynamic properties of solute $i$ deviate increasingly from ideal as a result of solute $i$ $\rightleftarrows$ solute $j$ interactions. This feature can be described [1,2] quantitatively using equation $(\mathrm{f})$ where $\beta_{1}, \beta_{2} \ldots$ describe the role of pairwise i-j , triplet $i-j-j \ldots \ldots$ solute-solute interactions.
$\ln \left(\gamma_{i}\right)=\beta_{1} \,\left(m_{j} / m^{0}\right)+\beta_{2} \,\left(m_{j} / m^{0}\right)^{2}+\ldots \ldots$
Depending on the signs of the $\beta$ - coefficients , added solute $j$ can either stabilise or destabilise solute-$i$; i.e. either lower or raise $\mu_{i}(a q)$ relative to that in a solution having ideal thermodynamic properties.
Footnotes
[1] Based on an analysis suggested by C. Wagner, Thermodynamics of Alloys, Addison-Wesley, Reading, Mass., 1952, pages 19-22.
[2] As quoted by G. N. Lewis and M. Randall, Thermodynamics, 2nd edn., revised by K. S. Pitzer and L Brewer, McGraw-Hill, New York, 1961, page 562.
1.1.11: Activity Coefficients
In descriptions of the properties of the components of liquid mixtures, rational activity coefficients are given the symbols $\mathrm{f}_{1}, \mathrm{f}_{2}, \mathrm{f}_{3} \ldots$. With reference to solvents their thermodynamic properties are described by rational activity coefficients $\mathrm{f}_{1}, \mathrm{f}_{2}, \mathrm{f}_{3} \ldots$ and (practical) osmotic coefficient, $\phi$. The properties of solutes in solutions are described using activity coefficients which are linked to the descriptions of the composition of solutions: molality scale, $\gamma_{j}$; concentration scale $\gamma_{j}$; mole fraction scale $\mathrm{f}_{\mathrm{j}}^{*}$.
These coefficients are intimately related to the concept of activity. Their significance is clarified by equations relating chemical potentials to the composition of a given system; e.g.
1. components in liquid mixtures,
2. solvents in solutions and
3. solutes in solutions.
In all cases they summarise the extent to which the thermodynamic properties of liquid mixtures and solutions are not ideal. The challenge for chemists is to understand in terms of molecular properties why the thermodynamic properties of real systems are not ideal. It has to be admitted that activity coefficients have a ‘bad press’ as far as most chemists are concerned. All too often their importance is ignored. But ‘learn to love activity coefficients! Perhaps the importance of activity coefficients can be understood in the following terms.
The chemical potential of urea as a solute in aqueous solutions, $\mu_{j}(\mathrm{aq})$ at temperature $\mathrm{T}$ and pressure $p$ ( $\approx$ the standard pressure $p^{0}$) is related to the molality of urea $\mathrm{m}_{j}$ using equation (a).
\begin{aligned} &\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T}, \mathrm{p})=\ &\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T}, \mathrm{p}, \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg} \mathrm{~kg}^{-1}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right) \end{aligned}
The term $\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T}, \mathrm{p}, \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{\textrm {kg } ^ { - 1 } )}\right.$ describes the chemical potential of solute, urea in an aqueous solution having unit molality where there are no urea - urea (i.e. solute – solute) interactions. Each urea molecule in these terms is unaware of the presence of other urea molecules in the aqueous solution. In a solution having ideal thermodynamic properties $\mu_{j}(\mathrm{aq} ; \mathrm{T}, \mathrm{p} ; \mathrm{id})$ depends on the molality of solute mj. Thus the osmotic pressure of this solution is a function of the molality of urea. For such an ideal solution there are no urea-urea interactions although there are important urea-water interactions, the hydration of urea. But in a real solution there are also solute-solute interactions. Each solute molecule ‘knows’ there are other solute molecules in the solution. Indeed the extent to which $\mu_{j}(\mathrm{aq} ; \mathrm{T}, \mathrm{p})$ differs from $\mu_{j}(\mathrm{aq} ; \mathrm{T}, \mathrm{p} ; \mathrm{id})$ is a function of the molality of the solute, $\mathrm{m}_{j}$.
1.1.12: Activity Coefficients- Salt Solutions- Ion-Ion Interactions
For most dilute aqueous salt solutions (at ambient temperature and pressure), mean ionic activity coefficients γ± are less than unity. Thus ion-ion interactions within a real solution lower chemical potentials below those of salts in the corresponding ideal solutions. Clearly a quantitative treatment of this stabilisation is enormously important. In fact for almost the whole of the 20th Century, scientists offered theoretical bases for expressing $\ln \left(\gamma_{\pm}\right)$ as a function of the composition of a salt solution, temperature, pressure and electric permittivity of the solvent.
In effect we offer as much information as demanded by the theory (e.g. molality of salt, nature of salt, permittivity of solvent, ion sizes, temperature, pressure, .....). We set the apparently simple task - please calculate the mean activity coefficient of the salt in this solution.
Many models and treatments have been proposed [1]. Most models start by considering a reference j-ion in an aqueous salt solution. In order to calculate the electric potential at the j-ion arising from all other ions in solution, we need to know the distribution of these ions about the j-ion. Unfortunately this distribution is unknown and so we need a model for this distribution. Further activity coefficients reflect the impact of ions on water-water interactions in aqueous solutions [2] .
Footnotes
[1] (a) R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edition revised,1965. (b) H. S. Harned and B. B. Owen., The Physical Chemistry of Electrolytic Solutions ,Reinhold, New York, 2nd. edn.,1950, chapter The analysis presented by Harned and Owen anticipates application to irreversible processes; e.g. electric conductance of salt solutions. Here we confine attention to equilibrium properties.
[2] H. S. Frank, Z. Phys. Chem., 1965,228,364.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.01%3A_Activity/1.1.10%3A_Activity_Coefficient-_Two_Neutral_Solutes-_Solute__Trace_Solute_i.txt
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A given closed system is prepared using ethyl ethanoate(aq) in an alkaline solution. The composition of the system changes spontaneously as a consequence of chemical reaction. The latter is described by the following chemical equation.
$\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{CH}_{3} \mathrm{COO}^{-}(\mathrm{aq})+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{aq})$
At each stage, the extent of chemical reaction is represented by the symbol $\xi$. The composition of the system varies with time as the reaction proceeds. At any given instant we characterise the rate of chemical reaction by $\mathrm{d} \xi / \mathrm{dt}$. We also ask ‘why did chemical reaction proceed in this direction?’ The answer is − the chemical reaction is driven in that direction by the affinity for spontaneous change, symbol $\mathrm{A}$ [1-3]. The affinity $\mathrm{A}$ for spontaneous chemical reaction is defined by the second law of thermodynamics which states that,
$\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi$
$\text { where } A \, d \xi \geq 0$
1.2.02: Affinity for Spontaneous Reaction- Chemi
For a closed system containing $\mathrm{k}$ − chemical substances, the differential dependence of Gibbs energy on temperature, pressure and chemical composition, is given by the following equation.
$\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}}\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}}$
The condition at constant $n(i \neq j)$ indicates that the amounts of each $i$ chemical substance except chemical substance $j$ is constant. The Gibbs energy of a closed system is a thermodynamic potential function; equation (b).
$\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi$
Here $\mathrm{A}$ is the affinity for spontaneous chemical reaction producing a change in extent of reaction, $\mathrm{d} \xi$, in this case a change in composition. Further the chemical potential of chemical substance $j$,
$\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}$
Comparison of equations (a) and (b) yields equation(d).
$-A \, d \xi=\sum_{j=1}^{j=k} \mu_{j} \, d n_{j}$
The stoichiometry in a chemical reaction for chemical substance $j$, $ν_{j}$ is defined such that $ν_{j}$ is positive for products and negative for reactants; a mnemonic is ‘P for P’.
$v_{j}=d n_{j} / d \xi$
Hence the affinity for spontaneous change,
$A=-\sum_{j=1}^{j=k} v_{j} \, \mu_{j}$
But at equilibrium, the affinity for spontaneous change $\mathrm{A}$ is zero.
$\text { Then, } \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{\mathrm{eq}}=0$
Equation (g) in terms of its simplicity is misleading. Chemists are experts at assaying a system at equilibrium in order to determine the chemical substances present and their amounts. For example, an assay of a given system yields (for defined temperature and pressure) the amounts of un-dissociated acid $\mathrm{CH}_{3}\mathrm{COOH}(\mathrm{aq})$, and the conjugate base $\mathrm{CH}_{3}\mathrm{COO}^{−} (\mathrm{aq})$ and hydrogen ions at equilibrium. We write equation (g) as follows.
$-\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COOH} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COO}^{-} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)=0$
Or, representing a balance of chemical potentials, (a useful approach)
$\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COOH} ; \mathrm{aq}\right)=\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COO}^{-} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)$
The concept of a balance of equilibrium chemical potentials at thermodynamic equilibrium is often the starting point for a description of the properties of closed systems.
1.2.03: Affinity for Spontaneous Chemical Reacti
A given system comprises two phases, I and II, both phases comprising i-chemical substances. We consider the transfer of one mole of chemical substance $j$ from phase I to phase II. The affinity for the transfer is given by equation (a).
$\mathrm{A}_{\mathrm{j}}=\mu_{\mathrm{j}}(\mathrm{I})-\mu_{\mathrm{j}}(\mathrm{II})$
If $\mu_{j}(\mathrm{II})<\mu_{\mathrm{j}}(\mathrm{I})$, $\mathrm{A}_{j}$ is positive and the process is spontaneous. If the system is at fixed $\mathrm{T}$ and pressure, the gradient of Gibbs energy is negative [1].
$\mathrm{A}_{\mathrm{j}}=-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}=\mu_{j}(\mathrm{I})-\mu_{\mathrm{j}}(\mathrm{II})$
We suppose the mole fraction of substance $–j$ in phases I and II are $x_{j}(\mathrm{I})$ and $x_{j}(\mathrm{II})$. We express the chemical potentials as functions of the mole fraction compositions of the two phases.
\begin{aligned} A_{j}=\mu_{j}^{*}(I)+& R \, T \, \ln \left[x_{j}(I) \, f_{j}(I)\right] \ &-\mu_{j}^{*}(I I)-R \, T \, \ln \left[x_{j}(I I) \, f_{j}(I I)\right] \end{aligned}
Here $\mathrm{f}_{j}(\mathrm{I})$ and $\mathrm{f}_{j}(\mathrm{II})$ are rational activity coefficients of substance $j$ in phases $\mathrm{I}$ and $\mathrm{II}$ respectively.
$\text { At all } T \text { and p, both } \operatorname{limit}\left(x_{j}(I) \rightarrow 1\right) f_{j}(I)=1$
$\text { and } \operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \text { (II) } \rightarrow 1\right) \mathrm{f}_{\mathrm{j}}(\text { II })=1$
$\text { By definition } \mu_{j}^{*}(\mathrm{II})-\mu_{\mathrm{j}}^{*}(\mathrm{I})=-\mathrm{R} \, \mathrm{T} \, \ln [\mathrm{K}(\mathrm{T}, \mathrm{p})]$
$\mathrm{K}_{j}(\mathrm{T}, p)$ is a measure of the difference in reference chemical potentials of substance $j$ in phases $\mathrm{I}$ and $\mathrm{II}$. If the two phases are in equilibrium, there is no affinity for substance $j$ to pass spontaneously between the two phases. At equilibrium, $\mathrm{A}_{j}$ is zero. Hence from equations (c) and (f), for the non-equilibrium state,
$A_{j}=R \, T \, \ln \left[K_{j}(T, p)\right]+R \, T \, \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right]$
$\frac{A_{j}}{T}=R \, \ln \left[K_{j}(T, p)\right]+R \, \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right]$
Equation (g) yields the affinity for chemical substance $j$ to pass between the phases in a non-equilibrium state. In applications of equation (h), we describe the dependence of $\left(\mathrm{A}_{j} / \mathrm{T}\right)$ on temperature, pressure and composition of the two phases. In other words we require the general differential of equation (h) which is written in the following form.
\begin{aligned} d\left(\frac{A_{j}}{T}\right)=R \,\left(\frac{\partial \ln K_{j}(T, p)}{\partial T}\right) \, d T \ &+R \,\left(\frac{\partial \ln K_{j}(T, p)}{\partial p}\right) \, d p+R \, d \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right] \end{aligned}
For the transfer process described by $\mathrm{K}_{j}(\mathrm{T}, p)$ we obtain equation (j) where $\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})$ and $\Delta_{\text {trans }} V_{j}^{0}(T, p)$ are the standard enthalpy and volume for transfer for chemical substance $j$.
\begin{aligned} \mathrm{d}\left(\frac{\mathrm{A}_{\mathrm{j}}}{\mathrm{T}}\right)=\left(\frac{\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{T}^{2}}\right) \, \mathrm{dT} \ &-\left(\frac{\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{T}}\right) \, \mathrm{dp}+\mathrm{R} \, \mathrm{d} \ln \left[\frac{\mathrm{x}_{\mathrm{j}}(\mathrm{I}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{I})}{\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})}\right] \end{aligned}
$\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})$ and $\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})$ are properties of pure chemical substance $j$; i.e. are not dependent on the composition of phases $\mathrm{I}$ and $\mathrm{II}$.
Footnotes
[1] Consider the freezing of water;
$\text { water }(\lambda) \rightarrow \text { water }(\mathrm{s})$
For this process, $v\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)=-1 ; v\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{s}\right)=1$
The general rule is — Positive for Products. The affinity for spontaneous change,
$A=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p}=-\sum_{j=1}^{j=i} v_{j} \, \mu_{j}=\mu^{*}\left(H_{2} \mathrm{O} ; \lambda\right)-\mu^{*}\left(H_{2} \mathrm{O} ; \mathrm{s}\right)$
At equilibrium (at fixed $\mathrm{T}$ and $p$), $\mathrm{A} = 0$.
$\text { Then, } \mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)=\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{s}\right)$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.02%3A_Affinity_for_Spontaneous_Chemical_Reaction/1.2.01%3A_Affinity_for_Spontaneous_Chemical_Reacti.txt
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A given closed system is prepared using $\mathrm{n}_{\mathrm{i}}^{0}$ moles of each chemical substance $i$. At extent of chemical reaction $\xi$ the ratio $(\mathrm{A}/\mathrm{T})$ where $\mathrm{A}$ is the affinity for spontaneous chemical reaction is defined by independent variables, $\mathrm{T}$, $p$ and $\xi$.
$(\mathrm{A} / \mathrm{T})=(\mathrm{A} / \mathrm{T})[\mathrm{T}, \mathrm{p}, \xi]$
The general differential of this equation has the following form.[1]
$\mathrm{d}(\mathrm{A} / \mathrm{T})=\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \, \mathrm{dT}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi} \, \mathrm{dp}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi$
Footnote
[1] Equation (b) forms the basis of equations describing the dependence of A on T at fixed p and on p at fixed T.
1.2.05: Affinity for Spontaneous Reaction- Depen
Using the definition of the Gibbs energy $\mathrm{G} [=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}=\mathrm{H}-\mathrm{T} \, \mathrm{S}]$, we form an equation for the entropy of a closed system. Thus $\mathrm{T} \, \mathrm{S}=-\mathrm{G}+\mathrm{H}$. The entropy of the closed system is perturbed by a change in composition/organisation, $\xi$ at fixed $\mathrm{T}$ and $p$.
$\text { Then, } T \,\left(\frac{\partial S}{\partial \xi}\right)_{T, p}=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p}+\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$
$\text { But the affinity for spontaneous reaction, } A=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p}$
A Maxwell equation requires that, $\left(\frac{\partial \mathrm{S}}{\partial \xi}\right)_{\mathrm{T}_{, \mathrm{p}}}=\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}$.
$\text { Hence, } \mathrm{T} \,\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}=\mathrm{A}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$
Equation (c) is rearranged to yield the following interesting equation.
$\mathrm{A}-\mathrm{T} \,\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}=-\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}] The affinity for spontaneous change and its dependence on temperature are simply related to the enthalpy of reaction at fixed $\mathrm{T}$ and $p$. We exploit this link by considering the derivative $\mathrm{d}(\mathrm{A} / \mathrm{T}) / \mathrm{dT}$ (at fixed $p$ and fixed $\xi$). \[\mathrm{d}(\mathrm{A} / \mathrm{T}) / \mathrm{dT}=(1 / \mathrm{T}) \,(\mathrm{dA} / \mathrm{dT})-\mathrm{A} / \mathrm{T}^{2}$
$\text { Hence }\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}=-\frac{1}{\mathrm{~T}^{2}} \,\left[\mathrm{A}-\mathrm{T} \,\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\right]$
$\text { Using equation }(\mathrm{d}),\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}=\frac{1}{\mathrm{~T}^{2}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$
The latter equation is an analogue of the Gibbs-Helmholtz Equation relating the change in Gibbs energy to the enthalpy of reaction, $\left(\frac{\partial H}{\partial \xi}\right)_{T, \mathrm{p}}$. The background to equation (g) is the definition of the dependent variable ($\mathrm{A} / \mathrm{T}$) in terms of independent variables, $\mathrm{T}$, $p$ and $\xi$.
$\text { Thus } \quad(\mathrm{A} / \mathrm{T})=(\mathrm{A} / \mathrm{T})[\mathrm{T}, \mathrm{p}, \xi]$
The general differential of the latter equation has the following form.
$\mathrm{d}(\mathrm{A} / \mathrm{T})=\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \, \mathrm{dT}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi} \, \mathrm{dp}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi$
But, from equation (e)
$\mathrm{d}(\mathrm{A} / \mathrm{T})=-\left(\mathrm{A} / \mathrm{T}^{2}\right) \, \mathrm{dT}+(1 / \mathrm{T}) \, \mathrm{dA}$
$\text { Or, } \mathrm{dA}=\mathrm{T} \, \mathrm{d}(\mathrm{A} / \mathrm{T})+(\mathrm{A} / \mathrm{T}) \, \mathrm{dT}$
We incorporate equation (i) for the term ($\mathrm{A} / \mathrm{T}$). Thus
$\mathrm{dA}=\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}+\frac{\mathrm{A}}{\mathrm{T}}\right] \, \mathrm{dT}+\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi} \, \mathrm{dp}+\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi$
Then using equation (g),
$\mathrm{dA}=\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right] \, \mathrm{dT}-\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{dp}+\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi$
The latter is a general equation for the change in affinity. We rearrange this equation as an equation for a change in extent of reaction.
\begin{aligned} \mathrm{d} \xi=-\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left[\frac{1}{\mathrm{~T}}\right.&\left.\,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right] \, \mathrm{dT}+\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dp} \ &+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA} \end{aligned}
The latter equation has the form of a general differential for the extent of reaction written as,
$\xi=\xi[T, \mathrm{p}, \mathrm{A}]$
$\text { Or, } \mathrm{d} \xi=\left(\frac{\partial \xi}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dT}+\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}} \, \mathrm{dp}+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA}$
Hence from equation (n),
$\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}}=-\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right]$
Equation (q) describes the dependence of extent of reaction on temperature at fixed pressure and affinity for spontaneous reaction. Then from equation (n),
$\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}}=+\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}}$
Equation (r) describes the dependence of extent of reaction at fixed temperature and fixed affinity for spontaneous change.
1.2.06: Affinity for Spontaneous Reaction - Depe
The Gibbs energy of a given closed system is defined by equation (a) where $\xi$ describes the chemical composition.
$\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi]$
We consider the dependence of Gibbs energy on pressure and extent of reaction at fixed temperature $\mathrm{T}$.
$\frac{\partial}{\partial p}\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)=\frac{\partial}{\partial \xi}\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)$
But volume $\mathrm{V}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}$ and affinity $A=-\left(\frac{\partial G}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$.
Volume $\mathrm{V}$ and affinity $\mathrm{A}$ are given by first differentials of the Gibbs energy, $\mathrm{G}$.
$\text { Then }-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}=\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$
Here $\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$ is the volume of reaction, being the increase volume accompanying unit increase in extent of reaction, $\xi$.
1.2.07: Affinity for Spontaneous Reaction - Stab
A given closed system at temperature $T$ and pressure $p$ undergoes a spontaneous change in chemical composition. Chemical reaction is driven by the affinity for spontaneous change such that the Gibbs energy decreases.
Thus
$\mathrm{A}=-\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \label{a}$
The plot of Gibbs energy $G$ against composition $\xi$ shows a gradual decrease until $\mathrm{G}$ reaches a minimum where the affinity $\mathrm{A}$ is zero at chemical equilibrium. An imagined plot beyond equilibrium would show an increase in Gibbs energy. In other words spontaneous chemical reaction stops at the point where $\mathrm{G}$ is a minimum (at fixed $\mathrm{T}$ and $p$). If the chemical reaction stops, the rate of chemical reaction is zero. We link the thermodynamic definition of chemical equilibrium and the definition of chemical equilibrium which emerges from the Law of Mass Action with reference to the kinetics of chemical reaction.
An accompanying plot shows a gradually decreasing affinity when plotted against $\xi$, passing zero at $\xi_{\mathrm{eq}}$. The gradient of the plot in the neighbourhood of equilibrium is negative;
$\text { i.e. }\left(\frac{\partial \mathrm{A}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}<0 \label{b}$
Equation \ref{b} is the thermodynamic condition for a stable chemical equilibrium. The composition does not change no matter how long we wait. Indeed that is the experience of chemists and Equation \ref{b} expresses quantitatively this observation.
One might ask--- how does the system ‘know’ it is at a minimum in Gibbs energy?
Within the system, any fluctuation in composition leads to an increase in Gibbs energy. This tendency is opposed spontaneously; i.e. these fluctuations are opposed.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.02%3A_Affinity_for_Spontaneous_Chemical_Reaction/1.2.04%3A_Affinity_for_Spontaneous_Reaction-_Gener.txt
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The differential change in Gibbs energy of a closed system $\mathrm{dG}$ is related to the change in chemical composition – organisation using equation (a) where $\mathrm{A}$ is the affinity for spontaneous chemical reaction such that $\mathrm{A} \, \mathrm{d} \xi \geq 0$.
$\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi \label{a}$
Spontaneous chemical reaction is driven by the affinity for spontaneous change, $\mathrm{A}$. Eventually the system reaches a minimum in Gibbs energy $\mathrm{G}$ where the affinity for spontaneous change is zero.
$\text { In general terms, } \mathrm{A}^{\mathrm{eq}}=-\sum \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{\mathrm{eq}}=0 \label{b}$
We identify chemical equilibrium as the state where the chemical potentials driving the chemical flow from reactants to products are balanced by the chemical potentials driving chemical flow from products to reactants [1]. The condition given in Equation \ref{b} is based on the first and second laws of thermodynamics [1]. These two laws do not lead to quantitative statements concerning the rate of change of chemical composition; i.e. the dependence of the concentration of reactants and products on time.
At a given time $\mathrm{t}$, the rate of change of composition $v$ is defined by equation (c).
$\mathrm{v}=\mathrm{d} \xi / \mathrm{dt}$
$\text { Hence, with } A \, d \xi>0, A \, V>0$
Therefore for chemical reaction in a closed system, the signs of $\mathrm{A}$ and $\mathrm{v}$ are identical. [1] Moreover if the system is at thermodynamic equilibrium such that $\mathrm{A}$ is zero, then $\mathrm{v}$ is zero. The latter sentence establishes a crucial link between chemical kinetics and chemical thermodynamics. However away from equilibrium we have no information concerning the rate of change of composition. The required property is the ratio $\mathrm{d} \xi / \mathrm{dt}$ at time $\mathrm{t}$ characterising the rate of change of the extent of chemical reaction. Intuitively one might argue that the rate depends on the affinity for spontaneous reaction—the greater the affinity for reaction the faster the reaction. The key equation might take the following form.
$\mathrm{d} \xi / \mathrm{dt}=\mathrm{L} \, \mathrm{A}$
Here $\mathrm{L}$ is a phenomenological parameter, describing the phenomenon of chemical reaction. Unfortunately no further progress can be made because we have no way of measuring the affinity $\mathrm{A}$ for chemical reaction; no affinity meter is available which we can plunge into the reacting system and ‘read off’ the affinity.[2] In these terms the analysis comes to a halt.
In the context of chemical kinetics, the rate of chemical reaction is defined by equation (f) where $n_{j}$ refers to the amount of chemical substance $j$ as either product or reactant; positive for product
$\mathrm{v}=\pm \mathrm{V}^{-1} \, \mathrm{dn} \mathrm{n}_{\mathrm{j}} / \mathrm{dt}$
For chemical reaction involving solutes in dilute solution the volume $\mathrm{V}$ of the system (at fixed $\mathrm{T}$ and $p$) is effectively independent of time.
$\text { Then, } v=\pm \mathrm{dc}_{\mathrm{j}} / \mathrm{dt}$
The kinetics of chemical reactions in solution are simpler than those for reactions in the gas phase [3] and we confine comment to the former.
The relationship between velocity $v$ and the chemical composition of a solution is described by the Law of Mass Action as proposed by Guldberg and Waage in 1867 [4]. The developments reported by Harcourt and Essen (1867), Bredig and Stern and by Lapworth in 1904 and by Goldschmidt in 1930 were important. Hammett [3] comments that by the 1930s the subject had emerged from the ‘dark ages’. Hammett draws attention to the contributions made by Bartlett, Ingold and Pedersen in the decades of 1920 and 1930. Effectively these authors showed that the rate of chemical reaction at time $t$ is a function of the concentrations of substances in the systems at that time, $t$. In the textbook case the spontaneous chemical reaction [5,6] between two chemical substances $\mathrm{X}$ and $\mathrm{Y}$ at fixed $\mathrm{T}$ and $p$ in aqueous solution, has the following form.
$\mathrm{x} . \mathrm{X}(\mathrm{aq})+\mathrm{y} . \mathrm{Y}(\mathrm{aq})->\text { products }$
$\text { Rate of reaction }=\mathrm{k} \,[\mathrm{X}]^{\alpha} \,[\mathrm{Y}]^{\beta}$
Here $\alpha$ and $\beta$ are orders of reaction with respect to substances $\mathrm{X}$ and $\mathrm{Y}$. These orders have to be determined from the experimental kinetic data because they do not necessarily correspond to the stoichiometric coefficients in the chemical equation [7].
We develop the argument by considering a chemical reaction in solution. An aqueous solution is prepared containing n1 moles of water(l) and $\mathrm{n}_{\mathrm{X}}^{0}$ moles of chemical substance $\mathrm{X}$ at time, $t = 0$ where $n_{X}^{0}>>n_{1}$. Spontaneous chemical reaction leads to the formation of product $\mathrm{Y}$, where at ‘$t=0$’, $\mathrm{n}_{\mathrm{Y}}^{0}$ is zero. Chemical reaction is described using equation (j).
$\begin{array}{llll} & \mathrm{X}(\mathrm{aq}) & \rightarrow & \mathrm{Y}(\mathrm{aq}) \ \mathrm{At} \mathrm{t}=0 & \mathrm{n}_{\mathrm{X}}^{0} & \mathrm{n}_{\mathrm{y}}^{0}=0 & \mathrm{~mol} \end{array}$
The convention is for the chemical reaction to be written in the form ‘reactants → products’, such that the affinity for reaction $\mathrm{A}$ is positive and hence $\mathrm{d}\xi$ and $\mathrm{d} \xi / \mathrm{dt}$ are positive. Many chemical reactions of the form shown in equation (j) go to completion.
$\text { Thus } \lim \mathrm{it}(\mathrm{t} \rightarrow \infty) \mathrm{n}_{\mathrm{X}}=0 ; \mathrm{n}_{\mathrm{Y}}=\mathrm{n}_{\mathrm{X}}^{0}$
The minimum in Gibbs energy (where $\mathrm{A} = 0$) is attained when all reactant has been consumed.
First Order Reactions
A given closed system contains $\mathrm{n}_{\mathrm{X}}^{0}$ moles of chemical substance $\mathrm{X}$ which $\mathrm{X}$ decomposes to form chemical substance $\mathrm{Z}$. At time $t$, $\xi$ moles of reactant $\mathrm{X}$ have formed product $\mathrm{Z}$.
$\begin{array}{lcc} \multicolumn{1}{c}{\text { Chemical Reaction }} & \mathrm{X} & \mathrm{Z} \ \text { Amounts at } \mathrm{t}=0 ; & \mathrm{n}_{\mathrm{X}}^{0} & 0 \mathrm{~mol} \ \text { Concentrations }(\mathrm{t}=0) & \mathrm{n}_{\mathrm{X}}^{0} / \mathrm{V} & 0 \mathrm{~mol} \mathrm{~m}^{-3} \ \text { At time } \mathrm{t} & \mathrm{n}_{\mathrm{X}}^{0}-\xi & \mathrm{mol} \ \text { Amounts } & \left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V} & \xi / \mathrm{V} \mathrm{mol} \mathrm{m} \end{array}$
The law of mass action is the extra-thermodynamic assumption which relates the rate of change of concentration t o the composition of the system.
$\text {Then, } -\frac{\mathrm{d}\left[\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V}\right]}{\mathrm{dt}}=\frac{\mathrm{d}[\xi / \mathrm{V}]}{\mathrm{dt}}=\mathrm{k} \, \frac{\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right]}{\mathrm{V}}$
The constant of proportionality, rate constant $\mathrm{k}$, in this case has units of $\mathrm{s}^{-1}$.
$\text { Then in terms of reactant } \mathrm{X}, \frac{\mathrm{d} \xi}{\mathrm{dt}}=\mathrm{k} \,\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right]$
Chemical reaction proceeds leading to a decrease in the Gibbs energy of the system until $\mathrm{n}_{\mathrm{X}}^{0}=\xi$ such that all reactant has been consumed. At this point $(\mathrm{d} \xi / \mathrm{dt})$ is zero, the system is at chemical equilibrium and the Gibbs energy is a minimum. Further from equation (m),
$\int_{\xi=0}^{\xi} \frac{d \xi}{\left(n_{X}^{0}-\xi\right)}=\int_{t=0}^{t} k \, d t$
The Law of Mass Action is the most important extra-thermodynamic equation in chemistry.
Second Order Reactions
A given closed system contains $\mathrm{n}_{\mathrm{X}}^{0}$ and $\mathrm{n}_{\mathrm{Y}}^{0}$ moles of chemical substances $\mathrm{X}$ and $\mathrm{Y}$ respectively at fixed $\mathrm{T}$ and $p$. Spontaneous chemical reaction produces chemical substance $\mathrm{Z}$. At time $t$, $\xi$ moles of product $\mathrm{Z}$ are formed from chemical substances $\mathrm{X}$ and $\mathrm{Y}$.
$\begin{array}{lccc} {c}{\text { Chemical Reaction }} & \mathrm{X} & +\mathrm{Y} & \mathrm{Z} \ \text { Amounts at } \mathrm{t}=0 ; & \mathrm{n}_{\mathrm{X}}^{0} & \mathrm{n}_{\mathrm{Y}}^{0} & 0 \mathrm{~mol} \ \text { Concentrations }(\mathrm{t}=0) & \mathrm{n}_{\mathrm{X}}^{0} / \mathrm{V} & \mathrm{n}_{\mathrm{Y}}^{0} / \mathrm{V} & 0 \mathrm{~mol} \mathrm{~m} \ \text { At time t } & & & \ \text { Amounts } & \mathrm{n}_{\mathrm{X}}^{0}-\xi & \mathrm{n}_{\mathrm{Y}}^{0}-\xi & \xi \mathrm{mol} \ \text { Concentrations } & \left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V} & \left(\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right) / \mathrm{V} & \xi / \mathrm{V} \mathrm{mol} \mathrm{m} \end{array}$
The Law of Mass Action is the extra-thermodynamic assumption, relating the rate of change of concentration to the composition of the system. Then,
$-\frac{\mathrm{d}\left[\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V}\right]}{\mathrm{dt}}=-\frac{\mathrm{d}\left[\left(\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right) / \mathrm{V}\right]}{\mathrm{dt}}=\frac{\mathrm{d}[\xi / \mathrm{V}]}{\mathrm{dt}}=\mathrm{k} \, \frac{\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right]}{\mathrm{V}} \, \frac{\left[\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right]}{\mathrm{V}}$
The unit of rate constant $\mathrm{k}$ is ‘$\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~s}^{-1}$’.
$\text { Then, } \frac{\mathrm{d} \xi}{\mathrm{dt}}=\mathrm{k} \, \mathrm{V}^{-1} \,\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right] \,\left[\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right]$
$\text { Or, } \quad \int_{\xi=0}^{\xi} \frac{\mathrm{d} \xi}{\left[n_{X}^{0}-\xi\right] \,\left[n_{Y}^{0}-\xi\right]}=\int_{t=0}^{t} k \, V^{-1} \, d t$
In applications of equation (q), rate constant $\mathrm{k}$ and volume $\mathrm{V}$ are usually treated as independent of time.[8]
The foregoing analysis of kinetics of chemical reactions illustrates the application of the variable $\xi$ in describing the composition of a closed system. Most accounts of chemical kinetics start out with a consideration of concentrations of chemical substances in a given system. [9,10]
Nevertheless for each and every chemical reaction, the form of the relevant ‘Law of Mass Action’ has to be determined from the observed dependence of composition on time. The latter sentence does not do justice to the skills of chemists in this context.
Reaction to Chemical Equilibrium
In the previous section we considered those cases where chemical reaction goes to completion in that one or more of the reactants are consumed. For many cases this is not the case. Here we imagine that a dilute solution has been prepared using $\mathrm{n}_{\mathrm{X}}^{0}$ and $\mathrm{n}_{\mathrm{Y}}^{0}$ moles of solute reactants $\mathrm{X}$ and $\mathrm{Y}$. Chemical reaction at fixed $\mathrm{T}$ and $p$ proceeds spontaneously. The Gibbs energy of the system decreases reaching a minimum where the affinity for spontaneous reaction is zero. Chemical analysis shows that the resulting system contains product $\mathrm{Z}$ together with reactants $\mathrm{X}$ and $\mathrm{Y}$, and that the chemical composition is independent of time; i.e. chemical kinetic equilibrium. Thus,
$\begin{gathered} \mathrm{X}(\mathrm{aq}) \quad+\quad \mathrm{Y}(\mathrm{aq}) \quad \Leftrightarrow \quad \mathrm{Z}(\mathrm{aq}) \ \mathrm{At} \mathrm{} \mathrm{t}=0 \quad \mathrm{n}_{\mathrm{X}}^{0} \quad \mathrm{n}_{\mathrm{Y}}^{0} \quad \mathrm{n}_{\mathrm{Y}}^{\mathrm{e}}=\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}} \quad \mathrm{n}_{\mathrm{eq}}^{\mathrm{eq}}=\xi^{\mathrm{eq}} \quad \mathrm{mol} \ \mathrm{At} \mathrm{} \mathrm{t}=\infty \quad \mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}} \quad \mathrm{n}_{\mathrm{eq}} \ \text { or, } \quad \mathrm{c}_{\mathrm{X}}^{\mathrm{eq}}=\left(\frac{\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \quad \mathrm{c}_{\mathrm{eq}}^{\mathrm{eq}}=\left(\frac{\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \quad \mathrm{c}_{\mathrm{Z}}^{\mathrm{eq}}=\left(\frac{\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \end{gathered}$
A number of assumptions are based on the Law of Mass Action. $\text { At time } t \text {, rate of forward reaction }=k_{f} \, c_{X}(t) \, c_{Y}(t) \text { and rate of the reverse reaction }=\mathrm{k}_{\mathrm{r}} \, \mathrm{c}_{\mathrm{Z}}(\mathrm{t})$
Rate constants $\mathrm{k}_{\mathrm{f}}$ and $\mathrm{k}_{\mathrm{r}}$ are initially assumed to be independent of the extent of reaction. A key conclusion is now drawn. Because at ‘$t \rightarrow$ infinity’, the properties of the system are independent of time, the system is ‘at chemical equilibrium where the rates of forward and reverse reactions are balanced.
$\text { Then, } \mathrm{k}_{\mathrm{f}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{X}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{Y}}^{\mathrm{eq}}=\mathrm{k}_{\mathrm{r}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{z}}^{\mathrm{eq}}$
$\text { Hence, } \quad \mathrm{d} \xi^{\mathrm{eq}} / \mathrm{dt}=0$
$\text { and } \mathrm{k}_{\mathrm{eq}}^{\mathrm{eq}} \,\left(\frac{\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \,\left(\frac{\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right)=\mathrm{k}_{\mathrm{r}}^{\mathrm{eq}} \, \frac{\xi^{\mathrm{eq}}}{\mathrm{V}}$
At this point we encounter a key problem -- we cannot determine $\mathrm{k}_{\mathrm{f}}^{\mathrm{eq}$ and $\mathrm{k}_{\mathrm{r}}^{\mathrm{eq}$ because at equilibrium the composition of the system is independent of time. Nevertheless we can express the ratio of rate constants as a function of the composition at equilibrium.
$\left(\mathrm{k}_{\mathrm{f}}^{\mathrm{eq}} / \mathrm{k}_{\mathrm{r}}^{\mathrm{eq}}\right) \,\left(\frac{\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \,\left(\frac{\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right)=\frac{\xi^{\mathrm{eq}}}{\mathrm{V}}$
The ratio $\mathbf{k}_{\mathrm{f}}^{\mathrm{eq}} / \mathbf{k}_{\mathrm{r}}^{\mathrm{eq}}$ is characteristic of the system (at defined $\mathrm{T}$ and $p$), defining what we might call a ‘Law of Mass Action equilibrium constant’, $\mathrm{K}(\operatorname{lma})$.
$\text { Thus } \mathrm{K}(\operatorname{lm} a)=\mathrm{k}_{\mathrm{f}}^{\mathrm{eq}} / \mathrm{k}_{\mathrm{r}}^{\mathrm{eq}}$
In other words, the property $\mathrm{K}(\operatorname{lma})$ is based on a balance of reaction rates whereas the thermodynamic equilibrium constant is based on a balance of chemical potentials. With reference to equation (v), at fixed $\mathrm{T}$ and $p$, the thermodynamic condition is given in equation (w).
$\mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq})+\mu_{\mathrm{Y}}^{\mathrm{eq}}(\mathrm{aq})=\mu_{\mathrm{Z}}^{\mathrm{eq}}(\mathrm{aq})$
$\text { For } \mathrm{p} \cong \mathrm{p}^{0}, \mathrm{~K}^{0}(\mathrm{~T})=\frac{\left(\mathrm{m}_{\mathrm{Z}} \, \gamma_{\mathrm{Z}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}}{\left(\mathrm{m}_{\mathrm{X}} \, \gamma_{\mathrm{X}} / \mathrm{m}^{0}\right)^{\mathrm{eq}} \,\left(\mathrm{m}_{\mathrm{Y}} \, \gamma_{\mathrm{Y}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}}$
Therefore the question is raised--- how is $\mathrm{K}(\operatorname{lma})$ related to $\mathrm{K}^{0}(\mathrm{T})$? In the absence of further information, a leap of faith by chemists sets $\mathrm{K}(\operatorname{lma})$ equal to $\mathrm{K}^{0}(\mathrm{T})$. We avoid debating the meaning of the phrase ‘rate constants at equilibrium’ [7].
Energy of Activation
Spontaneous chemical reaction involving a single solute $\mathrm{X}(\mathrm{aq}$) can be described using a rate constant $\mathrm{k}$ for a solution at fixed $\mathrm{T}$ and $p$ where the concentration of solute $\mathrm{X}$ at time $t$ is $\mathrm{c}_{\mathrm{X}}(\mathrm{aq})$.
If we can assume that in solution there are no solute-solute interactions, rate constant $\mathrm{k}$ is not dependent on the composition of the solution. In thermodynamic terms, the thermodynamic properties of solute $\mathrm{X}$ are ideal; for such a system rate constant $\mathrm{k}$ is independent of time and initial concentration of solute $\mathrm{X}$. In other words experiment yields the property $\mathrm{k}(\mathrm{T}, p)$ for a given solution indicating that the rate constant is a function of temperature and pressure. For nearly all chemical reactions in solution rate constants increase with increase in temperature, a dependence described by the Arrhenius equation. [11]
$\mathrm{k}=\mathrm{A} \, \exp \left(-\mathrm{E}_{\mathrm{A}} / \mathrm{R} \, \mathrm{T}\right)$
Then rate constant, $\mathrm{k}$ increases with increase in temperature. The idea emerges that spontaneous conversion of reactant to products is inhibited by an ‘energy’ barrier. Further in the $\operatorname{limit}(\mathrm{T} \rightarrow \infty) \ln (\mathrm{k})=\mathrm{A}$, the pre-exponential factor which has the same units as rate constant $\mathrm{k}$.
The assumption in the foregoing comments is that $\mathrm{E}_{\mathrm{A}}$ is independent of temperature such that for example a first order rate constant $\ln \left(\mathrm{k} / \mathrm{s}^{-1}\right)$ is a linear function of $\mathrm{T}^{-1}$.
$\text { Thus, } \quad \ln \left(\mathrm{k} / \mathrm{s}^{-1}\right)=\ln \left(\mathrm{A} / \mathrm{s}^{-1}\right)-\left(\mathrm{E}_{\mathrm{A}} / \mathrm{R} \, \mathrm{T}\right)$
The latter pattern is generally observed but there are many well-documented cases where the plot is not linear. In other words it is incorrect to conclude that equation (y) somehow predicts how rate constants depend on temperature. There is no substitute for actually measuring this dependence.
Transition State Theory
The law of mass action and the concept of an activation energy for a given chemical reaction are extrathermodynamic. This conclusion is unfortunate, implying that the treatment of kinetic data for reactions in solution is completely divorced from the thermodynamic treatment of the properties of solution. One can understand therefore why Transition Sate Theory (TST) attracts so much interest.[12,13] At the very least, this theory offers analysis of kinetic data a patina of thermodynamic respectability. We describe TST with respect to a chemical reaction where the dependence of composition on time is described using a first order rate constant.
For chemical reactions in the gas phase, statistical thermodynamics offers a reasonably straightforward approach to the description of both reactants and a transition state in which one vibrational mode for the transition state is transposed into translation along the reaction co-ordinate. The theory was re-expressed in terms of equations which could be directly related to the thermodynamics of the process of reaction in solutions.
Chemical reaction proceeds from reactant $\mathrm{X}(\mathrm{aq})$ to products through a transition state $\mathrm{X}^{\neq}(\mathrm{aq})$. As the reaction proceeds the amount of solute $\mathrm{X}$, $n_{\mathrm{X}}(\mathrm{aq})$ decreases but at all times reactant $\mathrm{X}(\mathrm{aq})$ and transition state $\mathrm{X}^{\neq}(\mathrm{aq})$ are in chemical equilibrium.
$\text { Thus } \mathrm{X}(\mathrm{aq}) \Leftrightarrow=\mathrm{X}^{7} \rightarrow \text { products }$
The condition ‘chemical equilibrium’ is quantitatively expressed in terms of chemical potentials.
$\mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\neq}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$
Conventionally where kinetics of reactions in solution are addressed, the composition of solutions is expressed in terms of concentrations using the unit, $\mathrm{mol dm}^{-3}$. Then equation (zc) is formed assuming that ambient pressure is close to the standard pressure, $p^{0}$; $\mathrm{c}_{\mathrm{r}} =1 \mathrm{~mol dm}^{-3}$. Hence,
$\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{X}} \, \mathrm{y}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right)^{\mathrm{eq}}=\mu_{\neq}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\neq} \, \mathrm{y}_{\neq} / \mathrm{c}_{\mathrm{r}}\right)^{\mathrm{eq}}$
The standard Gibbs energy of activation $\Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})$ is given by equation (zd) leading to the definition of an equilibrium constant ${ }^{*} \mathrm{~K}^{0}(\mathrm{aq} ; \mathrm{T}) ; \text { with } \mathrm{p} \approx \mathrm{p}^{0}$.
$\Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=\mu_{\neq}^{0}(\mathrm{aq} ; \mathrm{T})-\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{T})$
$\text { Then } \quad \Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\neq} \, \mathrm{y}_{\neq} / \mathrm{c}_{\mathrm{X}} \, \mathrm{y}_{\mathrm{X}}\right)^{\mathrm{eq}}$
$\text { By definition, } \Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=-\mathrm{R} \, \mathrm{T} \, \ln \left[{ }^{\neq} \mathrm{K}^{0}(\mathrm{aq} ; \mathrm{T})\right]$
$\text { Therefore, } \mathrm{c}_{\neq}={ }^{\neq} \mathrm{K}^{0}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{c}_{\mathrm{X}} \, \mathrm{y}_{\mathrm{X}} / \mathrm{y}_{\neq}$
Through the course of chemical reaction, as the concentration of reactant $\mathrm{X}(\mathrm{aq})$ decreases, the condition given in equation (zb) holds. Chemical reaction is not instantaneous because $\mu_{z}^{0}(\mathrm{aq} ; \mathrm{T})>\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{T})$; a barrier exists to chemical reaction. Consequently the concentration $\mathrm{c}_{\neq}$ is small. The analysis up to equation (zf) is based on a thermodynamic description of equilibrium between reactant and transition states. In the limit that the solution is very dilute, $y_{X}=y_{\neq}=1$ at all time $t$, $\mathrm{T}$ and $p$.
$\text { Thus at given } T \text { and } p,[14] \quad k=\frac{k_{B} \, T}{h} \, \exp \left(\frac{-\Delta^{\neq} G^{0}}{R \, T}\right)$
$\text { Then, } \mathrm{k}=\frac{\mathrm{k}_{\mathrm{B}} \, \mathrm{T}}{\mathrm{h}} \, \exp \left(-\frac{\Delta^{\neq} \mathrm{H}^{0}}{\mathrm{R} \, \mathrm{T}}+\frac{\Delta^{\neq} \mathrm{S}^{0}}{\mathrm{R}}\right)$
$\Delta^{\neq} \mathrm{G}^{0}$ is the standard Gibbs energy of activation defined in terms of reference chemical potentials of transition state and reactants. Here $\mathrm{k}_{\mathrm{B}}$ is the Boltzmann constant and $\mathrm{h}$ is Planck’s constant [15].
$\Delta^{\neq} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\frac{\mathrm{h} \, \mathrm{k}(\mathrm{T})}{\mathrm{k}_{\mathrm{B}} \, \mathrm{T}}\right)$
At temperature $\mathrm{T}$, $\Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})$ is re-expressed in terms of standard enthalpy and standard entropy of activation [16].
$\text { Then, } \Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=\Delta^{\neq} \mathrm{H}^{0}(\mathrm{aq} ; \mathrm{T})-\mathrm{T} \, \Delta^{\neq} \mathrm{S}^{0}(\mathrm{aq} ; \mathrm{T})$
The standard isobaric heat capacity of activation [17,18],
$\Delta^{\neq} \mathrm{C}_{\mathrm{p}}^{0}(\mathrm{aq})=\left(\frac{\partial \Delta^{\neq} \mathrm{H}^{0}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}}$
In summary transition state theory allows kinetic data to be analysed using the protocols and equations of thermodynamics.
Analysis of the dependence of rate constants at fixed temperature as a function of pressure yields standard volumes of activation $\Delta^{\neq} \mathrm{V}^{0}(\mathrm{aq} ; \mathrm{T})$. In further exercises this volumetric parameter is measured as functions of pressure at temperature $\mathrm{T}$ and of temperature at fixed pressure [20].
Nevertheless a word of caution is in order. Johnston [21] points out that diagrams describing the progress of chemical reaction through several intermediates are often misleading. Such diagrams should be based on reference chemical potentials otherwise misleading conclusions can be drawn; see also [22 - 28].
Footnotes
[1] I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London , 1953.
[2] In fact in the treatment of data obtained using fast reaction techniques (e.g. temperature jump and pressure-jump) where the displacement from equilibrium is small, it is assumed that the rate of response is linearly related to the affinity for chemical reaction. (a) E. F. Caldin, Fast Reactions in Solution, Blackwell, Oxford, 1964. (b) M. J. Blandamer, Introduction to Chemical Ultrasonics, Academic Press, London, 1973.
[3] L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 1970, 2nd. Edition.
[4] K. J. Laidler, [The World of Physical Chemistry, Oxford University Press, Oxford, 1993,page 115] comments
‘Neither did they ( i.e. Guldberg and Waage] make any contributions to kinetics since they worked in terms of forces and not of rates, although they did tentatively suggest that rates might be proportional to forces’.
[5] The condition ‘spontaneous change‘ signals a ‘natural’ direction but that does mean that the process is instantaneous. The properties of the system are dependent on time. We make this point to counter such statements as ‘Thermodynamics … deals exclusively with system showing no temporal change; reacting systems are outside its province’; cf. E. A. Moelwyn-Hughes, Kinetics of Reactions in Solution, Clarendon Press, Oxford, 1947, page 162.
[6] Thermodynamics in the form of the first and second laws offers no way forward. Intuitively one might argue that the rate of change of composition would be directly related to the affinity for spontaneous reaction, $\mathrm{A}$ where $\pi$ is a proportionality factor characteristic of the system, temperature and pressure.
$\text { Thus, } \mathrm{d} \xi / \mathrm{dt}=\pi \, \mathrm{A} \quad \text { where } \quad \lim \mathrm{it}(\mathrm{A} \rightarrow 0) \mathrm{d} \xi / \mathrm{dt}=0$
In fact we might draw an analogy with Ohm’s law whereby electric current $\mathrm{i}$ (= rate of flow of charge) is proportional to the electric potential gradient, $\Delta \phi$, the constant of proportionality being the electrical conductivity, $\mathrm{L}$; $\mathrm{i}=\mathrm{L} \, \Delta \phi$ where $\mathrm{L}$ is characteristic of the system, temperature and pressure. Indeed such a kinetic force-flow link might be envisaged. Indeed an link emerges with the phenomenon of electric potential driving an electric current through an electrical circuit.
Electric current = rate of flow of charge, unit = Ampere.
Electric potential has the unit, volt.
The product, $\mathrm{I}. \(\mathrm{V}=[\mathrm{A}] \,\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]=[\mathrm{W}]$
Here $\mathrm{W}$ is the SI symbol for the unit of power, watt.
Rate of chemical reaction $\mathrm{d} \xi / \mathrm{dt}=\left[\mathrm{mol} \mathrm{s} \mathrm{s}^{-1}\right]$
Affinity $\mathrm{A}=\left[\mathrm{J} \mathrm{mol}^{-1}\right]$
Then $\mathrm{A} \, \mathrm{d} \xi / \mathrm{dt}=\left[\mathrm{J} \mathrm{mol} \mathrm{m}^{-1}\right] \,\left[\mathrm{mol} \mathrm{s}^{-1}\right]=[\mathrm{W}]$
Interestingly chemists rarely refer to the 'wattage' of a chemical reaction.
[7] One cannot help be concerned with accounts which describe chemical equilibrium in terms of rates of chemical reaction. As we understand the argument runs along the following lines. For a chemical equilibrium having the following stochiometry, $\mathrm{X}+\mathrm{Y} \Leftrightarrow\mathrm{Z}$ at equilibrium the rates of forward and reverse reactions are balanced.
$\mathrm{k}_{\mathrm{f}} \,[\mathrm{X}] \,[\mathrm{Y}]=\mathrm{k}_{\mathrm{r}} \,[\mathrm{Z}]$
Here $\mathrm{k}_{\mathrm{f}}$ and $\mathrm{k}_{\mathrm{f}}$ are the forward and reverse rate constants.
$\text { Then } \mathrm{K}=\mathrm{k}_{\mathrm{f}} / \mathrm{k}_{\mathrm{r}}=[\mathrm{Z}] /[\mathrm{X}] \,[\mathrm{Y}]$
The argument loses some of its force if one turns to accounts dealing with chemical kinetics when questions of order and molecularity are raised. In any case one cannot measure rates of chemical reactions 'at equilibrium' because at equilibrium 'nothing is happening'. Even in those cases where the rates of chemical reactions 'at equilibrium ' are apparently measured the techniques rely on following the return to equilibrium when the system is perturbed.
[8] In nearly all applications of the law of mass action to chemical reactions in solution a derived rate constant is based on on a description of the composition on time at fixed T and p. Therefore the calculated rate constant is an isobaric-isothermal property of the system. Nevertheless the volume is usually taken as independent of time. Certainly in most applications in solution chemistry the solutions are quite dilute and so throughout the course of the reaction the volume is effectively constant. An interesting point now emerges in that a given rate constant is an isothermal-isobaric-isochoric property.
[9] K. J. Laidler and N. Kallay, Kem. Ind.,1988,37,183.
[10] M. J. Blandamer, Educ. Chemistry, 1999,36,78.
[11] $\mathrm{E}_{\mathrm{A}}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \text { such that } \mathrm{E}_{\mathrm{A}} / \mathrm{R} \, \mathrm{T}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] /\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=[1]$
[12] The account given here stresses the link with classical thermodynamics. The key equations should be developed using statistical thermodynamics; see H. Eyring, J. Chem. Phys., 1935, 3,107.
[13] S. Glasstone, K. J. Laidler and H. Eyring , The Theory of Rate Processes, McGraw-Hill, New York, 1941.
[14] $\frac{\mathrm{K}_{\mathrm{B}} \, \mathrm{T}}{\mathrm{h}}=\frac{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}{[\mathrm{J} \mathrm{s}]}=\left[\mathrm{s}^{-1}\right]$
[15] $\frac{\mathrm{k}_{\mathrm{B}} \, \mathrm{T}}{\mathrm{h}} \, \mathrm{c}_{\neq}=\left[\mathrm{s}^{-1}\right] \,\left[\mathrm{mol} \mathrm{dm}{ }^{-3}\right]=\left[\mathrm{mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}\right]$
[16]
1. K. J. Laidler, J. Chem. Educ.,1984,61,497.
2. S. R. Logan, J.Chem.Educ.,1982,59,278.
3. H. Maskill, Educ. Chem.,1985,22,154.
4. K. J. Laidler, J. Chem.Educ.,1988,65,540.
5. F. R. Cruikshank, A. J. Hyde and D. Pugh, J.Chem.Educ.,1977,54,288.
[17] R. E. Robertson, Prog. Phys. Org. Chem.,1967, 4, 213.
[18] M. J. Blandamer, J. M. W. Scott and R. E. Robertson, Prog. Phys. Org. Chem., 1985,15,149.
[19] M. J. Blandamer, Chemical Equilibria in Solution, Ellis Horwood, PTR Prentice Hall, New York,, 1992.
[20] H. S. Johnston, Gas Phase Reaction Rate Theory, Ronald Press, New York, 1966.
[21] R. B. Snadden, J.Chem.Educ.,1985,62,653.
[22] M. I. Page, Educ. Chem.,1981,18,52.
[23] R. D. Levine, J.Phys.Chem.,1979,83,159.
[24] I. H. Williams, Chem. Soc. Rev.,1993, 22,277.
[25] A. Williams, Chem. Soc. Rev.,1994,23,93.
[26] A. Drljaca, C. D. Hubbard, R. van Eldik, T. Asano, M. V. Basilevsky and W. J. Le Noble, Chem. Rev.,1998,98,2167.
[27] R. K. Boyd, Chem. Rev.,1977,77,93.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.02%3A_Affinity_for_Spontaneous_Chemical_Reaction/1.2.08%3A_Affinity_for_Spontaneous_Chemical_Reacti.txt
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According to the First and Second Laws of thermodynamics, the change in Helmholtz energy $\mathrm{dF}$ accompanying chemical reaction, change in volume and change in temperature is given by Equation \ref{a}.
$\mathrm{dF}=-\mathrm{S} \, \mathrm{dT}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi \label{a}$
where,
$\mathrm{A} \, \mathrm{d} \xi \geq 0$
At constant $\mathrm{T}$ and $\mathrm{V}$:
$\mathrm{dF}=-\mathrm{A} \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq 0$
If we monitor the change in composition of a closed system held at constant temperature and constant volume, equilibrium corresponds to a minimum in Helmholtz energy. In practice chemists concerned with spontaneous chemical reaction in solutions held at constant temperature, do not constrain the system to a constant volume. Rather they constrain the system to constant temperature and pressure. Therefore the following equation is the key.
$\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi$
$\text { At constant } \mathrm{T} \text { and } \mathrm{p}, \mathrm{dG}=-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0$
The rate at which chemical reaction drives the system to a minimum in Gibbs energy is described by the Law of Mass Action. Presumably a similar law would describe the approach to equilibrium of a system held at constant $\mathrm{T}$ and $\mathrm{V}$, the system moving spontaneously towards a lower Helmholtz energy. We do not explore this point. Nevertheless the isochoric condition is often invoked and we examine how this comes about.
(a) Chemical Kinetics
A key problem revolves around the role of the solvent in determining activation parameters; e.g. standard Gibbs energy of activation and standard enthalpy of activation. An extensive literature examines the role of solvents. But densities of solvents are a function of temperature and pressure. Then in attempting to understand the factors controlling kinetic activation parameters there is the problem that intermolecular distances (e.g. solvent-solvent and solvent-solute) are a function of temperature. In 1935 Evans and Polanyi [1] suggested that isochoric activation parameters for chemical reaction in aqueous solution might be more mechanistically informative than conventional isobaric activation parameters; i.e. $[\partial \ln (\mathrm{k}) / \partial \mathrm{T}]_{\mathrm{V}}$ rather than $[\partial \ln (\mathrm{k}) / \partial \mathrm{T}]_{\mathrm{p}}$ where $\mathrm{k}$ is the rate constant for spontaneous chemical reaction [2,3].
With reference to chemical reactions in dilute aqueous solution the isochoric standard internal energy of activation $\Delta^{\neq} \mathrm{U}_{\mathrm{V}}^{0}$ is related to the isobaric standard enthalpy of activation $\Delta^{\neq} \mathrm{H}_{\mathrm{p}}^{0}$ at temperature $\mathrm{T}$ and the standard volume of activation $\Delta^{\neq} V^{0}$ using equation (f) where $\alpha_{p 1}^{*}$ and $\kappa_{\mathrm{T} 1}^{*}$ are respectively the isobaric expansibilities and isothermal compressibilities of water.
$\Delta^{\neq} \mathrm{U}_{\mathrm{V}}^{0}=\Delta^{\neq} \mathrm{H}_{\mathrm{p}}^{0}-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*} / \kappa_{\mathrm{T} 1}^{*}\right] \, \Delta^{\neq} \mathrm{V}^{0}$
Baliga and Whalley [4] noted that the dependence on solvent mixture composition of $\Delta^{z} \mathrm{U}_{\mathrm{V}}^{0}$ is less complicated than that for $\Delta^{\neq} \mathrm{H}_{\mathrm{p}}^{0}$ for solvolysis of benzyl chloride in ethanol + water mixtures at $298.15 \mathrm{~K}$. A similar pattern was reported by Baliga and Whalley [5] for the hydrolysis of 2-chloro-2-methyl propane in the same mixture at $273.15 \mathrm{~K}$. [6]
The proposal concerning isochoric activation parameters sparked enormous interest and debate.[7-13] Much of the debate centred on the isochoric condition and the answer to the question ‘what volume is held constant?’ With reference to equation (f), $\alpha_{p l}^{*}$ and $\kappa_{\mathrm{T1}}^{*}$ depend on temperature. Then the volume identified by subscript $\mathrm{V}$ on $\Delta^{\neq} \mathrm{U}_{\mathrm{V}}^{0}$ is dependent on temperature. Further the molar volumes of binary liquid mixtures depend on $\mathrm{T}$ and $p$. In other words the isochoric condition is not global across the given data set. Haak et al. noted [14] that either side of the TMD of water there are pairs of temperatures where the molar volumes of water at ambient pressure are equal. Hence rates of reaction for chemical reactions in dilute aqueous solution at such temperatures would yield pairs of isochoric rate constants. Kinetic data for spontaneous hydrolysis of 1–benzoyl–1,2,4 triazole in aqueous solution at closely spaced temperatures close to the TMD of water reveal no unique features associated with the isochoric condition.[15]
The isochoric condition nevertheless remains interesting. There is however an important point to note. For the most part kinetic experiments investigate the rates of chemical reactions in solution at constant $\mathrm{T}$ and $p$. In other words spontaneous change is driven by the decrease in Gibbs energy. The latter is the operating thermodynamic potential function, both $\mathrm{T}$ and $p$ being held constant; the rate constant is an isobaric-isothermal property. Further the dependence of rate constant is often monitored on temperature at constant pressure, recognising that the volume of the system changes as the temperature is altered.
In principle spontaneous chemical reaction could be monitored, holding the system at constant $\mathrm{T}$ and volume $\mathrm{V}$. Here the direction of spontaneous chemical reaction would be towards a minimum in Helmholtz energy, $\mathrm{F}$. The dependence of the isochoric – isothermal rate constant on temperature could again in principle be measured. The technological challenge would be immense because one might expect enormous pressures to be required to hold the volume constant when the temperature was increased.
Similar concerns over the definition of isochoric emerge in the context of the dependence on $\mathrm{T}$ and $p$ of acid dissociation constants in aqueous solution of ethanoic acid [16].
(a) Electrical Mobilities
The migration of ions through a salt solution under the influence of an applied electric potential can be envisaged as a rate process, analogous to the rate of chemical reaction. In most studies the applied electric field is weak such that the ions are only marginally displaced from their ‘equilibrium’ positions. The derived property is the molar conductivity [17] characterising a given salt in solution at defined $\mathrm{T}$ and $p$.
$\text { Then, } \quad \ln (\Lambda)=\ln (\Lambda[\mathrm{T}, \mathrm{p}])$
The isobaric dependence of $\ln (\Lambda)$ on temperature yields the isobaric energy of activation $\mathrm{E}_{\mathrm{p}}[18] \quad\left[=\mathrm{R} \,\left\{\partial \ln \left(\Lambda^{0}\right) / \partial(1 / \mathrm{T})\right\}_{\mathrm{p}}\right]$. An extensive literature [19] describes the isochoric energies of activation defined by equation (h).
$\mathrm{E}_{\mathrm{V}}=-\mathrm{R} \, \mathrm{T}^{2}\left\{\partial \ln \left(\Lambda^{0}\right) / \partial\right\}_{\mathrm{V}}$
The property $\mathrm{E}_{\mathrm{V}}$ has attracted considerable attention and debate [18-,21]. Nevertheless the same question (i.e. which volume is held constant?) can be asked. The debate emerges partly form the observation that $\mathrm{T}$ and $p$ are intensive variables whereas $\mathrm{V}$ is an extensive variable.
Footnotes
[1] M. G. Evans and M. Polanyi, Trans. Faraday Soc., 1935, 31, 875.
[2] M. G. Evans and M. Polanyi, Trans. Faraday Soc.,1937,33,448.
[3] D. M. Hewitt and A. Wasserman, J. Chem. Soc.,1940,735.
[4] B. T. Baliga and E. Whalley, J. Phys. Chem., 1967, 71, 1166.
[5] B. T. Baliga and E. Whalley, Can. J. Chem., 1970, 48, 528.
[6] See also
1. D. L. Gay and E. Whalley, J. Phys. Chem.,1968,72,4145.
2. E. Whalley, Adv. Phys. Org. Chem.,1964,2,93.
3. E. Whalley, Ber. Bunsenges Phys. Chem.,1966,70,958.
4. G. Kohnstam, Prog. React. Kinet., 1970,15,335.
5. E. A. Moelwyn-Hughes, The Chemical Statics and Kinetics of Solutions, Academic Press, London, 1971.
6. B.T. Baliga, R. J. Withey, D. Poulton and E. Whalley, Trans. Faraday Soc.,1965,61,517.
7. G. J. Hills and C. A.Viana, in Hydrogen Bonded Solvent Systems, ed. A. K. Covington and P. Jones, Taylor and Francis, London, 1968, p. 261.
8. E. Whalley, Ann. Rev. Phys.Chem.,1967,18,205.
[7] E. Whalley, J. Chem. Soc. Faraday Trans.,1987,83,2901.
[8] L. M. P. C. Albuquerque and J. C. R. Reis, J. Chem. Soc., Faraday Trans. 1, 1989, 85, 207; 1991, 87,1553.
[9] H. A. J. Holterman and J. B. F. N. Engberts, J. Am. Chem. Soc., 1982, 104, 6382.
[10] P. G. Wright, J. Chem. Soc., Faraday Trans. 1, 1986, 82,2557.
[11] M. J. Blandamer, J. Burgess, B. Clarke, R. E. Robertson and J. M. W. Scott. J. Chem. Soc., Faraday Trans. 1, 1985, 81, 11.
[12] M. J. Blandamer, J. Burgess, B. Clarke, H. J. Cowles, I. M. Horn, J. F. B. N. Engberts, S. A. Galema and C. D. Hubbard, J. Chem. Soc. Faraday Trans. 1, 1989, 85, 3733.
[13] M. J. Blandamer, J. Burgess, B. Clarke and J. M. W. Scott, J. Chem. Soc., Faraday Trans. 1, 1984, 80, 3359.
[14] J. R. Haak, J. B. F. N. Engberts and M. J. Blandamer, J. Am. Chem. Soc., 1985, 107, 6031.
[15] M. J. Blandamer, N. J. Buurma, J. B. F. N. Engberts and J. C. R. Reis, Org. Biomol. Chem.,2003, 1,720.
[16] D. A. Lown, H. R. Thirsk and Lord Wynne-Jones, Trans. Faraday Soc., 1970, 66, 51.
[17] $\Lambda=\left[\mathrm{S} \mathrm{} \mathrm{m}^{2} \mathrm{~mol}^{-1}\right]$
[18] S. B. Brummer and G. J. Hills, Trans. Faraday Soc.,1961,57,1816,1823.
[19] [18] A. F. M. Barton, Rev. Pure Appl. Chem.,1971,21,49.
[20] S. B. Brummer, J. Chem. Phys.,1965,42,1636.
[21]
1. G. J. Hills, P. J. Ovenden and D. R.Whitehouse, Disc. Faraday Soc., 1965, 39, 207.
2. G. J. Hills, in Chemical Physics of Ionic Solutions, edited by B. E. Conway and R. G. Barradas, Wiley, New York, 1966, p. 521.
3. W. A. Adams and K. J. Laidler, Can. J.Chem.,1968,46,2005.
4. F. Barreira and G. J. Hills, Trans. Faraday Soc.,1968,64,1359.
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An isobaric calorimeter is designed to measure the heat accompanying the progress of a closed system from state (I) to state (II) at constant pressure. [1] It follows from the first law that if only ‘$p-\mathrm{V}$’ work is involved,
$\Delta \mathrm{U}=\mathrm{q}-\mathrm{p} \, \Delta \mathrm{V}$
By definition the enthalpy $\mathrm{H}$ of a closed system is given by equation (b);
$\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}$
$\text { Then, } \Delta \mathrm{H}=\Delta \mathrm{U}+\mathrm{p} \, \Delta \mathrm{V}+\Delta \mathrm{p} \, \mathrm{V}$
Hence from equations (a) and (c), at constant pressure,
$\Delta \mathrm{H}=\mathrm{q}$
$\text { Thus at constant pressure, } \Delta \mathrm{H}=\mathrm{H}(\mathrm{II})-\mathrm{H}(\mathrm{I})=\mathrm{q}$
Hence if we record the heat (exothermic or endothermic) at constant pressure we have the change in enthalpy, $\Delta \mathrm{H}$. [2] Equation (e) highlights the optimum thermodynamic equation. On one side of the equation is a measured property/change and on the other side of the equation is a change in a property of the system which we judge to be informative about the chemical properties of a system; e.g. $\Delta \mathrm{H}$. The problem is that the derived property is not the actual change in energy, $\Delta \mathrm{U}$.
Footnotes
[1] W. Zielenkiewicz, J.Therm. Anal.,1988, 33, 7.
[2] Hess’ Law. This law is a consequence of the observation that the enthalpy of a closed system is a state variable. $\Delta \mathrm{H}$ accompanying the change from state I to state II is independent of the number of intermediary states and of the general path between the two states and the rate of change.
[3] Isothermal calorimetry
1. I. Wadso, Chem. Soc. Rev.,1997,26,79.
2. I. Wadso, in Experimental Thermodynamics, IUPAC Data Series No. 39; volume 4, ed. K. N. Marsh and P. A. G. O’Hare, chapter 12, Blackwell, Oxford,1994.
3. I. Wadso, in Thermal and Energetic Studies of Cellular Systems, ed. A. M. James, Wright, Bristol, 1987, chapter 3.
4. J. B. Ott and C. J. Wormwald, in Experimental Thermodynamics, 2 IUPAC Data Series No. 39; ed. K. N. Marsh and P. A. G. O’Hare, chapter 8, Blackwell, Oxford,1994.
5. S. J. Gill, J. Chem.Thermodynamics, 1988,20,1361.
1.3.02: Calorimetry- Isobaric- General Operation
Calorimetry, particularly isobaric calorimetry, is a key technique in chemical thermodynamics, for studying the properties of liquid mixtures and solutions. Numerous designs for calorimeters have been published. [1,2] The operation of a classic calorimeter involves two key steps.
Step 1. Known amounts of two liquids (e.g. solvent and solution) are mixed in a thermally insulated reaction vessel at constant pressure. The rise in temperature is recorded.
Step 2. A known electric current is passed for a recorded length of time through an electric resistance in the reaction vessel to produce a comparable rise in temperature.
By proportion the required amount of energy to produce the measured rise in temperature in step 1 is obtained. [Complications emerge by the need to take account of spontaneous cooling in both steps when the temperature of the calorimeter exceeds ambient temperature; cf. Newton’s Law of Cooling.]
Another type of isobaric calorimeter involves injecting aliquots of one liquid (solution or solvent) into sample cell containing another liquid, recording the rise in temperature accompanying injection of each aliquot. The calorimeter is again calibrated electrically.
Footnotes
[1] M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, chapter 4.
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Classic (isobaric) calorimetric experiments often centre on the determination of the change in enthalpy ∆H for a given well-defined process. For example, the heat accompanying the mixing of known amounts of two liquids [e.g. water($\lambda$) and ethanol($\lambda$)] to form a binary liquid mixture yields the enthalpy of mixing, $\Delta_{\mathrm{mix}}\mathrm{H}$. Similarly enthalpies of solution are obtained by recording the heat accompanying the solution of a known amount of solute (e.g. urea) in a known amount of solvent; e.g. water($\lambda$). Key equations emerge from the following analysis.
The enthalpy $\mathrm{H}$ of a closed system is an extensive function of state which for a closed system is defined by the set of independent variables, $\mathrm{T}$, $p$ and $\xi$ where $\xi$ represents the chemical composition.
$\mathrm{H}=\mathrm{H}[\mathrm{T}, \mathrm{p}, \xi] \label{a}$
Equation \ref{b} is the complete differential of Equation \ref{a}.
$\mathrm{dH}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \, \mathrm{dp}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \label{b}$
If the closed system is held at constant pressure (e.g. ambient) the differential enthalpy $\mathrm{dH}$ equals the heat $\mathrm{dq}$.
$T\mathrm{dq}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi$
Here $\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}$, is the differential dependence of enthalpy $\mathrm{H}$ on temperature at constant pressure and composition whereas $\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$, is the differential dependence of enthalpy $\mathrm{H}$ on composition at fixed temperature and pressure.
1.3.04: Calorimetry- Solutions- Adiabatic
A general equation describes heat $q$ in terms of changes in temperature and composition at constant pressure; $\mathrm{dH} = q$.
$\mathrm{dq}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \label{a}$
In this application of Equation \ref{a}, the system is thermally insulated; i.e. $q$ is zero. An aliquot of solution containing a small amount of chemical substance $j$ is added to a solution held in a thermally insulated container. A rearranged Equation \ref{a} takes the following form.
$\mathrm{dT}=-\frac{(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \xi}} \, \mathrm{d} \xi\label{b}$
Chemical reaction occurs in the sample cell, the rate of chemical reaction being governed by the composition of the solution and appropriate rate constants. The differential isobaric dependence of temperature on time, $\mathrm{dT} / \mathrm{dt}$ is given by Equation \ref{c}.
$\frac{\mathrm{dT}}{\mathrm{dt}}=-\frac{(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \xi}} \, \frac{\mathrm{d} \xi}{\mathrm{dt}}\label{c}$
The calorimeter records the dependence of temperature on time. An equation based on the Law of Mass Action yields the rate of change of composition $\mathrm{d}\xi / \mathrm{dt}$. The integrated form of Equation \ref{c} yields a calculated dependence of $\mathrm{T}$ on time which can be compared with the recorded dependence. This subject is important in the context of thermal imaging calorimetry [1-4].
Footnotes
[1] M. J. Blandamer, P. M. Cullis, and P. T. Gleeson, Phys. Chem. Chem. Phys.,2002,4,765.
[2] B. Jandeleit, D. J. Schaefer, T. S. Powers, H. W. Turner and W. H. Weinbereg, Angew. Chem. Int. Ed. Engl.,1999,38,2495.
[3] M. T. Reetz, M. H. Becker, K. M. Kuling and A. Holzwarth, Angew. Chem. Int. Ed. Engl.,1998, 37,2647.
[4] G. C. Davies, R. S. Hutton, N. Millot, S. J. F. Macdonald, M. S. Hansom and I. B. Campbell, Phys. Chem. Chem.Phys.,2002,4,1791.
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At temperature $\mathrm{T}$ and pressure $p$, the enthalpy of a closed system having composition $\xi$ can be defined by Equation \ref{a}.[1]
$\mathrm{H}=\mathrm{H}[\mathrm{T}, \mathrm{p}, \xi] \label{a}$
The general differential of Equation \ref{a} takes the following form.
$\mathrm{dH}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \, \mathrm{dp}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi$
If the pressure is constant at, for example, ambient pressure, $\mathrm{dH}$ equals the differential heat $\mathrm{dq}$ passing between system and surroundings. In the application considered here, the temperature is held constant.
The following equation describes heat $\mathrm{dq}$ in terms of changes in composition at constant pressure and constant temperature.
Thus
$\mathrm{dq}=\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi$
Moreover $\mathrm{d}\xi$ is the extent of chemical reaction in the time period $\mathrm{dt}$.
$\text {Then } \left(\frac{\mathrm{dq}}{\mathrm{dt}}\right)=\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\mathrm{d} \xi}{\mathrm{dt}}\right)$
If the chemical reaction in the sample cell involves a single chemical reaction, $(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}$ is the enthalpy of reaction $\Delta_{\mathrm{r}}\mathrm{H}$.
$\text { Therefore, }\left(\frac{\mathrm{dq}}{\mathrm{dt}}\right)=\Delta_{\mathrm{r}} \mathrm{H} \,\left(\frac{\mathrm{d} \xi}{\mathrm{dt}}\right)$
In Heat Flow Calorimetry [2,3], a small closed reaction vessel is in contact with a heat sink so that the reaction vessel is held at constant temperature. The flow of heat between sample cell and heat sink is monitored such that the recorded quantity is the thermal power, the rate of heat production ($\mathrm{dq} / \mathrm{dt}$) as a result of chemical reaction. The property ($\mathrm{dq} / \mathrm{dt}$) is recorded as a function of time; also as the amount of reactants decreases, $\operatorname{limit}(\mathrm{t} \rightarrow \infty)(\mathrm{dq} / \mathrm{dt})$ is zero. Nevertheless because $(\mathrm{d} \xi / \mathrm{dt})$ is a function of time, $(\mathrm{dq} / \mathrm{dt})$ effectively monitors the progress of chemical reaction. Intuitively it is apparent that for an exothermic reaction $(\mathrm{dq} / \mathrm{dt})$ at time zero is also zero, rises rapidly and then decreases to zero as all reactants are consumed.
The Law of Mass Action relates $(\mathrm{d} \xi / \mathrm{dt})$ to the composition of the system at time $\mathrm{t}$. Because $(\mathrm{d} \xi / \mathrm{dt})$ depends on time, $(\mathrm{dq}/\mathrm{dt})$ also depends on time, approaching zero as reactants are consumed.
For example, in the case of a simple chemical reaction of the form $\mathrm{X} \rightarrow \mathrm{Y}$ where at $\mathrm{t} = 0$ the amount of chemical substance $\mathrm{X}$ is $n_{\mathrm{X}}^{0}$, the amounts of $\mathrm{X}$ and $\mathrm{Y}$ at time $t$ are $\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right)$ and $\xi$ moles respectively. If the volume of the sample cell is $\mathrm{V}$,
$(1 / \mathrm{V}) \, \mathrm{d} \xi / \mathrm{dt}=(1 / \mathrm{V}) \, \mathrm{k} \,\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right)$
$\text { Or, } \quad \mathrm{d} \xi / \mathrm{dt}=\mathrm{k} \, \mathrm{V} \,\left[\mathrm{c}_{\mathrm{x}}^{0}-(\xi / \mathrm{V})\right]$
$\text { Or, } \quad \mathrm{d} \xi / \mathrm{dt}=\mathrm{k} \, \mathrm{V} \, \mathrm{c}_{\mathrm{X}}^{0} \, \exp (-\mathrm{k} \, \mathrm{t})$
Hence using equation (e) and for a dilute solution,
$\mathrm{dq} / \mathrm{dt}=\Delta_{\mathrm{r}} \mathrm{H}^{0} \, \mathrm{k} \, \mathrm{V} \, \mathrm{c}_{\mathrm{X}}^{0} \, \exp (-\mathrm{k} \, \mathrm{t})$
The integral of equation (i) between $\mathrm{t} = 0$ and time $\mathrm{t}$ yields the amount of heat passing between system and heat sink.[3,4]
$\text { Thus, } \quad \int_{0}^{\mathrm{t}} \mathrm{dq}=\Delta_{\mathrm{r}} \mathrm{H}^{\circ} \, \mathrm{k} \, \mathrm{V} \, \mathrm{c}_{\mathrm{X}}^{0} \,[1-\exp (-\mathrm{k} \, \mathrm{t})]$
Hence the measured dependence of ($\mathrm{dq} / \mathrm{dt}$) is compared with that calculated using equations (i) and (j). The analysis is readily extended to second order reactions. [4,5]
The technique of heat flow calorimetry has been applied across a wide range of subjects (e.g. screening of catalysts [6]and characterising complex reactions[7]) and subject to different analytical approaches.
Footnotes
[1] I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London, 1954.
[2] I. Wadso, Chem. Soc. Rev.,1997,26,79.
[3] P. Backman, M. Bastos, D. Hallen, P. Lombro and I. Wadso, J. Biochem. Biophysic. Methods, 1994,28,85.
[4] M. J. Blandamer, P. M. Cullis and P. T. Gleeson, J. Phys. Org. Chem.,2002,15,343.
[5] A. Thiblin, J. Phys. Org. Chem.,2002,15,233.
[6] D. G. Blackmond, T. Rosner and A. Pfaltz, Organic Process Research and Development,1999,3,275.
[7] C. Le Blond, J . Wang, R. D. Larsen, C. J. Orella, A. L. Forman, R. N. Landau, J. Lequidara, J. R. Sowa Jr., D. G. Blackmond and Y.-K. Sun.,Thermochim Acta,1996,289.189.
[8] A. E. Beezer, A. C. Morris, M. A.A. O’Neil, R. J. Willson, A. K. Hills, J. C. Mitchell and J. A. O’Connor, J.Phys.Chem.B.,2001,105,1212.
[9] R. J. Willson, A. E. Beezer, J. C. Mitchell and W. Loh, J. Phys. Chem., 1995, 99.7108.
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In a common type of calorimeter, aliquots of one liquid (solution or solvent) are injected into a sample cell containing another liquid. The rise in temperature accompanying injection of each aliquot is recorded. The calorimeter is calibrated electrically.
With advances in microelectronics and calorimeter design the volume of liquids required in titration calorimetry has dropped so that only micro-litres of aliquots are injected into a sample cell having a volume of the order $1 \mathrm{~cm}^{3}$. Operation of the calorimeter is under the control of a mini-computer. The sensitivity of these calorimeters is such that recorded heats are of the order of $10^{–6} \mathrm{~J}$. In a typical experiment sample and reference cells, held in an evacuated enclosure, are heated such that the temperatures of both cells increase at the rate of a few micro-kelvin per second. The electronic heaters and thermistors are coupled so that these temperatures (plus that of the adiabatic shield) stay in step. Under computer control, aliquots of a given solution from a micro-syringe are injected into the sample cell at predetermined intervals. The operation of the calorimeter is readily understood where the chemical processes in the sample cell following injection of an aliquot are exothermic. In this case the temperature of the solution in the sample cell increases so heating of this cell is stopped. The reference cell continues to be heated until at some stage the temperatures of both sample and reference cells are again equal, when again both cells are heated in preparation for the next injection of an aliquot. The computer records how much heat was produced by the electric heaters in the reference cell to recover the situation of equal temperatures. This amount of heat must have been produced effectively by chemical processes in the sample cell.
Titration microcalorimetry [1] has had a major impact in biochemistry with respect to the study of enzyme - substrate binding [2-5].
The starting point of the thermodynamic analysis is the definition of the extensive variable enthalpy $\mathrm{H}$ of a closed system in terms of temperature, pressure and composition; equation (a). $\mathrm{H}=\mathrm{H}[\mathrm{T}, \mathrm{p}, \xi]$
The complete differential of equation (a) takes the following form. $\mathrm{dH}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \, \mathrm{dp}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi$
The key term in the present context is the last term in equation (b) which describes a change in enthalpy at constant $\mathrm{T}$ and $p$. $\mathrm{dH}=\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi$
In the present context the change in composition/organisation $\mathrm{d}\xi$ refers to the contents of the sample cell accompanying injection of an aliquot from the syringe. Heat $q$ is recorded following injection of $\mathrm{d}{n_{j}}^{0}$ moles of chemical substance $j$ into the sample cell on going from injection number $\mathrm{I}$ to injection number $\mathrm{I}+1$. $\left[\frac{\mathrm{q}}{\mathrm{dn}_{\mathrm{j}}^{0}}\right]_{1}^{1+1}=\left[\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{j}}^{0}}\right]_{1}^{1+1}$
Equation (d) is the key to titration microcalorimetry. The recorded quantity $q$ on the left-hand side of equation (d) is the recorded heat at injection number $\mathrm{I}+1$ when further $\mathrm{d}{n_{j}}^{0}$ moles of chemical substance $j$ are injected into the sample cell. The right-hand-side shows that the recorded ratio $\left[\frac{\mathrm{q}}{\mathrm{dn}_{\mathrm{j}}^{0}}\right]_{\mathrm{l}}^{\mathrm{l}+1}$ is related to the dependence of enthalpy $\mathrm{H}$ on composition, $\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$, and the dependence of composition/organisation on the amount of substance $j$ injected. Plots of $\left[\frac{\mathrm{q}}{\mathrm{dn}_{\mathrm{j}}^{0}}\right]_{\mathrm{l}}^{\mathrm{l}+1}$ as a function of injection number are called enthalpograms.
Equation (d) highlights an underlying problem in the analysis of experimental results. The recorded quantity is heat $q$ and no information immediately emerges concerning the chemical processes responsible although we note that the sign of heat $q$ is not predetermined; i.e. processes can be exo- or endo- thermic. The r.h.s. of equation (d) involves the product of two quantities, $\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$ and $\frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{j}}^{0}}$. We have no ‘a priori’ indication concerning how to pull these terms apart. In other words we require a model for the chemical processes in the sample cell.
Footnotes
[1] T. S. Wiseman, S. Williston, J.F.Brandts and Z.-N. Lim, Anal.Biochem., 1979, 179,131.
[2] Biocalorimetry, ed. J. E. Ladbury and B. Z. Chowdhry, Wiley Chichester, 1998.
[3] M. J. Blandamer, P. M. Cullis and J. B. F. N. Engberts, J. Chem. Soc. Faraday Trans., 1998, 94, 2261.
[4] J. Ladbury and B. Z. Chowdhry, Chemistry and Biology, 1996,3,79.
[5] M. J. Blandamer, P. M. Cullis and J. B. F. N. Engberts, Pure Appl. Chem.,1996,68,1577.
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The technique of titration microcalorimetry was developed with the aim of probing enzyme-substrate interactions.[1,2] At the start of the experiment, the sample cell contains an aqueous solution containing a known amount of a macromolecular enzyme $\mathrm{M}(\mathrm{aq})$. The injected aliquots contain a known amount of substrate $\mathrm{X}$ such that during the experiment the composition of the sample cell is described in terms of the following chemical equilibrium.
$\mathrm{M}(\mathrm{aq})+\mathrm{X}(\mathrm{aq}) \Leftrightarrow \mathrm{MX}(\mathrm{aq})$
[It is important to note that the term substrate refers to the chemical substance which is bound (adsorbed?) by the macromolecular enzyme. In treatments of adsorption, the macromolecular substrate adsorbs small molecules.] In the limit of strong binding (see below), most of the injected substrate at the start of the experiment is bound by the enzyme. But gradually as more substrate is added, the number of free binding sites decreases until eventually all sites are occupied and hence no heat $q$ is recorded. The plot of $\left[\mathrm{q} / \mathrm{dn}_{\mathrm{X}}^{0}\right]$ against injection number in the textbook case is sigmoidal.
The equilibrium established in the sample cell is described as follows.
$\begin{array}{llccc} & \mathrm{M}(\mathrm{aq}) & \mathrm{X}(\mathrm{aq}) \quad<==> & \mathrm{MX}(\mathrm{aq}) & \ \text { At } \mathrm{t}=0 & \mathrm{n}^{0}(\mathrm{M}) & \mathrm{n}^{0}(\mathrm{X}) & 0 & \mathrm{~mol} \ \text { At } \mathrm{t}=\infty & \mathrm{n}^{0}(\mathrm{M})-\xi & \mathrm{n}^{0}(\mathrm{X})-\xi & \xi & \mathrm{mol} \ & {\left[\mathrm{n}^{0}(\mathrm{M})-\xi\right] / \mathrm{V}} & {\left[\mathrm{n}^{0}(\mathrm{X})-\xi\right] / \mathrm{V}} & \xi / \mathrm{V} & \mathrm{mol} \mathrm{dm} \end{array}$
$\mathrm{V}$ is the volume of the sample cell. The analysis uses equilibrium constants defined in terms of the concentrations of chemical substances in the system. There are advantages in using equilibrium constants having the following form where $\mathrm{c}_{\mathrm{r} = 1 \mathrm{~mol dm}^{-3}$.
$\mathrm{K}=(\xi / \mathrm{V}) \, \mathrm{c}_{\mathrm{r}} /\left\{\left[\mathrm{n}^{0}(\mathrm{M})-\xi\right] / \mathrm{V}\right\} \,\left\{\left[\mathrm{n}^{0}(\mathrm{X})-\xi\right] / \mathrm{V}\right\}$
The latter is a quadratic in the extent of reaction, $\xi$. [3]
$\xi^{2}+b \, \xi+c=0$
where
$b=-n^{0}(M)-n^{0}(X)-V \, c_{r} \, K^{-1}$
and
$\mathrm{c}=\mathrm{n}^{0}(\mathrm{M}) \, \mathrm{n}^{0}(\mathrm{X})$
Therefore
$\xi=-(b / 2) \pm(1 / 2) \,\left(b^{2}-4 \, c\right)^{1 / 2}$
The negative root of the quadratic yields the required solution on the grounds that, with increase in $n^{0}(\mathrm{X}$\) in the sample cell, more substrate is bound by the enzyme. The required quantity is $\left(\mathrm{d} \xi / \operatorname{dn}_{\mathrm{X}}^{0}\right)$. We note that from equations (g) and (h), $\mathrm{db} / \mathrm{dn}_{\mathrm{X}}^{0}=-1 ; \quad \mathrm{dc} / \mathrm{dn}_{\mathrm{X}}^{0}=\mathrm{n}_{\mathrm{M}}^{0}$
In the experiment we control the ratio of total amounts of substrate to enzyme, $\mathrm{n}^{0}(\mathrm{X}) / \mathrm{n}^{0}(\mathrm{M})$, which increases as more substrate is added to the sample cell. A measure of the ‘tightness of binding’ is the fraction of substrate bound when this ratio is unity. $\text { By definition, } \quad X_{r}=n^{0}(X) / n^{0}(M)=\left[X_{\text {total }}\right] /\left[M_{\text {total }}\right]$
We define two variables $\mathrm{r}$ and $\mathrm{C}$; (note uppercase). $\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})=\mathrm{c}_{\mathrm{r}} / \mathrm{K} \,\left[\mathrm{M}_{\text {total }}\right]=\mathrm{r}=1 / \mathrm{C}$
$\text { From equation (e), }[\mathrm{X}]^{\mathrm{eq}} /[\mathrm{MX}]^{\mathrm{eq}}=\mathrm{c}_{\mathrm{r}} / \mathrm{K} \,[\mathrm{M}]^{\mathrm{eq}}$
If $\mathrm{K}$ is small, then $\mathrm{r}$ is large and only a small amount of the injected substrate is bound to the enzyme. $[\mathrm{X}]^{\mathrm{eq}}$ is, in relative terms, large and $[\mathrm{MX}]^{\mathrm{eq}$ is small. If $\mathrm{r}$ is large, $\mathrm{C}$ is small.
If on the other hand $\mathrm{K}$ is large, $\mathrm{r}$ is small, $\mathrm{C}$ is large and in the limit all substance $\mathrm{X}$ is bound to the enzyme $\mathrm{M}$.
We return to equation (i) because in order to calculate $\xi$ we require $b$. [4] $\mathrm{b}^{2}-4 \, \mathrm{c}=\left[\mathrm{n}^{0}(\mathrm{M})\right]^{2} \,\left[\mathrm{X}_{\mathrm{r}}^{2}-2 \, \mathrm{X}_{\mathrm{r}} \,(1-\mathrm{r})+(1+\mathrm{r})^{2}\right]$
We also require $\left[\mathrm{d} \xi / \operatorname{dn}^{0}(\mathrm{X})\right]$ which describes the dependence of extent of substrate binding on total amount of $\mathrm{X}$ in the sample cell, noting that we can control the latter through the concentration of the injected aliquots. We return to equation (i) making use of equation (n). [5] $\frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{X}}^{0}}=\frac{1}{2}+\frac{\left[1-(1 / 2) \,(1+\mathrm{r})-\mathrm{X}_{\mathrm{r}} / 2\right]}{\left[\mathrm{X}_{\mathrm{r}}^{2}-2 \, \mathrm{X}_{\mathrm{r}} \,(1-\mathrm{r})+(1+\mathrm{r})^{2}\right]^{1 / 2}}$
The enthalpy of the solution in the sample cell (assuming the thermodynamic properties of the solution are ideal) is given by equation (p). \begin{aligned} \mathrm{H}(\mathrm{aq} ; \mathrm{id})=\mathrm{n}_{1}(\lambda) \, \mathrm{H}_{1}^{*}(\lambda)+\left[\mathrm{n}^{0}(\mathrm{M})-\xi\right] \, \mathrm{H}^{\infty}(\mathrm{M} ; \mathrm{aq}) \ &+\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right] \, \mathrm{H}^{\infty}(\mathrm{X} ; \mathrm{aq})+\xi \, \mathrm{H}^{\infty}(\mathrm{MX} ; \mathrm{aq}) \end{aligned}
$[\partial \mathrm{H}(\mathrm{aq} ; \mathrm{id}) / \partial \xi \xi]=-\mathrm{H}^{\infty}(\mathrm{M} ; \mathrm{aq})-\mathrm{H}^{\infty}(\mathrm{X} ; \mathrm{aq})+\mathrm{H}^{\infty}(\mathrm{MX} ; \mathrm{aq})=\Delta_{\mathrm{B}} \mathrm{H}^{\infty}$
Hence the dependence of the ratio $\left[\mathrm{q} / \mathrm{dn}_{\mathrm{X}}^{0}\right]$ on composition of the sample cell is given by equation (r). $\frac{\mathrm{q}}{\mathrm{dn}_{\mathrm{X}}^{0}}=\Delta_{\mathrm{B}} \mathrm{H}^{\infty} \,\left[\frac{1}{2}+\frac{\left[1-(1 / 2) \,(1+\mathrm{r})-\mathrm{X}_{\mathrm{r}} / 2\right]}{\left[\mathrm{X}_{\mathrm{r}}^{2}-2 \, \mathrm{X}_{\mathrm{r}} \,(1-\mathrm{r})+(1+\mathrm{r})^{2}\right]^{1 / 2}}\right]$
The latter key equation describes the recorded enthalpogram. The latter falls into one of three general classes determined by the quantity $\mathrm{C}$ in equation ($\lambda$).
1. $\mathrm{C} = \infty$. All injected substrate is bound to the enzyme over the first few injections such that the ratio $\left[\mathrm{q} / \mathrm{dn}_{\mathrm{X}}^{0}\right]$ equals $\Delta_{\mathrm{B}} \mathrm{H}^{\infty}$. When all sites are occupied the ratio $\left[\mathrm{q} / \mathrm{dn}_{\mathrm{X}}^{0}\right]$ drops to zero and remains at zero for all further injections.
2. $40 < \mathrm{C} < 500$. This is the textbook case where the recorded enthalpogram has a sigmoidal shape. [6] The ratio $\left[\mathrm{q} / \mathrm{dn}_{\mathrm{X}}^{0}\right]$ at low injection numbers is close to $\Delta_{\mathrm{B}} \mathrm{H}^{\infty}$. The recorded enthalpogram is fitted using a non-linear least squares technique to yield estimates of $\mathrm{K}$ and $\Delta_{\mathrm{B}} \mathrm{H}^{\infty}$.
3. $0.1 < \mathrm{C} < 20$. Only a small fraction of the injected substrate is bound to the enzyme such that only poor estimates of $\mathrm{K}$ and $\Delta_{\mathrm{B}} \mathrm{H}^{\infty}$ are obtained.
Footnotes
[1] T. S. Wiseman, S. Williston, J.F.Brandts and Z.-N. Lim, Anal. Biochem., 1979,179,131.
[2] M. J. Blandamer, in Biocalorimetry, ed. J. E. Ladbury and B. Z. Chowdhry, Wiley Chichester, 1998, p5.
[3] $\mathrm{n}^{0}(\mathrm{M}) \, \mathrm{n}^{0}(\mathrm{X})-\xi \,\left[\mathrm{n}^{0}(\mathrm{M})+\mathrm{n}^{0}(\mathrm{X})\right]+\xi^{2}=\mathrm{V} \, \mathrm{c}_{\mathrm{r}} \, \mathrm{K}^{-1} \, \xi$
[4] $b^{2}=\left[-n^{0}(M)-n^{0}(X)-V \, c_{r} \, K^{-1}\right]^{2}$
$\text { or, } b^{2}=\left[n^{0}(M)\right]^{2} \,\left[1+n^{0}(X) / n^{0}(M)+V \, c_{r} / K \, n^{0}(M)\right]^{2}$
\begin{aligned} \mathrm{b}^{2}-4 \, \mathrm{c}=\left[\mathrm{n}^{0}(\mathrm{M})\right]^{2} \, & {\left[\mathrm{n}^{0}(\mathrm{X}) / \mathrm{n}^{0}(\mathrm{M})+1\right.} \ &\left.+\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})\right]^{2}-4 \, \mathrm{n}^{0}(\mathrm{M}) \, \mathrm{n}^{0}(\mathrm{X}) \end{aligned}
$\text { Or, } b^{2}-4 \, c=\left[n^{0}(M)\right]^{2} \,\left[X_{r}+1+r\right]^{2}-4 \, n^{0}(M) \, n^{0}(X)$
$\mathrm{b}^{2}-4 \, \mathrm{c}=\left[\mathrm{n}^{0}(\mathrm{M})\right]^{2} \,\left[\left(\mathrm{X}_{\mathrm{r}}+1\right)^{2}+\mathrm{r}^{2}+2 \, \mathrm{r} \,\left(\mathrm{X}_{\mathrm{r}}+1\right)\right]-4 \, \mathrm{n}^{0}(\mathrm{M}) \, \mathrm{n}^{0}(\mathrm{X})$
$\mathrm{b}^{2}-4 \, \mathrm{c}=\left[\mathrm{n}^{0}(\mathrm{M})\right]^{2} \,\left[\mathrm{X}_{\mathrm{r}}^{2}+2 \, \mathrm{X}_{\mathrm{r}}+1+\mathrm{r}^{2}+2 \, \mathrm{r} \, \mathrm{X}_{\mathrm{r}}+2 \, \mathrm{r}-4 \, \mathrm{X}_{\mathrm{r}}\right]$
[5] From equation (i) $\frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{x}}^{0}}=-\frac{1}{2} \frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{x}}^{0}}-\frac{1}{2} \, \frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \,\left[2 \, \mathrm{b} \, \frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{x}}^{0}}-4 \, \frac{\mathrm{dc}}{\mathrm{dn}_{\mathrm{x}}^{0}}\right]$
$\text { Or, } \quad \frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{X}}^{0}}=-\frac{1}{2} \frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{X}}^{0}}-\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \,\left[\mathrm{b} \, \frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{X}}^{0}}-2 \, \frac{\mathrm{dc}}{\mathrm{dn}_{\mathrm{X}}^{0}}\right]$
We ignore for the moment the term $\left(b^{2}-4 \, c\right)^{1 / 2}$ and concentrate attention on the two derivatives $\frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{X}}^{0}}$ and $\frac{\mathrm{dc}}{\mathrm{dn}_{\mathrm{X}}^{0}}$; equation (j). $\frac{\mathrm{d} \xi}{\operatorname{dn}_{\mathrm{X}}^{0}}=\frac{1}{2}-\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \,\left[-1 \,\left\{-\mathrm{n}^{0}(\mathrm{M})-\mathrm{n}^{0}(\mathrm{X})-\mathrm{V} \, \mathrm{c}_{\mathrm{r}} \, \mathrm{K}^{-1}\right\}-2 \, \mathrm{n}^{0}(\mathrm{M})\right]$
$\text { Or, } \frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{X}}^{0}}=\frac{1}{2}+\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \,\left[\mathrm{n}^{0}(\mathrm{M})-\mathrm{n}^{0}(\mathrm{X})-\mathrm{V} \, \mathrm{c}_{\mathrm{r}} \, \mathrm{K}^{-1}\right]$
\begin{aligned} &\text { Or, } \ &\frac{\mathrm{d} \xi}{\operatorname{dn}_{\mathrm{X}}^{0}}=\frac{1}{2}+\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \, \mathrm{n}^{0}(\mathrm{M}) \,\left[1-\mathrm{n}^{0}(\mathrm{X}) / \mathrm{n}^{0}(\mathrm{M})-\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})\right] \end{aligned}
\begin{aligned} &\text { Or, } \ &\frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{x}}^{0}}=\frac{1}{2}+\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \, \mathrm{n}^{0}(\mathrm{M}) \,\left[2-\left\{1+\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})\right\}-\mathrm{n}^{0}(\mathrm{X}) / \mathrm{n}^{0}(\mathrm{M})\right] \end{aligned}
\begin{aligned} &\text { Or } \ &\frac{d \xi}{\operatorname{dn}_{\mathrm{X}}^{0}}=\frac{1}{2} \ &+\frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \, \mathrm{n}^{0}(\mathrm{M}) \,\left[1-(1 / 2) \,\left\{1+\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})\right\}-\left\{\mathrm{n}^{0}(\mathrm{X}) / 2 \, \mathrm{n}^{0}(\mathrm{M})\right\}\right] \end{aligned}
Now consider the term $\left(b^{2}-4 \, c\right)^{1 / 2}$. From equation (n) $b^{2}-4 \, c=\left[n^{0}(M)\right]^{2} \,\left[X_{r}^{2}-2 \, X_{r} \,(1-r)+(1+r)^{2}\right]$
$\text { But } \frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{x}}^{0}}=\frac{1}{2}+\frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \, \mathrm{n}^{0}(\mathrm{M}) \,\left[1-(1 / 2) \,(1+\mathrm{r})-\mathrm{X}_{\mathrm{r}} / 2\right]$
[6] J. E. Ladbury and B. Z.Chowdhry, Chemistry and Biology, 1996, 3, 791.
1.3.08: Calorimetry- Titration Microcalorimetry- Micelle Deaggregation
Aliquots of a concentrated surfactant solution are injected into the sample cell of a titration microcalorimeter. The sample cell initially contains water($\lambda$). As more surfactant solution is injected into the sample cell a stage is reached where the concentration of surfactant in the sample cells exceeds the critical micellar concentration, $\mathrm{cmc}$. The magnitude of the recorded heat changes dramatically, leading to estimates of both the cmc and the enthalpy of micelle formation.
This calorimetric techniques has proved important in studies of ionic surfactants; e.g. hexadecyltrimethylammonium bromide (CTAB). For these surfactants the microcalorimeter signals a marked difference in recorded heats as the concentration of the surfactant changes from below to above the cmc. Titration microcalorimetric results for non-ionic surfactants are unfortunately not so readily interpreted. In addition to micelle formation, the monomers cluster in small aggregates below the cmc and the micelles cluster above the cmc.
Analysis
The volume of injected aliquot $\operatorname{vinj}$ is significantly less than the volume of the sample cell. The amount of surfactant in each aliquot is $\operatorname{ninj}$, the concentration of surfactant being $\operatorname{cinj}[=\operatorname{ninj} / \text { vinj }]$. If $\operatorname{cinj}$ is significantly above the $\mathrm{cmc}$, the contribution of the surfactant to the enthalpy of the injected aliquot is $\operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})$ where $\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})$ is the contribution of one mole of monomer to the molar enthalpy of a micelle. If the concentration of solution in the sample cell is below the cmc, the contribution of each monomer to the enthalpy of the solution equals $\mathrm{H}_{\mathrm{j}}^{0}(\text { mon })$. We concentrate attention on the contribution of the surfactant to the enthalpies of injected solution and the solution in the sample cell.
Enthalpy of the injected aliquot, $\mathrm{H}(\mathrm{inj})=\operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})$
The contribution of the surfactant to the enthalpy of the solution in the sample cell at injection number I is given by equation (b).
Enthalpy of solution in the sample cell at injection number I, $\mathrm{H}(\mathrm{I})=\mathrm{I} \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0} \text { (mon) }$
Enthalpy of solution in the sample cell at injection number (I+1), $\mathrm{H}(\mathrm{I}+1)=(\mathrm{I}+1) \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0} \text { (mon) }$
Recorded heat $\mathrm{q}=\mathrm{H}(!+1)-\mathrm{H}(\mathrm{I})-\mathrm{H}(\text { inj })$
$\text { or, }[\mathrm{q} / \text { ninj }]_{\text {lowl }}=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mon})-\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})=-\Delta_{\text {mic }} \mathrm{H}^{0}$
At high injection numbers the enthalpies of solution in the sample cell are $\mathrm{I} \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})$ and $(\mathrm{I}+1) \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})$. $[\mathrm{q} / \text { ninj }]_{\text {highl }}=(\mathrm{I}+1) \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})-\mathrm{I} \, \mathrm{H}_{\mathrm{j}}^{0}(\text { mic })-\mathrm{H}_{\mathrm{j}}^{0}(\text { mic })=0$
At low injection numbers the recorded $(q/\operatorname{ninj}$) is effectively the enthalpy of micelle formation. The recorded ratio $[q/\operatorname{ninj}]$ is effectively zero at high injection numbers, the switch in pattern of $(q/\operatorname{ninj}$) from $\Delta_{\operatorname{mic}} \mathrm{H}^{0}$ to zero marking the cmc of the surfactant.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.03%3A_Calorimeter/1.3.07%3A_Calorimeter-_Titration_Microcalorimetry-_Enzyme-Substrate_Interaction.txt
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A given closed system is prepared using $n_{1}$ moles of water ($\lambda$) and $n_{\mathrm{X}}^0$ moles of solute $\mathrm{X}$ at pressure $p$ ($\cong \mathrm{p}^{0}$, the standard pressure) and temperature $\mathrm{T}$. The thermodynamic properties of the solution are ideal such that, at some low temperature, the enthalpy of the solution $\mathrm{H}(\mathrm{aq} ; \text { low } \mathrm{T})$ is given by equation (a).
$\mathrm{H}(\mathrm{aq} ; \text { low } \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \text { low } \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{H}_{\mathrm{X}}^{\infty}(\text { aq } ; \text { low } \mathrm{T})$
The temperature of the solution is raised to high temperature such that the solution contains only solute $\mathrm{Y}$, all solute $\mathrm{X}$ having been converted to $\mathrm{Y}$.
$\mathrm{H}(\mathrm{aq} ; \text { high } \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \text { high } \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{H}_{\mathrm{Y}}^{\infty}(\text { aq; high } \mathrm{T})$
At intermediate temperatures, a chemical equilibrium exists between solutes $\mathrm{X}$ and $\mathrm{Y}$. At temperature $\mathrm{T}$, the chemical composition of the solution is characterised by extent of reaction $\xi(\mathrm{T})$.
$\begin{array}{llcc} & \mathrm{X}(\mathrm{aq})<\overline{=} & \mathrm{Y}(\mathrm{aq}) & \ \text { At low } \mathrm{T} & \mathrm{n}_{\mathrm{X}}^{0} & 0 & \mathrm{~mol} \ \text { At high } \mathrm{T} & 0 & \mathrm{n}_{\mathrm{x}}^{0} & \mathrm{~mol} \ \text { At intermediate } \mathrm{T} & \mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}(\mathrm{T}) & \xi^{\mathrm{eq}}(\mathrm{T}) & \mathrm{mol} \end{array}$
For a solution where the thermodynamic properties are ideal, we define an equilibrium constant $\mathrm{K}(\mathrm{T})$, at temperature $\mathrm{T}$.
$\mathrm{K}(\mathrm{T})=\xi^{\mathrm{eq}}(\mathrm{T}) /\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}(\mathrm{T})\right]$
By definition, the degree of reaction, $\alpha(T)=\xi^{e q}(T) / n_{x}^{0}$.
$\mathrm{K}(\mathrm{T})=\alpha(\mathrm{T}) \, \mathrm{n}_{\mathrm{X}}^{0} /\left[\mathrm{n}_{\mathrm{X}}^{0}-\mathrm{n}_{\mathrm{X}}^{0} \, \alpha(\mathrm{T})\right]$
$\text { Therefore, } \quad \alpha(\mathrm{T})=\mathrm{K}(\mathrm{T}) /[1+\mathrm{K}(\mathrm{T})]$
At temperature $\mathrm{T}$, the enthalpy of the aqueous solution is given by equation (f) where $\mathrm{H}_{1}^{*}(\lambda ; \mathrm{T})$ is the molar enthalpy of water($\lambda$) in the aqueous solution again assuming that the thermodynamic properties of the solution are ideal.
$\begin{array}{r} \mathrm{H}(\mathrm{aq} ; \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \,[1-\alpha(\mathrm{T})] \, \mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \ +\mathrm{n}_{\mathrm{X}}^{0} \, \alpha(\mathrm{T}) \, \mathrm{H}_{\mathrm{Y}}^{0}(\mathrm{aq} ; \mathrm{T}) \end{array}$
The limiting enthalpy of reaction, $\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq} ; \mathrm{T})$ is given by equation (g).
$\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq} ; \mathrm{T})=\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq} ; \mathrm{T})-\mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq} ; \mathrm{T})$
From equation (f),
\begin{aligned} \mathrm{H}(\mathrm{aq} ; \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \, & \mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \ &+\mathrm{n}_{\mathrm{x}}^{0} \, \alpha(\mathrm{T}) \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq} ; \mathrm{T}) \end{aligned}
We assume that $\Delta_{r} H^{\infty}(\mathrm{aq})$ is independent of temperature. The differential of equation (h) with respect to temperature yields the isobaric heat capacity of the solution.
$\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\lambda)+\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{C}_{\mathrm{pX}}^{\infty}(\mathrm{aq}) +\mathrm{n}_{\mathrm{X}}^{0} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \, \mathrm{d} \alpha / \mathrm{dT}$
The term $(\mathrm{d} \alpha / \mathrm{dT})$ signals the contribution of the change of composition with temperature to the isobaric heat capacity of the system, the ‘relaxational’ isobaric heat capacity $\mathrm{C}_{\mathrm{p}}(\text {relax})$.
Thus
$\mathrm{C}_{\mathrm{p}}(\text { relax })=\mathrm{n}_{\mathrm{X}}^{0} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \, \mathrm{d} \alpha / \mathrm{dT}$
$\text { From equation (e)[1] } \frac{\mathrm{d} \alpha(\mathrm{T})}{\mathrm{dT}}=\frac{\mathrm{K}(\mathrm{T})}{[1+\mathrm{K}(\mathrm{T})]^{2}} \, \frac{\mathrm{d} \ln [\mathrm{K}(\mathrm{T})]}{\mathrm{dT}}$
But according to the van’t Hoff Equation,
$\frac{\mathrm{d} \ln \mathrm{K}(\mathrm{T})}{\mathrm{dT}}=\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty}}{\mathrm{R} \, \mathrm{T}^{2}}$
$\text { Hence [2], } \mathrm{C}_{\mathrm{p}}(\text { relax })=\mathrm{n}_{\mathrm{x}}^{0} \, \frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}(\mathrm{T})}{[1+\mathrm{K}(\mathrm{T})]^{2}}$
The contribution to the molar isobaric heat capacity of the system from the change in composition of the solution is given by equation (n).
$C_{p m}(\text { relax })=\frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}(\mathrm{T})}{[1+\mathrm{K}(\mathrm{T})]^{2}}$
The dependence of $C_{p m}(\text {relax})$ on temperature has the following characteristics.
1. The plot forms a bell-shaped curve such that at very low and very high temperatures, $(C_{p m}(\text {relax})$ is zero [3-5].
2. $(C_{p m}(\text {max})$ occurs at the temperature $\mathrm{T}_{m}$.
3. The area under the ‘bell’ equals the enthalpy of reaction [6,7]. A scanning calorimeter is designed to raise the temperature of a sample (solution) in a controlled fashion. The calorimeter uses controlled electrical heating while monitoring the temperatures of a sample cell and a reference cell containing just solvent. If the heat capacity of the cell containing the sample under investigation starts to increase the calorimeter records the fact that more electrical energy is required to raise the temperature of the sample cell by say $0.1 \mathrm{~K}$ than required by the reference.
The tertiary structures of enzymes in aqueous solution are very sensitive to temperature. In the general case, an enzyme changes from, say, active to inactive form as the temperature is raised; i.e. the enzyme denatures. The change from active to inactive form is characterised by a ‘melting temperature’. The explanation is centred on the role of hydrophobic interactions in stabilising the structure of the active form. However the strength of hydrophobic bonding is very sensitive to temperature. Hence equation (n) forms the basis of an important application of modern differential scanning calorimeters [3] into structural reorganisation in biopolymers on changing the temperature [4-7]. The scans may also identify domains within a given biopolymer which undergo structural transitions at different temperatures [8].
Indeed there is evidence that a given enzyme is characterised by a temperature range within which the active form is stable. Outside this range, both at low and high temperatures the active form is not stable. In other words the structure of an enzyme may change to an inactive form on lowering the temperature [9,10]. The pattern can be understood in terms of the dependence of $\left[\mu_{\mathrm{j}}^{0} / \mathrm{T}\right]$ on temperature where $\left[\mu_{\mathrm{j}}^{0}$ is the reference chemical potentials for solute $j$. In this case we consider the case where in turn solute j represents the active and inactive forms of the enzyme. There is a strong possibility that the plots of two dependences intersect at two temperatures. The active form is stable in the window between the two temperatures.
The analysis leading to equation (n) is readily extended to systems involving coupled equilibria [11,12]. The impact of changes in composition is also an important consideration in analysing the dependence on temperature of the properties of weak acids in solution [13- 15].
Footnotes
[1] \begin{aligned} \frac{\mathrm{d} \alpha}{\mathrm{dT}}=\left[\frac{1}{1+\mathrm{K}}\right.&\left.-\frac{1}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}}=\left[\frac{1+\mathrm{K}-\mathrm{K}}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}}=\left[\frac{1}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}} \ &=\left[\frac{\mathrm{K}}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{d} \ln \mathrm{K}}{\mathrm{dT}} \end{aligned}
[2] $\mathrm{C}_{\mathrm{p}}(\text { relax })=[\mathrm{mol}] \, \frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]^{2}}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2}} \, \frac{[1]}{[1]}=\left[\mathrm{J} \mathrm{K}^{-1}\right]$
[3] V. V. Plotnikov, J. M. Brandts, L-V. Lin and J. F. Brandts, Anal. Biochem.,1997,250,237.
[4] C. O.Pabo, R. T. Sauer, J. M. Sturtevant and M. Ptashne, Proc. Natl. Acad.Sci. USA,1979, 76,1608.
[5] S. Mabrey and J. M. Sturtevant, Proc. Natl. Acad.Sci. USA,1976,73,3802.
[6] T. Ackerman, Angew. Chem. Int.Ed. Engl.,1989,28,981.
[7] J. M. Sturtevant, Annu. Rev.Phys.Chem.,1987,38,463.
[8] M. J. Blandamer, B. Briggs, P. M. Cullis, A. P Jackson, A. Maxwell and R. J. Reece, Biochemistry, 1994,33,7510.
[9] F. Franks, R. M. Hately and H. L. Friedman, Biophys.Chem.,1988,31,307.
[10] F. Franks and T. Wakabashi, Z. Phys. Chem., 1987,155,171.
[11] M. J. Blandamer, B. Briggs, J. Burgess and P. M. Cullis, J. Chem. Soc. Faraday Trans.,1990,86,1437.
[12] G. J. Mains, J. W. Larson and L. G. Hepler, J. Phys Chem.,1984,88,1257.
[13] J. K. Hovey and L.G. Hepler, J. Chem. Soc. Faraday Trans.,1990,86,2831.
[14] E. M. Woolley and L. G. Hepler, Can. J.Chem.,1977,55,158.
[15] J. K. Hovey and L.G. Hepler, J. Phys. Chem.,1990,94,7821.
1.3.10: Calorimetry- Solutions- Flow Microcalorimetry
An important event in experimental calorimetry was the development of the Picker flow microcalorimeter [1-3]. In this calorimeter, two liquids [e.g. water($\lambda$) and an aqueous solution] at the same temperature flow through two cells. The liquids are heated, the calorimeter recording the difference in power required to keep both liquids at the same temperature. The recorded difference is a function of the difference in isobaric heat capacities per unit volume. The isobaric heat capacity of the solvent [e.g. water($\lambda$)] per unit volume (or, heat capacitance) , $\sigma^{*}(\lambda)$ is the reference. The technique has been extended to measure enthalpies and rates of reaction [4].
Footnotes
[1] P. Picker, P-A. Leduc, P. R.Philip and J. E. Desnoyers, J. Chem. Thermodyn., 1971, 3, 631.
[2] J.-L. Fortier, P.-A. Leduc, P. Picker and J. E. Desnoyers, J. Solution Chem., 1973, 2, 467.
[3] J. E. Desnoyers, C. de Visser, G. Peron and P. Picker, J.Solution Chem., 1976, 5, 605.
[4] A. Roux, G. Peron, P. Picker and J. E. Desnoyers, J. Solution Chem.,1980,9,59.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.03%3A_Calorimeter/1.3.09%3A_Calorimetry-_Scanning.txt
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The conditions for chemical equilibrium in a closed system [1,2] at fixed temperature and pressure are as follows.
1. Minimum in Gibbs energy.
2. Affinity for spontaneous change, $\mathrm{A}$ equals zero.
3. Rate of chemical reaction, $\mathrm{d} \xi / \mathrm{dt}=0$.
4. For a chemical equilibrium involving i-chemical substances, the equilibrium chemical potentials of all substances in the system conform to the following condition.
$\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{e q}=0$
If all $i$-chemical substances are solutes in aqueous solution at temperature $\mathrm{T}$ and pressure $p$, the latter being close to the standard pressure $p^{0}$, the equilibrium chemical potentials are related to the composition of the system [3]. Hence,
$A^{\mathrm{cq}}=-\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=0=-\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]^{\mathrm{eq}}$
\text { Hence } \quad \sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{0}(a q)=-\sum_{j=1}^{j=i} v_{j} \, R \, T \, \ln \left(m_{j} \, \gamma_{j} / m^{0}\right)^{e q}\]
The left-hand-side of equation (c) defines the standard Gibbs energy of reaction, $\Delta_{\mathrm{r}} \mathrm{G}^{0}$ which in turn leads to the definition of an equilibrium constant $\mathrm{K}^{0}$.[1]
$\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}^{0}\right)=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}(\mathrm{aq})$
Combination of equation (c) and equation (d) yields an equation for $\mathrm{K}^{0}$ in terms of the equilibrium composition of the system [3].
$\mathrm{K}^{0}=\left[\Pi_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)^{\mathrm{v}(\mathrm{j})}\right]^{\mathrm{eq}}$
Equation (e) is remarkable. The right hand side describes the stoichiometry of the chemical equilibrium and the composition of the closed system at defined temperature and pressure. The left-hand-side in the form of $\mathrm{K}^{0}$ defined using equation (d) is related to the ideal thermodynamic process in terms of reference chemical potentials of reactants and products. If the solutes are non-ionic and the solution is dilute then a reasonable assumption sets $" \gamma_{j}^{e q}=1^{\prime \prime}$ for all $i$-solutes.
Footnotes
[1] From a thermodynamic standpoint, an equilibrium constant emerges from the idea of zero affinity for chemical reaction at a minimum in Gibbs energy. Accounts which treat equilibrium constants as the ratio of rate constants are unsatisfactory.
[2] The equations set out here describe the general case where substance $j$ is one of $i$-simple solutes in solution. In some cases one or more of the solutes are ionic and the solvent (e.g. water) is directly involved in the chemical reaction. In each case we assume that the systems have been assayed such that the composition of the system at equilibrium is known together with the stoichiometries.
[3] In general terms for a systems at pressure $p$,
$\mathrm{A}^{\mathrm{eq}}=-\left(\frac{\partial G}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=0$
Hence for a chemical equilibrium involving $i$-solutes in aqueous solution the following condition holds.
$0=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{p} \mathrm{~V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}\right]^{\mathrm{eq}}$
$\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} v_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)=-\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} v_{\mathrm{j}} \,\left[\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}\right]^{\mathrm{eq}}$
$\text { where } \Delta_{\mathrm{r}} \mathrm{G}^{0}\left(\mathrm{~T} ; \mathrm{p}^{0}\right)=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}^{0}\right)=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} v_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)$
1.4.02: Chemical Equilibria- Solutions- Derived Thermodynamic Parameter
A given closed system at fixed temperature and fixed pressure contains a number of chemical substances in chemical equilibrium. The composition of the system depends on temperature and pressure. Key equations describe the dependences of the equilibrium Gibbs energy on temperature and pressure. $\mathrm{H}^{\mathrm{eq}}=-\mathrm{T}^{2} \,\left(\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right)_{\mathrm{p}}^{\mathrm{eq}} ; \quad \mathrm{V}^{\mathrm{eq}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}}^{\mathrm{eq}}$ The situation is complicated by the fact that both $\mathrm{H}^{\mathrm{eq}}$ and $\mathrm{V}^{\mathrm{eq}}$ depend on the equilibrium composition of the system, $\xi^{\mathrm{eq}}$; $\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}$ and $\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}$. We consider the case where the chemical substances involved in the chemical equilibrium are solutes. Both partial derivatives are re-expressed in terms of the partial molar properties of each solute in the system.
$\left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{e q}=\sum_{j=1}^{j=i}\left(\frac{\partial H}{\partial n_{j}}\right)_{T, p, n(i \neq j)}^{e q} \,\left(\frac{\partial n_{j}}{\partial \xi}\right)^{e q}] \[\text { But partial molar enthalpy, } \quad \mathrm{H}_{\mathrm{j}}^{\mathrm{eq}}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}^{\mathrm{eq}}$
$\text { Further, } \quad\left(\partial \mathrm{n}_{\mathrm{j}} / \partial \xi\right)^{\mathrm{eq}}=\mathrm{v}_{\mathrm{j}}$
Here $ν_{j}$ is the stoichiometry associated with chemical substance $j$, being positive for products and negative for reactants.
$\text { Therefore, } \quad\left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{e q}=\sum_{j=1}^{j=i} v_{j} \, H_{j}^{e q}$
$\text { Therefore, } \quad \left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{V}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{\mathrm{eq}}$
${\mathrm{V}_{j}}^{\mathrm{eq}}$ is the partial molar volume of substance $j$ in the solution at equilibrium. The partial molar enthalpy of solute $j$ can be expressed in terms of a limiting molar enthalpy ${\mathrm{H}_{j}}^{\infty}$ and the dependence of activity coefficient $\gamma_{j}$ on temperature.
$\text { Therefore, } \quad\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}^{\mathrm{eq}}\right]$
In other words the dependence of enthalpy of the system on composition at equilibrium is a function of the limiting molar enthalpies of all chemical substances involved in the equilibrium and the dependences on temperature of their activity coefficients.
By definition, the limiting molar enthalpy of reaction,
$\Delta_{\mathrm{r}} \mathrm{H}^{\infty}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}$
$\text { Then [1] } \quad \left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{e q}=\Delta_{r} H^{\infty}-\sum_{j=1}^{j=i} v_{j} \, R \, T^{2} \,\left(\frac{\partial \ln \gamma_{j}}{\partial T}\right)_{p}^{e q}$
In some applications, the solutions are quite dilute and the assumption is made that at all temperatures and pressures $\gamma_{j}$ for chemical substance $j$ is unity.
$\text { Hence, }\left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{\text {eq }}=\Delta_{\mathrm{r}} H^{\infty}$
A similar analysis is possible in terms of partial molar volumes. From equation (e), we obtain the following equation for the volume of reaction.
$\left(\frac{\partial V}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{e q}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{V}_{\mathrm{j}} \,\left[\mathrm{V}_{\mathrm{j}}^{\infty}+\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}^{\mathrm{eq}}\right]$
$\text { The limiting volume of reaction, } \Delta_{\mathrm{r}} \mathrm{V}^{\infty}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{\infty}$
$\text { Thus [2], } \quad \left(\frac{\partial V}{\partial \xi}\right)_{T, p}^{e q}=\Delta_{r} V^{\infty}+\sum_{j=1}^{j=i} v_{j} \, R \, T \,\left(\frac{\partial \ln \left(\gamma_{j}\right)}{\partial p}\right)_{T}^{e q}$
If the solution is dilute, it can often be assumed that the activity coefficient of each chemical substance is independent of pressure. Then,
$\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}$
A slight complication to these general equations arises if one of the substances involved in the chemical equilibrium is the solvent. As an example we consider the following equilibrium.
$\mathrm{X}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\mathrm{aq}) \rightleftharpoons \mathrm{Y}(\mathrm{aq})$
\begin{aligned} &\text { Then, } \quad(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\left[\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{Y}} / \partial \mathrm{T}\right)_{\mathrm{p}}^{\mathrm{eq}}\right] \ &-\left[\mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{X}} / \partial \mathrm{T}\right)_{\mathrm{p}}^{\mathrm{eq}}\right] \ &-\left[\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1}\left(\mathrm{~m}_{\mathrm{X}}+\mathrm{m}_{\mathrm{Y}}\right)_{\mathrm{p}}^{\mathrm{eq}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}^{\mathrm{eq}}\right] \end{aligned}
If the properties of the solution are ideal (e.g. very dilute), equation (o) is written in the following form.
$(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq})-\mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda)$
With reference to the limiting volume of reaction, the analogue of equation (p) is as follows.
$(\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\mathrm{V}_{\mathrm{Y}}^{\infty}(\mathrm{aq})-\mathrm{V}_{\mathrm{X}}^{\infty}(\mathrm{aq})-\mathrm{V}_{1}^{*}(\lambda)$
Footnotes
[1] $\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}=\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2} \,\left(\frac{[1]}{[\mathrm{K}]}\right)\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]$
[2] $\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]+\left[[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{[1]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]}\right]\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.04%3A_Chemical_Equilibria/1.4.01%3A_Chemical_Equilibria-_Solutions.txt
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A given chemical equilibrium involves association of two solutes $\mathrm{X}(\mathrm{aq})$ and $\mathrm{Y}(\mathrm{aq})$ to form solute $\mathrm{Z}(\mathrm{aq})$.
$2 X(a q)+Y(a q) \rightleftharpoons 4 Z(a q)$
Phase Rule. The aqueous solution is prepared using two chemical substances: substance $\mathrm{Z}$ and solvent water. Hence $\mathrm{C} = 2$. There are 2 phases: vapour and solution so $\mathrm{P} = 2$. Then $\mathrm{F} = 2$. Hence at fixed temperature and in a system prepared using mole fraction $x_{\mathrm{Z}}$ of substance $\mathrm{Z}$ (an intensive composition variable), the equilibrium vapour pressure and the equilibrium amounts of $\mathrm{X}(\mathrm{aq})$, $\mathrm{Y}(\mathrm{aq})$ and $\mathrm{Z}(\mathrm{aq})$ are unique.
$\text { At equilibrium, } 2 \, \mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq})+\mu_{\mathrm{Y}}^{\mathrm{eq}}(\mathrm{aq})=4 \, \mu_{\mathrm{Z}}^{\mathrm{eq}}(\mathrm{aq})$
At fixed $\mathrm{T}$ and $p$, assuming ambient pressure is close to the standard pressure $p^{0}$,
\begin{aligned} &2 \,\left[\mu_{\mathrm{X}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{X}} \, \gamma_{\mathrm{X}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}\right] \ &\quad+\left[\mu_{\mathrm{Y}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{Y}} \, \gamma_{\mathrm{Y}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}\right] \ &\quad=4 \,\left[\mu_{\mathrm{Z}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{Z}} \, \gamma_{\mathrm{Z}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}\right] \end{aligned}
$\text { where } \Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}^{0}\right)=4 \, \mu_{\mathrm{Z}}^{0}(\mathrm{aq})-2 \, \mu_{\mathrm{X}}^{0}(\mathrm{aq})-\mu_{\mathrm{Y}}^{0}(\mathrm{aq})$
For this equilibrium at temperature $\mathrm{T}$ and pressure $p$,
$\mathrm{K}^{0}=\frac{\left(\mathrm{m}_{\mathrm{Z}}^{\mathrm{eq}} \, \gamma_{\mathrm{Z}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{4}}{\left(\mathrm{~m}_{\mathrm{X}}^{\mathrm{eq}} \, \gamma_{\mathrm{X}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{Y}}^{\mathrm{eq}} \, \gamma_{\mathrm{Y}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)}$
If the solution is quite dilute, $\gamma_{\mathrm{X}}^{\mathrm{eq}}$, $\gamma_{\mathrm{Y}}^{\mathrm{eq}}$ and \gamma_{\mathrm{Z}}^{\mathrm{eq}} are effectively unity in the real solution at equilibrium. Then
$\mathrm{K}^{0}=\left(\mathrm{m}_{\mathrm{eq}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{4} /\left(\mathrm{m}_{\mathrm{X}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{Y}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)$
$\mathrm{K}^{0}$ is dimensionless. But the latter statement signals a common problem in this subject because chemists find it more convenient and informative to define a quantity ${\mathrm{K}_{\mathrm{m}}}^{0}$ in which the m0 terms in equation (f) [or its equivalent] have been removed.
$\text { Thus, } \mathrm{K}_{\mathrm{m}}^{0}=\left(\mathrm{m}_{\mathrm{Z}}^{\mathrm{eq}}\right)^{4} /\left(\mathrm{m}_{\mathrm{X}}^{\mathrm{eq}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{Y}}^{\mathrm{eq}}\right)$
Hence the units for ${\mathrm{K}_{\mathrm{m}}}^{0}$ signal the stoichiometry of the equilibrium whereas the dimensionless $\mathrm{K}^{0}$ does not [1,2].
Footnotes
[1] As a consequence of the removal of the $\mathrm{m}^{0}$ terms, $\mathrm{K}^{0}$ quantities have units unless the equation for the chemical equilibrium is stoichiometrically balanced: e.g. n-moles of reactants form n-moles of products.
But from equation (g), $\mathrm{K}_{\mathrm{m}}^{0}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{4} \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right]^{-2} \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{-1}$ or $\mathrm{K}_{\mathrm{m}}^{0}=\left[\mathrm{mol} \mathrm{} \mathrm{kg}^{-1}\right]$
If we write, $\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{m}}^{0}\right)$
Then $\Delta_{\mathrm{r}} \mathrm{G}^{0}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \, \ln \left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right]$
ln There is clearly a slight problem in handling a logarithm of a composition unit. There are two approaches to this problem. The first approach ignores the problem, which is unsatisfactory practice. The second approach is to ask - what happened to the composition unit and trace the problem back through the equations.
[2] The impact of composition units in quantitative analysis of data was addressed by E. A. Guggenheim, Trans. Faraday Soc., 1937,33,607.
1.4.04: Chemical Equilibria- Solutions- Ion Association
A given equilibrium in aqueous solution involves association of two ions to form a neutral solute.
$\text { Thus, } \quad \mathrm{M}^{+}(\mathrm{aq})+\mathrm{X}^{-}(\mathrm{aq}) \rightleftharpoons \mathrm{Z}(\mathrm{aq})$
The chemical equilibrium is described in terms of chemical potentials using the following equation in which we recognise that the reactant is a 1:1 salt.
$\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right]^{\mathrm{eq}} = \mu^{0}(Z ; \mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}(\mathrm{Z}) \, \gamma(\mathrm{Z}) / \mathrm{m}^{0}\right]^{\mathrm{cq}}$
In the latter equation, $\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)$ is the mean ionic activity for salt $\mathrm{M}^{+} \mathrm{X}^{-}$ in the aqueous solution.
By definition, at fixed temperature $\mathrm{T}$ and pressure $p$ where this pressure is ambient and hence close to the standard pressure $p^{0}$,
$\Delta_{\mathrm{r}} \mathrm{G}^{0}=\mu^{0}(\mathrm{Z} ; \mathrm{aq})-\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}^{0}$
where,
$\mathrm{K}^{0}=\left[\frac{\mathrm{m}(\mathrm{Z}) \, \gamma(\mathrm{Z}) / \mathrm{m}^{0}}{\mathrm{~m}^{2}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}^{2}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) /\left(\mathrm{m}^{0}\right)^{2}}\right]^{e q}$
In many cases, particularly for dilute solutions $\gamma(\mathrm{Z})$ is approximately unity but rarely can one ignore the term $\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)$.
$\text { By definition, } \mathrm{K}^{0}(\mathrm{app})=\left\{\mathrm{m}(\mathrm{Z}) \, \mathrm{m}^{0} /\left[\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right]^{2}\right\}^{\mathrm{eq}}$
$\text { Then, } \ln \mathrm{K}^{0}(\text { app })=\ln \mathrm{K}^{0}+2 \ln \left[\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right]$
The solution may be so dilute that the mean ionic activity coefficient can be calculated using the Debye-Huckel Limiting Law (DHLL).
$\text { Hence } \quad \ln \mathrm{K}^{0}(\mathrm{app})=\ln \mathrm{K}^{0}-2 \, \mathrm{S}_{\gamma} \,\left[\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right] 11 / 2$
In this equation the negative sign signals that in real solutions the extent of ion association to form $\mathrm{Z}(\mathrm{aq})$ is less than in the corresponding ideal solution because charge - charge interactions in real solutions stabilise the ions.
1.4.05: Chemical Equilibria- Solutions- Sparingly Soluble Salt
A given aqueous solution contains a sparingly soluble 1:1 salt [e.g. $\operatorname{AgCl}(\mathrm{s})$] at fixed temperature and pressure. The following phase equilibrium is established.
$\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}) \rightleftharpoons \mathrm{M}^{+}(\mathrm{aq})+\mathrm{X}^{-}(\mathrm{aq}) \label{a}$
In terms of the Phase Rule, the system contains two components, water and sparingly soluble substance $\mathrm{MX}$; $\mathrm{C} = 2$. There are three phases: solution, vapour and solid. Then $\mathrm{F} = 1$. Hence if we define the temperature, the vapour pressure and the equilibrium composition of the liquid phase are defined.
A thermodynamic description of equilibrium (Equation \ref{a}) is based on equality of chemical potentials of reactants and products. The key point is that the solid, $\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s})$ is a reference state.
$\text { Hence, } \mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{s}\right)=\mu^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)$
Noting that $\mathrm{M}^{+} \mathrm{X}^{-}$ is a 1:1 salt,
\begin{aligned} &\mu^{0}(\mathrm{MX} ; \mathrm{s})= \ &\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right)^{\mathrm{eq}} \end{aligned}
The solubility product for salt $\mathrm{M}^{+} \mathrm{X}^{-}$, $\mathrm{K}_{\mathrm{s}}^{0}$ is defined as follows. [$\mathrm{K}_{\mathrm{s}}^{0}$ is dimensionless.]
$\Delta_{\mathrm{s}} \mathrm{G}^{0}=\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)-\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{s}\right)=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{s}}^{0}\right)$
$\text { Hence }[1], \mathrm{K}_{\mathrm{s}}^{0}=\left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} \, \gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right]^{2}$
$\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}}$ is the (equilibrium) solubility, a quantity obtained experimentally.
$\ln \left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]=-\ln \left(\gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right)+(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}^{0}\right)$
In many cases salt $\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}))$ is so sparingly soluble that $\ln \left(\gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right)$ can be calculated using the Debye-Huckel Limiting Law (DHLL). The DHLL relates $\ln \left(\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right)^{e q}$ to the ionic strength of the solution, I. The ionic strength is controlled by adding a second soluble salt $\mathrm{N}^{+} \mathrm{Y}^{-}$.
$\ln \left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]=\mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}+(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}^{0}\right)$
This is a classic equation [2] because in many cases $\ln \left[\mathrm{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]$ is a linear function of $\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}$ so that $\ln \left(\mathrm{K}_{\mathrm{S}}^{0}\right)$ is obtained from the corresponding intercept. We understand the form of equation (g) in terms of increasing stabilisation of the ions $\mathrm{M}^{+}(\mathrm{aq})$ and $\mathrm{X}^{-}(\mathrm{aq})$ in solution by the ion-ion interactions in the real solution which are enhanced when the ionic strength is increased by adding soluble salt $\mathrm{N}^{+} \mathrm{Y}^{-}$.
Footnotes
[1] $\mathrm{K}_{\mathrm{s}}^{0}$ is dimensionless. However in many reports a quantity $\mathrm{K}_{\mathrm{m}}^{0}$ is defined as follows.
$\mathrm{K}_{\mathrm{m}}^{0}=\left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{e q} \, \gamma_{\pm}^{\mathrm{eq}}\right]^{2}$
For a 1:1 salt, $\mathrm{K}_{\mathrm{m}}^{0}$ has units, $\left(\mathrm{mol kg}^{-1}\right)^{2}$. Or $\mathrm{K}_{\mathrm{s}}^{0}=\mathrm{K}_{\mathrm{m}}^{0} /\left(\mathrm{m}^{0}\right)^{2}$
[2] All three terms in equation (g) are dimensionless.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.04%3A_Chemical_Equilibria/1.4.03%3A_Chemical_Equilibria-_Solutions-_Simple_Solutes.txt
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We comment on the term ‘equilibrium constant‘ where the composition of the solution under examination is expressed in terms of solute molalities, solute concentrations and solute mole fractions. [1-5] We consider a closed system in which the solvent is water($\lambda$) at defined temperature and pressure where the pressure is ambient and close to the standard pressure $p^{0}$. The solution contains $n_{1}$ moles of water and ${n_{j}}^{\mathrm{eq}}$ moles of each chemical substance $j$, solutes, where $j$ the composition is described in terms of a chemical equilibrium. The latter is described in the following general terms where $ν_{j}$ is the stoichiometry for chemical substance $j$, being positive for products and negative for reactants.
$\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{e q}(a q)=0$
The sum is taken over all $i$-solutes in solution with respect to the equilibrium chemical potentials of each chemical substance $j$.
Molalities
The equilibrium molality of solute $j$ is given by the ratio $\left(\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right)$ where $\mathrm{M}_{1}$ is the molar mass of water. From equation (a),
$\sum_{j=1}^{j=i} v_{j} \,\left[\mu_{j}^{0}(a q ; T)+R \, T \, \ln \left(m_{j}^{\mathrm{eq}} \, \gamma_{j}^{e q} / m^{0}\right)\right]=0$
By definition for each solute $j$,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p}$
A standard equilibrium constant ${\mathrm{K}_{\mathrm{m}}}^{0}$ is defined using equation (d) where the $\mathrm{m}$ subscript ‘m’ is a reminder that we are using molalities to express the composition of the solution under examination.
$\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{~T} ; \mathrm{m}-\text { scale })=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})\right]=\sum_{\mathrm{j}=1}^{\mathrm{ji}} \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})$
At temperature $\mathrm{T}$, $\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})$ is related to the equilibrium composition of the solution using equation (e).
$\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})=\prod_{\mathrm{j}=1}^{\mathrm{ji}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}} \, \gamma_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{v(\mathrm{j})}$
$\text { Also by definition, } \mathrm{pK}_{\mathrm{m}}^{0}(\mathrm{~T})=-\lg \left[\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})\right]$
From the Gibbs –Helmholtz Equation ,
$\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{m}}^{0}=-\mathrm{T}^{2} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{m}}^{0}}{\mathrm{~T}}\right)\right]_{\mathrm{p}}=\mathrm{R} \, \mathrm{T}^{2} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{m}}^{0}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}=-\mathrm{R} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{m}}^{0}\right)}{\partial \mathrm{T}^{-1}}\right]_{\mathrm{p}}$
$\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{pm}}^{0}(\mathrm{~T})=\left[\partial \Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{m}}^{0} / \partial \mathrm{T}\right]_{\mathrm{p}}$
$\Delta_{\mathrm{r}} \mathrm{S}_{\mathrm{m}}^{0}=\mathrm{T}^{-1} \,\left[\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{m}}^{0}-\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{m}}^{0}\right]$
The algebra is a little tortuous but the points are clearly made if we confine attention to chemical equilibria in solutions having thermodynamic properties which are ideal. For a chemical equilibrium involving $j$ chemical substances in solution where the solvent is chemical substance 1, at fixed temperature and pressure,
$\mathrm{K}^{0}=\prod_{\mathrm{j}=2}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{\mathrm{v}_{\mathrm{i}}}$
$\text {Also } \quad \mathrm{K}_{\mathrm{m}}^{0}=\prod_{\mathrm{j}=2}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\right)^{v_{j}}$
$\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}^{0}$
$\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left[\prod_{\mathrm{j}=2}^{\mathrm{j}=\mathrm{i}}\left(\frac{\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}}{\mathrm{m}^{0}}\right)^{\mathrm{v}_{\mathrm{j}}}\right]$
$\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left[\prod_{\mathrm{j}=2}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\right)^{\mathrm{v}_{\mathrm{j}}}\right]+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)$
$\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}_{\mathrm{m}}^{0}+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)$
$\Delta_{\mathrm{r}} \mathrm{G}^{0}=\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{m}}^{0}+\mathrm{V} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)$
$\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{m}}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}_{\mathrm{m}}^{0}$
This rather dull analysis has merit in showing that the lost units in the equation $\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}_{\mathrm{m}}^{0}$ are found in the term $\left[V \, R \, T \, \ln \left(m^{0}\right)\right]$ where $V=\sum_{j=2}^{j=i} v_{j}$. This concern arises because for correct dimensions the logarithm operation should operate on a pure number. No such problems emerge if $ν$ is zero as is the case for a stoichiometrically balanced equilibrium. Moreover if we probe the dependence of $\mathrm{K}^{0}$ or ${\mathrm{K}_{\mathrm{m}}}^{0}$ on temperature we have that,
$\frac{\mathrm{d}}{\mathrm{dT}} \,\left[\frac{\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)}{\mathrm{T}}\right]=0$
If we are interested in the dependence of either $\mathrm{K}^{0}$ or ${\mathrm{K}_{\mathrm{m}}}^{0}$ on pressure, we have that,
$\frac{\mathrm{d}}{\mathrm{dp}} \,\left[\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)\right]=0$
Mole Fraction Scale
The total amount of all chemical substances in the closed system, an aqueous solution, at chemical equilibrium is given by equation (t).
$n_{\mathrm{T}}^{\mathrm{eq}}=\mathrm{n}_{1}+\sum \mathrm{n}_{\mathrm{j}}^{\mathrm{eq}}$
For a given chemical substance, solute $k$
$\mathrm{x}_{\mathrm{k}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{k}}^{\mathrm{eq}} /\left[\mathrm{n}_{1}+\sum \mathrm{n}_{\mathrm{j}}^{e q}\right]$
In terms of mole fractions, the equilibrium chemical potential for solute $j$, $\mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ is related to the equilibrium mole fraction ) $\mathrm{x}_{\mathrm{j}}^{e q}(\mathrm{aq})$ using equation (v).
$\mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{x}-\mathrm{scale} ; \mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{f}_{\mathrm{j}}^{*_{\mathrm{eq}}}\right)$
$\text { By definition, at all } \mathrm{T} \text { and } \mathrm{p} \text {, } \operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{f}_{\mathrm{j}}^{*}=1.0 \text {. }$
Further,
$\mu_{j}^{0}(\mathrm{x}-\text { scale; aq; } \mathrm{T})$
is the chemical potential of substance $j$ in aqueous solution at temperature $\mathrm{T}$ in a solution where the mole fraction of solute $j$ is unity. Here therefore $\mu_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale; aq; } \mathrm{T})$ is the reference chemical potential. From equation (a),
$\sum_{j=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale; aq; } \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{f}_{\mathrm{j}}^{* \mathrm{eq}}\right)\right]=0$
$\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{x}}^{0}(\mathrm{~T})=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})\right]=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} v_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale; aq; } \mathrm{T})$
(T) R T ln[K (T)] (x scale;aq;T) At temperature $\mathrm{T}$, $\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})$ is related to the equilibrium mole fractions of the solutes.
$\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})=\prod_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{x}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{f}_{\mathrm{j}}^{* \mathrm{eq}}\right)^{v(\mathrm{j})}$
From the Gibbs –Helmholtz Equation,
$\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{x}}^{0}=-\mathrm{T}^{2} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{x}}^{0}}{\mathrm{~T}}\right)\right]_{\mathrm{p}}=\mathrm{R} \, \mathrm{T}^{2} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{x}}^{0}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}=-\mathrm{R} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{x}}^{0}\right)}{\partial \mathrm{T}^{-1}}\right]_{\mathrm{p}}$
$\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{px}}^{0}(\mathrm{~T})=\left[\partial \Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{x}}^{0} / \partial \mathrm{T}\right]_{\mathrm{p}}$
$\Delta_{\mathrm{r}} \mathrm{S}_{\mathrm{x}}^{0}=\mathrm{T}^{-1} \,\left[\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{x}}^{0}-\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{x}}^{0}\right]$
We note a complication. We suppose that the composition of a given closed system, an aqueous solution, is described in terms of the formation of a dimer by a solute $\mathrm{Z}$ in aqueous solution at defined $\mathrm{T}$ and $p$
$2 Z(a q) \Longleftrightarrow Z_{2}(a q)$
At equilibrium, the solution contains n1 moles of water, $\mathrm{n}^{\mathrm{eq}}(\mathrm{Z})$ moles of monomer and $\mathrm{n}^{\mathrm{eq}}\left(Z_{2}\right)$ moles of dimer.
$x(Z)^{e q}=n(Z)^{e q} /\left[n_{1}+n(Z)^{e q}+n\left(Z_{2}\right)^{e q}\right]$
$x\left(Z_{2}\right)^{\text {eq }}=n\left(Z_{2}\right)^{\text {eq }} /\left[n_{1}+n(Z)^{e q}+n\left(Z_{2}\right)^{e q}\right]$
$\text { Further, } x_{1}^{e q}=n_{1} /\left[n_{1}+n(Z)^{e q}+n\left(Z_{2}\right)^{e q}\right]$
As a result of a change in temperature $\mathrm{n}^{\mathrm{eq}}(\mathrm{Z})$ and $\mathrm{n}^{\mathrm{eq}}\left(\mathrm{Z}_{2}\right)$ change; $n_{1}$ does not. For example $x(Z)^{e q}$ changes as a result of changes in both numerator and denominator in equation (ze). This unwelcome complication is not encountered if we use the molality scale. The way forward is to confine attention to dilute solutions such that at all temperatures, $\sum_{j=2}^{j=i} n_{j}^{e q}<<n_{1}$.
Mole Fraction and Molality Scales
The complication noted in conjunction with equation (zd) also emerges when equilibrium constants on these two scales are compared. If the thermodynamic properties of the solutions are ideal, we obtain the following two equations.
$\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})=\prod_{\mathrm{j=1}}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{v(\mathrm{j})}$
$\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})=\prod_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{x}_{\mathrm{j}}^{\mathrm{eq}}\right)^{\mathrm{v}(\mathrm{j})}$
$\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T}) / \mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})=\prod_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}^{\mathrm{eq}}\right)^{v(\mathrm{j})}$
But $\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{w}_{1}=\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{n}_{1} \, \mathrm{M}_{1}$
For dilute solutions $\mathrm{x}_{\mathrm{j}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{n}_{1}$ $\frac{m_{j}}{m^{0} \, x_{j}^{e q}}=\frac{n_{j}^{e q}}{n_{1} \, M_{1} \, m^{0}} \, \frac{n_{1}}{n_{j}^{e q}}$
$\prod_{\mathrm{j}=1}^{\mathrm{j}=i}\left[\frac{\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}}}{\mathrm{n}_{1} \, \mathrm{M}_{1} \, \mathrm{m}^{0}} \, \frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}}}\right]^{v(j)}=\prod_{\mathrm{j}=1}^{\mathrm{j}=1}\left[\frac{1}{\mathrm{M}_{1} \, \mathrm{m}^{0}}\right]^{\mathrm{v}(\mathrm{j})}$
The product $\left(\mathrm{M}_{1} \, \mathrm{m}^{0}\right)$ is dimensionless; i.e. $\left(\mathrm{kg} \mathrm{mol}^{-1}\right) \,\left(\mathrm{mol} \mathrm{kg}^{-1}\right)$.
$\text { Hence, } \mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})=\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T}) \, \prod_{\mathrm{j=1}}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{M}_{1} \, \mathrm{m}^{0}\right)^{v(j)}$
Consider a chemical equilibrium which has the following form,
$\mathrm{A}(\mathrm{aq}) \Leftrightarrow=\mathrm{B}(\mathrm{aq})$
$\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})=\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T}) \,\left(\mathrm{M}_{1} \, \mathrm{m}^{0}\right)^{+1} \,\left(\mathrm{M}_{1} \, \mathrm{m}^{0}\right)^{-1}$
$\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})=\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})$
In fact for all symmetric equilibria in dilute solution, the numerical values of $\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})$ and $\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})$ are equal.
Concentration Scale
The main advantage from a thermodynamic standpoint in expressing the composition of a solution in terms of molalities, $\mathrm{m}_{j}$ is the fact that the definition does not require specification of either temperature or pressure. The latter is a consequence of using a definition based on the masses of solvent and solute. Nevertheless from a practical standpoint there are many advantages in expressing the composition of a solution in terms of concentration $\mathrm{c}_{\mathrm{j}}\left(=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\right.$ where $\mathrm{V}$ is the volume of a solution). Many experimental techniques are based on ‘counting’ the amount of solute $j$ in a given volume of system. This is certainly true of uv/visible spectrophotometric methods based on Beer’s Law. Similarly electrical conductivities count the amount of conducting ions in a given volume of solution. Further when these methods are used, the actual densities of the solutions are only rarely measured so that conversion from concentration $\mathrm{c}_{j}$ to molality $\mathrm{m}_{j}$ is not straightforward.
We may find it convenient to characterise the composition of a given aqueous solution in terms of equilibrium concentrations of each solute $j$. Therefore the volume of the aqueous solution $\mathrm{V}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ is given by equation (zp).
$\mathrm{V}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ;)+\sum_{\mathrm{j}=2}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{V}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$
$\text { For solute } \mathrm{j}, \mathrm{c}_{\mathrm{j}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{V}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$
At this point a problem emerges. Even in the event that $\mathbf{n}_{\mathrm{j}}^{\mathrm{eq}}$ does not change when the temperature is changed, $\mathbf{c}_{\mathrm{j}}^{\mathrm{eq}}$ changes because for real solutions $\mathrm{V}^{e q}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ is dependent on temperature. Furthermore at fixed $\mathrm{T}$, $p$ and composition, the volume of the corresponding ideal solution differs from $\mathrm{V}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ and hence $\mathrm{c}_{\mathrm{j}}^{\mathrm{id}} \neq \mathrm{c}_{\mathrm{j}}^{\mathrm{eq}}$. The way forward explores chemical equilibria in very dilute solutions.
$\sum_{j=2}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{V}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) \ll \mathrm{n}_{1} \, \mathrm{V}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ;)$
$\text { Hence, } \quad V^{\text {eq }}(\mathrm{aq} ; T ; p)=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda ; \mathrm{T} ; \mathrm{p})$
$\text { Therefore, } \mathrm{c}_{\mathrm{j}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda)$
We express the chemical potential $\mu_{j}^{e q}(\mathrm{aq})$ using equation (zu).
$\mu_{j}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{c}-\text { scale; } \mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{y}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right)$
Here $\mathrm{c}_{\mathrm{r}}=1 \mathrm{~mol dm}^{-3}$, $\mathrm{c}_{\mathrm{j}}^{\mathrm{eq}}$ being expressed using the unit, $\text { mol } \mathrm{dm}^{-3}$; $\mathrm{y}_{\mathrm{j}}^{\mathrm{eq}}$ is the activity coefficient for solute $j$ on the concentration scale, describing the impact of solute-solute interactions in the aqueous solution. By definition at all $\mathrm{T}$ and $p$,
$\operatorname{limit}\left(c_{j} \rightarrow 0\right) y_{j}=1.0$
From equation (a),
$\sum_{j=1}^{j=i} v_{j} \,\left[\mu_{j}^{0}(c-s c a l e ; a q ; T)+R \, T \, \ln \left(c_{j}^{e q} \, y_{j}^{e q} / c_{r}\right)\right]=0$
By definition,
$\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{c}}^{0}(\mathrm{~T})=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{K}_{\mathrm{c}}(\mathrm{T})\right]=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}(\mathrm{c}-\text { scale; aq; } \mathrm{T})$
$K_{c}(T)=\prod_{j=1}^{j=i}\left(c_{j}^{e q} \, y_{j}^{e q} / c_{f}\right)^{v(j)}$
For dilute solutions we might assume that $\mathrm{y}_{\mathrm{j}}^{\mathrm{eq}}$ is unity for all solutes.
$K_{c}(T)=\prod_{j=1}^{j=1}\left(c_{j}^{e q} / c_{r}\right)^{v(j)}$
From the Gibbs - Helmholtz Equation,
$\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{c}}^{0}=-\mathrm{T}^{2} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{c}}^{0}}{\mathrm{~T}}\right)\right]_{\mathrm{p}}=\mathrm{R} \, \mathrm{T}^{2} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{c}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}=-\mathrm{R} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{c}}\right)}{\partial \mathrm{T}^{-1}}\right]_{\mathrm{p}}$
$\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{pc}}^{0}(\mathrm{~T})=\left[\partial \Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{c}}^{0} / \partial \mathrm{T}\right]_{\mathrm{p}}$
$\Delta_{\mathrm{r}} \mathrm{S}_{\mathrm{c}}^{0}=\mathrm{T}^{-1} \,\left[\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{c}}^{0}-\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{c}}^{0}\right]$
Molality and Concentration Scales
If the thermodynamic properties of the solutions are ideal, we obtain the following two equations.
$\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})=\prod_{\mathrm{j}=1}^{\mathrm{ji}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{v(\mathrm{j})}$
$K_{c}^{0}(T)=\prod_{j=1}^{j=i}\left(c_{j}^{e q} / c_{r}\right)^{v(j)}$
$\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T}) / \mathrm{K}_{\mathrm{c}}^{0}(\mathrm{~T})=\prod_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{c}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{m}^{0}\right)^{\mathrm{v}(\mathrm{j})}$
But $\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{w}_{1}=\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{n}_{1} \, \mathrm{M}_{1}$
$\text { And for dilute solutions, } \mathrm{c}_{\mathrm{j}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{V}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}$
$\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{j}}^{\mathrm{eq}}=\mathrm{V}_{1}^{*}(\lambda) / \mathrm{M}_{1}=1 / \rho_{1}^{*}(\lambda)$
By definition, $\rho^{0}=\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0}=\left[\mathrm{mol} \mathrm{dm}^{-3}\right] /\left[\mathrm{mol} \mathrm{kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{dm}^{-3}\right]$
By introducing the property $\rho^{0}$ we overcome the problem with units.
$\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T}) / \mathrm{K}_{\mathrm{c}}^{0}(\mathrm{~T})=\prod_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\rho_{1}^{*}(\lambda) / \rho^{0}\right)^{\mathrm{v}(\mathrm{j})}$
A scale factor of $10^{3}$ often occurs in these equations because concentrations are conventionally quoted using the unit, $\mathrm{mol dm}^{-3}$ and densities are expressed using the unit, $\mathrm{g cm}^{-3}$.
In summary care should be taken when using equilibrium constants based on different composition scales. [1-4] A classic example [5] from the field of kinetics shows how misleading conclusions can be drawn from, for example, comparison of standard entropies of activation and reaction expressed using different composition scales. [6]
Footnotes
[1] L. Hepler, Thermochim Acta,1981,50,69.
[2] E. A. Guggenheim, Trans. Faraday Soc.,1937,33,607.
[3] E. Euranto, J.J.Kankare and N.J.Cleve, J. Chem. Eng. Data,1969, 14, 455.
[4] R. W. Gurney, Ionic Processes in Solution , McGraw-Hill, New York, 1952.
[5] As Guggenheim remarked the units of $\ln (\mathrm{V})$ are $\ln \left(\mathrm{m}^{3}\right)$. If $\mathrm{V}=100 \mathrm{~m}^{3}$, then $\log (\mathrm{V})=\log (100)+\log \left(\mathrm{m}^{3}\right)$, $\log (\mathrm{V})-\log \left(\mathrm{m}^{3}\right)=2$; $\log \left(\mathrm{V} / \mathrm{m}^{3}\right)=2$.
[6] M. H. Abraham and A. Nasehzadeh, J. Chem. Soc. Chem. Commun., 1981, 905.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.04%3A_Chemical_Equilibria/1.4.06%3A_Chemical_Equilibria-_Cratic_and_Unitary_Quantities.txt
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The variable $\xi$ describes in quite general terms molecular composition – molecular organisation. For a given closed system at fixed $\mathrm{T}$ and $p$ there exists a composition-organisation $\xi_{\mathrm{eq}$ corresponding to a minimum in Gibbs energy where the affinity for spontaneous change is zero. In general terms there exists an extent of reaction $\xi$ corresponding to a given affinity $\mathrm{A}$ at defined $\mathrm{T}$ and $p$. In fact we can express $\xi$ as a dependent variable defined by the independent variables, $\mathrm{T}$, $p$ and $\mathrm{A}$.
$\xi=\xi[T, p, A]$
The general differential of equation (a) takes the following form.
$\mathrm{d} \xi=\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dT}+\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}} \, \mathrm{dp}+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA}$
Equation (b) describes the dependence of extent of reaction on changes in $\mathrm{T}$, $p$ and affinity $\mathrm{A}$.
$\text { Moreover, }\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}}=-\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right]$
At equilibrium where ‘$\mathrm{A} = 0$’ and $(\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0$, then
$\left(\frac{\partial \xi}{\partial T}\right)_{p, A=0} \text { takes the sign of }\left[\frac{1}{T} \,\left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{e q}\right]=\left[\frac{1}{T} \, \sum_{j=1}^{j=i} v_{j} \, H_{j}^{e q}\right]$
$\text { Similarly, }\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \text { takes the sign of }\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}}$
Again at equilibrium where ‘$\mathrm{A} = 0$’ and $(\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0$, then
$\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \text { takes the sign of }\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=-\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{\mathrm{eq}}$
Equations (d) and (f) are important being universally valid and forming the basis of important generalisations, the Laws of Moderation.
Equation (d) shows that the differential dependence of composition on temperature is related to the enthalpy of reaction. If the chemical reaction is exothermic {i.e. $\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}$ is negative}, the chemical equilibrium shifts to favour an increase in the amount of reactants. Whereas if the reaction is endothermic , the composition swings in a direction to favour the products.
In another experiment, the equilibrium system is perturbed by an increase in pressure. Equation (f) shows that the equilibrium composition swings to favour the reactants if the volume of reaction is positive. Alternatively if the volume of reaction is negative, the composition of the system changes to favour products [1].
Footnotes
[1] The conclusions reached here are called ‘Theorems of Moderation’. MJB was taught that the outcome is ‘Nature’s Laws of Cussedness’ [ = obstinacy]. An exothermic reaction generates heat to raise the temperature of the system, so the system responds, when the temperature is raised, by shifting the equilibrium in the direction for which the process is endothermic. The line of argument is not good thermodynamics but it makes the point.
1.4.08: Chemical Equilibrium Constants- Dependence on Temperature at Fi
A given set of data reports the dependence on temperature (at fixed pressure $p$, which is close to the standard pressure $p^{0}$) of $\mathrm{K}^{0}$ for a given chemical equilibrium.[1 - 3]
$\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \mathrm{T} \ln \mathrm{K}^{0}=\Delta \mathrm{H}^{0}-\mathrm{T} \Delta_{\mathrm{r}} \mathrm{S}^{0}$
If we confine our attention to systems where the chemical equilibria involve solutes in dilute solution in a given solvent, we can replace $\Delta_{\mathrm{r}} \mathrm{H}^{0}$ in this equation with the limiting enthalpy of reaction, $\Delta_{\mathrm{r}} \mathrm{H}^{\infty}$. According to the Gibbs - Helmholtz Equation, at fixed pressure,
$\dfrac{ \mathrm{d}\left[\Delta_{\mathrm{r}} \mathrm{G}^{0} / \mathrm{T}\right] }{ \mathrm{dT}} =- \dfrac{ \Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{T}^{2}}$
Hence
$\dfrac{ \mathrm{d} \ln \left(\mathrm{K}^{0}\right) }{\mathrm{dT}} = \dfrac{ \Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{R} \mathrm{T}^{2}}$
or, [4]
$\dfrac{ \mathrm{d} \ln \mathrm{K}^{0} }{\mathrm{dT}^{-1}} =- \dfrac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{R} }$
The latter two equations are equivalent forms of the van ’t Hoff Equation expressing the dependence of $\mathrm{K}^{0}$ on temperature. This equation does not predict how equilibrium constants depend on temperature. For example the van’t Hoff equation does not require that $\ln \left(\mathrm{K}^{0}\right)$ is a linear function of $\mathrm{T}-1$. In fact for simple carboxylic acids, the plots of $\ln (\text {acid dissociation constant})$ against temperature show maxima. For example, $\ln \left(\mathrm{K}^{0}\right)$ for the acid dissociation constant of ethanoic acid in aqueous solution at ambient pressure increases with increase in temperature, passes through a maximum near $295 \mathrm{~K}$ and then decreases. [5-7] At the temperature where $\mathrm{K}^{0}$ is a maximum, the limiting enthalpy of dissociation is zero. This pattern is possibly surprising at first sight but can be understood in terms of a balance between the standard enthalpy of heterolytic fission of the $\mathrm{O}-\mathrm{H}$ group in the carboxylic acid group and the standard enthalpies of hydration of the resulting hydrogen and carboxylate ions.
Thus the dependence of $\mathrm{K}^{0}$ on temperature can be obtained experimentally, the dependence being unique for each system [8]. Nevertheless these equations signal how the dependence forms the basis for determining limiting enthalpies of reaction. The analysis also recognises that $\Delta_{\mathrm{r}} \mathrm{H}^{0}$ is likely to depend on temperature. There is merit in expressing the dependence of $\mathrm{K}^{0}$ on temperature about a reference temperature $\theta$, chosen near the middle of the experimental temperature range [2,3,9]. Over the experimental temperature range straddling $\theta$, we express the dependence of $\mathrm{K}^{0}$ on temperature using the integrated form of equation (c).
$\ln \left[\mathrm{K}^{0}(\mathrm{~T})\right]=\ln \left[\mathrm{K}^{0}(\theta)\right]+\int_{\theta}^{\mathrm{T}}\left[\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\omega}}{\mathrm{RT}^{2}}\right] \mathrm{dT}$
By definition, the limiting isobaric heat capacity of reaction $\Delta_{\mathrm{r}} C_{\mathrm{p}}^{\infty}$ is given by equation (f).
$\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty}=\left( \dfrac{ \mathrm{d} \Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{dT}}\right)_{\mathrm{p}}$
The analysis becomes complicated because we recognise that $\Delta_{\mathrm{r}} C_{\mathrm{p}}^{\infty}$ depends on temperature. [9] In fact only in rare instances are experimental results sufficiently precise to warrant taking such a dependence into account. [10] A reasonable assumption is that $\Delta_{r} C_{p}^{\infty}$ is independent of temperature such that $\Delta_{\mathrm{r}} \mathrm{H}^{\infty}$ is a linear function of temperature over the experimental temperature range.[11]
$\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{T})=\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)+\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty}(\mathrm{T}-\theta)$
Hence,
\begin{aligned} &\ln \left[\mathrm{K}^{0}(\mathrm{~T})\right]= \ &\ln \left[\mathrm{K}^{0}(\theta)\right]+\frac{1}{\mathrm{R}} \int_{\theta}^{\mathrm{T}}\left[\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)}{\mathrm{T}^{2}}+\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty} \left(\frac{1}{\mathrm{~T}}-\frac{\theta}{\mathrm{T}^{2}}\right)\right] \mathrm{dT} \end{aligned}
Hence,
\begin{aligned} &\ln \left[\mathrm{K}^{0}(\mathrm{~T})\right]= \ &\ln \left[\mathrm{K}^{0}(\theta)\right]+\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)}{\mathrm{R}} \left[\frac{1}{\theta}-\frac{1}{\mathrm{~T}}\right]+\frac{\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty}}{\mathrm{R}} \left[\frac{\theta}{\mathrm{T}}-1+\ln \left(\frac{\mathrm{T}}{\theta}\right)\right] \end{aligned}
Numerical analysis uses linear least squares procedures with reference to the dependence of $\ln K^{0}(T)$ on temperature about the reference temperature $\theta$ in order to obtain estimates of $\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)$ and $\Delta_{r} C_{p}^{\infty}$. The coupling of estimates of derived parameters is minimal if θ is chosen near the centre of the measured temperature range. [2,3] Granted that the analysis yields $\Delta_{\mathrm{r}} \mathrm{H}^{\infty}$ at a given temperature and pressure, combination with the corresponding $\Delta_{\mathrm{r}} \mathrm{G}^{0}$ yields the entropy term, $\Delta_{\mathrm{r}} \mathrm{S}^{0}$.
Other methods of data analysis in this context use (a) orthogonal polynomials, [12] and (b) sigma plots. [13]
An extensive literature describes the thermodynamics of acid dissociation in alcohol + water mixtures. In these solvent systems the standard enthalpies and other thermodynamic parameters pass through extrema as the mole fraction composition of the solvent is changed. [14- 18]
Perlmutter-Hayman examines the related problem of the dependence on temperature of activation energies [19].
Enthalpies of dissociation for weak acids in aqueous solution can be obtained calorimetrically. [20]
Footnotes
[1] R. W. Ramette, J. Chem. Educ.,1977,54,280
[2] M. J. Blandamer, J. Burgess, R. E. Robertson and J. M. W. Scott, Chem. Rev., 1982, 82,259.
[3] M. J. Blandamer, Chemical Equilibria in Solution, Ellis Horwood PTR Prentice Hall, New York,1992.
[4] $\mathrm{d} \ln \mathrm{K}^{0} / \mathrm{dT}^{-1}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] /\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]=[\mathrm{K}]$
[5] H. S. Harned and N. D. Embree, J. Am. Chem. Soc.,1934,56,1050.
[6] H. S. Harned and R. W. Ehlers, J.Am.Chem.Soc.,1932,54,1350.
[7] See also ethanoic acid in D2O; M. Paabo, R. G. Bates and R. A. Robinson, J. Phys. Chem., 1966,70,2073; and references therein.
[8] Substituted benzoic acids(aq); L. E. Strong, C. L. Brummel and P. Lindower, J. Solution Chem., 1987, 16, 105; and references therein.
[9] E. C. W. Clarke and D. N. Glew, Trans. Faraday Soc.,1966,62,539.
[10] H. F. Halliwell and L. E. Strong, J. Phys. Chem.,1985,89,4137.
[11] $\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)=\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] [\mathrm{K}]$
[12] D. J. G. Ives and P. D. Marsden, J. Chem. Soc.,1965,649 and 2798.
[13] D. J. G. Ives, P. G. N. Moseley, J. Chem. Soc. Faraday Trans.1, 1976,72,1132.
[14] Anilinium ions in EtOH+water mixtures;W. van der Poel, Bull. Soc. Chim. Belges., 1971,80,401; and references therein.
[15] Enthalpies of transfer for carboxylic acids in water+ 2-methyl propan-2-ol mixtures; L. Avedikian, J. Juillard and J.-P. Morel, Thermochim. Acta, 1973,6,283.
[16] Benzoic acid in DMSO + water mixtures; F. Rodante, F. Rallo and P. Fiordiponti, Thermochim. Acta, 1974, 9,269.
[17] Tris in water + methanol mixtures; C. A. Vega, R. A. Butler, B. Perez and C. Torres, J. Chem. Eng. Data, 1985,30,376.
[18] F. J. Millero, C-h. Wu and L. G. Hepler, J. Phys. Chem., 1969, 73,2453.
[19] B. Perlmutter-Hayman, Prog. Inorg. Chem.,1976,20,229.
[20] F. Rodante, G. Ceccaroni and F. Fantauzzi, Thermochim. Acta,1983,70,91.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.04%3A_Chemical_Equilibria/1.4.07%3A_Chemical_Equilibria-_Composition-_Temperature_and_Pressure_Depe.txt
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A key quantity in the description of a chemical equilibrium is the equilibrium constant. In the majority of cases the symbol used is $\mathrm{K}^{0}$ indicating with the superscript ‘0’ a standard property. This symbol is used because, again in the majority of cases an equilibrium constant refers to a system at ambient pressure which is close to the standard pressure; i.e. $10^{5} \mathrm{~Pa}$. In reporting $\mathrm{K}^{0}$ therefore the temperature is stated but by definition $\mathrm{K}^{0}$ is not dependent on pressure. However the equilibrium composition of a closed system generally depends on pressure at fixed temperature $\mathrm{T}$. This problem over symbols and nomenclature is resolved as follows [1-7].
An aqueous solution contains $i$-chemical substances, solutes, in chemical equilibrium. For a given solute–$j$ the dependence of chemical potential $\mu_{j}(a q ; T ; p)$ on molality $\mathrm{m}_{j}$ is given by equation (a).
\begin{aligned} &\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})= \ &\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{\circ}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp} \end{aligned}
We define a reference chemical potential for solute-$j$ $\mu_{j}^{*}$ at temperature $\mathrm{T}$ and pressure $p$ using equation (b).
$\mu_{\mathrm{j}}^{\#}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}$
Combination of equations (a) and (b) yields equation (c).
$\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{\#}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
XK
Here $\mu_{j}^{H}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ is the chemical potential of solute-j in an ideal solution (i.e. $\gamma_{j}=1$) having unit molality (i.e. $\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}{ }^{-1}$) at specified $\mathrm{T}$ and $p$. At equilibrium at pressure $p$ and temperature $\mathrm{T}$,
$\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{\mathrm{eq}}(\mathrm{aq} ; T ; p)=0$
By definition,
$\Delta_{\mathrm{r}} \mathrm{G}^{*}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\sum \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{*}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}^{*}(\mathrm{~T}, \mathrm{p})$
$\text { and } \mathrm{K}^{\#}(\mathrm{~T}, \mathrm{p})=\prod_{\mathrm{j}=1}^{\mathrm{j}=1}\left[\left(\mathrm{~m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right) \, \gamma_{j}^{e q}\right]^{v(j)}$
The differential dependence on pressure [8,9] of $\mathrm{K}^{\#}(\mathrm{~T}, \mathrm{p})$ yields the limiting volume of reaction, $\Delta_{\mathrm{r}} \mathrm{V}^{\infty}$.
$\Delta_{r} V^{\infty}=\sum_{j=1}^{j=i} V_{j} \, V_{j}^{\infty}(a q ; T ; p)=0$
$\text { and [c.f. V } \left.=[\partial \mathrm{G} / \partial \mathrm{p}]_{\mathrm{T}}\right] \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\partial \mu_{\mathrm{j}}^{*}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{T}}$
$\text { Hence at pressure } p,\left(\frac{\partial \Delta_{\mathrm{r}} \mathrm{G}^{*}(\mathrm{~T})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{T}, \mathrm{p})$
$\text { or, }\left(\frac{\partial \ln \mathrm{K}^{\prime \prime}(\mathrm{T})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{T}, \mathrm{p})}{\mathrm{R} \, \mathrm{T}}$
The negative sign in equation (j) means that if $\ln \mathrm{K}^{\#}$ for a given chemical equilibrium increases with increases in pressure then $\Delta_{\mathrm{r}} \mathrm{V}^{\infty}$ is negative. But thermodynamics does not define how a given equilibrium constant depends on pressure. This dependence must be measured. Moreover we cannot assume that the limiting volume of reaction $\Delta_{\mathrm{r}} \mathrm{V}^{\infty}$ is independent of pressure. This dependence is described by the limiting isothermal compressions of reaction, $\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}$.
$\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}=-\left[\frac{\mathrm{d} \Delta_{\mathrm{r}} \mathrm{V}^{\infty}}{\mathrm{dp}}\right]_{\mathrm{T}}$
Indeed we cannot assume that $\mathrm{K}_{\mathrm{T}}^{\infty}$ is independent of pressure but in most cases the precision of the data is insufficient to obtain a meaningful estimate of this dependence. Hence we are often justified in assuming that $\Delta_{\mathrm{r}} \mathrm{V}^{\infty}$ is a linear function of pressure about a reference pressure $\pi$, the latter usually chosen as ambient pressure. [10]
$\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{p})=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)-\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}(\mathrm{p}-\pi)$
Hence, [11]
$\ln \left(\mathrm{K}^{\#}(\mathrm{p})\right)=\ln \left(\mathrm{K}^{\#}(\pi)\right)-(\mathrm{R} \, \mathrm{T})^{-1} \, \Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi) -(2 \, R \, T)^{-1} \, \Delta_{r} K_{T}^{\infty} \,\left((p-\pi)^{2}\right)$
Thus, $\ln \mathrm{K}^{\#}(\mathrm{p})$ is a quadratic in $(\mathrm{p}-\pi)$.
Alternatively we may express the dependence of $\ln \left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]$ on pressure using the following equation. [12,13]
\begin{aligned} &\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]= \ &\quad-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi)-0.5 \, \Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty} \,(\mathrm{p}-\pi)^{2} \end{aligned}
This equation shows how $\left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]$ may be calculated from estimates of $\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)$ obtained from independently obtained estimates of partial molar volumes and partial molar isothermal compressions of the chemical substances involved in the chemical equilibrium; e.g. acid dissociation of boric acid. [14,15]
Another approach expresses the ratio $\left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]$ as a function of solvent density at pressure $p$, $\rho(p)$ together with density $\rho(\pi)$ at pressure $\pi$ and a parameter $\beta$ using equation (o) [16].
$\ln \left[K^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]=(\beta-1) \, \ln [\rho(\pi) / \rho(\mathrm{p})]$
This approach is closely linked to numerical analysis based on equation (p). [17]
$\ln \left[K^{\#}(p) / K^{\#}(\pi)\right]=-\left[\Delta_{r} V^{\infty}(\pi) / R \, T\right] \,[p /(1+b \, p)]$
A rather different approach for chemical equilibria between solutes in aqueous solutions refers to equation (q). $\mathrm{A}$ and $\mathrm{B}$ are constants independent of pressure but dependent on temperature; these constants describe the dependence of the molar volume of water on pressure at fixed temperature; Tait’s isotherm [2,18-21].
$\mathrm{V}_{1}^{*}(\mathrm{p})=\mathrm{V}_{1}^{*}(\pi) \,\left[1-\mathrm{A} \, \ln \left(\frac{\mathrm{B}+\mathrm{p}}{\mathrm{B}+\pi}\right)\right]$
There are a few case where the experimental data warrant consideration of the dependence on pressure of $\Delta_{r} K^{\infty}$. Under these circumstances the Owen-Brinkley equation has the following form [2].
\begin{aligned} \mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{K}^{\#}(\mathrm{p})}{\mathrm{K}^{\#}(\pi)}\right]=-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi) \ &+\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}(\pi) \,\left[(\mathrm{B}+\pi) \,(\mathrm{p}-\pi)-(\mathrm{B}+\pi)^{2} \, \ln \left(\frac{\mathrm{B}+\mathrm{p}}{\mathrm{B}+\pi}\right)\right] \end{aligned}
Footnotes
[1] M. J. Blandamer, Chemical Equilibria in Solution, Ellis Horwood PTR, Prentice Hall, New York,1992.
[2] B. B. Owen and S. R. Brinkley, Chem. Rev.,1941,29,401.
[3] S. W. Benson and J. A. Person, J. Amer. Chem. Soc.,1962,84,152.
[4] S. D. Hamann, J. Solution Chem.,1982,11,63; and references therein.
[5] N. A. North, J.Phys.Chem.,1973,77,931.
[6] B. S. El’yanov and E. M. Vasylvitskaya, Rev. Phys. Chem. Jpn, 1980, 50, 169; and references therein.
[7] There is a strong link between this subject and analysis of the dependence of rate constants for chemical reactions on pressure at fixed temperature;
1. R. van Eldik and H. Kelm, Rev. Phys. Chem. Jpn,1980,50,145.
2. C. A. N. Viana and J. C. R. Reis, Pure Appl. Chem.,1996,68,1541.
3. E. Whalley, Adv. Phys.Org. Chem.,1964,2,93.
4. W. J. leNoble, J. Chem. Educ.,1967,44,729.
[8] By definition the standard equilibrium constant ${\mathrm{K}}_{\mathrm{m}}}^{0}$ describes the case where at temperature $\mathrm{T}$, the pressure is the standard pressure.
[9] $\frac{\mathrm{d} \ln \mathrm{K}^{*}}{\mathrm{dp}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}$
[10] $\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{p})=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]+\left(\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]}\right) \,\left[\mathrm{N} \mathrm{m}^{-2}\right]$
[11] $\ln \left(\mathrm{K}^{\#}(\mathrm{p})\right)=[1]+\frac{1}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right] +\frac{1}{[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~N}^{-1} \mathrm{~m}^{2}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{2}$
[12] D. A. Lown, H. R. Thirsk and Lord Wynne-Jones, Trans. Faraday Soc., 1968, 64, 2073.
[13] A. J. Read {J. Solution Chem.,1982, 11, 649;1988, 17, 213} uses a simpler form of the equation which has the general form, $y=m . x+c$. Thus, $\left[\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{p}-\pi}\right] \, \ln \left[\frac{\mathrm{K}^{\#}(\mathrm{p})}{\mathrm{K}^{\#}(\pi)}\right]=-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)-0.5 \,(\mathrm{p}-\pi) \, \Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}$
[14] G. K. Ward and F. J. Millero, J. Solution Chem.,1974,3,417.
[15] See also pH and pOH; Y. Kitamura and T. Itoh, J. Solution Chem., 1987, 16, 715.
[16] W. L. Marshall and R. E. Mesmer, J. Solution Chem., 1984, 13, 383; and references therein.
[17] B. S. El’yanov and S. D. Hamann, Aust. J. Chem.,1975,28,945.
[18] R. E. Gibson, J.Am.Chem.Soc.,1934,56,4.
[19] S. D. Hamann and F. E. Smith, Aust. J. Chem.,1971,24,2431.
[20] G. A. Neece and D. R. Squire, J. Phys. Chem.,1968,72,128.
[21] K. E. Weale, Chemical Reactions at High Pressure, Spon, London,1967; and references therein.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.04%3A_Chemical_Equilibria/1.4.09%3A_Chemical_Equilibria-_Dependence_on_Pressure_at_Fixed_Temperatur.txt
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In many Topics describing the thermodynamic properties of liquid mixtures and solutions, key equations relate the chemical potentials of components to the composition of a given system. For example in the case of a binary aqueous mixture the chemical potential of water $\mu_{1}(\mathrm{~T}, \mathrm{p}, \mathrm{mix})$ is related to the mole fraction of water $x_{1}$ at temperature $\mathrm{T}$ and pressure $p$ using equation (a).
$\mu_{1}(\mathrm{~T}, \mathrm{p}, \operatorname{mix})=\mu_{1}^{*}(\mathrm{~T}, \mathrm{p}, \ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)$
$\text { By definition, limit }\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{f}_{1}=1.0$
Here $\mu_{1}^{*}(\mathrm{~T}, \mathrm{p}, \ell)$ is the chemical potential of water($\ell$) at the same $\mathrm{T}$ and $p$; $\mathrm{f}_{1}$ is the rational activity coefficient of water in the mixture.
Similarly for solute $j$ in an aqueous solution at temperature $\mathrm{T}$ and pressure $p$, the chemical potential of solute $j$, $\mu_{j}(T, p, a q)$ is related to the molality mj using equation (c) where $\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{~kg}^{-1}$.
$\mu_{\mathrm{j}}(\mathrm{aq}, \mathrm{T}, \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq}, \mathrm{T}, \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
$\text { By definition, at all T and } p \text { limit }\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0$
Here $\mu_{\mathrm{j}}^{0}(\mathrm{aq}, \mathrm{T}, \mathrm{p})$ is the chemical potentials of solute $j$ in an aqueous solution at the same $\mathrm{T}$ and $p$ where $\mathrm{m}_{\mathrm{j}}=1.0 \mathrm{~mol} \mathrm{~kg}$ and $\gamma_{\mathrm{j}}=1.0$.
In equations (a) and (c) the parameter $\mathrm{R}$ is the Gas Constant, $8.314 \mathrm{~J mol}^{-1} \mathrm{~K}^{-1}$. The word ‘Gas’ in the latter sentence is interesting bearing in mind that equations (a) and (c) describe the properties of liquids, mixtures and solutions. Here we examine how this parameter emerges in these equations.
The starting point is a description of a closed system containing $i$–chemical substances, the amount of chemical substance $j$ being $n_{j}$.
$\text { Then, } \mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i}}\right]$
The chemical potential $\mu_{j}(T, p)$ of chemical substance $j$ is given by equation (f).
$\mu_{\mathrm{j}}(\mathrm{T}, \mathrm{p})=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}$
Moreover the partial molar volume $\mathrm{V}_{j}$ of chemical substance $j$ is given by equation (g).
$V_{j}=\left(\frac{\partial \mu_{j}}{\partial p}\right)_{T}$
We simplify the argument by considering a system comprising pure chemical substance 1.
$\text { Then } \quad \mathrm{V}_{1}^{*}=\left(\frac{\partial \mu_{1}^{*}}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
Thus $\mathrm{V}_{1}^{*}(\mathrm{~T}, \mathrm{p})$ is the molar volume of pure substance 1 at temperature $\mathrm{T}$ and pressure $p$. In the event that chemical substance 1 is a perfect (ideal) gas, the following equation describes the $p-\mathrm{V}-\mathrm{T}$ properties.
$p_{1}^{*} \, V_{1}^{*}(g)=R \, T$
We write equation (h) in the following form describing an ideal gas at constant temperature $\mathrm{T}$.
$d \mu_{1}^{*}(g)=V_{1}^{*}(g) \, d p$
Equations (i) and (j) yield equation (k).
$\mathrm{d} \mu_{1}^{*}(\mathrm{~g})=\mathrm{R} \, \mathrm{T} \, \mathrm{d} \ln \mathrm{p}_{1}^{*}$
We integrate equation (k) between limits $p_{1}^{*}$ and $p^{0}$ where $p^{0}$ is the standard pressure, $101325 \mathrm{~N m}^{-2}$.
$\text { Hence, at temperature } \mathrm{T}, \mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}_{1}^{*}\right)=\mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1}^{*} / \mathrm{p}^{0}\right)$
In a more complicated system, the gas phase is a gaseous mixture, comprising two components, component 1 and component 2 with partial pressures $\mathrm{p}_{1}$ and $\mathrm{p}_{2}$. We assume the thermodynamic properties of the gas phase in equilibrium with a liquid phase are ideal. Hence equation (l) takes the following form where $\mu_{1}\left(g ; \text { mix; } p_{1}\right)$ is the chemical potential of gas-1 at partial pressure $\mathrm{p}_{1}$.
$\mu_{1}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{mix} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}_{1}\right)=\mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1}^{\mathrm{eq}} / \mathrm{p}^{0}\right)$
Liquid Mixtures
A given closed system contains chemical substances 1 and 2, present in two phases, gas and a liquid mixture at temperature $\mathrm{T}$ and pressure $\mathrm{p}$. Thus $\mathrm{p}_{1}^{\text {eq }}$ is the equilibrium partial pressure of chemical substance 1 in the gas phase. At equilibrium the chemical potentials of chemical substance 1 in the vapour and liquid mixture phases are equal.
$\mu_{1}^{\mathrm{eq}}(\ell ; \operatorname{mix} ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{1}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{mix} ; \mathrm{T} ; \mathrm{p}_{1}^{\mathrm{eq}}\right)$
Thus $\mathrm{p}_{1}^{\text {eq }}$ is the partial pressure of chemical substance 1 in the gas phase, the superscript ‘eq’ indicating an equilibrium with the liquid phase at pressure $\mathrm{p}$; the complete system is at temperature $\mathrm{T}$.
Hence using equations (m) and (n) we obtain an equation for the equilibrium chemical potential of chemical substance 1 in an ideal liquid mixture at temperature $\mathrm{T}$ and pressure $\mathrm{p}$
$\mu_{1}^{\text {eq }}(\ell ; \operatorname{mix} ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{p}^{0} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1}^{\mathrm{eq}} / \mathrm{p}^{0}\right)$
The thermodynamic analysis calls on the results of experiments in which the partial pressure $\mathrm{p}_{i}$ of chemical substance-$i$ in a liquid mixture at temperature $\mathrm{T}$ is measured as a function of mole fraction $\mathrm{x}_{i}$. It turns out that for nearly all liquid mixtures at fixed temperature, $\mathrm{p}_{i}$ is approximately a linear function of the mole fraction $\mathrm{x}_{1}$ at low $\mathrm{x}_{1}$. We therefore define an ideal liquid mixture. By definition the (equilibrium) vapour pressure of chemical substance $i$, one component of a liquid mixture, is related to the mole fraction composition at temperature $\mathrm{T}$ using equation (p).
$\text { Thus } \mathrm{p}_{\mathrm{i}}^{\mathrm{eq}}(\mathrm{T} ; \text { mix } ; \mathrm{id})=\mathrm{x}_{\mathrm{i}} \, \mathrm{p}_{\mathrm{i}}^{*}(\ell ; \mathrm{T})$
Here $\mathrm{x}_{i}$ is the mole fraction of component-$i$ in the liquid mixture; $\mathrm{p}_{\mathrm{i}}^{*}(\ell ; \mathrm{T})$ is the vapour pressure of pure liquid substance 1 at temperature $\mathrm{T}$.
For example if $\mathrm{x}_{i}$ is $0.5$, the contribution to the vapour pressure of the (ideal) mixture is one-half of the vapour pressure of the pure liquid-$i$ at the same temperature. Equation (p) is Raoult’s law, describing the properties of an ideal liquid mixture having ideal thermodynamic properties. We note that the Gas Constant emerges in equation (o) because the r.h.s. of equation (o) describes the properties of chemical substance 1 in the vapour phase.
Combination of equations (o) and (p) yields equation (q).
$\mu_{\mathrm{i}}^{\mathrm{eq}}(\ell ; \mathrm{mix} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{i}}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{x}_{\mathrm{i}} \, \mathrm{p}_{\mathrm{i}}^{*}(\mathrm{~T}) / \mathrm{p}^{0}\right]$
$\text { Or, } \mu_{\mathrm{i}}^{\mathrm{eq}}(\ell ; \mathrm{mix} ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{\mathrm{i}}^{*}\left(\mathrm{~g} ; \mathrm{p}^{0} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{p}_{\mathrm{i}}^{*}(\mathrm{~T}) / \mathrm{p}^{0}\right]+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{i}}\right)$
For the pure liquid-$i$ at pressure $\mathrm{p}$,
$\mu_{\mathrm{i}}^{*}(\ell ; \mathrm{p} ; \mathrm{T})=\mu_{\mathrm{i}}^{*}\left(\mathrm{~g} ; \mathrm{p}^{0} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{p}(\mathrm{T}) / \mathrm{p}^{0}\right]$
$\text { Hence, } \mu_{\mathrm{i}}^{\mathrm{eq}}(\ell ; \text { mix } ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{\mathrm{i}}^{*}(\ell ; \mathrm{p} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{i}}\right)$
We notice that the Gas Constant in equation (t) emerged from equation (i) describing the properties of an ideal gas.
Solutions
A similar argument is used when we turn our attention to the thermodynamic properties of a solute, chemical substance $j$. In this case we use Henry’s Law as the link between theory and the properties of solutions. This law relates the equilibrium partial pressure $\mathrm{p}_{j}$ of solute $j$ to the molality of solute $j$, $\mathrm{m}_{j}|) for a solution at temperature \(\mathrm{T}$ and pressure $\mathrm{p}$. Experiment shows that certainly for dilute solutions, the partial pressure $\mathrm{p}_{j}$ is close to a linear function of molality $\mathrm{m}_{j}$. Taking this experimental result as a lead we state that, by definition, in the event that the thermodynamic properties of the solution are ideal, equation (u) relates the partial pressure $\mathrm{p}_{j}$ to the solute molality $\mathrm{m}_{j}$; $\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}$.
$\text { Thus, } \mathrm{p}_{\mathrm{j}}\left(\mathrm{s} \ln ; \mathrm{T} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{id}\right)=\mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Here $\mathrm{H}_{j}$ is Henry’s Law constant characteristic of solute, solvent, $\mathrm{T}$ and $\mathrm{p}$. $\mathrm{H}_{j}$ is a pressure being the partial pressure of solute $j$ where $\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}$. In other words equation (u) is not thermodynamic in the sense of being derived from the Laws of Thermodynamics. Rather the basis is experiment. We return to equation (n) but written for the equilibrium for solute in solution and in the vapour phase, a mixture of solute $j$ and solvent.
$\mu_{j}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{g} ; \operatorname{mix} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}^{\mathrm{eq}}\right)$
For the vapour phase, $\mu_{j}^{c q}\left(g ; \operatorname{mix} ; T ; \mathrm{p}_{\mathrm{j}}^{\mathrm{cq}}\right)$ is related to the partial pressure $\mathrm{p}_{\mathrm{j}}^{\mathrm{cq}}$ using equation (w).
$\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{mix} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}^{\mathrm{eq}}\right)=\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{p}^{0}\right)$
Hence using equations (u)-(w),
$\mu_{\mathrm{j}}^{\mathrm{cq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{H}_{\mathrm{j}}}{\mathrm{p}^{0}} \, \frac{\mathrm{m}_{\mathrm{j}}}{\mathrm{m}^{0}}\right]$
$\text { Or, } \mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\left\{\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{H}_{\mathrm{j}}}{\mathrm{p}^{0}}\right]\right\}+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
The term $\left\{\mu_{j}^{0}\left(g ; T ; p^{0}\right)+R \, T \, \ln \left[\frac{H_{j}}{p^{0}}\right]\right\}$ characterises solute $j$ in a solution at the same $\mathrm{T}$ and $\mathrm{p}$ when $\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}$. Thus we define a reference chemical potential for the solute-$j$,
$\mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln ; \mathrm{T} ; \mathrm{p}) \text { as given by }\left\{\mu_{\mathrm{j}}^{0}(\mathrm{~g} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{H}_{\mathrm{j}}}{\mathrm{m}_{\mathrm{j}}^{0}}\right]\right\}$
$\text { Therefore, } \mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Again we can trace the gas constant $\mathrm{R}$ in equation (za) to a description of the vapour state although the term $\mu_{j}^{\mathrm{cq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)$ describes the chemical potential of chemical substance $j$, the solute, in solution.
Finally we should note that for real as opposed to ideal liquid mixtures and ideal solutions, activity coefficients express the extent to which the properties of these systems differ from those defined as ideal.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.05%3A_Chemical_Potentials/1.5.01%3A_Chemical_Potentials_Composition_and_the_Gas_Constant.txt
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A given closed system contains gas $j$ at temperature $\mathrm{T}$ and pressure $\mathrm{p}$. The chemical potential $\mu_{\mathrm{j}}(\mathrm{g} ; \mathrm{T} ; \mathrm{p})$ is given by Equation \ref{a} where $\mathrm{p}^{0}$ is the standard pressure and $\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T}, \mathrm{p})$ is the molar volume of the gas $j$.
$\mu_{j}(g ; T ; p)= \mu_{j}^{0}(p f g ; T)+ R T \ln \left(\dfrac{p}{p^{0}}\right)+\int_{0}^{p}\left[V_{j}^{*}(T ; p)-\left(\dfrac{R T}{p}\right)\right] d p \label{a}$
$\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})$ is the molar volume at pressure $\mathrm{p}$ and temperature $\mathrm{T}$. In the event that gas $j$ has the properties of a perfect gas, the chemical potential is given by Equation \ref{b}.
$\mu_{j}(p f g ; T ; p)=\mu_{j}^{0}(p f g ; T)+R T \ln \left( \dfrac{p}{p^{0}} \right) \label{b}$
If gas $j$ exists at mole fraction $\mathrm{x}_{j}$ as one component of a mixture of $\mathrm{k}$ gases the chemical potential of gas $j$ is given by Equation \ref{c} where $\mathrm{x}_{\mathrm{k}}$ is the set of mole fractions defining the composition of the mixture [1].
$\mu_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{k}}\right)= \mu_{\mathrm{j}}^{0}(\mathrm{~g} ; \mathrm{T})+\mathrm{R} \mathrm{T} \ln \left(\mathrm{x}_{\mathrm{j}} \mathrm{p} / \mathrm{p}^{0}\right) +\int_{\mathrm{o}}^{\mathrm{p}}\left[\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{c}}\right)-\left(\dfrac{RT}{p}\right)\right] \mathrm{dp} \label{c}$
Here $\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{c}}\right)$ is the molar volume of gas $j$ in the gaseous mixture.
Footnote
[1] M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, page 184.
1.5.03: Chemical Potentials- Solutions- General Properties
A key quantity in chemical thermodynamics is the chemical potential of chemical substance $j$, $\mu_{j}$. The latter is the differential dependence of Gibbs energy on amount of substance $j$ at fixed $\mathrm{T}$, $\mathrm{p}$ and amounts of all other substances in the system [1].
$\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}$
An important point to note is that the conditions ‘fixed $T$ and fixed $p$’ on the partial differential refer to intensive variables. These conditions are called Gibbsian in recognition of the development by Gibbs of the concept of thermodynamic potential for changes in the properties of a closed system at fixed $\mathrm{T}$ and fixed $\mathrm{p}$.
In general terms, the chemical potential of substance $j$ is defined using analogous partial derivatives of the thermodynamic internal energy $\mathrm{U}$, enthalpy $\mathrm{H}$ and Helmholtz energy $\mathrm{F}$.
$\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{U}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{s}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{s}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{F}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}$
With reference to a given closed system, thermodynamics defines macroscopic properties including volume $\mathrm{V}$, Gibbs energy $\mathrm{G}$, enthalpy $\mathrm{H}$ and entropy $\mathrm{S}$. Nevertheless we need to “tell” these thermodynamic variables that a given system probably comprises different chemical substances. The analysis is reasonably straightforward if we define the system under consideration by the ‘Gibbsian’ set of independent variables; i.e. $\mathrm{T}$, $\mathrm{p}$ and amounts of each chemical substance [2]. The analysis leads to the definition of a chemical potential for each substance $j$, $\mu_{j}$, in a closed system [3,4].
Footnotes
[1] It might be argued that we have switched our attention from closed to open systems because we are considering a change in Gibbs energy when we add $\partial n_{j}$ moles of substance to the system. This comment is true in part. But what we actually envisage is something a little different. We take a closed system containing $n_{1}$ and $n_{j}$ moles of substances $1$ and $j$ respectively. We open the system, rapidly pop in $\delta \mathrm{n}_{\mathrm{j}}$ moles of substance $j$ and put the lid back on the system to return it to the closed state. Then the closed system contains $\left(\mathrm{n}_{\mathrm{j}}+\delta \mathrm{n}_{\mathrm{j}}\right)$ moles of substance j so changes in chemical composition and molecular organisation follow producing a change in Gibbs energy at, say, fixed $\mathrm{T}$ and fixed $\mathrm{p}$.
[2] G. N. Lewis, (with possibly one of the key papers in chemistry)
1. Z.Phys.Chem.,1907,61,129.
2. Proc. Acad. Arts Sci.,1907,43,259.
[Comment: Paper (a) is the German translation of paper (b).]
[3] The analysis presented here (a) is confined to bulk systems in the absence of magnetic and electric fields and (b) ignores surface effects.
[4] To quote E. Grunwald [J. Am. Chem. Soc., 1984, 106, 5414] “any first derivative with respect to any variable of state at equilibrium is isodelphic”; see also E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1997.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.05%3A_Chemical_Potentials/1.5.02%3A_Chemical_Potentials-_Gases.txt
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A given aqueous solution at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ (both near ambient) was prepared using $\mathrm{n}_{1}$ moles of water and $\mathrm{n}_{j}$ moles of urea (i.e. chemical substance $j$). The Gibbs energy $\mathrm{G}(\mathrm{aq})$, an extensive property (variable), is given by the sum of products of amounts of each chemical substance and chemical potentials [1].
$\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq}) \label{a}$
Equation \ref{a} is key although we cannot put number values to $\mathrm{G}(\mathrm{aq})$, $\mu_{1}(\mathrm{aq})$ and $\mu_{j}(\mathrm{aq}$. The latter two quantities are, respectively, the chemical potentials of the solvent, water and solute $j$ in the aqueous solution at the same temperature and pressure. Equation \ref{a} seems a strange starting point granted it contains three quantities which we can never know. Matters can only improve.
There is merit in turning attention to an intensive property describing the Gibbs energy of a solution prepared using $1 \mathrm{~kg}$ of solvent, $\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)$. Therefore, we do not have to worry about the size of the flask containing the solution. The same descriptor applies to $0.1 \mathrm{~cm}^{3}$ or $10 \mathrm{m}^{3}$ of a given solution [2,3].
$\text { By definition } \quad \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)=\mathrm{G}(\mathrm{aq}) / \mathrm{w}_{1}$
$\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})$
$\mathrm{M}_{1}$ is the molar mass of solvent, water, and mj is the molality of solute $j$. Again we cannot put number values to $\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)$, $\mu_{1}(\mathrm{aq})$ and $\mu_{\mathrm{j}}(\mathrm{aq})$. Faced with this situation, the well-established approach involves an examination of differences. With respect to $\mu_{1}(a q)$, the properties of water in an aqueous solution are compared with the properties of water at the same temperature and pressure. In these terms, we compare $\mu_{1}(\mathrm{aq}, \mathrm{T}, \mathrm{p})$ with $\mu_{1}^{*}(\ell, T, p)$. The superscript * in the latter term indicates that the chemical substance is pure and the symbol '$\ell$' indicates that this substance is a liquid. Hence, comparison is drawn with the chemical potential of pure liquid water at the same $\mathrm{T}$ and $\mathrm{p}$. In one sense we regard the solute as a controlled impurity perturbing the properties of the solvent. [We use the subscript '1' to indicate chemical substance 1 which in the convention used here refers to the solvent; water in the case of aqueous solutions.]
In considering the properties of, for example, urea in this aqueous solution molality $\mathrm{m}_{j}$, we need a reference state against which to compare the properties of urea in the real solution prepared by dissolving $\mathrm{n}_{j}$ moles of urea in $\mathrm{n}_{1}$ moles of water. There is little point in comparing the properties of solute, urea with those of solid urea, a hard crystalline solid. Instead, we identify a reference solution state.
In general terms chemists explore how the chemical potentials of solvent and solute in an aqueous solution are related to the composition of the solution. Equations which offer such relationships should satisfy two criteria [4]: in the limit of infinite dilution (i) the partial molar volumes $\mathrm{V}_{\mathrm{j}}(\mathrm{aq})$ and $\mathrm{V}_{1}(\mathrm{aq})$ are meaningful and (ii) the partial molar enthalpies $\mathrm{H}_{\mathrm{j}}(\mathrm{aq}$ and $\mathrm{H}_{1}(\mathrm{aq})$ are meaningful. In other words, these properties do not approach an asymptotic limit of either $+ \infty$ or $– \infty$ with increasing dilution. For this reason physical chemists usually favour expressing the composition of solutions in molalities.
In summary analysis of the properties of solutions and liquid mixtures is built around the somewhat abstract concept of the chemical potential introduced by J. Willard Gibbs and by Pierre Duhem. The task of showing chemists the significance and application of this concept was left to Lewis and Randall in their classic monograph [5] published in 1923 [6]. Chemical potentials are one example of a class of properties called partial molar which provide the key link between macroscopic thermodynamic descriptions of systems and molecular properties [7].
Footnotes
[1] $\mathrm{G}(\mathrm{aq})=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}{ }^{-1}\right]+[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=[\mathrm{J}]$
[2] \begin{aligned} &\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)= \ &\quad\left[1 / \mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{kg}^{-1}\right] \end{aligned}
[3] With reference to equations (a) and (c), we must avoid the temptation to write “at constant temperature and pressure”. This condition is implicit in the description of the system using the independent variables $\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]$ for the aqueous solution containing the solute with the added condition that $\mathrm{T}$ and $\mathrm{p}$ are intensive variables; i.e. the set of independent variables is Gibbsian. Nonetheless there is often merit in using a complete set of descriptions of a system even if we over-define the variable under discussion. In describing the Gibbs energy defined by equation (a), we might write $\mathrm{G}(\mathrm{T} ; \mathrm{p} ; \mathrm{aq})$. Similarly for the system described by equation (b) it is often helpful to write $\mathrm{G}\left(\mathrm{T} ; \mathrm{p} ; \mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}}\right)$. In reviewing the properties of solutions our interest, unless otherwise stated, centres on solutions at equilibrium where the affinity $\mathrm{A}$ is zero and the organisation characteristic of the equilibrium system, $\xi^{e q}$. We may find it helpful to write $\mathrm{G}\left(\mathrm{T} ; \mathrm{p} ; \mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{A}=0 ; \xi^{\mathrm{eq}}\right)$, replacing $\mathrm{G}$ by $\mathrm{H}$, $\mathrm{S}$ and $\mathrm{V}$ for the corresponding enthalpy, entropy and volume of this solution. Clearly this over-definition is somewhat silly. Nevertheless it is often preferable to over-define a system rather than under-define when mistakes can arise.
[4] J. E. Garrod and T. M. Herrington, J. Chem. Educ., 1969, 46, 165.
[5] G. N. Lewis and M. L. Randall, Thermodynamics and the Free Energy of Chemical Substances, McGraw-Hill, New York, 1923.
[6] See also, G. N. Lewis, Proc. Am. Acad. Arts Sci.,1907,43,259.
[7] L. Hepler, Thermochim. Acta, 1986, 100, 171.
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A given solution comprises $\mathrm{n}_{1}$ moles of solvent, liquid chemical substance 1, and $\mathrm{n}_{\mathrm{j}}$ moles of solute, chemical substance $\mathrm{j}$. We ask ---- What contributions are made by the solvent and by the solute to the volume of the solution at defined $\mathrm{T}$ and $\mathrm{p}$? In fact we can only guess at these contributions. This is disappointing. The best that we can do is to probe the sensitivity of the volume of a given solution to the addition of small amounts of solute and of solvent. This approach leads to a set of properties called partial molar. The starting point is the Gibbs energy of a solution. We develop an argument which places the Gibbs energy at the centre from which all other thermodynamic variables develop.
A given closed system comprises $\mathrm{n}_{1}$ moles of solvent (e.g. water) and $\mathrm{n}_{\mathrm{j}}$ moles of a simple solute j (e.g. urea) at temperature $\mathrm{T}$ and pressure $\mathrm{p}$. The Gibbs energy of the solution is defined by equation (a).
$\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{l}}, \mathrm{n}_{\mathrm{j}}\right] \label{a}$
We introduce a partial derivative having the following form: $\left(\frac{\partial G}{\partial n_{j}}\right)_{T, p, n_{1}}$. The latter partial differential describes the differential dependence of Gibbs energy $\mathrm{G}$ on the amount of chemical substance $\mathrm{j}$. By definition, the chemical potential of chemical substance $\mathrm{j}$,
$\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{l}}}$
We also envisage that displacement of the system by adding $\delta n_{j}$ moles of chemical substance $\mathrm{j}$ from the original state to a neighbouring state produces a change in Gibbs energy at temperature $\mathrm{T}$ and pressure $\mathrm{p}$. In one class of displacements the system moves along a path of constant affinity for spontaneous reaction $\mathrm{A}$. In another displacement the system moves along a path at constant organisation/composition, $\xi$; i.e. frozen. These two pathways are related by the following equation. For the system at fixed $\mathrm{T}$, $\mathrm{p}$ and $\mathrm{n}_{1}$
$\left[\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{n}_{\mathrm{j}}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{n}(\mathrm{j})} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{n}(\mathrm{j})}$
The conditions, constant $\mathrm{T}$ and $\mathrm{p}$, refer to intensive variables. We direct attention to a closed system at equilibrium where ‘$\mathrm{A} = 0$’ and the composition $\xi=\xi^{\mathrm{eq}}$. Moreover at equilibrium, $(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}$ is zero. Therefore the chemical potential of chemical substance $\mathrm{j}$ in a system at equilibrium is defined by the following equation. Hence from equation (c),
$\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{~A}=0}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1},, \mathrm{G}^{c_{q}}}$
A similar argument in the context of chemical substance 1 shows that,
$\mu_{1}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{j}}, \zeta^{\mathrm{eq}}}$
Equations (d) and (e) are key results. Similarly for a closed system at equilibrium at fixed $\mathrm{T}$ and fixed $\mathrm{p}$ (at a minimum in $\mathrm{G}$, $\mathrm{A} = 0$, $\xi=\xi^{\mathrm{eq}}$ ), for all $i$-substances,
$V_{j}(A=0)=V_{j}\left(\xi^{e q}\right)$
$\mathrm{S}_{\mathrm{j}}(\mathrm{A}=0)=\mathrm{S}_{\mathrm{j}}\left(\xi^{\mathrm{eq}}\right)$
$\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)=\mathrm{H}_{\mathrm{j}}\left(\xi^{e q}\right)$
$\mu_{j}(A=0)=\mu_{j}\left(\xi^{e q}\right)$
But in the case of, for example, isobaric expansions and isobaric heat capacities, $\mathrm{E}_{\mathrm{pj}}(\mathrm{A}=0) \neq \mathrm{E}_{\mathrm{pj}}\left(\xi^{\mathrm{eq}}\right)$ and $\mathrm{C}_{\mathrm{pj}}(\mathrm{A}=0) \neq \mathrm{C}_{\mathrm{pj}}\left(\xi^{e q}\right)$. The identifications, (f) to (i), arise because these variables are first derivatives of the Gibbs energy of a closed system at equilibrium where $(\partial \mathrm{G} / \partial \xi)$ at fixed $\mathrm{T}$ and $\mathrm{p}$ is zero.
1.5.06: Chemical Potentials- Liquid Mixtures- Raoult's Law
A given closed system contains two volatile miscible liquids. The closed system is connected to a pressure-measuring device which records that at temperature T the pressure is ptot. The composition of the liquid mixture is known; i.e. mole fractions $\mathrm{x}_{1}$ and $\mathrm{x}_{2}$ (where $\mathrm{x}_{2} = 1 - \mathrm{x}_{1}$). The system contains two components so that in terms of the Phase Rule, $\mathrm{C} = 2$ There are two phases, vapour and liquid so that $\mathrm{P} = 2$. From the rule, $\mathrm{P} + \mathrm{F} = \mathrm{C} + 2$, we have fixed the composition and temperature using up the two degrees of freedom. Hence the pressure ptot is fixed.
We imagine that the mixture under examination is a binary aqueous mixture; water is chemical substance 1. If we measure the partial pressure of, say, liquid 1, $\mathrm{p}_{1}$ we find that $\mathrm{p}_{1}$ is close to a linear function of mole fraction $\mathrm{x}_{1}$.
$\text { At equilibrium and temperature } \mathrm{T}, \quad \mathrm{p}_{1}^{\mathrm{eq}} \cong \mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1}$
As the mole fraction $\mathrm{x}_{1}$ approaches unity (i.e. the composition of the mixture approaches pure water) the equilibrium vapour pressure of water $\mathrm{p}_{1}^{\mathrm{eq}}$ approaches that of pure liquid water at the same temperature, $\mathrm{p}_{1}^{*}(\ell)$ We have linked the equilibrium vapour pressure of water to the composition of the liquid mixture.
In fact it turns out that as the composition of the mixture approaches pure liquid 1, the latter relationship becomes an equation. We assert that if the thermodynamic properties of the mixture were ideal then $\mathrm{p}_{1}$ would be related to mole fraction $\mathrm{x}_{1}$ using the following equation.
$\mathrm{p}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{p}_{1}^{*}(\ell)$
Returning to experiment, we invariably find that as a real solution becomes more dilute (i.e. as $\mathrm{x}_{1}$ approaches unity) $\mathrm{p}_{1}^{\mathrm{eq}}$ for real solutions approaches $p_{1}^{e q}(a q ; i d)$. Therefore we rewrite equation (b) as an equation for a real solution by introducing a new property called the (rational) activity coefficient $\mathrm{f}_{1}$.
$\mathrm{p}_{1}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{f}_{1} \, \mathrm{p}_{1}^{*}(\ell)$
Here $\mathrm{f}_{1}$ is the (rational) activity coefficient for liquid component 1 defined as follows.
$\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1$
$\text { Similarly for volatile liquid } 2 ; p_{2}=x_{2} \, f_{2} \, p_{2}^{*}$
$\operatorname{limit}\left(x_{2} \rightarrow 1\right) f_{2}=1$
Although equations (d) and (f) have simple forms, rational activity coefficients carry a heavy load in terms of information. For a given aqueous system, $\mathrm{f}_{1}$ describes the extent to which interactions involving water molecules in a real system differ from those in the corresponding ideal system. The challenge of expressing this information in molecular terms is formidable.
We carry over these ideas to the task of formulating an equation for the chemical potential of water in the liquid mixture at temperature $\mathrm{T}$ and pressure $\mathrm{p}$. We make the link between partial pressure and the tendency for liquid 1 to escape to the vapour phase, down a gradient of chemical potential
By definition (at temperature $\mathrm{T}$ and pressure $\mathrm{p}$),
$\mu_{1}(\operatorname{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)$
$\text { where, at all } \mathrm{T} \text { and } \mathrm{p}, \quad \operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1.0\right) \mathrm{f}_{1}=1.0$
$\mu_{1}^{*}(\ell)$ l is the chemical potential of pure liquid water (at the same $\mathrm{T}$ and $\mathrm{p}$). In other words, pure liquid water is the reference state against which we compare the properties of water in an aqueous mixture. For the pure liquid at temperature $\mathrm{T}$, $V_{1}^{*}(\ell)=\mathrm{d} \mu_{1}^{*}(\ell) / \mathrm{dp}$. If $\mathrm{p}^{0}$ is the standard pressure [1] $\left(10^{5} \mathrm{~N} \mathrm{~m}^{-2}\right)$,
$\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{d} \mu_{1}^{*}(\ell)=\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}$
$\text { Then }[2], \quad \mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mu_{1}^{0}(\ell ; \mathrm{T})=\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}$
$\mu_{1}^{0}(\ell ; \mathrm{T})$ is the standard chemical potential of water($\ell$) at temperature $\mathrm{T}$.
$\text { Therefore, } \quad \mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}$
This is an important equation although, at this stage, we can go no further. Without information concerning the dependence on pressure of $\mathrm{V}_{1}^{*}(\ell)$ {or density} we cannot evaluate the integral in equation (k). However, we can comment on possible patterns in these chemical potentials. If the thermodynamic properties of the liquid mixture are ideal then $\mathrm{f}_{1}$ equals $1.0$. Hence equation (k) takes the following simple form (at fixed $\mathrm{T}$ and $\mathrm{p}$).
$\mu_{1}(\operatorname{mix} ; \mathrm{id})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)$
In a solution the mole fraction x1 is less than unity and so $\ln \left(x_{1}\right)<0$ [3]. Hence $\mu_{1}(\operatorname{mix} ; \mathrm{id})<\mu_{1}^{*}(\ell)$ at the same $\mathrm{T}$ and $\mathrm{p}$ [4]
Footnotes
[1] J.D. Cox, Pure Appl. Chem., 1982,54, 1239; R.D. Freeman, Bull. Chem. Thermodyn., 1982,25, 523.
[2] $\mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{N} \mathrm{m} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]$
[3] With increase in the amount of component 2 so $\mathrm{x}_{1}$ tends to zero. In this limit $\mu_{1}(\mathrm{aq})$ is minus infinity.
[4] Note that $\mathrm{p}_{1}^{\mathrm{eq}}(\operatorname{mix} ; \mathrm{id})-\mathrm{p}_{1}^{*}<0$. Adding a solute lowers the vapour pressure of the water. However the total vapour pressure of a binary liquid mixture can be either increased or decreased by adding a small amount of the second component; G. Bertrand and C. Treiner, J. Solution Chem.,1984,13,43.
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A given closed system contains an aqueous solution; the solute is chemical substance $j$. The system is at equilibrium at temperature $\mathrm{T}$. The chemical potential of water in the aqueous solution is related to the mole fraction $\mathrm{x}_{1}$ of water using equation (a) which is based on Raoult’s Law for the solvent.
$\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}$
$\text { By definition, at all } \mathrm{T} \text { and } \mathrm{p}, \operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{f}_{\mathrm{1}}=1$
If ambient pressure is close to the standard pressure $\mathrm{p}^{0}$, the chemical potential of solvent water in the aqueous solution is given by equation (c).
$\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)$
For an ideal solution, $\mathrm{f}_{1} = 1$.
$\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)$
But for a solution $\mathrm{x}_{1} < 1.0$ and so $\ln \left(x_{1}\right)<0$. In other words, by adding a solute to water (forming an ideal solution) we stabilise the solvent. We define a quantity $\Delta(\ell \rightarrow a q) \mu_{1}(\mathrm{~T}, \mathrm{p})$ using equation (e) which measures the change in chemical potential of water when one mole of water is transferred from water($\ell$) to an ideal aqueous solution.
$\Delta(\ell \rightarrow \mathrm{aq}) \mu_{1}(\mathrm{~T}, \mathrm{p})=\mu_{1}(\mathrm{aq})-\mu_{1}^{*}(\ell)=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)$
i.e. $\Delta(\ell \rightarrow \mathrm{aq}) \mu_{1}(\mathrm{~T}, \mathrm{p} ; \mathrm{id})<0$
In the case of a real solution, the extent of stabilisation depends on whether $\mathrm{f}_{1}$ is either larger or smaller than unity. [Note that $\mathrm{f}_{1}$ cannot be negative]. This line of argument leads to an important theme in the description of the properties of aqueous solutions. We compare the chemical potentials of water in real and in the corresponding ideal solutions. The difference is the excess chemical potential, $\mu_{1}^{E}(a q ; T ; p)$.
$\text { By definition, } \quad \mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})$
$\text { Hence } \quad \mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{\mathrm{l}}\right)$
If $\mathrm{f}_{1} > 1.0$, then $\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})>0$; if $\mathrm{f}_{1} < 1.0$, then $\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})<0$. In the latter case, interactions involving solute and solvent are responsible for the fact that the properties of a given solution are not ideal and the fact that these interactions stabilise the solvent relative to that for an ideal solution.
1.5.08: Chemical Potentials- Solutions- Osmotic Coefficient
The chemical potential of (solvent) water in an aqueous solution can be related to the mole fraction composition of the solution. However, there is a possible disadvantage in an approach using the mole fraction scale to express the composition of a solution. We note that our interest is often in the properties of solutes in aqueous solutions, that the amount of solvent greatly exceeds the amount of solute in a solution, and that the sensitivity of equipment developed by chemists is sufficient to probe the properties of quite dilute solutions. Consequently the mole fraction scale for the solvent is not the most convenient method for expressing the composition of a given solution [1]. Hence another equation relating $\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ to the composition of a solution finds favour. By definition, for a solution containing a single solute, chemical substance $j$, [2],
$\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
In terms of the standard chemical potential for water at temperature $|mathrm{T}$,
$\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
$\mathrm{M}_{1}$ is the molar mass of water; $\phi$ is the practical osmotic coefficient which is characteristic of the solute, molality $\mathrm{m}_{j}$, temperature and pressure. By definition, $\phi$ is unity for ideal solutions at all temperatures and pressures.
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p}$
Further for ideal solutions, the partial differentials $(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}$, $\left(\partial^{2} \phi / \partial T^{2}\right)_{p}$ and $(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}$ are zero.
$\text { For an ideal solution [3], } \quad \mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
We rewrite equation (d) in the following form:
$\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})-\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
Hence with an increase in molality of solute in an ideal aqueous solution, the solvent is stabilised, being at a lower chemical potential than that for pure water($\ell$).We contrast the chemical potentials of the solvent in real and ideal solutions using an excess chemical potential, $\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$;
\begin{aligned} \mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}) \ &=(1-\phi) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \end{aligned}
The term $(1-\phi)$ is often encountered because it expresses succinctly the impact of a solute on the properties of a solvent. At a given molality (and fixed temperature and pressure), $\phi$ is characteristic of the solute.
Footnote
[1] Mole fractions of solvent $\mathrm{x}_{1}$ and solute $\mathrm{x}_{j}$ for aqueous solutions having gradually increasing molality of solute $\mathrm{m}_{j}$.
1. $\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}=10^{-3} ; \quad \mathrm{x}_{1}=0.999982 \quad \mathrm{x}_{\mathrm{j}}=1.8 \times 10^{-5}$
2. $\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}=10^{-2} ; \mathrm{x}_{1}=0.99982 \quad \mathrm{x}_{\mathrm{j}}=1.8 \times 10^{-4}$
3. $\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}=10^{-1} ; \mathrm{x}_{1}=0.9982 \quad \mathrm{x}_{\mathrm{j}}=1.8 \times 10^{-3}$
4. $\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}=0.5 ; \quad \mathrm{x}_{1}=0.9911 \quad \mathrm{x}_{\mathrm{j}}=8.9 \times 10^{-3}$
5. $\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}=1.0 ; \quad \mathrm{x}_{1}=0.9823 \quad \mathrm{x}_{\mathrm{j}}=1.77 \times 10^{-2}$
[2] \begin{aligned} &{\left[\mathrm{J} \mathrm{mol}^{-1}\right]=} \ &{\left[\mathrm{J} \mathrm{mol}^{-1}\right]-[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]} \end{aligned}
[3] The definitions of ideal solutions expressed here and in terms of mole fraction of solvent are not in conflict. For an ideal solution these equations require that, $-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=\ln \left(\mathrm{x}_{1}\right)$
But $\ln \left(\mathrm{x}_{1}\right)=\ln \left[\mathrm{M}_{1}^{-1} /\left(\mathrm{M}_{1}^{-1}+\mathrm{m}_{\mathrm{j}}\right)\right]=-\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)$
Bearing in mind that $\mathrm{M}_{1}=0.018 \mathrm{~kg} \mathrm{~mol}^{-1}$, for dilute solutions $\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)=\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}$.
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A given aqueous solution, at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ ($\cong \mathrm{p}^{0}$), contains a solute, chemical substance $j$. If the thermodynamic properties of the solution are ideal, the chemical potential of the solute is given by equation (a).
$\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
For the corresponding real solution,
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Here $\gamma_{j}$ is the activity coefficient. The excess chemical potential, $\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})$ is given by equation (c).
$\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=\mu_{\mathrm{j}}(\mathrm{aq})-\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})$
$\text { Then, } \mu_{j}^{E}(a q)=R \, T \, \ln \left(\gamma_{j}\right)$
Often an excess chemical potential $\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})$ is written in the form $\mathrm{G}_{j}^{\mathrm{E}}$. The latter notation stems from the fact that chemical potentials are partial molar Gibbs energies. In the case of the solvent, water($\ell$), the corresponding equations for the chemical potentials in solutions having either ideal or real thermodynamic properties are given by equations (e) and (f).
$\mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
$\mu_{1}^{\mathrm{E}}(\mathrm{aq})=(1-\phi) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
1.5.10: Chemical Potentials- Solutions- Henry's Law
A given aqueous solutions at temperature $\mathrm{T}$ contains a simple solute $j$, molality $\mathrm{m}_{j}$. Experiment shows that at equilibrium the partial pressure $\mathrm{p}_{j}$ is close to a linear function of molality $\mathrm{m}_{j}$, the constant of proportionality being the Henry’s Law constant for this particular solute in a defined solvent; equation(a).
$\mathrm{p}_{\mathrm{j}} \cong \mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
By definition $\mathrm{m}^{0}=1.0 \mathrm{~mol} \mathrm{~kg}{ }^{-1}$. Experiment shows that as a given real solution becomes more dilute so the relationship given in (a) can be written as an equation. The relationship in (a) is rewritten as an equation to describe the properties of a solution having thermodynamic properties which are ideal.
$p_{j}(\mathrm{id})=\mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
In other words $\mathrm{H}_{j}$ is the partial pressure of volatile solute in a solution having thermodynamic properties which are ideal and where the molality of the solute equals $1.0 \mathrm{~mol kg}^{-1}$.
For a real solution at equilibrium and at temperature $\mathrm{T}$, the partial pressure $\mathrm{p}_{\mathrm{j}}(\text { real })$ is related to molality mj using equation (c) where $\gamma_{j}$ is the activity coefficient describing the properties of solute $j$ in solution
$\mathrm{p}_{\mathrm{j}}(\text { real })=\mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \, \gamma_{\mathrm{j}}$
$\text { By definition, at all T and } p \lim \operatorname{it}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1$
As $\mathrm{m}_{j}$ decreases so $\mathrm{p}_{\mathrm{j}}(\text { real })$ approaches $\mathrm{p}_{\mathrm{j}}(\mathrm{id})$.
Henry’s Law forms the basis of equations which are used to related the chemical potential of solute $j$, $\mu_{j}$ to the composition of a solution.
1.5.11: Chemical Potentials- Solutes
A given aqueous solution is prepared using $1 \mathrm{~kg}$ of water at temperature $\mathrm{T}$ and pressure $\mathrm{p}$. The molality of solute $j$ is $\mathrm{m}_{j}$. The chemical potential of solute $j$, $\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ is related to $\mathrm{m}_{j}$ using equation (a).
$\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{\mathrm{a}}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp}$
$\text { By definition, } \operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0$
$\mu_{j}^{0}(\mathrm{aq} ; \mathrm{T})$ is the chemical potential of solute $j$ in an ideal solution (where $\gamma_{j} =1$) at temperature $\mathrm{T}$ and standard pressure $\mathrm{p}^{0}\left[=10^{5} \mathrm{~Pa}\right]$.
For solutions at ambient pressure which is close to $\mathrm{p}^{0}$, $\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ is related to $\mathrm{m}_{j}$ using equation (c).
$\mu_{j}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Henry’s law forms the basis of equations (a) and (c).
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A given aqueous solution comprises $\mathrm{n}_{1}$ moles of solvent (e.g. water) and $\mathrm{n}_{j}$ moles of solute (e.g. urea) at equilibrium, temperature $\mathrm{T}$ and pressure $\mathrm{p}$. Thus the molality of solute $j$ is given by equation (a).
$\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}$
$\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}$
1. $\mu_{j}(a q ; T ; p)=$ chemical potential of solute j in solution.
2. $\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=$ chemical potential of solute $j$ in the corresponding ideal solution (where $\gamma_{j} = 1.0$) at temperature $\mathrm{T}$ where $\mathrm{m}_{j} = 1 \mathrm{~mol kg}^{-1}$ and where the pressure is the standard pressure $\mathrm{p}^{0}$.
3. $\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{~kg}{ }^{-1}$
4. $\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})$ is the limiting partial molar volume of solute $j$ at temperature T.
5. By definition, $\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1$ at all $\mathrm{T}$ and $\mathrm{p}$.
Activity coefficient $\gamma_{j}$ takes account of the fact that the thermodynamic properties of real solutions are not ideal [1]. An important consideration in understanding the factors which affect $\gamma_{j}$ is the distance between solute molecules in solution. As we dilute the solution such that $\mathrm{m}_{j}$ approaches zero so inter solute distances approache infinity; i.e. in the limit of infinite dilution or zero molality.
If pressure is ambient and hence close to the standard pressure the integral term in equation (b) is negligibly small.
$\text { Hence, } \mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
At this stage we focus attention on the activity coefficient $\gamma_{j}$. We start with equation (c) and by split the logarithm term. For solutions where the pressure $\mathrm{p}$ is close to the standard pressure $\mathrm{p}^{0}$
$\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)$
If the properties of the solution are ideal, equation (c) takes the following form.
$\mu_{j}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
If $\mathrm{m}_{\mathrm{j}}<1.0 \mathrm{~mol} \mathrm{~kg}{ }^{-1}, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$ is $<0$. $\mu_{j}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})<\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})$. Solute-$j$ is stabilised relative to solute $j$ in the solution reference state, an ideal solution having unit molality. If $\mathrm{m}_{\mathrm{j}}>1.0 \mathrm{~mol} \mathrm{~kg}$, $\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$ is $>$ zero. Hence, $\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})>\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})$; solute-$j$ is destabilised relative to solute $j$ in the (ideal) solution reference state.
We also compare the chemical potentials of solute $j$ in real and ideal solutions at the same molality leading to the definition of an excess chemical potential for solute $j$, $\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})$.
$\mu_{j}^{\mathrm{E}}(\mathrm{aq})=\mu_{j}(\mathrm{aq})-\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})$
$\text { Hence, } \mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)$
$\text { where } \lim \operatorname{it}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1 \text { and } \ln \left(\gamma_{\mathrm{j}}\right)=0 \text { at all T and } \mathrm{p} \text {. }$
Equation (g) highlights the role played by activity coefficient $\gamma_{j}$; $\gamma_{j}$ can be neither zero nor negative; the range for $\gamma_{j}$ is from below to above unity. In contrast $\ln \left(\gamma_{j}\right)$ can be zero (as in an ideal solution) and be either greater or less than zero.
Activity coefficients are interesting quantities. For a given solute $j$ at molality $\mathrm{m}_{j}$ in an aqueous solution (at fixed temperature and pressure) $\gamma_{j}$ describes the impact on the chemical potential $\mu_{j}(\mathrm{aq})$ of solute - solute interactions. The basis of this conclusion follows from the definition given in equation (h) [3]. As a solution is diluted, so the mean distance of separation of solute molecules increases. In these terms a model for an ideal solution, molality mj is one in which each solute molecule contributes to the properties of a given solution independently of all other solutes in the system. In an operational sense, each solute molecule in unaware of the presence of other solute molecules in solution and in these terms the solute molecules are infinitely far apart.
We emphasise the point that activity coefficient $\gamma_{j}$ is an interesting and important quantity; $\gamma_{j}$ describes the impact on chemical potential $\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ of solute - solute interactions. These interactions can be cohesive (i.e. attractive) such that $\gamma_{j}<1, \ln \left(\gamma_{j}\right)<0$ and $\mu_{j}(\text { aq } ; \text { real; } T ; p)<\mu_{j}(\text { aq;ideal; } T ; p)$, a stabilising influence. On the other hand, solute - solute interactions may be repulsive. In view of the fact that molecules have a real size, this contribution is always present. Consequently the latter (together with other forms of solute – solute repulsions) contribute to cases where $\gamma_{j}>1.0$, $\ln \left(\gamma_{j}\right)>0$ and $\mu_{\mathrm{j}}(\text { aq; real; } \mathrm{T} ; \mathrm{p})>\mu_{\mathrm{j}}(\mathrm{aq} ; \text { ideal } ; \mathrm{T} ; \mathrm{p})$. In principle, activity coefficient $\gamma_{j}$ contains an enormous amount of information [4]. Many of the interesting properties of aqueous solutions are packed in the parameter $\gamma_{j}$. Unfortunately only in rare instances is it possible to dissect a given $\gamma_{j}$ into the several contributing interactions. A common though obviously dangerous procedure sets $\gamma_{j}$ equal to unity, assuming that the properties of a given solute $j$ are ideal. However in many cases we have no alternative but to make this assumption at least in initial stages of an analysis of experimental results.
Equation (e) is satisfactory for very dilute solutions of neutral solutes. Indeed this equation has enormous technological significance. The task of producing a very pure liquid requires lowering the molalities of solutes in a solution. With decreasing molality of a given solute, the chemical potential of a solute decreases; i.e. the solute is stabilised. So as more solute, an impurity, is removed, the trace remaining is increasingly stabilised [5].
We cannot put a number value to either $\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ or $\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})$. These quantities measure the contribution made by a solute $j$ to the total energy of a solution. One contribution to, for example, $\mu_{j}^{0}(\mathrm{aq} ; \mathrm{T})$ emerges from solvent-solute interactions. Interestingly, we can in general put a number value to the corresponding limiting partial molar volume, $V_{j}^{\infty}(a q)$ [6]. The concept of infinite dilution is extremely important in a practical sense. Nevertheless, we enter a word of caution. Returning to equation (b) for the chemical potential, we note that $\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \ln \left(\mathrm{m}_{\mathrm{j}}\right)$ tends to minus $\infty$.
$\text { Hence (at all T and p) limit }\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mu_{\mathrm{j}}(\mathrm{aq})=-\infty$
The practical significance of equation (i) is that with increasing dilution so the chemical potential of a solute decreases - the solute is increasingly stabilised. That is why the challenge of removing the last traces of unwanted solute presents such a formidable task, particularly to those industries where very high solvent purity is essential; e.g. the pharmaceutical industry [5].
Footnotes
[1] Activity coefficients have a ‘bad press’. They are not ‘loved’ except by a minority of chemists. Nevertheless these coefficients contain information concerning the way in which solute molecules ‘communicate’ to each other in solution.
[2] M. Spiro, Educ. Chem.,1966, 3,139.
[3] The concept of an activity coefficient for a solute tending to unity at infinite dilution was proposed by A.A. Noyes and W.C. Bray: J. Am. Chem. Soc., 1911, 33,1643.
[4] E. Wilhelm, Thermochim. Acta, 1987,119,17; Interactions in Ionic and Non-Ionic Hydrates, ed. H. Kleenberg, Springer –Verlag, Berlin, 1987,p.118.
[5] S. F. Sciamanna and J. M. Prausnitz, AIChE J., 1987, 33, 1315.
[6] Throughout this subject, it is good practice to examine equations describing the dependence of partial molar properties of solvent and solute in the limit that the composition of the solution tends to increasingly dilute solutions; see J. E. Garrod and T. M. Herringon, J. Chem. Educ., 1969, 46, 165.
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For the most part we use either the molality scale or the concentration scale to express the composition of aqueous solutions. Nevertheless, the mole fraction scale is often used. Hence we express the chemical potential of solute $j$, $\mu_{j}$ as a function of mole fraction of solute $j$, $\mathrm{x}_{j} \left[=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right)\right]$. Note that we are relating the property $\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ to the composition of the solution using a different method from that used where the composition is expressed in terms of the molality or concentration of a solutes. By definition at fixed temperature and fixed pressure,
\begin{aligned} &\mu_{j}(a q ; T ; p)= \ &\quad \mu_{j}^{0}(a q ; T ; x-\text { scale })+R \, T \, \ln \left(x_{j} \, f_{j}^{*}\right)+\int_{p^{0}}^{p} V_{j}^{\infty}(a q ; T) \, d p \end{aligned}
$\text { By definition, } \operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{f}_{\mathrm{j}}^{*}=1 \text { at all } \mathrm{T} \text { and } \mathrm{p} \text {. }$
$f_{j}^{*}$ is the asymmetric solute activity coefficient on the mole fraction scale. The word 'asymmetric', although rarely used, emphasises the difference between $f_{j}^{*}$ and the rational activity coefficients.
For solutions at ambient pressure, the integral term in equation (a) is negligibly small. At pressure $\mathrm{p}$ and temperature $\mathrm{T}$,
$\mu_{j}(a q)=\mu_{j}^{0}(a q ; x-\text { scale })+R \, T \, \ln \left(x_{j} \, f_{j}^{*}\right)$
For an ideal solution, $f_{j}^{*}$ is unity. The reference state for the solute is the solution where the mole fraction of solute-$j$ is unity. This is clearly a hypothetical solution but we assume that the properties of the solute j in this solution can be obtained by extrapolating from the properties of solute-$j$ at low mole fractions [1]. For an ideal solution at ambient pressure and temperature,
$\mu_{j}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}\right)$
$\text { or, } \mu_{j}(\mathrm{aq} ; \mathrm{id})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}\right)$
Because, $\mathrm{x}_{\mathrm{j}}<1.0, \quad \ln \left(\mathrm{x}_{\mathrm{j}}\right)<0$ Hence, $\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})<\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })$. The solute is at a lower chemical potential than in the solution reference state [1].
Footnote
[1] The extrapolation defining the reference state as a solution wherein the mole fraction of solute is unity might seem strange. In fact such long extrapolations are common in chemical thermodynamics. For example at $0.1 \mathrm{~MPa}$ and $298.15 \mathrm{~K}$, liquid water is the stable phase. At $0.1 \mathrm{~MPa}$ and $273.15 \mathrm{~K}$ both water($\ell$) and ice-1h are the stable phases. Nevertheless we know that, assuming water($g$) is an ideal gas, the volume occupied by $0.018 \mathrm{~kg}$ of water($g$) at $273 \mathrm{~K}$ and $0.1 \mathrm{~MPa}$ equals $22.4 \mathrm{~dm}^{3}$. The fact that this involves a long extrapolation into a state where water vapour is not the stable phase does not detract from the usefulness of the concept.
1.5.14: Chemical Potentials Solute Concentration Scale
Both molalities and mole fractions are based on the masses of solvent and solute in a solution. Hence neither the molality $\mathrm{m}_{j}$ nor mole fraction $\mathrm{x}_{j}$ of a non-reacting solute depend on temperature and pressure. In fact, when we differentiate the equations for $\mu_{j}(\mathrm{aq} ; \mathrm{T})$ with respect to pressure we take advantage of the fact that $\mathrm{m}_{j}$ and $\mathrm{x}_{j}$ do not depend on pressure. In addition, molalities and mole fractions are precise; there is no ambiguity concerning the amount of solvent and solute in the solution.
However, when we describe the meaning and significance of the activity coefficient $\gamma_{j}$ and the meaning of the term 'infinite dilution' we refer to the distance between solute molecules. In fact, in reviewing the properties of solutions, chemists are often more interested in the distance between solute molecules than in their masses. [The same can be said about the interest shown by humans in the behaviour of other human beings!] Therefore, chemists often use concentrations to express the composition of solutions.
The concentration of solute $\mathrm{c}_{j}$ describes the amount of chemical substance $j$ in a given volume of solution; $\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}$. The common method for expressing $\mathrm{c}_{j}$ uses the unit '$\mathrm{mol} \mathrm{dm}$' [1,2]. By definition [at temperature $\mathrm{T}$ and pressure $p\left(\cong p^{0}\right)$]
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\mathrm{scale})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)$
$\mathrm{c}_{\mathrm{r}}$ is the reference concentration, $1 \mathrm{~mol dm}^{-3}$; $\mathrm{c}_{j}$ is expressed using the same unit; $\mathrm{y}_{j}$ is the activity coefficient for the solute $j$ on the concentration scale.
$\text { By definition, } \quad \lim \operatorname{it}\left(c_{j} \rightarrow 0\right) y_{j}=1 \quad \text { (at all } T \text { and } p \text { ) }$
$\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })$ is the chemical potential of solute $j$ in an ideal $\left(\mathrm{y}_{\mathrm{j}}=1.0\right)$ aqueous solution (at the same $\mathrm{T}$ and $\mathrm{p}$) where the concentration of solute $\mathrm{c}_{\mathrm{j}}=1.0 \mathrm{~mol} \mathrm{dm}^{-3}$
Footnotes
[1] Using the base SI units, concentration is given by the ratio $\left(n_{j} / V\right)$ where $\mathrm{V}$ is expressed using the unit $\mathrm{m}^{3}$; $\mathrm{n}_{j}$ is the amount of chemical substance $j$, the unit being the mole. Nevertheless in the present context, general practice uses the reference concentration $1 \mathrm{~mol dm}^{-3}$; $\mathrm{c}_{j}$ is expressed using the unit, $\mathrm{mol dm}^{-3}$. This practice emerges from the fact that for dilute aqueous solutions at ambient $\mathrm{T}$ and $\mathrm{p}$, unit concentration of solute, $1 \mathrm{~mol dm}^{-3}$ is almost exactly unit molality, $1 \mathrm{~mol kg}^{-1}$.
[2] For comments on standard states see E. M. Arnett and D. R. McKelvey, in Solute-Solvent Interactions, ed. J. F. Coetzee, M. Dekker, New York, 1969, chapter 6.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.05%3A_Chemical_Potentials/1.5.13%3A_Chemical_Potentials-_Solutes-_Mole_Fraction_Scale.txt
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For a given solution we can express the chemical potential of solute $j$, $\mu_{\mathrm{j}}(\mathrm{aq})$ in an aqueous solution at temperature $\mathrm{T}$ and pressure $\mathrm{p}\left(\approx \mathrm{p}^{0}\right)$ using two equations. Therefore, at fixed $\mathrm{T}$ and $\mathrm{p}$,
\begin{aligned} &\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)= \ &\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right) \end{aligned}
Therefore,
$\ln \left(\mathrm{y}_{\mathrm{j}}\right)=\ln \left(\gamma_{\mathrm{j}}\right)+\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) +(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })\right]$
In the latter two equations the composition variables $\mathrm{m}_{j}$ and $\mathrm{c}_{j}$ are expressed in the units ‘$\mathrm{mol kg}^{-1}$’ and ‘$\mathrm{mol dm}^{-3}$’ respectively [1]. The ratio ‘$\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}$’ equals the density expressed in the unit ‘$\mathrm{kg dm}^{-3}$’. For dilute solutions, $\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}=\rho_{1}^{*}(\ell)$, the density of the pure solvent.
$\text { Also, } \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0}=\left[\mathrm{mol} \mathrm{d \textrm {dm } ^ { - 3 }}\right] /\left[\mathrm{mol} \mathrm{kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{dm}^{-3}\right]$
For dilute aqueous solutions at ambient pressure and $298.2 \mathrm{~K}$ [2,3],
$\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right)=-\ln (0.997)$
With reference to equation (b), with increasing dilution, $\mathrm{y}_{\mathrm{j}} \rightarrow 1, \gamma_{\mathrm{j}} \rightarrow 1,\left(\mathrm{~m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) \rightarrow \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)$ Hence,
$\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\mathrm{scale})-\mu_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\right]$
We combine equations (b) and (e).
$\ln \left(\mathrm{y}_{\mathrm{j}}\right)=\ln \left(\gamma_{\mathrm{j}}\right)+\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right)-\ln \left[\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\right]$
$\ln \left(\mathrm{y}_{\mathrm{j}}\right)=\ln \left(\gamma_{\mathrm{j}}\right)+\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right)-\ln \left[\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\right]$
Footnotes
[1] A given solution is prepared by adding $\mathrm{n}_{j}$ moles of solute $j$ to $\mathrm{w}_{1} \mathrm{~kg}$ of solvent.
Molality of solute $\mathrm{j} / \mathrm{mol} \mathrm{kg}{ }^{-1}=\left(\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}\right)$
Total mass of solution/kg $=w_{1}+n_{j} \, M_{j}$ where molar mass of solute/kg $\mathrm{mol}^{-1}=\mathrm{M}_{\mathrm{j}}$
Volume of solution/$\mathrm{m}^{3} = \mathrm{V}$
Density of solution $\rho / \mathrm{kg} \mathrm{m}^{-3}=\left[\frac{\mathrm{w}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\mathrm{V}}\right]$
By convention chemists express the composition of solutions in terms of (i) concentration using the unit ‘$\mathrm{mol dm}^{-3}$’ and (ii) molality using the unit, ‘$\mathrm{mol kg}^{-1}$’. These composition scales stem from the fact that at $298.15 \mathrm{~K}$, $1 \mathrm{~dm}^{3}$ of water has a mass of approx. $1 \mathrm{~kg}$. So as we swap composition scales a conversion factor is often required .
For dilute solutions $w_{1}>n_{j} \, M_{j}$ and density of solution $\rho$ equals the density of the pure solvent (at same temperature and pressure), i.e. density $\rho=\rho 1(\ell) \mathrm{kg} \mathrm{m} \mathrm{m}^{-3}$
[2] A typical conversion takes the following form for water at $298.2 \mathrm{~K}$ and ambient pressure.
\begin{aligned} \text { Density }=0.997 \mathrm{~g} \mathrm{~cm}^{-3} &=0.997\left(10^{-3} \mathrm{~kg}\right)\left(10^{-2} \mathrm{~m}^{-3}\right.\ &=0.997 \mathrm{X} \mathrm{} 10^{3} \mathrm{~kg} \mathrm{~m}^{-3} \ =& 997 \mathrm{~kg} \mathrm{~m}^{-3}=0.997 \mathrm{~kg} \mathrm{\textrm {dm } ^ { - 3 }} \end{aligned}
$\text { Then } \frac{\mathrm{c}_{\mathrm{j}} / \mathrm{mol} \mathrm{dm}^{-3}}{\mathrm{~m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}}=\frac{\mathrm{n}_{\mathrm{j}} / \mathrm{mol}}{\mathrm{V} / \mathrm{dm}^{3}} \, \frac{\mathrm{w}_{1} / \mathrm{kg}}{\mathrm{n}_{\mathrm{j}} / \mathrm{mol}}=\frac{\mathrm{w}_{1} / \mathrm{kg}}{\mathrm{V} / \mathrm{dm}^{3}}=\rho / \mathrm{kg} \mathrm{dm}^{-3}$
[3] \begin{aligned} \ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) &=\ln \left[\left(\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0}\right) /\left(\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\right)\right] \ =& \ln \left[\left(\mathrm{kg} \mathrm{d \textrm {m } ^ { - 3 } ) / \rho ]}=-\ln \left(\rho / \mathrm{kg} \mathrm{d \textrm {dm } ^ { - 3 } )}\right.\right.\right. \end{aligned}
1.5.16: Chemical Potentials- Solute- Molality and Mole Fraction Scales
The chemical potential of solute j in aqueous solution at temperature $\mathrm{T}$ and at close to ambient pressure is related to the molality $\mathrm{m}_{j}$ and mole fraction $\mathrm{x}_{j}$ [1].
\begin{aligned} \mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \ &=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}-\mathrm{scale})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}} \, \mathrm{f}_{\mathrm{j}}^{*}\right) \end{aligned}
Therefore,
\begin{aligned} \ln \left(\mathrm{f}_{\mathrm{j}}^{*}\right)=\ln \gamma_{\mathrm{j}} &+\ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{x}_{\mathrm{j}} \, \mathrm{m}^{0}\right) \ &+(\mathrm{l} / \mathrm{R} \, \mathrm{T})\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x} \text { - scale })\right] \end{aligned}
For dilute solutions [1], $\left(1 / \mathrm{M}_{1}\right)>\mathrm{m}_{\mathrm{j}}$. Hence $\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}\right)$ equals $\left(m^{0} \, M_{1}\right)^{-1}$, a dimension-less quantity. Therefore,
\begin{aligned} \ln \left(f_{j}^{*}\right)=\ln \gamma_{j} &-\ln \left(m^{0} \, M_{1}\right) \ &+(1 / R \, T)\left[\mu_{j}^{0}(a q ; T ; p)-\mu_{j}^{0}(a q ; T ; p ; x-s c a l e)\right] \end{aligned}
It is unrealistic to expect that $\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ equals $\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}-\mathrm{scale})$ because the two reference states for solute-$j$ are quite different. In general terms, $\mathrm{f}_{\mathrm{j}}^{*}$ does not equal $\gamma_{j}$ for the same solution. Nevertheless, both $\mathrm{f}_{\mathrm{j}}^{*}$ and $\gamma_{j}$ tend to the same limit, unity, as the solution approaches infinite dilution.
Hence as $\mathrm{n}_{j}$ tends to zero,
$\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}-\mathrm{scale})=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0} \, \mathrm{M}_{\mathrm{l}}\right)$
For example, in the case of aqueous solutions at $298.15 \mathrm{~K}$, $\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0} \, \mathrm{M}_{1}\right)$ equals $\left(-9.96 \mathrm{~kJ} \mathrm{~mol}^{-1}\right)$ meaning that, with respect to the reference states for the two solutions, the chemical potential of solute $j$ is higher on the mole fraction scale than on the molality scale. Combination of equations (b) and (d) yields an equation relating the two activity coefficients with the two terms describing the composition of the solution.
$\ln \left(f_{j}^{*}\right)=\ln \gamma_{j}+\ln \left(m_{j} \, M_{1} / x_{j}\right)$
The term ‘unitary’ is sometimes used to describe reference chemical potentials on the mole fraction scale, $\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{j}}=1\right)$. The term $\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0} \, \mathrm{M}_{1}\right)$ in equation (d) is called cratic [2] because it refers to different amounts of solute and solvent which are mixed to form reference states for the solute on molality, mole fraction and concentration scales. The impression is sometimes given that standard states for solutes based on the mole fraction scale (sometimes identified as the unitary scale) are more fundamental but there is little experimental evidence to support this view.
Footnotes
[1] For a solution prepared using $\mathrm{w}_{1}$ kg of water and $\mathrm{n}$ moles of solute $j$,
$\begin{gathered} \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\left(\mathrm{w}_{1} / \mathrm{M}_{1}\right)+\mathrm{n}_{\mathrm{j}}\right] \text { and } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1} \, \ \mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \,\left[\left(\mathrm{w}_{\mathrm{l}} / \mathrm{M}_{\mathrm{l}}\right)+\mathrm{n}_{\mathrm{j}}\right] / \mathrm{w}_{1} \, \mathrm{m}^{0} \, \mathrm{n}_{\mathrm{j}} \ \mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}=\left[\left(1 / \mathrm{M}_{1}\right)+\mathrm{m}_{\mathrm{j}}\right] / \mathrm{m}^{0} \end{gathered}$
[2] The terms 'unitary' and 'cratic' were suggested by R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.05%3A_Chemical_Potentials/1.5.15%3A_Chemical_Potentials-_Solute-_Concentration_and_Molality_Scales.txt
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We consider a salt j having the following general formula;
$v_{+} \mathrm{M}^{\mathrm{z}+} v_{-} \mathrm{X}^{\mathrm{z}-} .$
Here $ν_{+}$ and $ν_{-}$ are the (integer) stoichiometric coefficients; $\mathrm{z}$, and $\mathrm{z}_{-}$ are the (integer) charge numbers. We assume the salt $j$ is completely dissociated in aqueous solution. Hence the solution contains (apart from solvent) two chemical substances. With complete dissociation each mole of salt produces $v\left(=v_{+}+v_{-}\right)$ moles of ions. The condition of electric neutrality is expressed by equation (b).
$\left|v_{+} \, z_{+}\right|=\left|v_{-} \, z_{-}\right|$
For a solution molality $\mathrm{m}_{j}$, the molalities of cations and anions are $ν_{+} \, \mathrm{m}_{j}$ and $ν_{−} \, \mathrm{m}_{j}$ respectively. If the chemical potentials of cations and anions are $\mu_{+}(\mathrm{aq})$ and $\mu_{-}(\mathrm{aq})$ respectively, the chemical potential of salt $j$ in aqueous solution (at molality, $\mathrm{m}_{j}$ temperature $\mathrm{T}$ and ambient pressure) is given by equation (c).
$\mu_{j}(a q)=v_{+} \, \mu_{+}(a q)+v_{-} \, \mu_{-}(a q)$
In an ideal solution (at the same $\mathrm{T}$ and ambient pressure) where the molality of the salt $j$ is $1 \mathrm{~mol kg}^{–1}$.
$\mu_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{v}_{+} \, \mu_{+}^{0}(\mathrm{aq})+\mathrm{v}_{-} \, \mu_{-}^{0}(\mathrm{aq})$
For each ionic substance $\mathrm{i}$, the chemical potential (at the same $\mathrm{T}$ and $\mathrm{p}$) is given by equation (e) where $\gamma_{\mathrm{i}}$ is the single ionic activity coefficient.
$\mu_{\mathrm{i}}(\mathrm{aq})=\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)$
Hence for the salt with $\mathrm{m}_{+}=v_{+} \, \mathrm{m}_{\mathrm{j}}$ and $\mathrm{m}_{-}=v_{-} \, \mathrm{m}_{\mathrm{j}}$.
\begin{aligned} \mu_{\mathrm{j}}(\mathrm{aq})=\left[\mathrm{v}_{+}\right.&\left.\, \mu_{+}^{0}(\mathrm{aq})+\mathrm{v}_{-}-\mu_{-}^{0}(\mathrm{aq})\right] \ &+\mathrm{v}_{+} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{v}_{+} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{+} / \mathrm{m}^{0}\right) \ &+\mathrm{v}_{-} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{v}_{-} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{-} / \mathrm{m}^{0}\right) \end{aligned}
We draw together the logarithm terms.
\begin{aligned} \mu_{j}(\mathrm{aq})=\left[v_{+}\right.&\left.\, \mu_{+}^{0}(\mathrm{aq})+\mathrm{v}_{-} \, \mu_{-}^{0}(\mathrm{aq})\right] \ &+\mathrm{R} \, \mathrm{T} \, \ln \left[\left(\mathrm{v}_{+} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{+} / \mathrm{m}^{0}\right)^{\mathrm{v}+} \,\left(\mathrm{v}_{-} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{-} / \mathrm{m}^{0}\right)^{\mathrm{v}-}\right] \end{aligned}
The latter far from elegant equation contains all the parameters we expect to be present. Here it is convenient to introduce a parameter $\mathrm{Q}$ [1,3].
$\mathrm{Q}^{v}=v_{+}^{v+} \, v_{-}^{v-}$
The (geometric) mean activity coefficient is defined by equation (i).
$\gamma_{\pm}^{v}=\gamma_{+}^{v+} \, \gamma_{-}^{v-}$
Hence equation (g) can be rewritten in the following form.
$\mu_{j}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+v \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right]$
The quantity $\mathrm{Q}$ takes account of the stoichiometric composition of the salt. In preparing the salt solution we target the molality but this does not take account of how many moles of each ionic substance are produced by one mole of salt; the quantity $ν$ only records how many moles of ionic substances are produced by each mole of salt. For the salt $\mathrm{M}^{2+} 2 \mathrm{X}^{-}$ [e.g. \mathrm{Mg Br}_{2}\)] $v_{+}=1$, $v_{-}=2$ where $\mathrm{Q}^{3} =1 ^{2}$. $2^{2} = 4$. For this salt,
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+3 \, \mathrm{R} \, \mathrm{T} \, \ln \left[4^{1 / 3} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right]$
The complex algebra associated with a thermodynamic description of salt solutions stems from a conflict of interests and practical chemistry. The ground rules in this subject are quite simple--- measurements are made on electrically neutral solutions; e.g. $\mathrm{NaCl}(\mathrm{aq})$. In the latter case chemists often favour a description of this system in terms of an aqueous solution of two solutes, sodium ions and chloride ions. Rather than using an activity coefficient for the solute [e.g; $\gamma(\mathrm{NaCl})$], we define a mean activity coefficient $\gamma_{\pm}$ which recognises the presence of two ionic substances. An indication of the presence of two solutes (i.e. $\mathrm{Na}^{+}$ and $\mathrm{Cl}^{-}$) rather than one solute (e.g. \mathrm{NaCl}\)) is the stoichiometric parameter $ν$ in the equation for the chemical potential of the solvent. This parameter is readily determined from the depression of solvent freezing points ( i.e. cryoscopy) and osmotic pressures. Both properties are directed at the properties of solvents. The cryoscopic technique is based on measurement of the temperature at which solvent in a solution is in equilibrium (at fixed pressure) with pure solid solvent. The osmotic pressure $\pi$ characterises the equilibrium (a fixed temperature) between the solution at pressure $\mathrm{p} + \pi$ and pure solvent at pressure $\mathrm{p}$. Both techniques determine $ν$ or, in colloquial terms, count solute particles. The molar mass of $\mathrm{NaCl}$ and urea are roughly equal; the number of solute particles in $\mathrm{NaCl}(\mathrm{aq})$ is twice that in urea($\mathrm{aq}$) for the same mass of solute in $1 \mathrm{~kg}$ of water. We confirm this observation by measuring the depressions of freezing points or the osmotic pressures of two solutions.
Equation (j) signals an important challenge. If we could separate out ionic contributions to $\mu_{\mathrm{j}}^{0}(\mathrm{aq})$ for salt $j$, we could probe the contributions made by ion-water interactions, the hydration properties for a given ion $\mathrm{i}$ at defined $\mathrm{T}$ and $\mathrm{p}$ to $\mu_{j}^{0}(\mathrm{aq})$. In this exercise we might then extend the analysis to single ion enthalpies, $\mathrm{H}_{\mathrm{i}}^{0}(\mathrm{aq})$, volumes $\mathrm{V}_{\mathrm{i}}^{0}(\mathrm{aq})$, and entropies $\mathrm{S}_{\mathrm{i}}^{0}(\mathrm{aq})$. Unfortunately the story is not simple. Indeed we cannot measure these chemical potentials and then obtain absolute estimates for the above derived properties.
Footnotes
[1] R. H. Stokes and R. H. Robinson, J. Am. Chem.Soc.,1948,70,1870.
[2] R. G. Bates, Pure Appl. Chem.,1973,36,407.
[3] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1997,93,2171.
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There is an important point to consider in the context of salt solutions. For a dilute aqueous solution containing sodium chloride, osmotic and colligative properties confirm that for each mole of sodium chloride the aqueous solution contains (almost exactly) two moles of solutes. These observations result in an added complexity in that chemists describe the solute, sodium chloride in two ways. In one description there is one solute - 'sodium chloride'. In another description there are two solutes sodium ions and chloride ions. The latter description is certainly attractive because we can ring the changes through a series of solutes; $\mathrm{NaCl } \rightarrow \mathrm{ NaBr } \rightarrow \mathrm{ KBr } \rightarrow \mathrm{ KCl } \rightarrow$ etc. Here we change in stepwise fashion one chemical substance in the salt to produce a new solute. There is, however, one crucial condition. Aqueous solutions are electrically neutral although the solutions contain ions. Therefore, the total charge on all cations equals in magnitude the total charge on all anions in the same solution. There is, therefore, a major problem. We cannot examine the properties of aqueous solutions containing, for example, just cations. We can only examine the properties of electrically neutral solutions. How can we obtain the properties of ionic substance (e.g. $\mathrm{Na}^{+}$) in aqueous solutions at defined temperature and pressure? The frustrating answer is that we cannot measure the thermodynamic properties of single ions in solution. This realisation does not stop us speculating about such properties. In fact, a common procedure involves estimating the properties of single ions but then in the last stage of the analysis we pull the derived single ion properties together to describe the properties of a given salt solution.
The chemical potential of each ionic substance i in solution is related to its molality mi using equation (a) for a solution at fixed temperature and fixed pressure. We assume that the latter is ambient pressure and therefore close to the standard pressure $\mathrm{p}^{0}$.
$\mu_{i}(a q)=\mu_{i}^{0}(a q)+R \, T \, \ln \left(m_{i} \, \gamma_{i} / m^{0}\right)$
Here $\mu_{\mathrm{i}}^{0}(\mathrm{aq})$ is the standard chemical potential of ion i in an aqueous solution where both the molality $\mathrm{m}_{\mathrm{i}}$ and single ion activity coefficient $\gamma_{\mathrm{i}$ are unity (at the same $\mathrm{T}$ and $\mathrm{p}$). However, the terms $\mu_{i}(\mathrm{aq})$, $\mu_{\mathrm{i}}^{0}(\mathrm{aq})$ and $\gamma_{\mathrm{i}}$ have no practical significance because, we cannot prepare a solution containing just one ionic chemical substance. The way forward involves using equation (a) for all ionic substances in the solution. In order to show how the argument develops we consider an aqueous solution containing a 1:1 salt (e.g. $\mathrm{NaCl}$) which we assert is fully dissociated into ions. We make two (extrathermodynamic) assumptions.
1. The chemical potential of salt $j$ in an ideal solution at unit salt molality ($\mathrm{m}_{j} = 1.0 \mathrm{~mol kg}^{-1}$) is given by the sum of the corresponding reference chemical potentials of cations and anions (at the same temperature and pressure).
\begin{aligned} &\mu_{j}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\right)= \ &\mu_{+}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{+}=1 \mathrm{~mol} \mathrm{~kg}{ }^{-1}\right)+\mu_{-}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{-}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\right) \end{aligned}
As demanded by the analysis, such an ideal solution would be electrically neutral. But we have no information concerning the contributions which the ions make to the overall chemical potential, $\mu_{\mathrm{j}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}\right)$. We anticipate that such contributions are characteristic of the ions. For a 1:1 salt combination of the three previous equations yields for a solution at fixed $\mathrm{T}$ and $\mathrm{p}$, equation (d).
\begin{aligned} &\mu_{j}(\mathrm{aq})= \ &\mu_{j}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg} \mathrm{~kg}^{-1}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{+} \, \mathrm{m}_{-} \, \gamma_{+} \, \gamma_{-} / \mathrm{m}^{0} \, \mathrm{m}^{0}\right) \end{aligned}
A (geometric) mean ionic activity coefficient $\gamma_{\pm}$ is defined by equation (e).
$\gamma_{\pm}^{2}=\gamma_{+} \, \gamma_{-}$
Also for a 1: 1 salt $\mathrm{m}_{\mathrm{j}}^{2}=\mathrm{m}_{+} \, \mathrm{m}_{-}$. Therefore (at fixed temperature and pressure),
$\mu_{j}(a q)=\mu_{j}^{0}(a q)+2 \, R \, T \, \ln \left(m_{j} \, \gamma_{\pm} / m^{0}\right)$
where $\mu_{j}^{0}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}{ }^{-1} ; \mathrm{p} \equiv \mathrm{p}^{0}\right)$
$\text { By definition limit }\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p}$
The origin of the integer '2' in equation (f) is the stoichiometry of the salt; each mole of salt forms two moles of ions assuming complete dissociation. Hence $\mu_{\mathrm{j}}^{0}(\mathrm{aq})$ ( ) is the chemical potential of salt $j$ in an ideal solution at the same $\mathrm{T}$ and $\mathrm{p}$_ ($\cong \mathrm{p}^{0}$) where the molality of the salt is $1 \mathrm{~mol kg}^{-1}$. If the properties of the salt are ideal the chemical potential of the salt is given by equation (h).
$\mu_{j}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
When $\mathrm{m}_{\mathrm{j}}>\mathrm{m}^{0}$, the chemical potential of the salt in the ideal solution $\mu_{j}(\mathrm{aq} ; \mathrm{id})>\mu_{\mathrm{j}}^{0}(\mathrm{aq})$; the salt is at a higher chemical potential than in the reference state. When $\mathrm{m}_{\mathrm{j}}<1.0 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}$, the chemical potential of the salt in the ideal solution is lower than in the reference solution where $\mathrm{m}_{\mathrm{j}}=1.0 \mathrm{~mol} \mathrm{~kg}^{-1}$.
Returning to the equation (f), there is merit in writing the equation in the following form.
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\pm}\right)$
$\text { Or, } \quad \mu_{j}(\mathrm{aq})=\mu_{j}(\mathrm{aq} ; \mathrm{id})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\pm}\right)$
The difference $\left[\mu_{j}(\mathrm{aq})-\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\right]$ is a measure of the extent to which the chemical potential of a salt in a real salt solution differs from the chemical potential of the same salt in an ideal solution. For $\mathrm{KCl}$($\mathrm{aq}$; $298.2 \mathrm{~K}$; $0.1 \mathrm{~mol kg}^{-1}$) the mean ionic activity coefficient $\gamma_{\pm}$ equals $0.769$; $0.769 ; 2 \, \mathrm{R} \, \mathrm{T} \, \ln (0.769) =-1.30 \mathrm{~kJ} \mathrm{~mol}^{-1}$. In other words, $\mathrm{KCl}$ in this solution is at a lower chemical potential than in the corresponding ideal solution. In fact, $\gamma_{\pm}$ for most dilute aqueous salt solution is $< 1.0$ at ambient $\mathrm{T}$ and $\mathrm{p}$, and so this pattern in chemical potentials is quite common. However, even though we know $\gamma_{\pm}$ for these systems we are not in a position to comment on the single ion activity coefficients for the reasons discussed above [1].
The difference described by equation (j) prompts the definition of an excess chemical potential, $\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})$.
$\text { Thus } \quad \mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\pm}\right)$
A key contribution to $\mu_{j}^{0}(\mathrm{aq})$ emerges from cation-water and anion-water interactions, namely ionic hydration. In contrast $\gamma_{\pm}$ is determined by ion-ion interactions in real solutions. For a solution at pressure $\mathrm{p}$, equation (f) takes the following form.
$\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{p}^{0}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}$
Footnotes
[1] For comparison of $\gamma_{\pm}$ for dilute salt solutions in aqueous solution and in $\mathrm{D}_{2}\mathrm{O}$ see O. D. Bonner, J. Chem. Thermodyn., 1971, 3,837; and references therein.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.05%3A_Chemical_Potentials/1.5.18%3A_Chemical_Potentials-_Solutions-_1-1_Salts.txt
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An aqueous solution is prepared using $\mathrm{n}_{j}$ moles of salt $\mathrm{MX}$ and $\mathrm{n}_{1}$ moles of water. The properties of the system are accounted for using one of two possible Descriptions.
Description I
The solute $j$ comprises a 1:1 salt MX molality $\mathrm{m}(\mathrm{MX})\left[=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}\right. \text { where } \mathrm{w}_{1} \text { is the mass of water} \right]$.
The single ion chemical potentials, are defined in the following manner
\begin{aligned} &\mu\left(\mathrm{M}^{+}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{x}^{-}\right)} \ &\mu\left(\mathrm{X}^{-}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{M}^{+}\right)} \end{aligned}
The total Gibbs energy (at fixed $\mathrm{T}$ and $\mathrm{p}$ where $p \approx p^{0}$) is given by equation (b).
\begin{aligned} &\mathrm{G}(\mathrm{aq} ; \mathrm{I})=\mathrm{n}_{1} \, \mu_{1}^{\mathrm{eq}}(\mathrm{aq}) \ &\quad+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \ &\quad+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{X}^{-}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \end{aligned}
Description II
According to this Description each mole of cations is hydrated by $\mathrm{h}_{\mathrm{m}\left(\mathrm{H}_{2}\mathrm{O}\right)$ moles of water and each mole of anions is hydrated by $\mathrm{h}_{\mathrm{x}\left(\mathrm{H}_{2}\mathrm{O}\right)$ moles of water.
The single ion chemical potentials are defined as follows.
$\mu\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\left[\partial \mathrm{G} / \partial\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)\right]$
at constant $\mathrm{T}$, $\mathrm{p}$, $\mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right) \mu\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)\right]$
$\text { at constant } \mathrm{T}, \mathrm{p}, \mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right)$
$\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] ;$
$\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] .$
The (equilibrium) Gibbs energy (at defined $\mathrm{T}$ and $\mathrm{p}$) is given by the following equation.
\begin{aligned} &\mathrm{G}(\mathrm{aq} ; \mathrm{II})=\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right] \, \mu_{1}(\mathrm{aq}) \ &+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{0} \,\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \ &+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \end{aligned}
But the Gibbs energies defined by equations ( b) and (g) are identical (at equilibrium at defined $\mathrm{T}$ and $\mathrm{p}$). After all, it is the same solution. Hence, (dividing by $\mathrm{n}_{j}$)
\begin{aligned} &{\left[\mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]} \ &\quad+\left[\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]= \ &\quad-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}^{\mathrm{eq}}(\mathrm{aq})+ \ &\quad\left[\mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \ &+\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \ &\text { Then, } \mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\} \ &+\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}(\mathrm{X} \, ; \mathrm{I}) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\} \ &=-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \ &+\left\{\mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \ &+\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \mathrm{h}_{\mathrm{X}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \end{aligned}
We use the latter equation to explore what happens in the limit that $\mathrm{n}_{j}$ approaches zero. Thus,
\begin{aligned} &\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right) \gamma_{+}(\mathrm{I})=1 \quad \gamma_{-}(\mathrm{I})=1 \ &\gamma_{+}(\mathrm{II})=1 \quad \gamma_{-}(\mathrm{II})=1 \ &\mathrm{~m}_{\mathrm{j}}=0 \ &\mathrm{~m}\left(\mathrm{M}^{+} \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) / \mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)=1.0 \end{aligned}
$\begin{gathered} \text { Hence, } \mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)= \ \mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right) \ -\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}^{*}(\ell) \end{gathered}$
We obtain an equation linking the ionic chemical potentials. Thus,
$\ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+\ln \left\{\gamma_{+}(\mathrm{II})\right\}+\ln \left\{\gamma_{-} \text {(II) }\right\}$
\begin{aligned} &\text { But } \ln \left\{\gamma_{+}(\mathrm{I})\right\}+\ln \left\{\gamma_{-}(\mathrm{I})\right\}=2 \,\left\{\ln \gamma_{\pm}(\mathrm{I})\right\} \ &\text { Then, } 2 \, \ln \left\{\gamma_{\pm}(\mathrm{I})\right\}=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+2 \, \ln \left\{\gamma_{\pm} \text {(II) }\right\}
We identify relationships between single ion activity coefficients in an extra-thermodynamic analysis. Thus from equation (l),
$\ln \left\{\gamma_{+} \text {(II) }\right\}=\ln \left\{\gamma_{+} \text {(I) }\right\}-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{m}}$
$\ln \left\{\gamma_{-}(\mathrm{II})\right\}=\ln \left\{\gamma_{-}(\mathrm{I})\right\}-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{x}}$
It is noteworthy that in these terms the solution can be ideal using description I where $\gamma_{\pm} = 1.0$ but non-ideal using description II. Nevertheless, these equations show how the activity coefficient of the hydrated ion (description II) is related to the activity coefficient of the simple ion (description I).
1.5.20: Chemical Potentials- Salt Solutions- Ion-Ion Interactions
For most dilute aqueous salt solutions (at ambient temperature and pressure), mean ionic activity coefficients $\gamma_{\pm}$ are less than unity. Ion-ion interactions within a real solution lower chemical potentials below those of salts in the corresponding ideal solutions. A quantitative treatment of this stabilisation is enormously important. In fact for almost the whole of the 20th Century, scientists have offered theoretical bases for expressing $\ln \left(\gamma_{\pm}\right)$ as a function of the composition of a salt solution, temperature, pressure and electrical permittivity of the solvent.
In effect we offer as much information as demanded by the theory (e.g. molality of salt, nature of salt, permittivity of solvent, ion sizes, temperature, pressure, .....). We set the apparently simple task - please calculate the mean activity coefficient of the salt in this solution.
Many models and treatments have been proposed [1]. Most models start by considering a reference j-ion in an aqueous salt solution. In order to calculate the electric potential at the $j$-ion arising from all other ions in solution, we need to know the distribution of these ions about the $j$-ion. Unfortunately this distribution is unknown and so we need a model for this distribution. Further activity coefficients also reflect the impact of ions on water-water interactions in aqueous salt solutions. [2]
Footnotes
[1]
1. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edition revised,1959.
2. H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions ,Reinhold, New York,1950, 2nd. edn.,1950, chapter 2. The analysis presented by Harned and Owen anticipates application to irreversible processes; e.g. electrolytic conductance of salt solutions. We confine attention to equilibrium properties.
[2] H. S. Frank, Z. Phys. Chem, 1965,228,364.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.05%3A_Chemical_Potentials/1.5.19%3A_Chemical_Potentials-_Solutions-_Salt_Hydrates_in_Aqueous_Soluti.txt
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The chemical potential of salt j in an aqueous solution at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ (which is close to the standard pressure $\mathrm{p}^{0}$) is related to the molality of salt $\mathrm{m}_{j}$ using equation (a).
$\mu_{j}(a q)=\mu_{j}^{0}(a q)+v \, R \, T \, \ln \left(Q \, m_{j} \, \gamma_{\pm} / m^{0}\right)$
$\text { Here } Q^{v}=v_{+}^{v(+)} \, v_{-}^{v(-)}$
In equation (b), $ν_{+}$ and $ν_{-}$ are the number of moles of cations and anions respectively produced on complete dissociation by one mole of salt; $ν = ν_{+} + ν_{-}$. Here $\gamma_{\pm}$ is the mean ionic activity coefficient where by definition, at all $\mathrm{T}$ and $\mathrm{p}$,
$\operatorname{limit}\left(m_{j} \rightarrow 0\right) \gamma_{\pm}=1$
If the thermodynamic properties of the solution are ideal than $\gamma_{\pm}$ is unity. However the thermodynamic properties of salt solutions, even quite dilute solutions, are not ideal as a consequence of strong long-range charge-charge interactions between ions in solution. The challenge is therefore to come up with an equation for $\gamma_{\pm}$ granted that the temperature, pressure and properties of the solvent and salt are known together with the composition of the solution. The first successful attempt to meet this challenge was published by Debye and Huckel in 1923 and 1924 [1,2].
In most published accounts, the CGS system of units is used. However here we use the SI system and trace the units through the treatment. The solvent is a dielectric (structureless) continuum characterised by its relative permittivity, $\mathcal{\varepsilon}_{\mathrm{r}}$. The solute (salt) comprises ions characterised by their charge and radius; e.g. for ion-$j$, charge $\mathrm{z}_{j} \, e$ and radius $\mathrm{r}_{j}$ such that for cations $\mathrm{z}_{\mathrm{j}} \geq 1$ and for anions $\mathrm{z}_{j} \leq -1$ where $\mathrm{z}_{j}$ is an integer.
The analysis combines two important physical chemical relationships; Boltzmann’s Law and Poisson’s Equation.
We consider an aqueous salt solution containing $\mathrm{i}$-ionic substances, each substance having molality mi. The solution contains cations and anions. A KEY condition requires that the electric charge on the solution is zero.
$\text { Thus, } \sum_{j=1}^{j=i} m_{j} \, z_{j}=0$
Published accounts of the Debye-Huckel equation almost always use the concentration scale because the analysis concentrates on the distances between ions in solution rather than their mass. The concentration of $j$ ions in a solution, volume $\mathrm{V}$, is given by equation (e)
$\text { Thus, } \sum_{j=1}^{j=i} c_{j} \, z_{j}=0$
$\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}$
Here $\mathrm{n}_{j}$ is the amount of solute $j$ (expressed using the unit, mole) and $\mathrm{V}$ is the volume of solution.
These equations describe the solution as seen from the standpoint of a chemist interested in the properties of a given solution. However the ‘view’ from the standpoint of, for example, a cation in the solution is quite different. The neutrality condition in equation (d) requires that the electric charge on the solution surrounding the cation $j$ with charge $+\left|z_{j} \, e\right|$ equals $-\left|z_{j} \, e\right|$; i.e. equal in magnitude but opposite in sign. This is the electric charge on the rest of the solution and constitutes the ‘ion atmosphere’ of the $j$ ion. Every ion in the solution has its own atmosphere having a charge equal in magnitude but opposite in sign. Moreover interaction between a j ion and its atmosphere stabilises the $j$ ion in solution. The task of the theory is to obtain an equation for this stabilisation of the salt (i.e. the lowering of its chemical potential in solution). Intuitively we might conclude that this stabilisation is a function of the ionic strength of the salt solution and the dielectric properties of the solvent.
We consider a reference $j$ ion, radius $\mathrm{r}_{j}$, in solution together with a small volume element, $\mathrm{dV}$, a distance not more than say ($50 \times \mathrm{r}_{j}$) from the $j$ ion. In terms of probabilities, if the $j$ ion is a cation the probability of finding an anion in the reference volume is greater than finding a cation. Again with the $j$ ion as reference, we identify a time averaged electric potential $\psi_{j}$ at the volume element. The distribution of ions about the cation $j$ is assumed to follow the Boltzmann distribution law. The time average number of cations $\mathrm{dn}_{+}$ and anions $\mathrm{dn}_{-}$ in the volume element is given by equation (g) where ion $\mathrm{i}$ is, in turn, taken as a cation and then as an anion.
$\mathrm{dn}_{\mathrm{i}}=\mathrm{p}_{\mathrm{i}} \, \exp \left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \boldsymbol{\psi}_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right) \, \mathrm{dV}$
Here $\mathrm{p}_{\mathrm{i}}$ is the number of $\mathrm{i}$ ions in unit volume of the solution [3]. Each i ion has electric charge $z_{i} \, e$. Hence the electric charge on the volume $\mathrm{dV}$ is obtained by summing over the charge on the time average number of all ions. The charge density $\rho_{j}$ is given by equation (h), where the subscript $j$ on $\rho_{j}$ stresses that the charge is described with respect to the charge on the $j$ ion [4].
$\rho_{\mathrm{j}}=\sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \exp \left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right)$
The subscript $j$ on $\rho_{j}$ and $\psi_{j}$ identifies the impact of ion $j$ on the composition and electric potential of the reference volume $\mathrm{dV}$ distance $\mathrm{r}$ from the $j$ ion. At this point some simplification is welcomed. We expand the exponential in equation (h) [5].
$\text { Hence, } \quad \rho_{j}= \sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e}-\sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e} \,\left(\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right)+\sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e} \,(1 / 2) \,\left(\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right)^{2}-\ldots \ldots$
The solution as a whole has zero electric charge.
$\text { Hence } \quad \sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e}=0$
$\text { Also for dilute solutions, }\left(\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right) \ll<1$
Hence the third and all subsequent terms in equation (i) are negligibly small.
$\text { Therefore } \rho_{j}=-\sum \frac{p_{i} \,\left(z_{i} \, e\right)^{2} \, \psi_{j}}{k \, T}$
The approximation leading to equation (l) is welcome for an important reason. Equation (l) satisfies a key condition which requires a linear interdependence between $\rho_{j}$ and $\psi_{j}$.
Equation (l) relates charge density $\rho_{j}$ and electric potential $\psi_{j}$. These two properties are also related by Poisson’s theorem [6]:
$\nabla^{2} \psi_{j}=-\rho_{j} / \varepsilon_{0} \, \varepsilon_{\mathrm{r}}$
Here $\varepsilon_{0}$ is the permittivity of free space; $\varepsilon_{\mathrm{r}$ is the relative permittivity of the solvent [6]; $\rho_{j}$ is the charge density per unit volume.
In the case considered here, the electric charges (ions) are spherically distributed about the reference $j$ ion. Then Poisson’s equation takes the following form [7].
$\left[\frac{1}{r^{2}}\right] \, \frac{d}{d r}\left(r^{2} \, \frac{d \psi_{j}}{d r}\right)=-\frac{\rho_{j}}{\varepsilon_{0} \, \varepsilon_{r}}$
Combination of equations (l) and (n) yields the key equation (o).
$\left[\frac{1}{r^{2}}\right] \, \frac{d}{d r}\left(r^{2} \, \frac{d \psi_{j}}{d r}\right)=\frac{e^{2}}{\varepsilon_{0} \, \varepsilon_{r} \, k \, T} \, \sum p_{i} \, z_{i}^{2} \, \psi_{j}$
$\text { Or, } \quad\left[\frac{1}{\mathrm{r}^{2}}\right] \, \frac{\mathrm{d}}{\mathrm{dr}}\left(\mathrm{r}^{2} \, \frac{\mathrm{d} \psi_{\mathrm{j}}}{\mathrm{dr}}\right)=\kappa^{2} \, \psi_{\mathrm{j}}$
$\text { where }[8] \kappa^{2}=\frac{e^{2}}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}}^{2}$
Property $\kappa$ has the unit ‘reciprocal distance’. Equation (p) is a second order differential equation [9] having the general solution given by equation (p).
$\psi_{j}=A_{1} \, \exp (-\kappa \, r) / r+A_{2} \, \exp (\kappa \, r) / r$
However $\operatorname{limit}(r \rightarrow \infty) \exp (\kappa \, r) / r$ is very large where $\psi_{j}$ is zero. Hence $\mathrm{A}_{2}$ must be zero.
$\text { Therefore [10], } \Psi_{j}=\mathrm{A}_{1} \, \exp (-\kappa \, \mathrm{r}) / \mathrm{r}$
We combine equations (l) and (s) [11].
$\rho_{j}=-A_{1} \, \frac{\exp (-\kappa \, r)}{r} \, \sum \frac{p_{i} \,\left(z_{i} \, e\right)^{2}}{k \, T}$
Using the definition of $\kappa^{2}$ in equation (q) [12],
$\rho_{j}=-A_{1} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \kappa^{2} \, \frac{\exp (-\kappa \, r)}{r}$
At this point, a geometric condition is taken into account. Charge density $\rho_{j}$ describes the electrical properties of the solution ‘outside ‘ the $j$ ion. No other ions can approach the $j$ ion closer than a ‘distance of closest approach’ $\mathrm{a}_{j}$. The total charge on the solution ‘outside’ the $j$ ion equals in magnitude but opposite in sign that on the $j$ ion. Hence,
$4 \, \pi \, \int_{a(j)}^{\infty} \rho_{j} \, r^{2} \, d r=-z_{j} \, e$
$\text { Or, } 4 \, \pi \, \int_{a(j)}^{\infty}\left[-A_{1} \, \varepsilon_{0} \, \varepsilon_{r} \, K^{2} \, \frac{\exp (-K \, r)}{r}\right] \, r^{2} \, d r=-z_{j} \, e$
This integration yields an equation for $\mathrm{A}_{1}$.
$A_{1}=\frac{\left(z_{j} \, e\right) \, \exp \left(\kappa \, a_{j}\right)}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \,\left(1+\kappa \, a_{j}\right)}$
$\text { Hence } \psi_{\mathrm{j}}=\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right) \, \exp \left(\kappa \, \mathrm{a}_{\mathrm{j}}\right)}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \,\left(1+\kappa \, \mathrm{a}_{\mathrm{j}}\right)} \, \frac{\exp (-\kappa \, \mathrm{r})}{\mathrm{r}}$
We recall that $\psi_{j}$ is the electric potential at distance $\mathrm{r}$ from the $j$ ion. In the event that the solution contains just the $j$ ion (i.e. an isolated $j$ ion) with charge $z_{j} \, e$, the electric potential $\psi(\text { iso })$, distance $\mathrm{r}$ from the $j$ ion, is given by equation (z) [13].
$\psi_{j}(i S 0)=\frac{z_{j} \, e}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{r} \, r}$
The electric potential $psi_{j}$ given by equation (y) is the sum of $\psi(\text { iso })$ and the electric potential produced by all other ions in solution $\psi(\text { rest })$.
$\text { Then } \quad \psi_{j}=\psi_{j}(\text { iso })+\psi_{j}(\text { rest })$
$\text { Hence, } \quad \psi_{\mathrm{j}}(\text { rest })=\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\left[\frac{\exp \left(\kappa \, \mathrm{a}_{\mathrm{j}}\right) \, \exp (-\kappa \, \mathrm{r})}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}-1\right]$
Equation (zb) is valid for all values of $\mathrm{r}$, including for $\mathrm{r}=\mathrm{a}_{\mathrm{j}}$. Then from equation (zb), $\psi_{\mathrm{j}}(\text { rest })$ at $\mathrm{r} = \mathrm{a}_{j}$ is given by equation (zc).
$\psi_{j}(\text { rest })=-\frac{z_{j} \, e}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{r}} \, \frac{\kappa}{1+\kappa \, a_{j}}$
We imagine that the $j$ ion is isolated in solution and that the electrical interaction with all other $i$ ions is then switched on at fixed $\mathrm{T}$ and $\mathrm{p}$. The change in chemical potential of single $j$ ion is given by equation (zd),
$\Delta \mu_{j}(\mathrm{elec})=-\frac{\mathrm{z}_{\mathrm{j}}^{2} \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}$
For one mole of $j$ ions, $\Delta \mu_{\mathrm{j}}(\mathrm{elec})$ is given by equation (ze) where an additional factor of ‘2’ is introduced into the denominator. Otherwise each ion would be counted twice; i.e. once as the $j$ ion and once in solution around the $j$ ion [14].
$\Delta \mu_{\mathrm{j}}(\mathrm{elec} ; \text { one mole })=-\frac{\mathrm{z}_{\mathrm{j}}^{2} \, \mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}$
The chemical potential of single ion $j$ in an aqueous solution, $\mu_{\mathrm{j}}(\mathrm{aq})$ is related to molality $\mathrm{m}_{j}$ and single ion activity coefficient $\gamma_{j}$ using equation (zf).
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)$
Comparison of equations (ze) and (zf) yields equation (zg).
$\ln \left(\gamma_{\mathrm{j}}\right)=-\frac{\mathrm{z}_{\mathrm{j}}^{2} \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}$
The mean ionic activity coefficient $\gamma_{\pm}$ for the salt in solution is given by equation (zh); i.e. for a simple salt where each mole of salt contains $ν_{+}$ moles of cations and $ν_{-}$ moles of anions.
$\gamma_{\pm}=\left(\gamma_{+}^{v+} \, \gamma_{-}^{v-}\right)^{1 / v}$
$\text { Or, } \quad\left(v_{+}+v_{-}\right) \, \ln \left(\gamma_{\pm}\right)=v_{+} \, \ln \left(\gamma_{+}\right)+v_{-} \, \ln \left(\gamma_{-}\right)$
We envisage closest approaches only between differently charged ions. Then for a given salt, $\mathrm{a}_{+}=\mathrm{a}_{-}=\mathrm{a}_{\mathrm{j}}$. Hence from equation (zg),
$\ln \left(\gamma_{\pm}\right)=-\frac{\mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left(\frac{\kappa}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}\right) \,\left(\frac{\mathrm{v}_{+} \, \mathrm{z}_{+}^{2}+\mathrm{v}_{-} \, \mathrm{z}_{-}^{2}}{\mathrm{v}_{+}+\mathrm{v}_{-}}\right)$
But the salt is overall electrically neutral.
$\text { Or, } \quad \mathrm{V}_{+} \, \mathrm{Z}_{+}=-\mathrm{V}_{-} \, \mathrm{Z}_{-}$
$\text { Whence, } \quad v_{+}=-v_{-} \, z_{-} / z_{+}$
$\text { So }[15], \frac{\mathrm{v}_{+} \, \mathrm{z}_{+}^{2}+\mathrm{v}_{-} \, \mathrm{z}_{-}^{2}}{\mathrm{v}_{+}+\mathrm{v}_{-}}=-\mathrm{z}_{+} \, \mathrm{z}_{-}$
Hence we arrive at an equation for the mean ionic activity coefficient, $\gamma_{\pm}$ [16].
$\ln \left(\gamma_{\pm}\right)=\frac{z_{+} \, z_{-} \, e^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{s} \, k \, T} \,\left(\frac{K}{1+\kappa \, a_{j}}\right)$
At this point we return to equation (q) and recall that $\mathrm{p}_{\mathrm{i}$ is the number of ions in unit volume of solution. If the concentration of $\mathrm{i}$ ions equals $\mathrm{c}_{\mathrm{i}}$, (with $\mathrm{N}_{\mathrm{A}} =$ the Avogadro constant),
$\text { then [17] } \mathrm{p}_{\mathrm{i}}=\mathrm{N}_{\mathrm{A}} \, \mathrm{c}_{\mathrm{i}}$
$\text { Therefore }[18], \quad \kappa^{2}=\frac{e^{2} \,\left(N_{A}\right)^{2}}{\varepsilon_{0} \, \varepsilon_{r} \, R \, T} \, \sum c_{i} \, z_{i}^{2}$
The convention is to express concentrations using the unit, $\mathrm{mol dm}^{-3}$ for which we use the symbol, $\mathrm{c}^{\prime}$.
$\text { Hence } \quad \kappa^{2}=\frac{\mathrm{e}^{2} \,\left(\mathrm{N}_{\mathrm{A}}\right)^{2}}{10^{3} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}} \, \sum \mathrm{c}_{\mathrm{i}}^{\prime} \, \mathrm{z}_{\mathrm{i}}^{2}$
For dilute solutions, the following approximation is valid where
$\text { ionic strength } \mathrm{I}=(1 / 2) \, \sum \mathrm{m}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}}^{2}$
$\kappa^{2}=\frac{2 \, \mathrm{e}^{2} \,\left(\mathrm{N}_{\mathrm{A}}\right)^{2} \, \rho_{1}^{*}(\ell)}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}} \, \mathrm{I}$
From equations (zn) and (zs) [19],
$\ln \left(\gamma_{\pm}\right)=\frac{\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left[\frac{2 \, \mathrm{e}^{2}\left(\mathrm{~N}_{\mathrm{A}}\right)^{2} \, \rho_{1}^{*}(\ell)}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2} \, \frac{(\mathrm{I})^{1 / 2}}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}$
For very dilute solutions, the Debye Huckel Limiting Law (DHLL) is used where it is assumed that $1+\kappa \, \mathrm{a}_{\mathrm{j}}=1.0$. Hence,
$\ln \left(\gamma_{\pm}\right)=\frac{\mathrm{e}^{3} \,\left[2 \, \mathrm{N}_{\mathrm{A}} \, \rho_{1}^{*}(\ell)\right]^{1 / 2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left[\frac{\mathrm{N}_{\mathrm{A}}}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2} \, \mathrm{z}_{+} \, \mathrm{z}_{-} \,(\mathrm{I})^{1 / 2}$
Equation (zu) may be written in the following form.
$\ln \left(\gamma_{\pm}\right)=\frac{\mathrm{e}^{3} \,\left[2 \, \mathrm{N}_{\mathrm{A}} \, \rho_{1}^{*}(\ell)\right]^{1 / 2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left[\frac{\mathrm{N}_{\mathrm{A}}}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2} \, \mathrm{z}_{+} \, \mathrm{z}_{-} \,(\mathrm{I})^{1 / 2}$
For aqueous solutions at ambient pressure and $298.15 \mathrm{~K}$, $\rho_{1}^{*}(\ell)=997.047 \mathrm{~kg} \mathrm{~m}^{-3}$ and $\varepsilon_{\mathrm{r}}=78.36$.
$\text { Hence [20] } \ln \left(\gamma_{\pm}\right)=(1.1749) \, \mathrm{z}_{+} \, \mathrm{z}_{-} \,\left(\mathrm{I} / \mathrm{mol} \mathrm{kg}^{-1}\right)^{1 / 2}$
We note that with $Z_{+} \, Z_{-}=-\left|Z_{+} \, Z_{-}\right|$, $\ln \left(\gamma_{\pm}\right)<0$.
In other words $\ln \left(\gamma_{\pm}\right)$ is a linear function of the square root of the ionic strength I. Most authors choose to write equation (zx) using logarithms to base 10.
$\text { Then } \log \left(\gamma_{\pm}\right)=\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{A}_{\gamma} \,\left(\mathrm{I} / \mathrm{mol} \mathrm{kg}^{-1}\right)^{1 / 2}$
Here[22] $\mathrm{A}_{\gamma}=0.510$. Certainly the latter constant is readily remembered as ‘one-half’. Slight disagreements between published estimates of $\mathrm{A}_{\gamma}$ are a result of different estimates of $\varepsilon_{\mathrm{r}}$ and $\rho_{1}^{*}(\ell)$. Harned and Owen [1d] published a useful Table for $\mathrm{A}_{\gamma}$ as a function of temperature for aqueous solutions.
The full equation for $\ln \left(\gamma_{\pm}\right)$ following on from equation (zt) takes the following form [22].
$\ln \left(\gamma_{\pm}\right)=\frac{-\left|z_{+} \, z_{-}\right| \, S_{\gamma} \,\left(\mathrm{I} / \mathrm{mol} \mathrm{kg}^{-1}\right)^{1 / 2}}{\left.1+\beta \, a_{j} \,(\mathrm{I} / \mathrm{mol} \mathrm{kg})^{-1}\right)^{1 / 2}}$
For aqueous solutions at ambient pressure and $298.15 \mathrm{~K}$ [5], $\mathrm{S}_{\gamma}=1.175$ and $\beta=3.285 \mathrm{~nm}^{-1}$. Adam [23] suggests that aj can be treated as a variable in fitting the measured dependence of $\ln \left(\gamma_{\pm}\right)$ on ionic strength for a given salt.
Footnotes
[1] P. Debye and E. Huckel, Physik. Z.,1923, 24,185,334;1924,25,97.
[2] For accounts of the theory see--
1. K. S. Pitzer. Thermodynamics, McGraw-Hill, New York, 3rd. edition,1995, chapter 16.
2. S. Glasstone, An Introduction to Electrochemistry, D Van Nostrand, New York, 1942, chapter III.
3. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London , 2nd. edn. Revised 1965.
4. H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, 2nd edn. Revised and Enlarged, 1950, chapter 2.
5. P. A. Rock, Chemical Thermodynamics, MacMillan, Toronto, 1969, section 13.9.
6. A. Prock and G. McConkey, Topics in Chemical Physics (based on The Harward Lectures by Peter J. W. Debye), Elsevier, Amsterdam, 1962, chapter 5.
7. J. O’M. Bockris and A.K. N. Reddy, Modern Electrochemistry: Ionics, Plenum Press, New York, 2nd. edn.,1998,chapter 3.
8. R. J. Hunter, J.Chem.Educ.,1966,43,550.
9. For comments on the role of water-water and water-ion interactions in aqueous salt solutions see H. S. Frank, Z. fur physik. Chemie,1965,228,364.
[3] In equation (g), dni describes the number of ions in volume $\mathrm{dV}$; $\mathrm{p}_{\mathrm{i}}$ describes the number of ions in unit volume of solution. [In other words the units used to express $\mathrm{p}_{\mathrm{i}}$ and $\mathrm{dn}_{\mathrm{i}}$ differ.
$\begin{gathered} \frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}=\frac{[1] \,[\mathrm{C}] \,[\mathrm{V}]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{[1] \,[\mathrm{A} \mathrm{s}] \,\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=[1] \ \mathrm{p}_{\mathrm{i}} \, \exp \left(-\frac{\mathrm{z}_{1} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right) \, \mathrm{dV}=\left[\frac{1}{\mathrm{~m}^{3}}\right] \,[1] \,\left[\mathrm{m}^{3}\right]=[1] \ \mathrm{dn}_{\mathrm{i}}=[1] \quad \mathrm{p}_{\mathrm{i}}=\left[\mathrm{m}^{-3}\right] \end{gathered}$
[4] $\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi}{\mathrm{k} \, \mathrm{T}}=\frac{[1] \,[\mathrm{C}] \,[\mathrm{V}]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{[\mathrm{A} \mathrm{s}] \,\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]}{[\mathrm{J}]}=[1]$ Then $\rho_{\mathrm{j}}=\left[\frac{1}{\mathrm{~m}^{3}}\right] \,[1] \,[\mathrm{C}]=\left[\frac{\mathrm{C}}{\mathrm{m}^{3}}\right]$; i.e. charge per unit volume
[5] $\exp (x)=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots \ldots$
$\rho_{\mathrm{j}}=\frac{\left[\mathrm{m}^{-3}\right] \,[1]^{2} \,[\mathrm{C}]^{2} \,[\mathrm{V}]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{\left[\mathrm{m}^{-3}\right] \,[\mathrm{A} \mathrm{s}]^{2} \,\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]}{[\mathrm{J}]}=\left[\mathrm{C} \mathrm{m}^{-3}\right]$
i.e. charge per unit volume
[6] $\nabla^{2} \psi_{\mathrm{j}}=\frac{1}{\left[\mathrm{~m}^{2}\right]} \,[\mathrm{V}]=\left[\frac{\mathrm{V}}{\mathrm{m}^{2}}\right]$
$\rho_{\mathrm{j}} / \varepsilon_{0} \, \varepsilon_{\mathrm{r}}=\frac{\left[\mathrm{C} \mathrm{m}^{-3}\right]}{\left[\mathrm{Fm}^{-1}\right] \,[1]}=\frac{\left[\mathrm{As} \mathrm{} \mathrm{m}^{-3}\right]}{\left[\mathrm{AsV^{-1 } \mathrm { m } ^ { - 1 } ]}\right.}=\left[\frac{\mathrm{V}}{\mathrm{m}^{2}}\right]$
[7] $\left[\frac{1}{r^{2}}\right] \, \frac{d}{d r}\left(r^{2} \, \frac{d \psi_{j}}{d r}\right)=\left[\frac{1}{m^{2}}\right] \, \frac{1}{[m]} \,\left[m^{2}\right] \, \frac{\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]}{[m]}=\left[\frac{\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}}{\mathrm{~m}^{2}}\right]$
$\frac{\rho}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}}}=\left[\frac{\mathrm{C}}{\mathrm{m}^{3}}\right] \, \frac{1}{\left[\mathrm{Fm}^{-1}\right]} \, \frac{1}{[1]}=\left[\frac{\mathrm{As}}{\mathrm{m}^{3}}\right] \,\left[\frac{1}{\mathrm{As} \mathrm{V}^{-1} \mathrm{~m}^{-1}}\right]=\left[\frac{\mathrm{V}}{\mathrm{m}^{2}}\right]=\left[\frac{\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}}{\mathrm{~m}^{2}}\right]$
[8]
\begin{aligned} &\kappa^{2}=\frac{[\mathrm{C}]^{2}}{\left[\mathrm{Fm}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]} \,\left[\frac{1}{\mathrm{~m}^{3}}\right] \,[1]^{2}\ &\kappa=\left[\mathrm{m}^{-1}\right] \end{aligned}
[9]
\begin{aligned} &\frac{\mathrm{d}}{\mathrm{dr}}\left(\mathrm{r}^{2} \, \frac{\mathrm{d} \psi_{\mathrm{j}}}{\mathrm{dr}}\right)=2 \, \mathrm{r} \, \frac{\mathrm{d} \psi_{\mathrm{j}}}{\mathrm{dr}}+\mathrm{r}^{2} \, \frac{\mathrm{d}^{2} \psi_{\mathrm{j}}}{\mathrm{dr}^{2}}=\kappa^{2} \, \psi_{\mathrm{j}} \ &\text { Then, } \mathrm{r}^{2} \, \frac{\mathrm{d}^{2} \psi_{\mathrm{j}}}{\mathrm{dr}}+2 \, \mathrm{r} \, \frac{\mathrm{d} \psi_{\mathrm{j}}}{\mathrm{dr}}-\mathrm{\kappa}^{2} \, \psi_{\mathrm{j}}=0 \end{aligned}
[10] $\psi_{j}=[\mathrm{V}]$ and $\psi_{\mathrm{j}}=\mathrm{A}_{1} \, \exp \left([\mathrm{m}]^{-1} \,[\mathrm{m}]\right) /[\mathrm{m}] \quad \mathrm{A}_{1}=[\mathrm{Vm}]$
[11] $\rho_{\mathrm{j}}=\left[\frac{\mathrm{C}}{\mathrm{m}^{3}}\right]$
[12]
$\mathrm{A}_{1} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \kappa^{2} \, \frac{\exp (-\mathrm{K} \, \mathrm{r})}{\mathrm{r}}=[\mathrm{V} \mathrm{m}] \,\left[\mathrm{Fm} \mathrm{m}^{-1}\right] \,[1] \,[\mathrm{m}]^{-2} \, \frac{[1]}{[\mathrm{m}]}$
[13] $\psi_{j}(\text { iso })=\frac{[1] \,[\mathrm{C}]}{[1] \,[1] \,\left[\mathrm{Fm}^{-1}\right] \,[1] \,[\mathrm{m}]}=\frac{[\mathrm{As}]}{\left[\mathrm{As} \mathrm{V}^{-1}\right]}=[\mathrm{V}]$
[14]
\begin{aligned} &\Delta \mu_{j}(\mathrm{elec} ; \text { one mole })=\frac{[1]^{2} \,[\mathrm{C}]^{2} \,[\mathrm{mol}]^{-1}}{[1] \,\left[\mathrm{F} \mathrm{m}^{-1}\right] \,[1]} \,\left[\frac{[\mathrm{m}]^{-1}}{1+[\mathrm{m}]^{-1} \,[\mathrm{m}]}\right] \ &=\frac{[\mathrm{A} \mathrm{s}]^{2} \,\left[\mathrm{mol}^{-1}\right] \,[\mathrm{m}]^{-1}}{\left[\mathrm{~A} \mathrm{~s} \mathrm{} \mathrm{V}^{-1} \mathrm{~m}^{-1}\right]}=\frac{[\mathrm{A} \mathrm{s}] \,[\mathrm{mol}]^{-1}}{\left[\mathrm{~J}^{-1} \mathrm{~A} \mathrm{~s}\right]}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \end{aligned}
[15]
\begin{aligned} &{ \frac{\mathrm{v}_{+} \, \mathrm{z}_{+}^{2}+\mathrm{v}_{-} \, \mathrm{z}_{-}^{2}}{\mathrm{v}_{+}+\mathrm{v}_{-}}=\frac{-\left(\mathrm{v}_{-} \, \mathrm{z}_{-} \, \mathrm{z}_{+}\right)+\left(\mathrm{v}_{-} \, \mathrm{z}_{-}^{2}\right)}{-\left(\mathrm{v}_{-} \, \mathrm{z}_{-} / \mathrm{z}_{+}\right)+\mathrm{v}_{-}}} \ &=-\mathrm{z}_{+} \, \mathrm{z}_{-}\left[\frac{\mathrm{v}_{-}-\left(\mathrm{v}_{-} \, \mathrm{z}_{-} / \mathrm{z}_{+}\right)}{-\left(\mathrm{v}_{-} \, \mathrm{z}_{-} / \mathrm{z}_{+}\right)+\mathrm{v}_{-}}\right]=-\mathrm{z}_{+} \, \mathrm{z}_{-} \end{aligned}
[16]
\begin{aligned} &\ln \left(\gamma_{\pm}\right)=\frac{[1] \,[\mathrm{C}]^{2}}{[1] \,\left[\mathrm{F} \mathrm{m}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]} \,\left[\frac{[\mathrm{m}]^{-1}}{1+[\mathrm{m}]^{-1} \,[\mathrm{m}]}\right]\ &=\frac{[\mathrm{A} \mathrm{s}]^{2}}{\left[\mathrm{As} \mathrm{} \mathrm{V}^{-1}\right]} \, \frac{1}{[\mathrm{~J}]}=\frac{[\mathrm{As}]}{\left[\mathrm{J}^{-1} \mathrm{As}\right] \,[\mathrm{J}]}=[1] \end{aligned}
[17] $\mathrm{N}_{\mathrm{A}} \, \mathrm{c}_{\mathrm{i}}=\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{m}^{-3}\right]=\left[\mathrm{m}^{-3}\right]$
[18]
\begin{aligned} &\kappa^{2}=\frac{[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]^{2}}{\left[\mathrm{~F} \mathrm{~m}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]} \,\left[\mathrm{mol} \mathrm{m}^{-3}\right] \ &=\frac{[\mathrm{A} \mathrm{s}]^{2} \,[\mathrm{m}]^{-2}}{\left[\mathrm{~A} \mathrm{~s} \mathrm{} \mathrm{V}^{-1}\right] \,[\mathrm{J}]}=\frac{[\mathrm{As}] \,[\mathrm{m}]^{-2}}{\left[\mathrm{~J}^{-1} \mathrm{~A} \mathrm{~s}\right] \,[\mathrm{J}]}=[\mathrm{m}]^{-2} \end{aligned}
[19]
\begin{aligned} &\frac{\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}=\frac{[1] \,[\mathrm{C}]^{2}}{[1] \,[1] \,\left[\mathrm{Fm}^{-1}\right] \,\left[\mathrm{JK}^{-1}\right] \,[\mathrm{K}]}\ &=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right]}{\left[\mathrm{A} \mathrm{s} \mathrm{} \mathrm{J}^{-1} \mathrm{As} \mathrm{} \mathrm{m}^{-1}\right] \,[\mathrm{J}]}=[\mathrm{m}]\ &\left[\frac{2 \, \mathrm{e}^{2} \,\left(\mathrm{N}_{\mathrm{A}}\right)^{2} \, \rho_{1}^{*}(\ell)}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2}=\left[\frac{[1] \,[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]^{2} \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]}{\left[\mathrm{Fm}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]}\right]^{1 / 2}\ &=\left[\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{mol}^{-1}\right]^{2} \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]}{\left[\mathrm{As} \mathrm{J} \mathrm{A} \mathrm{s} \mathrm{}{ }^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]}\right]^{1 / 2}=\left[\left[\mathrm{mol}^{-1}\right] \,[\mathrm{kg}] \,\left[\mathrm{m}^{-2}\right]\right]^{1 / 2}\ &=\frac{\left[\mathrm{m}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{1 / 2}}\ &\kappa \, \mathrm{a}_{\mathrm{j}}=\left[\mathrm{m}^{-1}\right] \,[\mathrm{m}]=[1]\ &\frac{\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left[\frac{2 \, \mathrm{e}^{2}\left(\mathrm{~N}_{\mathrm{A}}\right)^{2} \, \rho_{1}^{*}(\ell)}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2} \, \frac{(\mathrm{I})^{1 / 2}}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}\ &=[\mathrm{m}] \, \frac{\left[\mathrm{m}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{1 / 2}} \, \frac{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{1 / 2}}{[1]}=[1] \end{aligned}
As required $\ln \left(\gamma_{\pm}\right)=[1]$
[20] J. S. Winn, Physical Chemistry, Harper Collins, New York, 1995,page 315.
[21] M. L. McGlashan, Chemical Thermodynamics, Academic Press, 1979, p 304.
[22] Using
1. $\rho_{1}^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \ell ; 2981.5 \mathrm{~K}\right)=997.0474$; G.S.Kell, J. Chem. Eng.Data, 1975,20,97.
2. $\varepsilon_{\mathrm{r}}\left(\mathrm{H}_{2} \mathrm{O} ; \ell ; 2981.5 \mathrm{~K}\right)=78.36 \pm 0.05$ H. Kienitz and K. N. Marsh, Pure Appl. Chem.,1981,53,1874.
3. CODATA 1986 fundamental constants
[23] N. K. Adam, Physical Chemistry, Oxford, 1956, page 395.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.05%3A_Chemical_Potentials/1.5.21%3A_Chemical_Potentials-_Salt_Solutions-_Debye-Huckel_Equation.txt
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A solution comprises at least two different chemical substances where at least one substance is in vast molar excess. The term ‘solution’ is used to describe both solids and liquids. Nevertheless the term ‘solution’ in the absence of the word ‘solid’ refers to a liquid. Chemists are particularly expert at identifying the number and chemical formulae of chemical substances present in a given closed system. Here we explore how the chemical composition of a given system is expressed. We consider a simple system prepared using water($\ell$) and urea(s) at ambient temperature and pressure. We designate water as chemical substance 1 and urea as chemical substance $j$, so that the closed system contains an aqueous solution. The amounts of the two substances are given by $\mathrm{n}_{j} \left(=\mathrm{w}_{1} / \mathrm{M}_{1}\right)$ and $\mathrm{n}_{\mathrm{j}}\left(=\mathrm{w}_{\mathrm{j}} / \mathrm{M}_{\mathrm{j}}\right)$ where $\mathrm{w}_{1}$ and $\mathrm{w}_{j}$ are masses; $\mathrm{M}_{1}$ and $\mathrm{M}_{j}$ are the molar masses of the two chemical substances. In these terms, $\mathrm{n}_{1}$ and $\mathrm{n}_{j}$ are extensive variables.
$\text { Mass of solution, } \mathrm{w}=\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}$
$\text { Mass of solvent, } \mathrm{w}_{1}=\mathrm{n}_{1} \, \mathrm{M}_{1}$
For water, $\mathrm{M}_{1} = 0.018 \mathrm{~kg mol}^{-1}$. However in reviewing the properties of solutions, chemists prefer intensive composition variables.
Mole Fraction
The mole fractions of the two substances $\mathrm{x}_{1}$ and $\mathrm{x}_{j}$ are given by the following two equations:
$\mathrm{x}_{1}=\mathrm{n}_{\mathrm{l}} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right) \quad \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right)$
Here $x_{1}+x_{j}=1.0$. In general terms for a system comprising $\mathrm{i}$-chemical substances, the mole fraction of substance $\mathrm{k}$ is given by equation (d).
$\mathbf{x}_{\mathrm{k}}=\mathrm{n}_{\mathrm{k}} / \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}}$
Hence $\sum_{j=1}^{j=i} x_{j}=1.0$. The advantage of the dimensionless mole fraction scale is that in the absence of chemical reaction, the mole fraction $\mathrm{x}_{\mathrm{k}}$ of chemical substance $\mathrm{k}$ is independent of both temperature and pressure.
Molality
For liquid systems, the chemical substance in vast excess is called the solvent whereas the other substances are called solutes. In the urea + water system, water is the solvent if $\mathrm{n}_{1}>>\mathrm{n}_{\mathrm{j}}$. We as observers of the properties of this system draw a distinction between the two substances, identifying urea as the solute. The molality of solute $\mathrm{m}_{j}$ is given by the amount of solute in $1 \mathrm{~kg}$ of solvent [1,2].
$\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}$
Because the molality of solute $j$ is defined in terms of masses of solute and solvent, $\mathrm{m}_{j}$ is independent of temperature and pressure. Hence for precise characterization of the properties of solutes the molality scale is preferred [2].
Concentration
The concentration of chemical substance $j$ in a system volume $\mathrm{V}$,
$c_{j}=n_{j} / V$
The latter statement is quite general. However chemists interested in the properties of solutions normally use the term ‘concentration’ with reference to the properties of a solute, substance $\mathrm{j}$. Several problems are associated with equation (f). The major problem is that the volume of a solution depends on both temperature and pressure so both these intensive variables should be stated when $\mathrm{c}_{j}$ is quoted. A further problem emerges when there is a need to specify the precise composition of a given solution. Many chemists prepare a solution by dissolving a known mass of solute $j$ in small volume of solvent. The volume of the solution is then ‘made up to the mark’ for a given volume (e.g. $250 \mathrm{~cm}^{3}$). But often chemists do not record precisely how much solvent is used. In these terms we see why the molality is often the preferred composition scale for solutions because the amounts of solvent and solute are precisely defined. However when thinking about the properties of solutions, chemists consider the distance between solute molecules rather than their masses. An interesting calculation offers insight into the dependence of intermolecular separation for solutes as a function of solute concentration, $\mathrm{c}_{j}$_ [2-4]. As the concentration of the solute (e.g. urea) in an aqueous solution increases so the mean distance between the solute molecules decreases. In the event that the solute is a 1:1 salt (e.g. potassium bromide, $\mathrm{KBr}$), the calculation takes account of the fact that each mole of salt produces, with complete dissociation two moles of solute ions [3,4]. We gain insight into the problem by considering a solution of $\mathrm{KBr}(\mathrm{aq}, 1 \mathrm{~mol dm}^{-3}$). The calculated distance between ion centres is $0.94 \mathrm{~nm}$. The radii of these ions are approx. $0.15 \mathrm{~nm}$. The diameter of a water molecule is around $0.4 \mathrm{~nm}$. [5]. So there are relatively few water molecules between the ions at this concentration.
These calculations are important because they indicate how solute-solute distances change on increasing the concentration of solute. [6] Chemists often want to know how solute - solute molecular interactions affect the properties of solutions. [7-11] Certainly the distance between solute molecules is a key consideration in reviewing the properties of solutes in aqueous solutions. The task of understanding the properties of aqueous solutions is usually divided into two parts. For the first part we use the term hydration to describe solute - solvent interactions. We imagine a molecule of solute, chemical substance $j$, in an infinite expanse of solvent and direct attention to the organisation of solvent molecules surrounding each solute molecule, the cosphere. [7] The term hydration number often refers to the number of water molecules contiguous to each solute molecule but the term 'hydration shell' often extends to include solvent molecules outside the immediate sheath.
With increase in solute concentration the mean separation between solute molecules decreases. In responding to the task of understanding the properties of real solutions we define the properties of ideal solutions [10]. In the case of salt solutions, strong and long - range ion - ion interactions contribute to marked deviations from the properties of ideal solutions [11].
Footnotes
[1] $\mathrm{m}_{\mathrm{j}}=\left[\mathrm{mol} \mathrm{} \mathrm{kg}^{-1}\right]$
[2] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn. revised, 1965, page 15.
[3] Consider a solution in which each solute molecule is placed at the centre of a cube, edge d metres. Then distance between solute molecules$/\mathrm{m} = \mathrm{d}$ Volume of one $text{cube}/\mathrm{m}^{3} = \mathrm{d}^{3}$; Volume of $\mathrm{n}_{j}$ moles of cubes $/\mathrm{m}^{3} = \mathrm{n}_{j} \,s \mathrm{N}_{\mathrm{A} \,s \mathrm{d}^{3}$ where \mathrm{N}_{\mathrm{A}}\) is the Avogadro constant. Thus $V(s \ln )=n_{j} \, N_{A} \, d^{3} \text { or } d=\left(c_{j} \, N_{A}\right)^{-1 / 3}$ where $\mathrm{c}_{j}$ is expressed in $\mathrm{mol m}^{-3}$. $\mathrm{d}=\left\{\left[\mathrm{mol} \mathrm{m}^{-3}\right] \,\left[\mathrm{mol}^{-1}\right]\right\}^{-1 / 3}=[\mathrm{m}]$ If $\mathrm{c}_{j}$ is expressed, as is conventional, in $\mathrm{mol dm}^{-3}$, $\mathrm{d}=\left(10^{3} \, \mathrm{c}_{\mathrm{j}} \, \mathrm{N}_{\mathrm{A}}\right)^{-1 / 3}$.
[4]
$\begin{array}{lll} \mathrm{c}_{\mathrm{j}} / \mathrm{mol} \mathrm{dm}^{-3} & \text { Single Solute } & 1: 1 \text { salt } \ & \mathrm{d} / \mathrm{nm} & \mathrm{d} / \mathrm{nm} \ 10^{-4} & 25.5 & 20.2 \ 10^{-3} & 11.8 & 9.4 \ 10^{-2} & 5.5 & 4.4 \ 10^{-1} & 2.6 & 2.0 \ 1 & 1.2 & 0.94 \ 5 & 0.69 & 0.55 \end{array}$
[5] N. E. Dorsey, Properties of Ordinary Water Substance, Reinhold, New York, 1940. The estimate quoted in the main text is based on the estimates given in Table 15 of this fascinating monograph. The latter offers information concerning, for example, the load which ice will support. Apparently, ice having a thickness of 20 cm will support a battery of artillery with carriages and horses (see p. 458).
[6] Molalities are based on mass, and concentrations on distances.
[7] R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.
[8] H. L. Friedman and C. V. Krishnan,in Water - A Comprehensive Treatise, ed. F. Franks, Plenum Press, New York, 1973, Vol. 3, Chapter 1.
[9] J.-Y. Huot and C. Jolicoeur, in The Chemical Physics of Solvation, Part (a) Theory of Solvation,ed. J.Ulstrup, Elsevier, New York, 1985.
[10] One interesting feature is common throughout aqueous chemistry. If a given water molecule is strongly hydrogen bonded to four other water molecules, that water molecule exists in a local state which has low density (high volume). In other words, strong cohesion implies low density; a pattern contrary to that encountered in nearly all natural systems (and in human activities); A. Ben-Naim, Chem. Phys. Lett., 1972,13, 406. [Exceptions to this statement are found in the structures of high pressure ice polymorphs - but then there are always exceptions to general statements.]
[11] The seminal paper in this subject, aqueous chemistry, is probably that written by J. D. Bernal and R. H. Fowler, J. Chem. Phys., 1933,1, 515. This paper is remarkable in that despite the subjects covered (i.e. ice, liquid water and aqueous solutions), the term 'hydrogen-bond' is not used. The authors refer to the tendency of water to group in 'tetrahedral coordination’.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.06%3A_Composition/1.6.01%3A_Composition-_Mole_Fraction-_Molality-_Concentration.txt
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For a solution in a single solvent, chemical substance 1, containing solute $j$,
$\text { Molaity, } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}$
Here $\mathrm{n}_{j}$ is the amount of solute and $\mathrm{w}_{1}$ is the mass of solvent [1]
Mole Faction and Molality
For the same system, the amount of solvent,
$\mathrm{n}_{1}=\mathrm{n}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}$
But mole fraction, $\mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{\mathrm{1}}+\mathrm{n}_{\mathrm{j}}\right)$
$\text { Thus } \quad x_{j}=\frac{m_{j} \, n_{1} \, M_{1}}{n_{1}+m_{j} \, n_{1} \, M_{1}} \quad \text { or } \quad x_{j}=\frac{m_{j} \, M_{1}}{1+m_{j} \, M_{1}}$
For dilute solutions, $1>>m_{j} \, M_{1}$.
$\text { Then } x_{j}=m_{j} \, M_{1}$
For water($\ell$), $\mathrm{M}_{1}=0.018 \mathrm{~kg} \mathrm{~mol}^{-1}$. From equation (c) $x_{j}+x_{j} \, m_{j} \, M_{1}=m_{j} \, M_{1}$
$\text { Then } \quad x_{j}=m_{j} \, M_{1} \,\left(1-x_{j}\right) \quad \text { or } \quad m_{j}=x_{j} /\left[M_{1} \,\left(1-x_{j}\right)\right]$
For dilute solutions $1-x_{j} \approx 1.0$. and we recover equation (d). In short, equations (c) and (e) provide exact conversions between $\mathrm{m}_{j}$ and $\mathrm{x}_{j}$ whereas equation (d) is only valid for dilute solutions.
Concentration and Molality
We consider a solution having volume $\mathrm{V}(\mathrm{s} \ln )$. $\text { Mass of solution }=\rho(s \ln ) \, V(s \ln )$ where (at defined $\mathrm{T}$ and $\mathrm{p}$), density $=\rho(\mathrm{sln})$. If amount of substance $j$ in this solution is $\mathrm{n}_{j}$ mol then mass of solute $\mathrm{w}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}$, where $\mathrm{M}_{j} =$ molar mass of solute.
Mass of solvent in system $=\rho(s \ln ) \, V(s \ln )-n_{j} \, M_{j}$
$\text { Hence molality } \quad m_{j}=\frac{n_{j}}{\rho(s \ln ) \, V(s \ln )-n_{j} \, M_{j}}$
$\text { and concentration } \quad c_{j}=\frac{n_{j}}{V(s \ln )}$
From (f) and (g) $\mathrm{m}_{\mathrm{j}}=\frac{\mathrm{n}_{\mathrm{j}}}{\frac{\rho(\mathrm{s} \ln ) \, \mathrm{n}_{\mathrm{j}}}{\mathrm{c}_{\mathrm{j}}}-\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}=\frac{1}{\frac{\rho(\mathrm{s} \ln )}{\mathrm{c}_{\mathrm{j}}}-\mathrm{M}_{\mathrm{j}}}$
$\text { Or, } m_{j}=\frac{c_{j}}{\rho(s \ln )-M_{j} \, c_{j}}$
$\text { Or, } \quad c_{j}=\frac{m_{j} \, \rho(s \ln )}{1+m_{j} \, M_{j}}$
For dilute solutions $\rho(\mathrm{s} \ln )>>\mathrm{M}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{j}}$.
$\text { Then[2] } c_{j}(s \ln )=m_{j} \, \rho(s \ln )$
Therefore the exact conversion is given by equations (h) and (i) which reduce to equation (j) for dilute solutions.
Molality and Concentration
An elegant conversion is possible between $\mathrm{m}_{j}$ and $\mathrm{c}_{j}$ scales. The volume of a simple solution is given by equation (k).
$\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
Here $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$ is the apparent molar volume of solute $j$.
$\text { Or, } \quad \mathrm{V}(\mathrm{aq}) / \mathrm{n}_{\mathrm{j}}=\left(\mathrm{n}_{1} / \mathrm{n}_{\mathrm{j}}\right) \,\left[\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right)$
$\text { Or, } \quad 1 / \mathrm{c}_{\mathrm{j}}=\left[1 / \mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right)$
$\text { Then, } 1 / \mathrm{m}_{\mathrm{j}}=\left[\rho_{1}^{*}(\ell) / \mathrm{c}_{\mathrm{j}}\right]-\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \rho_{1}^{*}(\ell)\right]$
Footnotes
[1] Amount of solvent, molar mass $\mathrm{M}_{1}$, $\mathrm{n}_{1}=\mathrm{w}_{1} / \mathrm{M}_{1}$ Then, $\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}$ Units $\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}=[\mathrm{mol}] /[\mathrm{mol}] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right]=\left[\mathrm{mol} \mathrm{kg}^{-1}\right]$
[2] $c_{j}=\left[\mathrm{mol} \mathrm{m}^{-3}\right]=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]$
1.6.03: Composition- Scale Conversion- Solvent Mixtures
A given mixed solvent is prepared (at fixed $\mathrm{T}$ and $\mathrm{p}$) by mixing $\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \mathrm{m}^{3}$ of liquid 1 and $\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell) \mathrm{m}^{3}$ of liquid 2. We will assume that the thermodynamic properties of the mixture are ideal.
$\text { Then volume } \mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)$
Then volume% of liquid 2 in the mixture is given by equation (b).
$\mathrm{V}_{2} \%=\left[10^{2} \, \mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right] /\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]$
The mass of a given mixed solvent system equals $\mathrm{w}_{\mathrm{s}}$. Further mass% of liquid 2 is $\mathrm{w}_{2}$%.
$\text { Thus } \mathrm{w}_{2} \%=\mathrm{w}_{2} \, 10^{2} /\left(\mathrm{w}_{1}+\mathrm{w}_{2}\right)$
$\text { Mole fraction } \mathrm{x}_{2}=\left(\mathrm{w}_{2} \% / \mathrm{M}_{2}\right) /\left[\frac{\left(10^{2}-\mathrm{w}_{2} \%\right)}{\mathrm{M}_{1}}+\frac{\mathrm{w}_{2} \%}{\mathrm{M}_{2}}\right]$
$\text { Also } \left.\mathrm{V}_{2} \%(\mathrm{mix} ; \text { id })=\left[10^{2} \, \mathrm{w}_{2} / \rho_{2}^{*}(\ell)\right] / \frac{\mathrm{w}_{1}}{\rho_{1}^{*}(\ell)}+\frac{\mathrm{w}_{2}}{\rho_{2}^{*}(\ell)}\right]$
If $\left(\mathrm{w}_{1}+\mathrm{w}_{2}\right)=100 \mathrm{~kg}$,
$\mathrm{V}_{2} \%(\operatorname{mix} ; \mathrm{id})=\left[10^{2} \, \mathrm{w}_{2} \% / \rho_{2}^{*}(\ell)\right] /\left[\frac{\left(10^{2}-\mathrm{w}_{2} \%\right)}{\rho_{1}^{*}(\ell)}+\frac{\mathrm{w}_{2} \%}{\rho_{2}^{*}(\ell)}\right]$
Molality and Mole fraction
A given solvent mixture has mass $10^{2} \mathrm{~kg}$ is prepared using $\mathrm{w}_{2} \mathrm{~kg}\left[=\mathrm{w}_{2} \% \right]$ of liquid 2; nj moles of solute are dissolved in this mixture.
$\text { Molality } \mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{} \mathrm{kg}^{-1}=\mathrm{n}_{\mathrm{j}} / 10^{2}$
$\text { Mole fraction, } \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right) / \mathrm{M}_{1}\right]+\left[\mathrm{w}_{2} \% / \mathrm{M}_{2}\right]+\mathrm{n}_{\mathrm{j}}\right\}$
For dilute solutions, $\mathrm{n}_{\mathrm{j}}<<\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)$
$\text { Then, } \quad \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right) / \mathrm{M}_{1}\right]+\left[\mathrm{w}_{2} \% / \mathrm{M}_{2}\right]\right\}$
$\text { Or, } \left.\mathrm{x}_{\mathrm{j}}=10^{2} \, \mathrm{m}_{\mathrm{j}} /\left\{\left[10^{2}-\mathrm{w}_{2} \%\right] / \mathrm{M}_{1}\right]+\left[\mathrm{w}_{2} \% / \mathrm{M}_{2}\right]\right\}$
Concentration and Molality
A given solution is prepared (at fixed $\mathrm{T}$ and $\mathrm{p}$) using $\mathrm{n}_{1}$ moles of liquid 1, $\mathrm{n}_{2}$ moles of liquid 2 and $\mathrm{n}_{j}$ moles of a simple solute (e.g. urea) where $n_{j}<<\left(n_{1}+n_{2}\right)$.
$\text { Mass of mixed solvent } \mathrm{w}_{\mathrm{s}}=\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}$
$\text { Mass of system, } w=n_{j} \, M_{j}+n_{1} \, M_{1}+n_{2} \, M_{2}$
$\text { Molality of solute } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{w}_{1}+\mathrm{w}_{2}\right]$
$\text { Or, } \quad \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{\mathrm{s}}$
Density of solution $= \rho$ Mass of solution $= \mathrm{w}$
$\text { Volume of solution } \mathrm{V}=\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}+\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right] / \rho$
$\text { Concentration of solute } j, \mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \rho /\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}+\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]$
$\text { For dilute solutions, } \mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}} \ll\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]$
$\text { Then, } \mathrm{c}_{\mathrm{j}} \cong \mathrm{n}_{\mathrm{j}} \, \rho /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]$
If the solution is dilute, the density of the solution is approx. equal to density of the solvent $\rho_{s}$ at the same $\mathrm{T}$ and $\mathrm{p}$.
$\text { Hence } \mathrm{c}_{\mathrm{j}} \cong \mathrm{n}_{\mathrm{j}} \, \rho_{\mathrm{s}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]$
$\text { Molality of solute } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]$
Then $c_{j} \cong m_{j} \, p_{s}$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.06%3A_Composition/1.6.02%3A_Composition-_Scale_Conversions-_Molality.txt
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The isothermal compressions of solutions and liquids have been extensively studied and the subject has a remarkable history. The term compression, symbol $\mathrm{K}$ describes the sensitivity of the volume of a system to an isothermal change in pressure, $(\partial V / \partial p)$. Reference is usually made to the voyage made by HMS Challenger and the report of experiments undertaken by Tait into the compression of water [1-3]. Kell summarises various equations which have been proposed describing the isothermal dependence of the molar volume of water on pressure [4]; see also references [5,6].
The dependence of the volume of water($\ell$) at low pressures and at a given temperature on pressure can be represented by equation (a) where $\mathrm{A}$ and $\mathrm{B}$ are constants.
$[\mathrm{V}(\text { ref })-\mathrm{V}] / \mathrm{V}(\text { ref }) \, \mathrm{p}=\mathrm{A} /(\mathrm{B}+\mathrm{p})$
Here $\mathrm{V}(\text{ref})$ is the volume ‘at zero pressure’, usually ambient pressure (i.e. approx $105 \mathrm{~N m}^{-2}$). This equation often called the Tait equation [4] has the form shown in equation (b).
$-\left(1 / \mathrm{V}^{0}\right) \,(\partial \mathrm{V} / \partial \mathrm{p})=\mathrm{A} /(\mathrm{B}+\mathrm{p})$
$\text { Alternatively [4] } \mathrm{V}=\mathrm{V}^{0}\{1-\mathrm{A} \, \ln [(\mathrm{B}+\mathrm{p}) / \mathrm{B}]\}$
The challenge of measuring the isothermal compression of liquids has been taken up by many investigators; e.g. references [7-12]. The isothermal compressions of a liquid $\mathrm{K}_{\mathrm{T}}$ is defined by equation (d) [13].
$\mathrm{K}_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}$
The isothermal compressibility is given by equation (e) [14].
$\kappa_{\mathrm{T}}=-\mathrm{V}^{-1} \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}$
For all thermodynamic equilibrium states, both $\mathrm{K}_{\mathrm{T}}$ and $\kappa_{\mathrm{T}}$ are positive variables. A related variable is the isochoric thermal pressure coefficient, $(\partial p / \partial T)_{v}$.[15]
We develop the story in the context of systems containing two liquid components. For a closed system containing $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$ moles of chemical substances 1 and 2, the Gibbs energy is a dependent variable and the variables $\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]$ are the independent variables. Temperature $\mathrm{T}$ is the thermal potential; pressure $\mathrm{p}$ is the mechanical variable. The number of thermodynamic variables necessary to define the system is established using the Gibbs Phase Rule [16]. For a closed system (at defined $\mathrm{T}$ and $\mathrm{p}$) at thermodynamic equilibrium the composition/organisation is represented by $\xi^{e q}$. The affinity for spontaneous change is zero consistent with the Gibbs energy being a minimum; equation (f).
$\mathrm{A}=-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=0$
The Gibbs energy, volume and entropy of a solution at equilibrium are state variables. We contrast these properties with those properties which are associated with a process (pathway). Thus we contrast the state variable V with an unspecified compression of a solution. We need to define the path followed by the system when the pressure is changed. The Gibbs energy of a closed system at thermodynamic equilibrium (where the affinity for spontaneous change is zero and where the molecular composition/organisation is characterised by $\xi^{e q}$) is described by equation (g).
$\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]$
The same state is characterised by the equilibrium volume and equilibrium entropy by equations (h) and (i) respectively.
$\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]$
$\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]$
We use two intensive variables, $\mathrm{T}$ and $\mathrm{p}$, in the definition of extensive variables $\mathrm{G}$, $\mathrm{V}$ and $\mathrm{S}$. When the pressure is increased by finite increments from $\mathrm{p}$ to ($\mathrm{p} + \Delta \mathrm{p}$), the volume changes in finite increments from $\mathrm{V}$ to ($\mathrm{V} + \Delta \mathrm{V}$). For an important pathway, the temperature is constant. However to satisfy the condition that the affinity for spontaneous change $\mathrm{A}$ is zero, the molecular organisation/composition $\xi$ changes. The volume at pressure ($\mathrm{p} + \Delta \mathrm{p}$) is defined using equation (j).
$\mathrm{V}=\mathrm{V}\left[\mathrm{T},(\mathrm{p}+\Delta \mathrm{p}), \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]$
In principle we plot the volume as a function of pressure at constant temperature, $\mathrm{n}_{1}$, $\mathrm{n}_{2}$, and at ‘$\mathrm{A} = 0$’. The gradient of the plot defined by equation (h) yields the equilibrium isothermal compression, $\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)$; equation (k)
$\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}=0}$
$\mathrm{K}_{\mathrm{T}}(\mathrm{A} = 0)$ characterises the state defined by the set of variables, $\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]$.
We turn our attention to another property starting with a system having a volume defined by equation (h). The system is perturbed by a change in pressure from $\mathrm{p}$ to ($\mathrm{p} + \Delta \mathrm{p}$) in an equilibrium displacement. However on this occasion we require that the entropy of the system remains constant at a value defined by equation (i). In principle we plot the volume $\mathrm{V}$ as a function of pressure at constant $\mathrm{n}_{1}$, $\mathrm{n}_{2}$, at ‘$\mathrm{A}=0$’ and at a constant entropy defined by equation (i). The gradient of the plot at the point where the volume is defined by equation (g) yields the equilibrium isentropic compression $\mathrm{K}_{\mathrm{S}} (\mathrm{A}=0)$; equation (l) where isentropic = adiabatic and ‘at equilibrium’.
$\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{S}, \mathrm{A}=0}$
The equilibrium state characterised by $\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)$ is defined by the variables $\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]$. In other words an isentropic volumetric property describes a solution defined in part by the intensive variables $\mathrm{T}$ and $\mathrm{p}$. Significantly the condition on the partial derivative in equation (l) is an extensive variable, entropy. For a stable phase $\mathrm{K}_{\mathrm{S}}$ is positive.
The arguments outlined above are repeated with respect to both isobaric equilibrium expansions $\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)$ and isentropic equilibrium expansions, $\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)$; equations (m) and (n).
$\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}, \mathrm{A}=0}$
$\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{S}, \mathrm{A}=0}$
The (equilibrium) volume intensive isothermal $\kappa_{\mathrm{T}}$ and isentropic $\kappa_{\mathrm{S}}$ compressibilities are defined by equations (o) and (p) .
$\kappa_{\mathrm{T}}=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}=\mathrm{K}_{\mathrm{T}} \, \mathrm{V}^{-1}$
$\kappa_{\mathrm{s}}=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=\mathrm{K}_{\mathrm{s}} \, \mathrm{V}^{-1}$
In 1914 Tyrer reported isentropic and isothermal compressibilities for many liquids [9]. Equations (q) and (r) define two (equilibrium) expansibilities, isentropic and isobaric, volume intensive properties.
$\alpha_{\mathrm{s}}=(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{s}}=\mathrm{E}_{\mathrm{S}} \, \mathrm{V}^{-1}$
$\alpha_{p}=(1 / V) \,(\partial \mathrm{V} / \partial \mathrm{T})_{p}=\mathrm{E}_{\mathrm{p}} \, \mathrm{V}^{-1}$
Rowlinson and Swinton state that the property $\alpha_{\mathrm{S}}$ is ‘of little importance’ [17]. The isobaric heat capacity per unit volume $\sigma$ is the ratio $\left[\mathrm{C}_{\mathrm{p}} / \mathrm{V}\right]$. A property of some importance is the difference between compressibilities, $\delta$; equation (s).
$\delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}=\mathrm{T} \,\left[\alpha_{\mathrm{p}}\right]^{2} \, \mathrm{V} / \mathrm{C}_{\mathrm{p}}=\mathrm{T} \,\left[\alpha_{p}\right]^{2} / \sigma$
The property $\sigma$ is given different symbols and names; e.g. volumetric specific heat. Here we identify $\sigma$ as the thermal (or, heat) capacitance. The property $\varepsilon$ is the difference between isobaric and isentropic expansibilities; equation (t).
$\varepsilon=\alpha_{p}-\alpha_{s}=\kappa_{T} \, \sigma / T \, \alpha_{p}$
The Newton–Laplace equation is the starting point for the determination of isentropic compressibilities of liquids using sound speeds and densities; equation (u).
$u^{2}=\left(\kappa_{\mathrm{s}} \, \rho\right)^{-1}$
The isentropic condition on $\kappa_{\mathrm{S}}$ means that as a sound wave passes through a liquid the pressure and temperature fluctuate within each microscopic volume but the entropy remains constant.
Footnotes
[1] P. G. Tait, ‘Voyage of HMS Challenger’ (Physics and Chemistry), 1888, Volume II, Part IV, 76pp.
[2] P. G. Tait, ‘Scientific Papers’, The University Press, Cambridge, 1898, Volume I, p.261.
[3] See also N. E. Dorsey, Properties of Ordinary Water Substance, Reinhold, New York , 1940, pp. 207-253.
[4] G.S Kell, Water A Comprehensive Treatise, ed. F Franks, Plenum Press, New York, 17972, Volume 1, pp. 382-383.
[5] J. H. Dymond and R. Malhotra, Int. J. Thermophys., 1988, 9,941.
[6] A. T. J. Hayward, Brit.J. Appl. Phys., 1967, 18,965.
[7] G. A. Neece and D. R. Squire, J.Phys.Chem.,1968,72,128.
[8] J. H. Hildebrand, Phys.Rev.,1929,34,649.
[9] D. Tyrer, J. Chem. Soc., 1914,105,2534.
[10] H. E. Eduljee, D. M. Newitt and K. E. Weale, J.Chem.Soc.,1951,3086.
[11] L. A. K. Staveley, W. I. Tupman, and K. R. Hart, Trans. Faraday Soc.,1955,51,323.
[12] D. N. Newitt and K.Weale, J.Chem. Soc.,1951,3092.
[13] $\mathrm{K}_{\mathrm{T}}=\left[\mathrm{m}^{3}\right] /\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right]$
[14] $\mathrm{K}_{\mathrm{T}}=\frac{1}{\left[\mathrm{~m}^{3}\right]} \, \frac{\left[\mathrm{m}^{3}\right]}{[\mathrm{Pa}]}=\left[\mathrm{Pa}^{-1}\right]$
[15] $(\partial \mathrm{p} / \partial \mathrm{T})_{\mathrm{V}}=\left[\mathrm{Pa} \mathrm{K}{ }^{-1}\right]$
[16] Phase Rule; $\mathrm{P} = 1$; $\mathrm{C} = 2$. Hence $\mathrm{F} = 3$. Then we define $\mathrm{T}$, $\mathrm{p}$ and mole fraction composition.
[17] J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, Butterworths, London, 3rd edn., 1982, pp 16-17.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.01%3A_Compressions_and_Expansions-_Liquids.txt
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The (equilibrium) isothermal compressibility of a closed system containing a condensed phase is given by equation (a).
$\kappa_{\mathrm{T}}=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
$\text { Or, } \quad \kappa_{\mathrm{T}}=-\left(\frac{\partial \ln (\mathrm{V})}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
Here we assume that over a range of pressures of interest here , $\kappa_{\mathrm{T}}$ is independent of pressure.
$\text { Hence at fixed temperature, } \int_{p=0}^{p} d \ln (V)=-K_{T} \, \int_{p=0}^{p} d p$
We define a property $V(p=0)$, the volume of the system under consideration extrapolated to zero pressure at fixed temperature.
$\text { Therefore } \ln [\mathrm{V}(\mathrm{p}) / \mathrm{V}(\mathrm{p}=0)]=-\kappa_{\mathrm{T}} \, \mathrm{p}$
$\text { Or, } \mathrm{V}(\mathrm{T}, \mathrm{p})=\mathrm{V}(\mathrm{T}, \mathrm{p}=0) \, \exp \left(-\mathrm{K}_{\mathrm{T}} \, \mathrm{p}\right)$
For systems at ordinary pressures, $\kappa_{\mathrm{T}} \, \mathrm{P}<<1$.
$\text { Hence [1] } \mathrm{V}(\mathrm{T}, \mathrm{p})=\mathrm{V}(\mathrm{T}, \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T}} \, \mathrm{p}\right]$
For example, in the case of a pure liquid , chemical substance 1 [e.g. water]
$\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T}, \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]$
$\text { But for water }(\ell),\left\lfloor\partial \mu_{1}^{*}(\ell) / \partial \mathrm{p}\right\rfloor=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})$
$\text { Hence } \quad \left[\frac{\partial \mu_{1}^{*}(\ell)}{\partial \mathrm{p}}\right]=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T1}}^{*}(\ell) \, \mathrm{p}\right]$
Or, following integration between limits ‘$\mathrm{p}=0$’ and $\mathrm{p}$,
\begin{aligned} &\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \ &\quad+\mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right] \end{aligned}
The latter equation relates the chemical potential of a liquid at pressure p to the isothermal compressibility of the liquid [2].
Footnote
[1] With $\exp (x)=1+x+\left(x^{2} / 2 !\right)+\left(x^{3} / 3 !\right)+\ldots \ldots$ At small $\mathrm{x}$, $\exp (x) \approx 1+x$
[2] I. Prigogine and R. Defay, Chemical Thermodynamics, transl D. H. Everett, Longmans Green, London, 1953.
1.7.03: Compressions- Isentropic- Solutions- General Comments
At fixed $\mathrm{T}$ and $\mathrm{p}$, the equilibrium state for an aqueous solution is a minimum in Gibbs energy, $\mathrm{G}^{\text{eq}}$. The first derivative of $\mathrm{G}^{\text{eq}}$ with respect to temperature at constant pressure yields the equilibrium enthalpy $\mathrm{H}^{\text{eq}}$. The first derivative of $\mathrm{H}^{\text{eq}}$ with respect to temperature also at constant pressure yields the equilibrium isobaric heat capacity ${\mathrm{C}_{\mathrm{p}}}^{\text{eq}}$. Alternatively we can track the pressure derivatives of $\mathrm{G}^{\text{eq}}$. The first derivative of $\mathrm{G}^{\text{eq}}$ with respect to pressure at fixed temperature is the equilibrium volume $\mathrm{V}^{\text{eq}}$. The first derivative of $\mathrm{V}^{\text{eq}}$ with respect to pressure at fixed temperature yields the equilibrium isothermal compression $\mathrm{K}_{\mathrm{T}}^{\mathrm{eq}}$, the ratio $\mathrm{K}_{\mathrm{T}}^{\mathrm{eq}} / \mathrm{V}^{\mathrm{eq}}$ yielding the equilibrium isothermal compressibility $\kappa_{\mathrm{T}}^{\mathrm{eq}}$. Concentrating attention on equilibrium properties of aqueous solutions, an extensive literature concerns $\mathrm{V}(\mathrm{aq})$ in terms of the corresponding densities, $\rho(\mathrm{aq})$. An extensive literature describes isobaric heat capacities $\mathrm{C}_{\mathrm{p}}(\mathrm{aq}$, effectively the second derivative of $\mathrm{G}(\mathrm{aq}$. Rather less literature describes $\kappa_{\mathrm{T}}(\mathrm{aq}$, a second derivative of $\mathrm{G}(\mathrm{aq})$ with respect to pressure. However an extensive literature reports isentropic compressibilities, $\kappa_{\mathrm{S}(\mathrm{aq})$; equation (a).
$\kappa_{\mathrm{S}}=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{S}}=\mathrm{K}_{\mathrm{S}} \, \mathrm{V}^{-1}$
This perhaps surprising observation is accounted for by the fact that speeds of sound (at low frequency, e.g. $1 \mathrm{~MHz}$) in aqueous solutions are conveniently and precisely measured using either the ‘sing-around; [1] or ‘pulse-echo-overlap’ [2] methods {for a summary of the ‘History of Sound’ see reference 3.) Then using the Newton-LaPlace equation $\kappa_{\mathrm{S}}(\mathrm{aq})$ is obtained [5]; equation (b).
$\mathrm{u}^{2}=\left(\kappa_{\mathrm{s}} \, \rho\right)^{-1}$
The speed of sound at zero frequency is a thermodynamically defined property [5,6]. The isentropic compressibility of water($\ell$) at ambient $\mathrm{T}$ and $\mathrm{p}$ can be calculated using either the speed of sound $\kappa_{\mathrm{s}}^{*}(\ell ; \text { acoustic })$ or using $\kappa_{\mathrm{T}}^{*}(\ell)$, $\alpha_{\mathrm{P}}^{*}(\ell)$ and ${\sigma}^{*}(\ell)$ to yield $\kappa_{\mathrm{s}}^{*}(\ell ; \text { thermodynamic })$. The two estimates agree lending support to the practice of calculating isentropic compressibilities of solutions using the Newton-Laplace equation. We equate the isentropic condition with adiabatic, provided that the compression is reversible.
An important quantity is the difference $\delta$ between compressibilities; equation (c).
$\delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}=\mathrm{T} \,\left(\alpha_{\mathrm{p}}\right)^{2} / \sigma$
The property $\sigma$ is given a number of different names but here we use the term, heat (or, thermal ) capacitance. The ratio of isothermal to isentropic compressions equals the ratio of isobaric to isochoric heat capacities [8].
$\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{S}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}$
Interest in the isentropic compressibilities of solutions was stimulated by Gucker and co-workers [9,10] and, in particular, by Harned and Owen [11]. The latter authors defined a property of the solute, here called $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ using equation (e) where the composition of a given aqueous solution is expressed using concentration $\mathrm{c}_{j}$.
$\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right) \equiv\left[\kappa_{\mathrm{S}}(\mathrm{aq})-\kappa_{\mathrm{Sl}}^{*}(\ell)\right] \,\left[\mathrm{c}_{\mathrm{j}}\right]^{-1}+\kappa_{\mathrm{S} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
Also
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right) \equiv \ &\quad\left[\kappa_{\mathrm{S}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{Sl}}^{*}(\ell) \, \rho(\mathrm{aq})\right] \,\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{S} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} \,\left[\rho_{1}^{*}(\ell)\right]^{-1} \end{aligned}
Similar equations relate $\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)$ to the molality of the solute, $\mathrm{m}_{j}$.
$\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right) \equiv\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right) \equiv \ &{\left[\kappa_{\mathrm{S}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{Sl}}^{*}(\ell) \, \rho(\mathrm{aq})\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1}} \ &+\kappa_{\mathrm{S}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}} \,[\rho(\mathrm{aq})]^{-1} \end{aligned}
The latter four equations are stated by analogy with those relating $\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)$ to the composition of a solution. In these terms equations (e) to (h) are said to describe the same property of a given solute. A crucial feature of equations (e) - (h) is the equivalence symbol (i.e.. $\equiv$). In this sense Harned and Owen [11] defined an apparent isentropic compression of solute-$j$ in terms of the quantities on the r.h.s. of equation (a). They recognised that $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ does not have thermodynamic basis. The target quantity is the apparent molar isentropic compression defined by equation (i) which, however, is not a description of an isentropic process as its name might suggest.
$\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)=\left(1 / \mathrm{n}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{s}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\left(\mathrm{n}_{1} / \mathrm{n}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})$
In fact $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)$ is a measure of the change in the isentropic compression of a solvent when solute $j$ is added under isothermal-isobaric conditions. The equivalence symbol in equations (e) - (h) is important [12,13]. In fact reservations are often expressed especially when estimates of $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ are discussed, particularly the dependence of $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ on solution composition. Franks and co-workers [14] recognised that the lack of isobaric heat capacity data forces the adoption of an approach in which $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ is often effectively assumed equal to $\phi\left(\mathrm{K}_{\mathrm{Tj}^{\mathrm{j}}}\right)$. Owen and Simons [15] comment that overlooking the difference between $\kappa_{\mathrm{S}}(\mathrm{aq})$ and $\kappa_{\mathrm{T}}(\mathrm{aq})$ causes errors of approximately 7.5% in estimates of $\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}$ for $\mathrm{NaCl}(\mathrm{aq})$ and $\mathrm{KCl}(\mathrm{aq})$ at $298 \mathrm{~K}$.
In terms of the development of the theory, a problem is encountered with the differential dependence of the molar volume of the solvent on pressure at constant entropy of the solution. The task is to describe how the molar volume of the solvent would depend on pressure if it were held at the same entropy of the solution.
Footnotes
[1] R. Garnsey, R. J. Boe, R. Mahoney and T. A. Litovitz, J. Chem. Phys., 1969, 50, 5222.
[2] E. P. Papadakis, J.Acoust. Soc. Am.,1972,52,843.
[3] R Taton Science in the Nineteenth Century, trans., A J Pomerans, Basic Books, New York, 1965,chapter 3.
[4] G. Horvath-Szabo, H. Hoiland and E. Hogseth, Rev. Sci.Instrum.,1994,65,1644.
[5] G. Douheret, M. I. Davis, J. C. R. Reis and M. J. Blandamer, Chem. Phys. Phys Chem., 2001, 2, 148.
[6] J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, Butterworths, London, 3rd. edn., 1982, pp.16-17.
[7] J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954, chapters 5 and 11.
[8] We use several calculus operations. Thus, $(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}$ And, $(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}}=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}$ Then, $(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} /(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{S}}=(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} /(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}$ Further $\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}$ and $\mathrm{S}=-(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}$ Hence, $\mathrm{H}=\mathrm{G}-\mathrm{T} \,(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}$ Then $(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\partial^{2} \mathrm{G} / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}$ Similarly $(\partial \mathrm{U} / \partial \mathrm{T})_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}$ Therefore, $\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{S}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}$
[9] F. T. Gucker, Chem. Rev.,1933,13,127.
[10] F. T. Gucker, F. W. Lamb, G. A. Marsh and R. M. Haag, J. Amer. Chem. Soc., 1950, 72, 310; and references therein.
[11] H.S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, 3rd. edn., 1958, section 8.7.
[12] M. J. Blandamer, J. Chem. Soc. Faraday Trans., 1998, 94,1057.
[13] M. J. Blandamer, M. I. Davis, G. Douheret, and J. C. R. Reis, Chem.Revs.,2001,30,8.
[14] F. Franks, J. R. Ravenhill and D. S. Reid, J. Solution Chem., 1972, 1, 3.
[15] B. B. Owen and H. L. Simons, J. Phys. Chem., 1957, 61, 479.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.02%3A_Compressibilities_%28Isothermal%29_and_Chemical_Potentials-_Liquids.txt
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A given closed system at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ contains chemical substances 1 and $j$. The system at specified $\mathrm{T}$ and $\mathrm{p}$ is at equilibrium where the affinity for spontaneous change is zero. We describe the volume and the entropy of the system using the following two equations.
$\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0\right]$
$\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0\right]$
The system is perturbed by a change in pressure. We envisage two possible paths tracked by the system accompanying a change in volume. In the first case the temperature is constant along the path for which ‘$\mathrm{A}=0$’. The isothermal equilibrium dependence of volume on pressure, namely the equilibrium isothermal compression $\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)$, is defined by equation (c).
$K_{T}(A=0)=-\left(\frac{\partial V}{\partial p}\right)_{T, A=0}$
In the second case the entropy remains constant along the path travelled by the system where ‘$\mathrm{A}=0$’. The differential equilibrium isentropic compression is given by equation (d); isentropic = adiabatic + equilibrium
$\mathrm{K}_{\mathrm{s}}(\mathrm{A}=0)=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}$
For all stable phases the volume of a given system decreases with increase in pressure at fixed temperature. The minus signs in equations (c) and (d) mean that compressions are positive variables. Neither $\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)$ or $\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)$ are strong functions of state because both variables describe pathways between states. The partial differentials in equations (c) and (d) differ in an important respect. The isothermal condition refers to an intensive variable whereas the isentropic condition refers to an extensive variable. The two properties $\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)$ and $\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)$ are related using a calculus operation.
$\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{S}}\right)_{\mathrm{p}, \mathrm{A}=0} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{P}, \mathrm{A}=0}$
Hence, [1]
$\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}+\left[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}\right]^{2} \, \frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)}$
But the (equilibrium) isobaric expansibility,
$\alpha_{p}(A=0)=\frac{1}{V} \,\left(\frac{\partial V}{\partial T}\right)_{p, A=0}$
$\operatorname{Then}\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}+\left[\alpha_{\mathrm{p}}(\mathrm{A}=0)\right]^{2} \, \frac{\mathrm{V}^{2} \, \mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)}$
By definition, the equilibrium isobaric heat capacity per unit volume [2] {also called heat capacitance [3]},
$\sigma(A=0)=C_{p}(A=0) / V$
In terms of compressions,
$\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)=\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)-\left[\alpha_{\mathrm{p}}(\mathrm{A}=0)\right]^{2} \, \frac{\mathrm{V} \, \mathrm{T}}{\sigma(\mathrm{A}=0)}$
Three terms in equation (j), $\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)$, $\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)$ and $\mathrm{V}$, are volume extensive variables. However it is convenient to rewrite these equations using volume intensive variables. Two equations define the isentropic equilibrium compressibility $\kappa_{\mathrm{S}}(\mathrm{A}=0)$ and isothermal equilibrium compressibility $\kappa_{\mathrm{T}}(\mathrm{A}=0)$ of a given system.
$\kappa_{\mathrm{T}}(\mathrm{A}=0)=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}=\frac{\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)}{\mathrm{V}}$
$\kappa_{\mathrm{S}}(\mathrm{A}=0)=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{s}, \mathrm{A}=0}=\frac{\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)}{\mathrm{V}}$
$\text { Therefore } \kappa_{\mathrm{S}}(\mathrm{A}=0)=\kappa_{\mathrm{T}}(\mathrm{A}=0)-\left[\alpha_{\mathrm{p}}(\mathrm{A}=0)\right]^{2} \, \frac{\mathrm{T}}{\sigma(\mathrm{A}=0)}$
$\text { By definition, } \delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}$
$\text { Then } \delta(A=0)=\left[\alpha_{p}(A=0)\right]^{2} \, \frac{T}{\sigma(A=0)}$
Footnotes
[1] From a Maxwell relationship for the condition at ‘$\mathrm{A}=0$’; i.e. at equilibrium, $\partial^{2} \mathrm{G} / \partial \mathrm{T} \, \partial \mathrm{p}=\partial^{2} \mathrm{G} / \partial \mathrm{p} \, \partial \mathrm{T}$. Then, $\mathrm{E}_{\mathrm{p}}=(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}$ From the Gibbs - Helmholtz equation, we combine the equations, $\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}$ and $\mathrm{S}=-(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}$. Hence, $\mathrm{H}=\mathrm{G}-\mathrm{T} \,(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}$ Then, $(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\partial^{2} \mathrm{G} / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}$
[2] $\kappa_{\mathrm{S}}(\mathrm{A}=0)=[\mathrm{Pa}]^{-1} \quad \kappa_{\mathrm{T}}(\mathrm{A}=0)=[\mathrm{Pa}]^{-1}$
\begin{aligned} &{\left[\alpha_{p}(\mathrm{~A}=0)\right]^{2} \, \frac{\mathrm{T}}{\sigma(\mathrm{A}=0)}=\left[\mathrm{K}^{-1}\right]^{2} \,[\mathrm{K}] \,\left[\mathrm{J} \mathrm{K}{ }^{-1} \mathrm{~m}^{-3}\right]^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=\mathrm{Pa}^{-1}} \ &\sigma(\mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0) / \mathrm{V}=\left[\mathrm{J} \mathrm{K}{ }^{-1}\right] \,[\mathrm{m}]^{-3} \end{aligned}
[3] M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis, Chem. Soc Rev., 2001, 30,8.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.04%3A_Compressibilities-_Isentropic-_Related_Properties.txt
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Isentropic properties of aqueous solutions are defined in a manner analogous to that used to define isothermal compressions and isothermal compressibilities. The assertion is made that a system (e.g. an aqueous solution) can be perturbed along a pathway where the affinity for spontaneous change is zero by a small change in pressure $\delta \mathrm{p}$, to a neighbouring state having the same entropy. The (equilibrium) isentropic compression is defined by equation (a).
$\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=-[\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{p}]_{\mathrm{S}(\mathrm{aq}), A=0}$
The constraint on this partial differential refers to 'at constant $\mathrm{S}(\mathrm{aq})$'. The definition of $\mathrm{K}_{\mathrm{S}}(\mathrm{aq})$ uses non-Gibbsian independent variables. In other words, isentropic parameters do not arise naturally from the formalism which expresses the Gibbs energy in terms of independent variables in the case of, for example, a simple solution, $\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{l}}, \mathrm{n}_{\mathrm{j}}\right]$ [1]. The isothermal compression of a solution $\mathrm{K}_{\mathrm{T}}(\mathrm{aq})$ and partial molar isothermal compressions of both solvent $\mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})$ and solute $\mathrm{K}_{\mathrm{T} j}(\mathrm{aq})$ are defined using Gibbsian independent variables. Unfortunately the corresponding equations cannot be simply carried over to the isentropic property $\mathrm{K}_{\mathrm{S}}(\mathrm{aq})$. The volume of a solution is expressed in terms of the amounts of solvent $\mathrm{n}_{1}$ and solute $\mathrm{n}_{j}$.
$\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})$
The latter equation is differentiated with respect to pressure at constant entropy of the solution $\mathrm{S}(\mathrm{aq})$. The latter condition includes the condition that the system remains at equilibrium where the affinity for spontaneous change is zero. We emphasize a point. The entropy which remains constant is that of the solution.
$\mathrm{K}_{\mathrm{s}}(\mathrm{aq})=-\mathrm{n}_{1} \,\left[\partial \mathrm{V}_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq}) ; \mathrm{A}=0}-\mathrm{n}_{\mathrm{j}} \,\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq}) ; \mathrm{A}=0}$
$\mathrm{K}_{\mathrm{S}}(\mathrm{aq})$ is an extensive property of the aqueous solution. $\mathrm{K}_{\mathrm{S}}(\mathrm{aq})$ may also be re-expressed using Euler’s theorem as a function of the composition of the solution.
$\mathrm{K}_{\mathrm{s}}(\mathrm{aq})=\mathrm{n}_{1} \,\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}+\mathrm{n}_{\mathrm{j}} \,\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})}$
Because $\mathrm{K}_{\mathrm{S}}(\mathrm{aq})$ is defined using non-Gibbsian independent variables, two important inequalities follow.
$-\left[\partial \mathrm{V}_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} \neq\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}$
$-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} \neq\left[\partial \mathrm{K}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)}$
$\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}$ and $\left[\partial \mathrm{K}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)}$ are respectively the partial molar properties of the solvent and solute. Because partial molar properties should describe the effects of a change in composition on the properties of a solution, we write equation (d) for an aqueous solution in the following form.
$\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{S} 1}(\mathrm{aq} ; \text { def })+\mathrm{n}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \text { def })$
$\text { Hence, } \quad \mathrm{K}_{\mathrm{sj}}(\mathrm{aq} ; \text { def }) \neq-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}$
In view of the latter inequality $\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \mathrm{def})$ is a non-Lewisian partial molar property [2]. We could define a molar isentropic compression of solute $j$ as (minus) the isentropic differential dependence of partial molar volume on pressure. This alternative definition is consistent with equation (g) expressing a summation rule analogous to that used for partial molar properties. However some other thermodynamic relationships involving partial molar properties would not be valid in this case. Therefore, $-\left[\partial V_{j}(a q) / \partial p\right]_{S(a q)}$ is a semi-partial molar property. A similar problem is encountered in defining an apparent molar compression for solute $j$, $\phi\left(\mathrm{K}_{\mathrm{Sj}_{\mathrm{j}}}\right)$ in a solution where the solute has apparent molar volume $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$; cf. equation (h) [3,4]. We might assert that $\phi\left(\mathrm{K}_{\mathrm{Sj}_{\mathrm{j}}}\right)$ is related to the isentropic differential dependence of $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$ on pressure, $-\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}$. Alternatively, using as a guide the apparent molar properties $\phi\left(\mathrm{E}_{\mathrm{pj}}\right)$ and $\phi\left(\mathrm{K}_{\mathrm{Tj}^{\mathrm{j}}}\right)$, we could define $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)$ using equation (i).
$\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\mathrm{l})+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{S} \mathrm{j}} ; \text { def }\right)$
$\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \mathrm{def})$ as given by equation (d) and $\phi\left(K_{S j} ; \text { def }\right)$ are linked; equation (j).
$\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \operatorname{def})=\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)+\mathrm{n}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})}$
Equation (j) is of the general form encountered for other apparent and partial molar properties. This form is also valid in the case of partial and apparent molar isobaric expansions, isothermal compressions and isobaric heat capacities. On the other hand, the semi-partial molar isentropic compression defined by $-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{s}(\mathrm{aq})}$ and the semi-apparent molar isentropic compression defined by $-\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}$ are related. The isentropic pressure dependence of $\mathrm{V}_{\mathrm{j}}(\mathrm{aq})$ is given by equation (k).
\begin{aligned} &-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}= \ &-\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}-\mathrm{n}_{\mathrm{j}} \,\left\{\partial\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})} / \partial \mathrm{p}\right\}_{\mathrm{s}(\mathrm{aq})} \end{aligned}
However,
$\left\{\partial\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} / \partial \mathrm{p}\right\}_{\mathrm{S}(\mathrm{aq})} \neq\left\{\partial\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} / \partial \mathrm{n}_{\mathrm{j}}\right\}_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)}$
Hence, the analogue of equation (j) does not hold for these 'semi' properties. The inequalities (e) and (f) highlight the essence of non-Lewisian properties. Their origin is a combination of properties defined in terms of Gibbsian and non-Gibbsian independent variables as in equations (e) and (f). This combination is also the reason for the inequality (l). We stress that the isentropic condition in equations (e) and (f) refers to the entropy $\mathrm{S}(\mathrm{aq})$ of the solution defined as is the volume $\mathrm{V}(\mathrm{aq})$ by the Gibbsian independent variables $\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]$. But this is not the entropy $\mathrm{S}_{1}^{*}(\ell)$ of the pure solvent having volume $\mathrm{V}_{1}^{*}(\ell)$. $\mathrm{S}(\mathrm{aq})$ at fixed composition is not simply related to $\mathrm{S}_{1}^{*}(\ell)$ as, for example, linear functions of temperature and pressure.
The isentropic condition is involved in the definitions of isentropic compression, $\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)$ and isentropic compressibility $\kappa_{\mathrm{S} 1}^{*}(\ell)$ of the solvent.
$\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)=-\left[\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right] \text { at constant } \mathrm{S}_{1}^{*}(\ell)$
\begin{aligned} &\kappa_{\mathrm{Sl}}^{*}(\ell)=\mathrm{K}_{\mathrm{Sl}}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\ &=-\left[\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right] / \mathrm{V}_{1}^{*}(\ell) \text { at constant } \mathrm{S}_{1}^{*}(\ell) \end{aligned}
The different isentropic conditions in equation (a) and in equations (m) and (n) signal a complexity in the isentropic differentiation of equation (o) with respect to pressure [5,6].
$\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
Footnotes
[1] J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douheret, Chem. Phys. Phys. Chem., 2001, 3,1465.
[2] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,2,1982, 78,1565.
[3] M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis, Chem. Soc. Rev.,2001,30,8.
[4] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998,94,2385.
[5] M. J. Blandamer, J. Chem. Soc. Faraday Trans.,1998,94,1057.
[6] M. J. Blandamer, Chem. Soc. Rev.,1998,27,73.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.05%3A_Compressions-_Isentropic-_Solutions-_Partial_and_Apparent_Molar.txt
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Granted that $\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)$ has been measured for solutions containing neutral solutes (at defined $\mathrm{T}$ and $\mathrm{p}$), interesting patterns emerge for the dependences of $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)$ on molality $\mathrm{m}_{j}$ and on solute $j$. Further these dependences are readily extrapolated (geometrically) to infinite dilution to yield estimates of $\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}$. These comments apply to solutions of neutral solutes in both aqueous and non-aqueous solutions; e.g. solutions in propylene carbonate [1] and aqueous solutions of carbohydrates [2].
For dilute solutions of neutral solutes $\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)$ is often approximately a linear function of the molality $\mathrm{m}_{j}$.
$\text { Thus } \phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}} ; \text { def }\right)=\phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}} ; \text { def }\right)^{\infty}+\mathrm{b}_{\mathrm{KS}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
For aqueous solutions containing ureas, acetamides and $\alpha,\omega$-alkanediols, the slope $b_{\mathrm{KS}}$ is positive. For dextrose(aq), sucrose(aq), urea(aq) and thiourea(aq) φ(; ) K def Sj ∞ is negative. In contrast φ(; ) K def Sj ∞ is positive for dioxan(aq) and acetamide(aq). In other words $\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}$ is characteristic of the solute [3,4]. Group additivity schemes are discussed for $\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}$ with respect to glycylpeptides(aq) [5], amino acids(aq) [6-8] and alcohols [9-11]. With increase in temperature $\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}$ for amino acids(aq) [8] and glycyl dipeptides(aq) [12,13] increases. Particularly interesting in terms of solute-water interactions is the study reported by Galema et al [14, 15] who comment on the calculation of $\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)$ for solute-$j$ using equation (b).
$\mathrm{K}_{\mathrm{sj}}(\mathrm{aq} ; \operatorname{def})=\phi\left(\mathrm{K}_{\mathrm{sj}} ; \operatorname{def}\right)+\mathrm{m}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{K}_{\mathrm{s} j} ; \operatorname{def}\right) / \partial \mathrm{m}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}}$
This study confirmed the importance of the stereochemistry of carbohydrates on their hydration. A clear contrast is drawn between those solutes where the hydrophilic groups match and mismatch into the three dimensionally hydrogen - bonded structure of liquid water. With increase in solute concentration, the dependence of $\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \text { def })$ on composition is non-linear [16]. For amines(aq) $\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \text { def })$ passes through minima [16].
Chalikian discusses the isentropic compression of a wide range of solutes with reference to group contributions [17], the discussion being extended to proteins [18] and oligopeptides[19].
Footnotes
[1] H. Høiland, J. Solution Chem., 1977, 6, 291.
[2] P. J. Bernal and W. A. Van Hook, J. Chem. Thermodyn., 1986,18,955.
[3] A. Lo Surdo, C. Shin and F. J. Millero, J. Chem. Eng. Data, 1978, 23, 197.
[4] F. Franks, J. R. Ravenhill and D. S. Reid, J Solution Chem.,1972, 1,3.
[5] M. Iqbal and R. E. Verrall, J.Phys.Chem.,1987,91,967.
[6] D. P. Kharakoz, J.Phys.Chem.,1991,95,5634.
[7] F. J. Millero, A. Lo Surdo and C. Shin, J.Phys.Chem.,1978,82,784.
[8] T. V. Chalikian, A. P. Sarvazyan, T. Funck, C. A. Cain and K. J. Breslauer, J. Phys. Chem., 1994, 98, 321.
[9] M. Kikuchi, M. Sakurai and K. Nitta, J. Chem. Eng. Data, 1995,40,935
[10] M. Sakurai, K. Nakamura, K. Nitta and N. Takenaka, J. Chem. Eng. Data, 1995,40,301.
[11] T. Nakajima, T. Komatsu and T. Nakagawa, Bull. Chem. Soc Jpn., 1975,48,788.
[12] G. R. Hedwig, H. Hoiland and E. Hogseth, J. Solution Chem.,1996,25,1041.
[13] G. R. Hedwig, J. D. Hastie and H. Hoiland, J. Solution Chem.,1996,25,615.
[14] S. A. Galema and H. Høiland, J. Phys. Chem., 1991, 95, 5321.
[15] S. A. Galema, J. B. F. N. Engberts, H. Hoiland and G. M. Forland, J. Phys. Chem., 1993, 97, 6885.
[16] M. Kaulgud and K. J. Patil, J. Phys. Chem., 1974,78,714.
[17] T. V. Chalikian, J. Phys. Chem.B,2001,105,12566.
[18] N Taulier and T.V. Chalikian, Biochem. Biophys. Acta, 2002,1595,48.
[19] A. W.Hakin, H. Hoiland and G. R. Hedwig, Phys. Chem. Chem. Phys.,2000,2,4850.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.06%3A_Compressions-_Isentropic-_Neutral_Solutes.txt
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An extensive literature describes the isentropic compressibilities of salt solutions prompted by earlier studies by Passynski [1] described by Owen [2].
The isentropic compression of a given aqueous salt solution $\mathrm{K}_{\mathrm{s}}(\mathrm{aq})$ is determined using the Newton-Laplace Equation in conjunction with speeds of sound and densities. An apparent molar compression of salt $j \phi\left(K_{s} ; \text { def }\right)$ is calculated using equation (a).
$\mathrm{K}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{s}} ; \operatorname{def}\right)$
Here $\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)$ is the isentropic compression of the solvent at the same $\mathrm{T}$ and $\mathrm{p}$. For salt solutions, particularly aqueous salt solutions, the dependence of $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ on the molality of the salt is generally examined in the light of equations describing the role of ion-ion interactions [3; see also reference 4]. For dilute salt solutions, equation (b) forms the basis for examining the dependence of $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ on $\left(\mathrm{m}_{\mathrm{j}}\right)^{1 / 2}$ where $\mathrm{m}_{j}$ is the molality of the salt-$j$.
$\text { Then, } \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)=\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)^{\infty}+\mathrm{S}_{\mathrm{KS}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}$
The form of the equation (b) has all the hallmarks of a pattern required by the $\mathrm{DHLL}$. In practice $\mathrm{S}_{\mathrm{KS}}$ cannot be calculated because the required isentropic dependence of the relative permittivity of the solvent on pressure is generally not known. However a plot is obtained using equation (b) yielding an estimate of $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}$.
For a large range of 1:1 salts $\phi\left(K_{S_{j}} ; \operatorname{def}\right)^{\infty}$ is negative, a pattern attributed to electrostriction of neighbouring solvent molecules by electric charges on the ions [3-10]. $\phi\left(\mathrm{K}_{\mathrm{S}_{j}} ; \text { def }\right)^{\infty}$ is more negative for solutions in $\mathrm{D}_{2}\mathrm{O}$ than in $\mathrm{H}_{2}\mathrm{O}$ as a consequence of more intense electrostriction in $\mathrm{D}_{2}\mathrm{O}$ [5]. Further on the basis of the Desnoyers-Philip Equation, the difference $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}-\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}$ is small but not negligible, amounting to approx. 10%. For alkylammonium ions in aqueous solutions $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}$ decreases with increase in the hydrophobic character, matching a general increase in $\phi\left(V_{j}\right)^{\infty}$ [11]. Group and ionic contributions to $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}$ have been estimated [10,11]. Indeed $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ is approximately a linear function of $\left(m_{j} / m^{0}\right)^{1 / 2}$ for a wide range of aqueous and non-aqueous salt solutions [11-13]; e.g. salts in $\mathrm{DMSO}$ [14] and in propylene carbonate [15]. $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ for copper(I) and sodium perchlorates in cyanobenzene, pyridine and cyanomethane show almost no dependence on salt molality [16].
A problem is further complicated by the fact that the $\mathrm{DHLL}$ for $\phi\left(K_{\mathrm{S} j} ; \text { def }\right)$ is itself a complicated function of salt molality [17], $\mathrm{S}_{\mathrm{KS}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}$ being however the leading term. A problem is encountered with the differential dependence of the molar volume of the solvent $\mathrm{V}_{1}^{*}(\ell)$ on pressure at constant $\mathrm{S}(\mathrm{s} \ln )$ describing how the volume of the solution would depend on pressure if it were held at the same entropy of the solution. Thus [18]
\begin{aligned} -\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{s}(\mathrm{aq})} &=\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \ &+\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, \mathrm{T} \, \alpha_{1}^{*}(\ell) \,\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq}) / \sigma(\mathrm{aq})\right]-\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right]\right\} \end{aligned}
By definition,
$\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{S} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
Then,
\begin{aligned} &-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}=\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right) \ &\quad+\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, \mathrm{T} \, \alpha_{1}^{*}(\ell) \,\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq}) / \sigma(\mathrm{aq})\right]-\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right]\right\} \end{aligned}
Consequently the difference between $-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{s}(\mathrm{aq})}$ and $\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)$ is determined by the property $\Delta \phi$, defined in equation (f).
$\Delta \phi=\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq}) / \sigma(\mathrm{aq})\right]-\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right]\right\}$
However $\Delta \phi / \mathrm{m}_{\mathrm{j}}$ is indeterminate at infinite dilution . But using L’Hospital‘s rule,
$\operatorname{Limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \Delta \phi / \mathrm{m}_{\mathrm{j}}= \left[\left[\rho_{1}^{*}(\ell) \, \alpha_{p 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right] \,\left\{\frac{\phi\left(E_{p j}\right)^{\infty}}{\alpha_{p 1}^{*}(\ell)}\right]-\left[\frac{\phi\left(C_{p j}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\right]\right\}$
Despite the thermodynamic polish given to the analysis of isentropic compressions, the problem of contrasting conditions ‘at constant $\mathrm{S}(\mathrm{aq})$’ and ‘at constant $\mathrm{S}_{1}^{*}(\ell)$’ underlies the analysis.
Footnotes
[1] A. Passynski, Acta Physicochim. URSS, 1938,8,385.
[2] B. B. Owen and P. L. Kronick, J. Am. Chem.Soc.,1961,65,84.
[3] F. J. Millero, in Activity Coefficients of Electrolyte Solutions, ed. R. M. Pytkowicz, CRC Press, Boca Raton, Fl,1979 , chapter 13.
[4] F. J. Millero, F. Vinokurova, M. Fernandez and J. P. Hershey, J. Solution Chem.,1987,16,269.
[5] J. G. Mathieson and B. E. Conway, J. Chem. Soc Faraday Trans.1, 1974, 70,752.
[6] B. B. Owen and H. L. Simons, J. Am. Chem. Soc.,1957,61,479
[7] Transition metal chlorides(aq); A. Lo Surdo and F. J. Millero, J. Phys. Chem.,1980,84,710.
[8] Nitroammino cobalt(III) complexes(aq); F. Kawaizumi, K. Matsumoto and H. Nomura, J. Phys. Chem., 1983, 87,3161.
[9] Sodium nitrate(aq) and sodium thiosulfate(aq); N. Rohman, and S. Mahiuddin, J. Chem. Soc. Faraday Trans.,1997,93,2053.
[10] Bipyridine and phenanthroline complexes of Fe(II), Cu(II), Ni(II) and Cu(II) chlorides(aq); F. Kawaizumi, H. Nomura and F. Nakao, J. Solution Chem.,1987, 16,133
[11] R. Buwalda, J. B. F. N. Engberts, H. Høiland and M. J. Blandamer, J. Phys. Org. Chem., 1998, 11, 59.
[12] E. Ayranci and B. E. Conway, J. Chem. Soc. Faraday Trans.1, 1983,79,1357.
[13] G. Peron, G. Trudeau and J.E.Desnoyers, Can J.Chem.,1987,65,1402.
[14] J. I. Lankford, W. T. Holladay and C. M. Criss, J. Solution Chem.,1984,13,699.
[15] J. I. Lankford and C. M. Criss, J. Solution Chem.,1987,16,753.
[16] D. S. Gill, P. Singh, J. Singh, P. Singh, G. Senanayake and G. T. Hefter, J. Chem. Soc., Faraday Trans., 1995, 91, 2789.
[17] J. C. R. Reis and M. A. P. Segurado, Phys.Chem.Chem.Phys.,1999,1,1501.
[18] M. J. Blandamer, J. Chem. Soc. Faraday Trans.,1998,94,1057.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.07%3A_Compressions-_Isentropic-_Salt_Solutions.txt
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A given aqueous solution is prepared using $\mathrm{n}_{1}$ moles of water and $\mathrm{n}_{j}$ moles of solute $j$. The thermodynamic properties of this solution are ideal.
$\text { Then, } \quad V_{m}(a q ; \text { id })=x_{1} \, V_{1}^{*}(\ell)+x_{j} \, \phi\left(V_{j}\right)^{\infty}$
$\text { Here } \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})=\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\partial \mathrm{V}(\mathrm{aq} ; \mathrm{id})}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)}$
The molar entropy of the ideal solution is given by equation (c).
$\mathrm{S}_{\mathrm{m}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{S}_{1}^{*}(\ell)-\mathrm{x}_{1} \, \mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})$
$\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})$ is the partial molar entropy of solute $j$ at the same $\mathrm{T}$ and $\mathrm{p}$. The solution is perturbed by a change in pressure and displaced to a neighbouring state having the same entropy, $\mathrm{S}_{\mathrm{m}}(\mathrm{aq} ; \mathrm{id})$.
$\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{aq} ; \mathrm{id})}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{jd}), \mathrm{x}(\mathrm{j})}$
From equation (a),
$\mathrm{K}_{\mathrm{Sm}(\mathrm{a} ; ; \mathrm{dd})}=-\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}-\mathrm{x}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; ; \mathrm{dd}), \mathrm{x}(\mathrm{j})}$
On these partial differentials, the isentropic condition is not the most convenient because it refers to the entropy of an ideal solution. Using the technique adopted for liquid mixtures, we obtain in equation (f), an expression for the unconventional isentropic compression of the solvent .
\begin{aligned} &\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{dd}), \mathrm{x}(\mathrm{j})}= \ &-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)-\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell) \, \mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id}) / \mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id}) \end{aligned}
Or,
\begin{aligned} &\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{qq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}= \ &-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)-\mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell) \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)-\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id}) / \mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})\right] \end{aligned}
Except for the different ideal reference state, equation (g) for the solvent is formally identical to the corresponding equation for liquid mixtures. However, in this case we need to follow a different approach for chemical substance $j$. The appropriate choice for isentropic conditions on solute properties is the entropy of the pure solvent at same $\mathrm{T}$ and $\mathrm{p}$. Hence,
$\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}=$
Or,
\begin{aligned} &\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}=\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}_{1}^{*}(\ell)} \ &-\mathrm{T} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \, \frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}+\mathrm{T} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})} \end{aligned}
Or,
\begin{aligned} &\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \times(\mathrm{j})}=\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}_{1}^{*}(\ell)} \ &-\mathrm{T} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \,\left[\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}\right] \end{aligned}
The isentropic pressure dependences of apparent and partial molar volumes are complicated functions. We are interested in obtaining an expression for $\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})$ in terms of the limiting apparent or partial molar isentropic compression of solute $j$, $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}$. We use the following expression [1-3].
$\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}_{1}^{*}(\ell)}=-\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}-\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right) \,\left(\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right)$
We combine the results in equations (e), (g), (j) and (k) to obtain an equation for $\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})$; equation (l).
\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})= \ &\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{s} 1}^{*}(\ell)+\mathrm{x}_{1} \, \mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}\right] \ &+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{sj}}\right)^{\infty}+\mathrm{x}_{\mathrm{j}} \, \mathrm{T} \,\left[\frac{\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}\right] \,\left[\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right] \ &+\mathrm{x}_{\mathrm{j}} \, \mathrm{T} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \,\left[\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}\right] \end{aligned}
Finally, by noting that $\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}$ after slight simplification we arrive at an expression for $\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; 1 \mathrm{~d})$; equation (m).
\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \,\left\{\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)+\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\right\} \ &+\mathrm{x}_{\mathrm{j}} \,\left\{\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}+\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right) \,\left(\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell)}\right)\right. \ &-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})} \end{aligned}
The complexity of equation (m) for solutions can be attributed to a combination of the non-Gibbsian character of $\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; 1 \mathrm{~d})$ with the non-Lewisian character of $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}$. Clearly $\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; 1 \mathrm{~d})$ and $\mathrm{K}_{\mathrm{mix}}(\mathrm{aq} ; 1 \mathrm{~d})$ are not equal because the reference states for chemical substance $j$ differ. We are interested in the apparent molar isentropic compression of solute $j$ in ideal aqueous solutions $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})$, which is defined in equation (n) and expressed by equation (o).
$\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{S} I}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})$
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{sj}}\right)(\mathrm{aq} ; \mathrm{id})=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty} \ &+\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right) \,\left[\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right] \ &+\mathrm{T} \,\left(\frac{\mathrm{x}_{1} \,\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}\right) \, \frac{1}{\mathrm{x}_{\mathrm{j}}} \end{aligned}
Limiting values for $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})$ are interesting. For the ideal solution at $\mathrm{x}_{j} = 0$, which is the same state as the real solution at infinite dilution, we naturally obtain $\phi\left(\mathrm{K}_{\mathrm{Sj}_{j}}\right)^{\infty}$ although using equation (o) for this purpose requires solving an indeterminate form. For the ideal solution at $\mathrm{x}_{j} = 0$ we obtain equation (p), which yields equation (q) after major reorganisation.
\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 1\right) \phi\left(\mathrm{K}_{\mathrm{sj}}\right)(\mathrm{aq} ; \mathrm{id})=\phi\left(\mathrm{K}_{\mathrm{s} j}\right)^{\infty} \ &+\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right) \,\left[\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]-\mathrm{T} \,\left(\frac{\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}\right]^{2}}{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}\right) \end{aligned}
\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 1\right) \phi\left(\mathrm{K}_{\mathrm{s} j}\right)(\mathrm{aq} ; \mathrm{id})=\phi\left(\mathrm{K}_{\mathrm{sj}}\right)^{\infty} \ &-\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}\right) \,\left[\frac{\phi\left(\mathrm{E}_{\mathrm{p} j}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right] \end{aligned}
The latter equation expresses the molar isentropic compression of solute $j$ in a standard state of unit mole fraction in terms of properties for the pure solvent and for the solute at infinite dilution. The ideal aqueous solution may be described as a non-ideal liquid mixture. An excess property is defined by equation (r).
$\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}}=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})-\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\text { mix} ; \mathrm{id})$
$\text { Or, } \phi\left(K_{S j}\right)^{E}=\left[K_{S m}(\mathrm{aq} ; i d)-K_{S m}(\operatorname{mix} ; i \mathrm{~d})\right] / x_{j}$
A working equation for $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}}$ can be generated from equation (o). After little reorganisation, we obtain equation (t).
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}}=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}-\mathrm{K}_{\mathrm{Sj}}^{*}(\ell) \ &+\mathrm{T} \,\left[\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right] \,\left[\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}^{\infty}\right)}{\mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)} \ &-\mathrm{T} \,\left[\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right] \, \frac{1}{\mathrm{x}_{\mathrm{j}}} \end{aligned}
Interestingly, the first four terms on the right end side of equation (t) express the difference $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}-\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{*}(\ell)$. For solution chemists the important reference state is at infinite dilution. The limiting excess property $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}$ is given by equation (u) [4].
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{S} \mathrm{j}}\right)^{\mathrm{E}, \infty}=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell) \ &-\mathrm{T} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pj}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} j}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]^{2} \end{aligned}
Estimates of $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}$ using $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}$ data for binary liquid mixtures often neglect the last term in equation (u).
Equation (u) works in two ways. A solution chemist will estimate $\phi\left(K_{S j}\right)^{\infty}$ from data reporting $\phi\left(K_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}$. A chemist interested in the properties of liquid mixtures will estimate $\phi\left(K_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}$ from data reporting $\phi\left(K_{S j}\right)^{\infty}$.
Footnotes
[1] J. C. R. Reis, J. Chem. Soc., Faraday Trans.2. 1982, 78, 1595.
[2] M. J. Blandamer, J. Chem. Soc., Faraday Trans., 1998, 94, 1057.
[3] J. C. R. Reis, J. Chem. Soc., Faraday Trans., 1998, 94, 2395.
[4] M. I. Davis, G. Douheret, J. C. R. Reis and M. J. Blandamer, Phys. Chem. Chem. Phys., 2001,3,4555.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.08%3A_Compresssions-_Isentropic-_Aqueous_Solution.txt
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The Newton Laplace Equation relates the speed of sound $\mathrm{u}$ in an aqueous solution, density $\rho(\mathrm{aq})$ and isentropic compressibility $\kappa_{\mathrm{S}}(\mathrm{aq})$; equation (a).
$\mathrm{u}^{2}=\left[\kappa_{\mathrm{s}}(\mathrm{aq}) \, \rho(\mathrm{aq})\right]^{-1}$
The differential dependence of sound velocity $\mathrm{u}$ on $\kappa_{\mathrm{S}}(\mathrm{aq})$ and $\rho(\mathrm{aq})$ is given by equation (b).
\begin{aligned} &2 \, u(a q) \, d u(a q)= \ &\quad-\frac{1}{\left[\kappa_{\mathrm{s}}(a q)\right]^{2} \, \rho(a q)} \, d \kappa_{s}(a q)-\frac{1}{\left.\kappa_{s}(a q)\right] \,[\rho(a q)]^{2}} \, d \rho(a q) \end{aligned}
We divide equation (b) by equation (a).
$2 \, \frac{\mathrm{du}(\mathrm{aq})}{\mathrm{u}(\mathrm{aq})}=-\frac{\mathrm{d} \kappa_{\mathrm{s}}(\mathrm{aq})}{\kappa_{\mathrm{s}}(\mathrm{aq})}-\frac{\mathrm{d} \rho(\mathrm{aq})}{\rho(\mathrm{aq})}$
We explore three approaches based on equation (c)
Analysis I
Two extra-thermodynamic assumptions are made.
1. Sound velocity $\mathrm{u}(\mathrm{aq})$ is a linear function of solute concentration, $\mathrm{c}_{j}$.
$\text { Thus[1] } \quad \mathrm{u}(\mathrm{aq})=\mathrm{u}_{1}^{*}(\ell)+\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}$
$\text { By definition, } \quad \mathrm{du}(\mathrm{aq})=\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)=\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}$
2. Density $\rho(\mathrm{aq})$ is a linear function of concentration $\mathrm{c}_{j}$.
$\text { Thus[1] } \quad \rho(\mathrm{aq})=\rho_{1}^{*}(\ell)+\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}$
Hence from equations (c)-(f),
$2 \, \frac{\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}(\mathrm{aq})}=-\frac{\mathrm{d} \kappa_{\mathrm{S}}(\mathrm{aq})}{\mathrm{K}_{\mathrm{s}}(\mathrm{aq})}-\frac{\mathrm{A}_{\mathrm{\rho}} \, \mathrm{c}_{\mathrm{j}}}{\rho(\mathrm{aq})}$
$\frac{\mathrm{d} \kappa_{\mathrm{S}}(\mathrm{aq})}{\kappa_{\mathrm{S}}(\mathrm{aq})}=-2 \, \frac{\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}(\mathrm{aq})}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho(\mathrm{aq})}$
In principle the change in $\kappa_{\mathrm{S}}(\mathrm{aq})$ resulting from addition of a solute $j$ to form a solution concentration $\mathrm{c}_{j}$ can be obtained from the experimentally determined parameters $\mathrm{A}_{\rho}$ and $\mathrm{A}_{\mathrm{u}}$.
Analysis II
Another approach expresses the two dependences using a general polynomial in $\mathrm{c}_{j}$.
$\text { By definition, } \quad \mathrm{A}_{\mathrm{u}}^{\infty}=\operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\partial \mathrm{u}(\mathrm{aq})}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}$
$\text { and } \mathrm{A}_{\rho}^{\infty}=\operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}$
The assumption is made that both $\mathrm{A}_{\mathrm{u}}^{\infty}$ and $\mathrm{A}_{\rho}^{\infty}$ are finite.
$\text { Similarly } \operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{S}}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}=\left(\frac{\partial \kappa_{\mathrm{S}}(\mathrm{aq})}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}^{\infty}$
Analysis III
The procedures described above are incorporated into the following equation for $\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)$.
$\text { Thus } \phi\left(\mathrm{K}_{\mathrm{s} j} ; \text { def }\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{s} 1}^{*}(\ell)$
Hence using equation (h) with $\mathrm{d}_{\mathrm{s}}(\mathrm{aq})=\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)$
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)= \ &\qquad\left[\kappa_{\mathrm{S}}(\mathrm{aq}) / \mathrm{c}_{\mathrm{j}}\right] \,\left[-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}(\mathrm{aq})}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho(\mathrm{aq})}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{S} 1}^{\mathrm{*}}(\ell) \end{aligned}
If we assume that $\kappa_{\mathrm{S}}(\mathrm{aq})$ is close to $\kappa_{\mathrm{S} 1}^{*}(\ell)$, then [2]
$\phi\left(\mathrm{K}_{\mathrm{Sj}_{j}} ; \operatorname{def}\right)=\kappa_{\mathrm{s}}(\mathrm{aq}) \,\left[-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}(\mathrm{aq})}-\frac{\mathrm{A}_{\rho}}{\rho(\mathrm{aq})}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]$
Equation (n) is complicated in the sense that the properties $\kappa_{\mathrm{S}}(\mathrm{aq})$, $\mathrm{u}(\mathrm{aq})$, $\rho(\mathrm{aq})$ and $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$ depend on concentration $\mathrm{c}_{j}$. With respect to $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$, the following equation is exact.
$\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(\mathrm{aq})\right]+\mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell)$
$\text { Using equation }(f), \phi\left(V_{j}\right)=-\frac{A_{\rho}}{\rho_{1}^{*}(\ell)}+\frac{M_{j}}{\rho_{1}^{*}(\ell)}$
$\text { Or, }-\frac{A_{\rho}}{\rho_{1}^{*}(\ell)}=\phi\left(V_{j}\right)-\frac{M_{j}}{\rho_{1}^{*}(\ell)}$
Equation (q) is multiplied by the ratio, $\rho_{1}^{*}(\ell) / \rho(\mathrm{aq})$.
$\text { Thus }-\frac{\mathrm{A}_{\rho}}{\rho(\mathrm{aq})}=\frac{\rho_{1}^{*}(\ell)}{\rho(\mathrm{aq})} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}$
Combination of equations (n) and (r) yields equation (s).
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)= \ &\kappa_{\mathrm{s}} \,\left[-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}(\mathrm{aq})}+\frac{\rho_{1}^{*}(\ell)}{\rho(\mathrm{aq})} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \end{aligned}
The argument is advanced that $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)$ can be meaningfully extrapolated to infinite dilution.
$\operatorname{limit}\left(c_{j} \rightarrow 0\right) \phi\left(K_{\mathrm{Sj}_{j}} ; \operatorname{def}\right)=\phi\left(\mathrm{K}_{\mathrm{sj}} ; \operatorname{def}\right)^{\infty}$
In the same limit $\rho_{1}^{*}(\ell) / \rho(\mathrm{aq})=1.0$ and $\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=\mathrm{K}_{\mathrm{S}}^{*}(\ell)$.
$\phi\left(\mathrm{K}_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}=\kappa_{\mathrm{s} 1}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}-\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}\right]$
$\text { But from equation }(\mathrm{d}), \mathrm{A}_{\mathrm{u}}=\left[\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)\right] / \mathrm{c}_{\mathrm{j}}$
$\phi\left(\mathrm{K}_{\mathrm{sj}} ; \operatorname{def}\right)^{\infty}=\kappa_{\mathrm{sl}}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}-2 \, \mathrm{U}-\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]$
where (cf. equation (v)),
$\mathrm{U}=\left[\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)\right] /\left[\mathrm{u}_{1}^{*}(\ell) \, \mathrm{c}_{\mathrm{j}}\right]$
The symbol $\mathrm{U}$ identifies the relative molar increment of the speed of sound [3-9]. Equation (w) shows $\phi\left(K_{S_{j}} ; \operatorname{def}\right)^{\infty}$ is obtained from $\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}$ and the speed of sound in a solution concentration $\mathrm{c}_{j}$.
$\text { In this approach we assume that }\left(\frac{\partial \mathrm{u}}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}=\frac{\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}$
$\text { Then, } U=\frac{1}{\mathrm{u}_{1}^{*}(\ell)} \,\left(\frac{\mathrm{du}(\mathrm{aq})}{\mathrm{dc}_{\mathrm{j}}}\right)$
However $\left(\frac{\mathrm{du}(\mathrm{aq})}{\mathrm{dc}}\right)$ and similarly $\left(\frac{\mathrm{du}(\mathrm{aq})}{\mathrm{dm}_{\mathrm{j}}}\right)$ are obtained using experimental results for real concentrations. Hence the estimated $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}$ is likely to be poor.
Analysis IV
The apparent molar isothermal compression of solute $j$ is related to the concentration $\mathrm{c}_{j}$ using the following exact equation.
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell)$
$\text { By definition. } \quad \delta(a q)=\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{S}}(\mathrm{aq})$
$\text { and } \delta_{1}^{*}(1)=\kappa_{\mathrm{T} 1}^{*}(\ell)-\kappa_{\mathrm{S} 1}^{*}(\ell)$
$\text { For an aqueous solution, } \kappa_{\mathrm{T}}(\mathrm{aq})=\delta(\mathrm{aq})+\kappa_{\mathrm{s}}(\mathrm{aq})$
According to the Newton-Laplace Equation.
$[u(\mathrm{aq})]^{2}=\left[\kappa_{\mathrm{s}}(\mathrm{aq}) \, \rho(\mathrm{aq})\right]^{-1}$
$\text { From equation }(\mathrm{zd}), \kappa_{\mathrm{T}}(\mathrm{aq})=\delta(\mathrm{aq})+\left\{[\mathrm{u}(\mathrm{aq})]^{2} \, \rho(\mathrm{aq})\right\}^{-1}$
At this stage, assumptions are made concerning the dependences of $\kappa_{\mathrm{T}}(\mathrm{aq})$ and $\delta(\mathrm{aq})$ on concentration $\mathrm{c}_{j}$.
$\text { Thus } \quad \kappa_{\mathrm{T}}(\mathrm{aq})=\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{KT}} \, \mathrm{c}_{\mathrm{j}}$
$\text { and } \quad \delta(\mathrm{aq})=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}}$
Using equations (d), (f) and (zf),
\begin{aligned} \kappa_{\mathrm{Tl}}^{*}(\ell)+\mathrm{A}_{\kappa \mathrm{T}} \, \mathrm{c}_{\mathrm{j}}=& \delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \ &+\frac{1}{\left\{\mathrm{u}_{1}^{*}(\ell)+\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}\right\}^{2} \,\left\{\rho_{1}^{*}(\ell)+\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}\right\}} \end{aligned}
Or,
\begin{aligned} &\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\kappa \mathrm{T}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \ &+\frac{1}{\left[\mathrm{u}_{1}^{*}(\ell)\right]^{2} \,\left\{1+\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}} / \mathrm{u}_{1}^{*}(\ell)\right\}^{2} \, \rho_{1}^{*}(\ell) \,\left\{1+\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}} / \rho_{1}^{*}(\ell)\right\}} \end{aligned}
Assuming $\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}} / \mathrm{u}_{1}^{*}(\ell)<<1$ and $A_{\rho} \, c_{j} / \rho_{1}^{*}(\ell)<<1$,
\begin{aligned} &\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{kT}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \ &+\frac{1}{\left[\mathrm{u}_{1}^{*}(\ell)\right]^{2} \, \rho_{1}^{*}(\ell)} \,\left[1-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}\right] \,\left[1-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right] \end{aligned}
$\text { We assume that }\left[\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}\right] \,\left[\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]<<1$
\begin{aligned} &\text { Therefore, } \ &\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{KT}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \ &+\frac{1}{\left[\mathrm{u}_{1}^{*}(\ell)\right]^{2} \, \rho_{1}^{*}(\ell)} \,\left[1-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right] \end{aligned}
$\text { But } \kappa_{\mathrm{s} 1}^{*}(\ell)=\left\{\left[u_{1}^{*}(\ell)\right]^{2} \, \rho_{1}^{*}(\ell)\right\}^{-1}$
$\text { and } \kappa_{\mathrm{T} 1}^{*}(\ell)=\delta_{1}^{*}(\ell)+\kappa_{\mathrm{S} 1}^{*}(\ell)$
Then,
\begin{aligned} &\delta_{1}^{*}(\ell)+\kappa_{\mathrm{S} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{KT}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \ &+\kappa_{\mathrm{Sl}}^{*}(\ell) \,\left[1-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right] \end{aligned}
$\text { Or } \mathrm{A}_{\mathrm{K}}=\mathrm{A}_{\delta}-\kappa_{\mathrm{S} 1}^{*}(\ell) \,\left[\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{A}_{\rho}}{\rho_{1}^{*}(\ell)}\right]$
From equations (za) and (zg),
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\mathrm{KT}}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{Tl}}^{*}(\ell)$
Equations (zq) and (zr) yield equation (as),
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\delta}-\kappa_{\mathrm{S} 1}^{*}(\ell) \,\left[\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{A}_{\rho}}{\rho_{1}^{*}(\ell)}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell)$
Or, using equation (q)
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\delta}-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell) \,[&\left.\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \ &+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \end{aligned}
Using equation (zc),
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}\right)=& \mathrm{A}_{\delta}-\kappa_{\mathrm{S} 1}^{*}(\ell) \,\left[\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \ &+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell)+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{S} 1}^{*}(\ell) \end{aligned}
Or,
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\delta}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell) \ &+\kappa_{\mathrm{Sl}}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{M}_{\mathrm{j}}}{\left.\rho_{1}^{*} \ell\right)}-\frac{2 \, \mathrm{A}_{u}}{\left.\mathrm{u}_{1}^{*} \ell\right)}\right] \end{aligned}
The latter is the Owen-Simons Equation[4] which takes the following form in the limit of infinite dilution.
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\left[\mathrm{A}_{\delta}\right.&\left.+\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \, \delta_{1}^{*}(\ell)\right] \ &+\kappa_{\mathrm{Sl}}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}-\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}\right] \end{aligned}
The term $\left[\mathrm{A}_{\delta}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \, \delta_{1}^{*}(\ell)\right]$ is not negligibly small. Using equation (u), equation (zw) takes the following form,
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\left[\mathrm{A}_{\delta}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \, \delta_{1}^{*}(\ell)\right]+\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}$
Clearly the approximation which sets $\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}$ equal to $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}$ is poor although often made. In fact Hedwig and Hoiland [10] show that for N-acetylamino acids in aqueous solution at $298.15 \mathrm{~K} \mathrm{} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}$ and $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}$ can have different signs, offering convincing evidence that the assumption is untenable.
Footnotes
[1] \begin{aligned} &A_{u}=\left[\frac{m}{s}\right] \,\left[\frac{m^{3}}{m o l}\right]=\left[m^{4} \mathrm{~s}^{-1} \mathrm{~mol}^{-1}\right] \ &A_{\rho}=\left[\frac{k g}{m^{3}}\right] \,\left[\frac{m^{3}}{m o l}\right]=\left[k g \mathrm{~mol}^{-1}\right] \end{aligned}
[2] \begin{aligned} &2 \, \frac{\mathrm{A}_{\mathrm{u}}}{\mathrm{u}} \, \mathrm{K}_{\mathrm{S}}(\mathrm{aq})=[1] \, \frac{1}{\left[\mathrm{~m} \mathrm{~s}^{-1}\right]} \,\left[\mathrm{m}^{4} \mathrm{~s}^{-1} \mathrm{~mol}^{-1}\right] \, \frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]} \ &\frac{\mathrm{A}_{\rho}}{\rho} \, \kappa_{\mathrm{S}}(\mathrm{aq})=\frac{\left[\mathrm{kg} \mathrm{m}^{-3}\right]}{\left[\mathrm{mol} \mathrm{m}^{-3}\right]} \, \frac{1}{\left.\mathrm{~kg} \mathrm{~m}^{-3}\right]} \, \frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{Nm}^{-2}\right]} \ &\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{s}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \, \frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]} \end{aligned}
[3] S. Barnatt, J. Chem. Phys.,1952,20,278.
[4] B. B. Owen and H. L. Simons, J. Phys.Chem.,1957,61,479.
[5] H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, 1958, 3rd. edn., section 8.7.
[6] D. P. Kharakov, J. Phys.Chem.,1991,95,5634.
[7] T. V. Chalikian, A. P .Sarvazyan, T. Funck, C. A.Cain, and K. J. Breslauer, J. Phys.Chem.,1994,98,321.
[8] T. V. Chalikian, A.P.Sarvazyan and K. J. Breslauer, Biophys. Chem.,1994,51,89.
[9] P. Bernal and J. McCluan, J Solution Chem.,2001,30,119.
[10] G. R. Hedwig and H. Hoiland, Phys. Chem. Chem. Phys.,2004,6,2440.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.09%3A_Compressions-_Isentropic_and_Isothermal-_Solutions-_Approximate_Limiti.txt
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In terms of isentropic and isothermal compressibilities the Desnoyers-Philip Equation is important. A key equation expresses the difference between two apparent properties, $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{J}}}\right)^{\infty}$ and $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}$ [1]. We develop the proof in the general case starting from equation (a).
$\delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}=\mathrm{T} \,\left(\alpha_{\mathrm{p}}\right)^{2} / \sigma$
$\text { Hence, for an aqueous solution, } \delta(\mathrm{aq})=T \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})$
$\text { For water }(\ell) \text { at the same } \mathrm{T} \text { and } \mathrm{p}, \delta_{1}^{*}(\ell)=\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)$
We formulate an equation for the difference, $\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)$
$\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=\mathrm{T} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)$
We add and subtract the same term. With some slight reorganisation,
\begin{aligned} \delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=& \mathrm{T} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma(\mathrm{aq}) \ &-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)+\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma(\mathrm{aq}) \end{aligned}
Or,
\begin{aligned} \delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=& \mathrm{T} \,[\sigma(\mathrm{aq})]^{-1} \,\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}-\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}\right\} \ &-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} \,\left[\frac{1}{\sigma_{1}^{*}(\ell)}-\frac{1}{\sigma(\mathrm{aq})}\right] \end{aligned}
We identify the term $\left\{\left[\alpha_{p}(a q)\right]^{2}-\left[\alpha_{p 1}^{*}(\ell)\right]^{2}\right\}$ as ‘a square minus a square’.
\begin{aligned} \delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=& \mathrm{T} \,[\sigma(\mathrm{aq})]^{-1} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right\} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right\} \ &-\mathrm{T} \, \frac{\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\sigma(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)} \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right] \end{aligned}
We use equations (b) for $\delta(\mathrm{aq})$ and (c) for $\delta_{1}^{*}(\ell)$ to remove explicit reference to temperature in equation (g).
\begin{aligned} &\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)= \ &\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{-2} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right\} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right\} \ &-\delta_{1}^{*}(\ell) \,[\sigma(\mathrm{aq})]^{-1} \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right] \end{aligned}
$\text { But } \phi\left(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(\mathrm{c}_{\mathrm{j}}\right)^{-1} \,\left[\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\right]+\delta_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
We insert equation (h) for the difference $\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)$ into equation (i).
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)= \ &\frac{\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right]}{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}} \, \frac{\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]}{\mathrm{c}_{\mathrm{j}}} \ &-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \, \frac{\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]}{\mathrm{c}_{\mathrm{j}}}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell) \end{aligned}
$\text { But, } \phi\left(E_{p j}\right)=\left[c_{j}\right]^{-1} \,\left[\alpha_{p}(a q)-\alpha_{p l}^{*}(\ell)\right]+\alpha_{p 1}^{*}(\ell) \, \phi\left(V_{j}\right)$
We identify the difference $\left[\alpha_{p}(a q)-\alpha_{p 1}^{*}(\ell)\right]$.
$\text { Then } \phi\left(E_{p j}\right)-\alpha_{p 1}^{*}(\ell) \, \phi\left(V_{j}\right)=\left[c_{j}\right]^{-1} \,\left[\alpha_{p}(a q)-\alpha_{p 1}^{*}(\ell)\right]$
$\text { Similarly } \phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]$
Then from equations (k), (l), (m) and (n),
$\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\frac{\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right]}{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \ -\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell) \end{gathered}$
We collect the $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$ terms.
$\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)= \ \frac{\delta(\mathrm{aq})}{\alpha_{\mathrm{p}}(\mathrm{aq})} \,\left\{1+\frac{\alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right\} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \ +\left\{-\frac{\delta(\mathrm{aq}) \, \alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}-\frac{\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}}+\frac{\delta_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}{\sigma(\mathrm{aq})}+\delta_{1}^{*}(\ell)\right\} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{gathered}$
We note that in the second {----} bracket, the product term of $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$. By using equations (b) for $\delta(\mathrm{aq})$ and (c) for $\delta_{1}^{*}(\ell)$ the second and third terms are together equal to zero.
Hence
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)= \ &\qquad \frac{\delta(\mathrm{aq})}{\alpha_{\mathrm{p}}(\mathrm{aq})} \,\left\{1+\frac{\alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right\} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \ &+\left\{\delta_{1}^{*}(\ell)-\frac{\delta(\mathrm{aq}) \, \alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right\} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}
The latter is the full Desnoyers–Philip equation [1]. But
$\begin{gathered} \operatorname{limit}\left(c_{j} \rightarrow 0\right) \alpha_{p}(a q)=\alpha_{p l}^{*}(\ell) \ \phi\left(E_{p j}\right)=\phi\left(E_{p j}\right)^{\infty}, \ \phi\left(C_{p j}\right)=\phi\left(C_{p j}\right)^{\infty}, \delta(a q)=\delta_{1}^{*}(\ell) \ \text { and } \sigma(a q)=\sigma_{1}^{*}(\ell) \end{gathered}$
$\text { Then } \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)^{\infty}=\delta_{1}^{*}(\ell) \,\left\{\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\right\}$
Footnotes
[1] J. E. Desnoyers and P. R. Philip, Can. J. Chem, 1972, 50,1094.
[2] M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis, Chem. Soc. Rev., 2001, 30, 8.
1.7.11: Compression- Isentropic- Apparent Molar Volume
A given liquid system is prepared using $\mathrm{n}_{1}$ moles of water, molar mass $\mathrm{M}_{1}$, and $\mathrm{n}_{j}$ moles of substance $j$. The closed system is at equilibrium, at temperature $\mathrm{T}$ and pressure $\mathrm{p}$. The volume of the system is given by equation (a).
$\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{\mathrm{1}}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
Here $V_{1}^{*}(\ell)$ is the molar volume of pure water and $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$ is the apparent molar volume of substance $j$ in the system; $\mathrm{V}(\mathrm{aq})$ and $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$ depend on the composition of the system, but $\mathrm{V}_{1}^{*}(\ell)$ does not.
The solution is perturbed to a local equilibrium state by a change in pressure along a path for which the entropy remains constant at $\mathrm{S}(\mathrm{aq})$. At a specified molality $\mathrm{m}_{j the change in volume is characterised by the isentropic compressibility, \(\mathrm{K}_{\mathrm{s}}(\mathrm{aq})$ defined in equation (b).
$\kappa_{\mathrm{s}}(\mathrm{aq})=-\frac{1}{\mathrm{~V}(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}$
Hence,
$\mathrm{V}(\mathrm{aq}) \, \mathrm{K}_{\mathrm{s}}(\mathrm{aq})=-\mathrm{n}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}-\mathrm{n}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}$
The isentropic condition on the first partial differential in equation (c) refers to the entropy of an aqueous solution at molality, $\mathrm{m}_{j}$. There is interest in relating this partial differential to the isentropic compressibility of the pure liquid substance 1 at the same $\mathrm{T}$ and $\mathrm{p}$, which is defined in equation (d).
$\kappa_{\mathrm{s} 1}^{*}(\ell)=-\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\ell)}$
For substance 1 the different isentropic conditions are related by equation (e).
$\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\mathrm{aq}) \mathrm{m}(\mathrm{j})}=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\ell)}+\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) / \mathrm{m}(\mathrm{j})} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(\mathrm{l})}\right)_{\mathrm{p}^{*}}$
In the latter equation we identify $\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\ell)}$ and $\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(1)}\right)_{\mathrm{p}^{*}}$ with, respectively, $-\mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{S} 1}^{*}(\ell)$ and $\mathrm{T} \, \alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)$, which are thermodynamic properties of water ($\ell$). Here $\sigma_{1}^{*}(\ell)$ is the heat capacitance (or heat capacity per unit volume) of water ($\ell$) Using the same calculus operation, the remaining partial differential is related to an isothermal property in equation (f).
$\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}=\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})} \,\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}^{*}}$
Since $\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{pl}}^{*}(\ell),\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq})) \mathrm{m}(\mathrm{j})}=\mathrm{T} \, \frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\sigma(\mathrm{aq})}$, and $\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}^{*}}=\frac{\mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}{\mathrm{T}}$, we combine these results with equation (f) to express equation (e) as equation (g).
\begin{aligned} &\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}= \ &-\kappa_{\mathrm{S} 1}^{*}(\ell)-\mathrm{T} \, \frac{\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\sigma_{1}^{*}(\ell)}+\mathrm{T} \, \frac{\alpha_{\mathrm{p} 1}^{*}(\ell) \, \alpha_{\mathrm{p}}(\mathrm{aq})}{\sigma(\mathrm{aq})} \end{aligned}
We return to equation (c). Using equation (a) for $\mathrm{V}(\mathrm{aq})$, equation (c) yields equation (h).
\begin{aligned} &-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}= \ &{\left[\left(\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right) \, \mathrm{V}_{1}^{*}(\ell)+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \, \mathrm{K}_{\mathrm{s}}(\mathrm{aq})+\left(\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right) \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}} \end{aligned}
We note that $\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}=\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \mathrm{M}_{1}}$ And that density $\rho_{1}^{*}(\ell)=\frac{\mathrm{M}_{1}}{\mathrm{~V}_{1}^{*}(\ell)}$. Then combining equations (g) and (h) leads to equation (i) after slight simplification.
\begin{aligned} &-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}= \ &{\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)} \ &+\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, \mathrm{T} \, \alpha_{\mathrm{p} 1}^{*}(\ell) \,\left[\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\sigma(\mathrm{aq})}-\frac{\alpha_{\mathrm{p} 1}^{*}(\ell)}{\sigma_{1}^{*}(\ell)}\right] \end{aligned}
An equivalent derivation of equation (i) has been given [1].
Footnotes
[1] M. J. Blandamer, J. Chem. Soc., Faraday Trans., 1998, 94, 1057.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.10%3A_Compressions-_Desnoyers_-_Philip_Equation.txt
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A given liquid mixture is prepared using $\mathrm{n}_{1}$ moles of water and $\mathrm{n}_{j}$ moles of liquid substance $j$. The closed system is at equilibrium, at temperature $\mathrm{T}$ and pressure $\mathrm{p}$. The volume of the system is defined by the following equation.
$\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]$
Similarly the entropy of the system is defined by equation (b).
$\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]$
The volume of the system is given by equation (c).
$\mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}$
Here $\mathrm{V}_{1}$ and $\mathrm{V}_{j}$ are the partial molar volumes of the two substances in the system; $\mathrm{V}$, $\mathrm{V}_{1}$ and $\mathrm{V}_{j}$ depend on the composition of the system. Similarly the entropy of the system is given by equation (d).
$\mathrm{S}=\mathrm{n}_{1} \, \mathrm{S}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}$
The molar volume of the system $\mathrm{V}_{\mathrm{m}}$ is given by the ratio $\mathrm{V} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right)$.
$\text { Hence, } \quad \mathrm{V}_{\mathrm{m}}=\mathrm{x}_{1} \, \mathrm{V}_{1}+\mathrm{x}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}$
$\text { Similarly } \quad S_{m}=x_{1} \, S_{1}+x_{j} \, S_{j}$
The system under examination is a binary liquid mixture such that the thermodynamic properties of the mixture are ideal.
$\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{*}(\ell)$
$\mathrm{S}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{S}_{1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}^{*}(\ell)+\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})$
In other words the two reference states are the pure substances at the initially fixed $\mathrm{T}$ and $\mathrm{p}$. In equation (h) the term $\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})$ describes the ideal molar entropy of mixing, which is a function of composition only. The liquid mixture is perturbed to a local equilibrium state by a change in pressure along a path for which the entropy remains at that given by equation (h). At a specified mole fraction $\mathrm{x}_{j}$ the change in volume is characterised by the isentropic compression, $\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})$ which is $\mathrm{K}_{\mathrm{m}}\left(\text { at constant } \mathrm{S}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\right)$ defined in equation (i).
$\mathrm{K}_{\mathrm{Sm}}\left(\operatorname{mix} ; \mathrm{x}_{\mathrm{j}} ; \mathrm{id}\right)=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}$
Hence,
\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}\left(\mathrm{mix} ; \mathrm{x}_{\mathrm{j}} ; \mathrm{id}\right)= \ &\quad-\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} ; \mathrm{dd}), x(\mathrm{j})}-\mathrm{x}_{\mathrm{j}} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} ; \mathrm{id}), x(j)} \end{aligned}
We note that the isentropic condition on the partial differentials in equation (j) refers to the entropy of an ideal mixture at mole fraction $\mathrm{x}_{j}$. There is merit in relating these partial differentials to the isentropic compressions of the pure liquid substances 1 and $j$ at the same $\mathrm{T}$ and $\mathrm{p}$, which are defined in equations (k) and (l).
$\mathrm{K}_{\mathrm{si}}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(1)^{*}}$
$\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(j)^{*}}$
The required relationship is obtained using a calculus operation. For substance 1 the different isentropic conditions are related by equation (m).
\begin{aligned} &\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{jd}) \times(\mathrm{j})}= \ &\left.\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(1)^{*}}+\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; ; \mathrm{d}) \times(j)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(\ell)}\right)_{\mathrm{p}}\right)^{*} \end{aligned}
In the latter equation we identify $\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(1)^{*}} \text { and }\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(\ell)}\right)_{\mathrm{p}}$ with, respectively, $-\mathrm{K}_{\mathrm{Sl}}^{*}(\ell)$ and $\mathrm{T} \, \frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}$ which are thermodynamic properties of water($\ell$). Using the same calculus operation, the remaining partial differential is related to an isothermal property in equation (n).
\begin{aligned} &\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} \text {;id }) \times(\mathrm{j})}= \ &\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{id}) \times(\mathrm{j})} \,\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \end{aligned}
But $\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{E}_{\mathrm{pl}}^{*}(\ell)$, $\left(\frac{\partial T}{\partial p}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{jd}) \mathrm{x}(\mathrm{j})}=\mathrm{T} \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{A} ; \mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{A} ; \mathrm{mix} ; \mathrm{id})}$, and $\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\frac{\mathrm{C}_{\mathrm{pm}}^{*}(\ell)}{\mathrm{T}}$,
We combine these results with equation (n) to re-express equation (m) as equation (o).
\begin{aligned} &\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} ; \mathrm{id}) \mathrm{x}(\mathrm{j})}= \ &-\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell) \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})} \end{aligned}
Similarly, for the substance j we obtain equation (p).
\begin{aligned} &\left(\frac{\partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{dd}) \times(\mathrm{j})}= \ &-\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}+\mathrm{T} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell) \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})} \end{aligned}
Equations (o) and (p) can be used to recast equation (j) in the form of equation (q).
\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{Sl} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+ \ &\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}\right. \ &\left.-\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{pl}}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right] \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right\} \end{aligned}
$\mathrm{K}_{\mathrm{Sm}}(\mathrm{A} ; \text { mix; id })$ is the ideal molar isentropic compression, which is commonly denoted by id $\mathrm{K}_{\mathrm{Sm}}^{\mathrm{id}}$. The sum, $\left[x_{1} \, E_{p 1}^{*}(\ell)+x_{j} \, E_{p j}^{*}(\ell)\right]$ is the ideal molar isobaric thermal expansion $E_{p m}^{\text {id }}$ for the binary liquid mixture, here denoted as $\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})$, and analogously for the ideal molar isobaric heat capacity $\mathrm{C}_{\mathrm{pm}}^{\mathrm{id}}= \mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})$. Thus,
\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{s} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+ \ &\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right\} \end{aligned}
The last term of equation (r) expresses a mixing property. In general terms,
$\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \text { id })=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{sl}}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+\Delta_{\text {mix }} \mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})$
From equations (r) and (s) we obtain the following expression.
\begin{aligned} &\Delta_{\text {mix }} \mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})= \ &\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})}\right\} \ &\Delta_{\text {mix }} \mathrm{K}_{\mathrm{Sm}}(\mathrm{A} ; \operatorname{mix} ; \mathrm{id})=\mathrm{T}\left\{\mathrm{x}_{1}\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\right. \ &\left.+\mathrm{x}_{\mathrm{j}}\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{pj}}^{*}(\ell)-\left[\mathrm{E}_{\mathrm{pm}}(\text { mix } ; \mathrm{id})\right]^{2} / \mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right\} \end{aligned}
This is an important, albeit frequently neglected term, in the calculation of isentropic compressions of thermodynamically ideal liquid mixtures. In general ideal molar mixing values are non-zero for non-Gibbsian properties, the origin of which has been discussed [1].
We recall the definition of an apparent molar property. Hence,
$\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{S} j}\right)(\mathrm{mix} ; \mathrm{id})$
Here $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})$ is the apparent molar isentropic compression of chemical substance $j$ in the ideal liquid mixture. Combination of equations (r) and (u) yields equation (v) [2].
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}}\right)(\operatorname{mix} ; \mathrm{id})= \ &\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right\} / \mathrm{x}_{\mathrm{j}} \end{aligned}
The limiting values for $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)$ at $\mathrm{x}_{j} = 1$ and $\mathrm{x}_{j} = 0$ are of particular interest. For the pure liquid substance $j$, equation (w) is readily obtained.
$\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 1\right) \phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})=\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)$
The latter is the expected property. However, the infinite dilution limit of equation (v) is not immediately obvious. In fact, both the numerator and denominator in the last term approach zero as we approach infinite dilution ($\mathrm{x}_{j} = 0$). What emerges is equation (x) [3], which is an example of the unusual formalism for non-Lewisian properties [4].
\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})= \ &\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+\mathrm{T} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pj}}^{*}(\ell)}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]^{2} \end{aligned}
Thus for ideal liquid mixtures $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}$ is given by a combination of properties for both pure liquid components. In other words, the chemical nature of component 1 affects the non-Lewisian properties of its mate j in the ideal mixture. This is in contrast with apparent molar Lewisian properties, such as $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$, for which the values in ideal mixtures are the same as in the pure liquid state of substance $j$.
An extensive literature describes isentropic compressions of binary liquid mixtures [5]
Footnotes
[1] G. Douhéret, M. I. Davis, J. C. R. Reis and M. J. Blandamer, ChemPhysChem, 2001, 2, 148.
[2] M. I. Davis, G. Douhéret, J. C. R. Reis and M. J. Blandamer, Phys. Chem. Chem. Phys., 2001, 3, 4555.
[3] It is instructive to show how the limiting value for $\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})$ at $\mathrm{x}_{j} = 0$ (and hence $\mathrm{x}_{1} = 1$) is obtained from equation (v). In this limit the last term (without $\mathrm{T}$) of equation (v) becomes,
\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right)\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})}\right\} / 0\ &=0 / 0 \end{aligned}
We apply L'Hospital's rule which asserts that this limit is equal to the ratio of limits (bb) and (cc).
\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \ &\mathrm{d}\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)}\right\} / \mathrm{dx} \mathrm{x}_{\mathrm{j}} \end{aligned}
$\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{dx}_{\mathrm{j}} / \mathrm{dx}_{\mathrm{j}}$
The latter limit is unity. The former is obtained from the following differential.
\begin{aligned} &\mathrm{d}\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)}\right\} / \mathrm{dx} \mathrm{x}_{\mathrm{j}} \ &=-\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)} \ &-\left\{2 \, \frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]}{\left[\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)\right]}\right\} \,\left[-\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right] \ &+\left\{\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\left[\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}\right\} \,\left[-\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{C}_{\mathrm{pj}}^{*}(\ell)\right] \end{aligned}
The limiting value of this differential is,
$\operatorname{limit}\left(x_{1} \rightarrow 0\right) \frac{\left.\mathrm{d}_{\{} \ldots\right\}}{\mathrm{dx}_{\mathrm{j}}}=\mathrm{C}_{\mathrm{pj}}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pj}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} j}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]^{2}$
This is the value for the limit of equation (aa), which was used to obtain equation (x) above.
[4] J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douhéret, Phys. Chem. Chem. Phys., 2001, 3, 1465.
[5]
1. methanol + alcohols; H. Ogawa and S. Murakami, J. Solution Chem.,1987, 16, 135.
2. γ-caprolactam + alcohols; S. K. Mehta, R. K. Chauhan and R. K. Dewan, J. Chem. Soc. Faraday Trans.,1996,92,4463.
3. ethanol + water; G. Onori, J. Chem. Phys.,1988, 89,4325.
4. alkoxyethanol + water; G.Douheret, A. Pal and M. I. Davis, J. Chem.Thermodyn.,1990,22,99.
5. 2-butoxyethanol(aq); 2-butanone(aq) Y. Koga, K..Tamura and S. Murakami, J. Solution Chem.,1995,24,1125.
6. pyrrolidin-2-one + alkanols; S. K. Mehta, R. K. Chauhan and R. F. Dewan, J. Chem. Soc. Faraday Trans.,1996,92,1167.
7. isomeric 2-butoxyethanols+water; G. Douheret, M. I. Davis, J. C. R. Reis, I. J. Fjellanger, M. B. Vaage and H. Hoiland, Phys. Chem. Chem. Phys., 2002,4,6034.
8. benzene + cyclohexane; G. C.Benson and O. Kiyohara, J. Chem. Thermodyn.,1979,11,1061.
9. water +_ ethanol; G. C. Benson + M. K. Kumaran, J. Chem. Thermodyn., 1983,15,799.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.12%3A_Compressions-_Isentropic-_Binary_Liquid_Mixtures.txt
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The volume of a given closed system is defined by the set of independent variables $\mathrm{T}$, $\mathrm{p}$ and composition $\xi$; $\mathrm{V}=\mathrm{V}[\mathrm{T}, \mathrm{p}, \xi]$. We assert that in this state the affinity for spontaneous chemical reaction is $\mathrm{A}$. The system is perturbed by a change in pressure such that the system can track one of two pathways; (i) at constant $\mathrm{A}$ or (ii) at constant $\xi$. The differential dependences of volume on pressure are related using equation (a).
$\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$
The two differentials expressing the dependence of volume on pressure define the equilibrium isothermal compression and the frozen isothermal compression respectively [1]. For a system at equilibrium, (i.e. minimum in $\mathrm{G}$ at fixed $\mathrm{T}$ and $\mathrm{p}$) following perturbation by a change in pressure,
$\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \quad \mathrm{~K}_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi^{\mathrm{a}}}$
The negative signs recognise that for all thermodynamically stable systems, the volume decreases with increase in pressure. Nevertheless there is merit in thinking of compression (and compressibility) as a positive feature of a system. Both $\mathrm{K}_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)$ and $\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)$ are extensive variables characterising two possible pathways. From equation (a) [2],
$\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)=\mathrm{K}_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)-\left[\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\right]^{2} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}}$
But at equilibrium, $(\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0$. Hence, irrespective of the sign of $(\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}$, $\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)>\mathrm{K}_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)$. Equation (c) is rewritten in terms of compressibilities [3].
$\kappa_{\mathrm{T}}(\mathrm{A}=0)=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}=0} \quad \kappa_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \xi^{\text {eq }}}$
$\kappa_{\mathrm{T}}(\mathrm{A}=0)=\kappa_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)-(1 / \mathrm{V}) \,\left[(\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\right]^{2} \,(\partial \xi / \partial \mathrm{A})_{\mathrm{T}, \mathrm{p}}$
Because $(\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0$, for all stable systems, $\kappa_{\mathrm{T}}(\mathrm{A}=0)>\kappa_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)$. According therefore to equation (e) the volume decrease accompanying a given change in pressure is more dramatic under condition that $\mathrm{A} = 0$ than under the condition where $\xi$ remains constant at $\xi^{\mathrm{eq}}$ [4]. Both $\kappa_{\mathrm{T}}(\mathrm{A}=0)$ and $\kappa_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)$ are volume intensive properties of a solution [5].
Footnotes
[1] The contrast between the two conditions is familiar to anyone who has dived into a swimming pool and “got it wrong”. Hitting the wall of water is similar to the conditions for$\mathrm{K}_{\mathrm{T}}(\xi)$ whereas for a good dive the conditions resemble $\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)$; the water molecules move apart to allow a smooth entry into the water.
[2] Consider $\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{p} \, \mathrm{d} \xi}\right)=\left(\frac{\partial^{2} \mathrm{G}}{\partial \xi \, \mathrm{dp}}\right)$. Then, $-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}=\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$
[3] $\mathrm{K}_{\mathrm{T}}=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right] ; \kappa_{\mathrm{T}}=\left[\mathrm{Pa}^{-1}\right]$
[4] Equation (e) forms the basis of the pressure-jump fast reaction technique. A rapid change in pressure produces a “frozen” system which relaxes to the equilibrium state at a rate characteristic of the system.
[5] For information concerning $\mathrm{D}_{2}\mathrm{O}(\ell)$, see R. A. Fine and F. J. Millero, J.Chem.Phys.,1975,63,89.
1.7.14: Compressions- Ratio- Isentropic and Isothermal
Using a calculus operation, we obtain equations relating isothermal and isentropic dependencies of volume on pressure.
Thus,
\begin{aligned} (\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}} \ &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} \end{aligned}
and,
\begin{aligned} (\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=&-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{v}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \ &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}} \end{aligned}
Then,
$(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} /(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} /(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}$
The Gibbs-Helmholtz equation requires that
$\mathrm{H}=\mathrm{G}-\mathrm{T} \,(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}$
Also
$(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\partial^{2} \mathrm{G} / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}$
Similarly,
$(\partial \mathrm{U} / \partial \mathrm{T})_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}$
Hence,
$\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{s}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.13%3A_Compressions-_Isothermal-_Equilibrium_and_Frozen.txt
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The density of an aqueous solutions at defined $\mathrm{T}$ and $\mathrm{p}$ and solute molality $\mathrm{m}_{j}$ yields the apparent molar volume of solute $j$, $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$. The dependence of $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$ on $\mathrm{m}_{j}$ can be extrapolated to yield the limiting (infinite dilution) property $\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}$. The isothermal dependence of densities on pressure can be expressed in terms of an analogous infinite dilution apparent molar isothermal compression, $\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}$. Similarly the isentropic compressibilities of solutions are characterised by $\phi\left(\mathrm{K}_{\mathrm{S} j} ; \operatorname{def}\right)^{\infty}$ which is accessible via the density of a solution and the speed of sound in the solution. Nevertheless the isothermal property $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{J}}}\right)^{\infty}$ presents fewer conceptual problems in terms of understanding the properties of solutes and solvents which control volumetric properties. The challenge is to use $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}$ in order to obtain $\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}$. The linking relationship is the Desnoyers-Philip equation [1]. The apparent molar isothermal compression for solute $j \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ is related to the concentration $\mathrm{c}_{j}$ of solute using equation (a) where $\phi\left(V_{j}\right)$ is the apparent molar volume of the solute.
$\phi\left(K_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{Tl}}^{*}(\ell)\right] \,\left(\mathrm{c}_{\mathrm{j}}\right)^{-1}+\kappa_{\mathrm{Tl}}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
The corresponding isentropic compression for solute $j$, $\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)$ is related to the concentration $\mathrm{c}_{j}$ using equation (b).
$\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right) \equiv\left[\kappa_{\mathrm{S}}(\mathrm{aq})-\kappa_{\mathrm{S} 1}^{*}(\ell)\right] \,\left(\mathrm{c}_{\mathrm{j}}\right)^{-1}+\kappa_{\mathrm{S} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
[We replace the symbol ≡ by the symbol = in the following account.]
$\text { By definition } \delta(\mathrm{aq})=\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{S}}(\mathrm{aq})$
$\text { And } \delta_{1}^{*}(\ell)=\kappa_{\mathrm{T} 1}^{*}(\ell)-\kappa_{\mathrm{S} 1}^{*}(\ell)$
Hence $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ and $\phi\left(K_{\mathrm{Sj}} ; \operatorname{def}\right)$ are related by equation (e).
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(\mathrm{c}_{\mathrm{j}}\right)^{-1} \,\left[\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\right]+\delta_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
The difference $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}} ; \mathrm{def}\right)$ depends on the concentration of the solute $\mathrm{c}_{j}$. Further $\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)$ is not zero. In fact,
\begin{aligned} \delta(\mathrm{aq})-& \delta_{1}^{*}(\ell)=\ &\left\{\mathrm{T} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})-\left\{\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)\right\}\right. \end{aligned}
Using the technique of adding and subtracting the same quantity, equation (f) is re-expressed as follows.
\begin{aligned} &\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)= \ &\begin{aligned} \left\{\delta(\mathrm{aq}) /\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}\right\} \,\left[\, \alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right] \ &-\left[\delta_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right] \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right] \end{aligned} \end{aligned}
The difference $\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]$ is related to $\phi\left(\mathrm{E}_{\mathrm{pj}}\right)$ using equation (h).
$\phi\left(E_{p j}\right)=\left[\alpha_{p}(a q)-\alpha_{p 1}^{*}(\ell)\right] \,\left(c_{j}\right)^{-1}+\alpha_{p 1}^{*}(\ell) \, \phi\left(V_{j}\right)$
Similarly, $\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]$ is related to $\phi\left(C_{\mathrm{p} j}\right)$ using equation (i).
$\phi\left(C_{p j}\right)=\left[\sigma(a q)-\sigma_{1}^{*}(\ell)\right] \,\left(c_{j}\right)^{-1}+\sigma_{1}^{*}(\ell) \, \phi\left(V_{j}\right)$
Using equations (g) - (i), we express equation (e) as follows.
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)=& \ {\left[\delta(\mathrm{aq}) / \alpha_{\mathrm{p}}(\mathrm{aq})\right] \,\left\{1+\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \alpha_{\mathrm{p}}(\mathrm{aq})\right]\right\} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) } \ &-\left[\delta_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right] \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)+\left\{\delta_{1}^{*}(\ell)\right.\ &\left.-\left[\delta(\mathrm{aq}) \, \alpha_{\mathrm{p} 1}^{*}(\ell) / \alpha_{\mathrm{p}}(\mathrm{aq})\right]\right\} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}
Equation (j) was obtained by Desnoyers and Philip [1] who showed that if $\phi\left(K_{T_{j}}\right)^{\infty}$ and $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}$ are the limiting (infinite dilution) apparent molar properties, the difference is given by equation (k).
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}-\phi\left(\mathrm{K}_{\mathrm{Sj}} ;\right.&\operatorname{def})^{\infty}=\ \delta_{1}^{*}(\ell) \,\left\{\left[2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} / \alpha_{\mathrm{pl} 1}^{*}(\ell)\right]-\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty} / \sigma_{1}^{*}(\ell)\right]\right\} \end{aligned}
Using equation (b), $\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)$ is plotted as a function of cj across a set of different solutions having different entropies. $\operatorname{Limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)$ defines $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}$. Granted two limiting quantities, $\phi\left(E_{p j}\right)^{\infty}$ and $\phi\left(C_{p j}\right)^{\infty}$ are available for the solution at the same $\mathrm{T}$ and $\mathrm{p}$, equation (k) is used to calculate $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{J}}}\right)^{\infty}$ using $\phi\left(K_{\mathrm{S}_{\mathrm{j}}} ; \text { def }\right)^{\infty}$.
An alternative form of equation (j) refers to a solution, molality mj [2].
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)= \ &\quad \delta_{1}^{*}(\ell) \,\left\{\left[2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) / \alpha_{\mathrm{p} 1}^{*}(\ell)\right]-\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma_{1}^{*}(\ell)\right]\right. \ &\quad+\left[\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\right]^{2} /\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}\right\} \,\left\{1+\left[\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma_{1}^{*}(\ell)\right]\right\}^{-1} \end{aligned}
The fact that $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}$ can be obtained from $\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)^{\infty}$ indicates the importance of the Desnoyers-Philip equation. Bernal and Van Hook [3] used the Desnoyers-Philip equation to calculate $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}$ for glucose(aq), sucrose(aq) and fructose(aq) at $348 \mathrm{~K}$. Similarly Hedwig et. al. used the Desnoyers –Philip equation to obtain estimates of $\phi\left(K_{\mathrm{Tj}}\right)^{\infty}$ for glycyl dipeptides (aq) at $298 \mathrm{~K}$ [4].
Footnotes
[1] J. E. Desnoyers and P. R. Philip, Can. J.Chem.,1972, 50,1095.
[2] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998,94,2385.
[3] P. J. Bernal and W. A. Hook, J.Chem.Thermodyn.,1986,18,955.
[4] G. R. Hedwig, J. D. Hastie and H. Hoiland, J. Solution Chem.,1996, 25, 615.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.15%3A_Compression-_Isentropic_and_Isothermal-_Solutions-_Limiting_Estimates.txt
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A given solution is perturbed by a change in pressure to a neighbouring state at constant affinity, $\mathrm{A}$.
$(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}} \,(\partial \mathrm{T} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}$
But for the pure solvent (at constant affinity $\mathrm{A}$) S * 1 * S1 K (l) = −(∂V (l)/ ∂p) and T * 1 * T1 K (l) = −(∂V (l)/ ∂p) (b)
We confine attention to perturbation at ‘$\mathrm{A} = 0$’; i.e. an equilibrium process. [Note the change in sign.]
$\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\left(\partial \mathrm{S}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}} \,\left(\partial \mathrm{T} / \partial \mathrm{S}_{1}^{*}(\ell)\right)_{\mathrm{p}} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}$
From a Maxwell Equation,
$\left(\partial \mathrm{S}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}}=-\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}$
From the Gibbs-Helmholtz Equation,
$\left(\partial \mathrm{S}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}=\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{T}$
$C_{p 1}^{*}(\ell)$ is the molar ( equilibrium) isobaric heat capacity of the solvent at defined $\mathrm{T}$ and $\mathrm{p}$. From equation (c), [Note change of sign.]
$\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)-\left[\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}\right]^{2} \, \mathrm{T} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)$
But
$\alpha_{1}^{*}(\ell) \, V_{1}^{*}(\ell)=\left(\partial V_{1}^{*}(\ell) / \partial T\right)_{p}$
$\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)-\left[\alpha_{1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)\right]^{2} \, \mathrm{T} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)$
But the ratio of isobaric heat capacity of the solvent to its molar volume,
$\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\sigma_{1}^{*}(\ell) .$
$\left.\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)-\left[\alpha_{1}^{*}(\ell)\right]^{2} \, \mathrm{V}_{1}^{*}(\ell)\right] \, \mathrm{T} / \sigma_{1}^{*}(\ell)$
But
$\kappa_{\mathrm{s} 1}^{*}(\ell)=\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell) \text { and } \kappa_{\mathrm{T} 1}^{*}(\mathrm{l})=\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)$
$\kappa_{\mathrm{S} 1}^{*}(\ell)=\kappa_{\mathrm{T} 1}^{*}(\ell)-\left[\alpha_{1}^{*}(\ell)\right]^{2} \, \mathrm{T} / \sigma_{1}^{*}(\ell)$
Similarly for an aqueous solution, molality $\mathrm{m}_{j}$,
$\kappa_{\mathrm{S}}(\mathrm{aq})=\kappa_{\mathrm{T}}(\mathrm{aq})-[\alpha(\mathrm{aq})]^{2} \, \mathrm{T} / \sigma(\mathrm{aq})$
Also
$\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left(\mathrm{n}_{\mathrm{j}}\right)^{-1} \,\left\{\mathrm{K}_{\mathrm{T}}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]$
We convert from compressions to compressibilities.
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left(\mathrm{n}_{\mathrm{j}}\right)^{-1} \,\left\{\mathrm{V}(\mathrm{aq}) \, \kappa_{\mathrm{T}}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{\mathrm{T} 1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell)\right]$
But we know $\kappa_{\mathrm{T}}(\mathrm{aq})$ in terms of ) $\kappa_{\mathrm{S}}(\mathrm{aq})$ (aq [see equation (m)] and $\kappa_{\mathrm{T} 1}^{*}(\ell)$ in terms of $\kappa_{\mathrm{s} 1}^{*}(\ell)$. Then [NB change of sign]
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\frac{\mathrm{V}(\mathrm{aq})}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\mathrm{K}_{\mathrm{S}}(\mathrm{aq})+\frac{\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\sigma(\mathrm{aq})}\right] \ &-\left[\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\kappa_{\mathrm{Sl}}^{*}(\ell)+\frac{\left\{\alpha_{1}^{*}(\ell)\right\}^{2} \, \mathrm{T}}{\sigma_{1}^{*}(\ell)}\right] \end{aligned}
We introduce densities into equation (q). For a solution having mass $\mathrm{w}$,
\begin{aligned} \mathrm{V}(\mathrm{aq}) / \mathrm{n}_{\mathrm{j}} &=\left[1 / \mathrm{n}_{\mathrm{j}}\right] \,[\mathrm{w} / \rho(\mathrm{aq})]=\left[1 / \mathrm{n}_{\mathrm{j}} \, \rho(\mathrm{aq})\right] \,\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}+\mathrm{n}_{1} \, \mathrm{M}_{1}\right] \ &=\left[1 / \mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})\right]+\left[\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})\right] \end{aligned}
Also,
$\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) / \mathrm{n}_{\mathrm{j}}=\left[\mathrm{n}_{1} / \mathrm{n}_{\mathrm{j}}\right] \,\left[\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\right]=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}$
From equations (q), (r) and (s),
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\right] \,\left[\kappa_{\mathrm{s}}(\mathrm{aq})+\frac{\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\sigma(\mathrm{aq})}\right] \ &-\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \,\left[\kappa_{\mathrm{s} 1}^{*}(\ell)+\frac{\left\{\alpha_{1}^{*}(\ell)\right\}^{2} \, \mathrm{T}}{\sigma_{1}^{*}(\ell)}\right] \end{aligned}
We factor out the six terms . The order in which we write these terms anticipates the next but one step.
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\frac{\kappa_{\mathrm{s}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})}\right]-\left[\frac{\kappa_{\mathrm{Sl}}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right]+\left[\frac{\mathrm{M}_{\mathrm{j}} \, \kappa_{\mathrm{s}}(\mathrm{aq})}{\rho(\mathrm{aq})}\right] \ &+\left[\frac{\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \sigma(\mathrm{aq})}\right] \ &-\left[\frac{\left\{\alpha_{1}^{*}(\ell)\right\}^{2} \, \mathrm{T}}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}\right]+\left[\frac{\mathrm{M}_{\mathrm{j}} \,\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\rho(\mathrm{aq}) \, \sigma(\mathrm{aq})}\right] \end{aligned}
Hence,
$\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left\{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right\}^{-1} \,\left[\left\{\mathrm{K}_{\mathrm{s}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right\}-\left\{\kappa_{\mathrm{s} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right\}\right] \ +\left\{\kappa_{\mathrm{S}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})+\mathrm{A}+\mathrm{B}\right. \end{gathered}$
where,
$\mathrm{A}=\left[\frac{\mathrm{T}}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}\right] \,\left[\left(\frac{\{\alpha(\mathrm{aq})\}^{2} \, \rho_{1}^{*}(\ell)}{\sigma(\mathrm{aq})}\right)-\left(\frac{\left\{\alpha_{1}^{*}(\ell) \, \rho(\mathrm{aq})\right.}{\sigma_{1}^{*}(\ell)}\right)\right]$
and
$\mathrm{B}=\mathrm{M}_{\mathrm{j}} \,\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T} / \rho(\mathrm{aq}) \, \sigma(\mathrm{aq})$
With reference to solutions we compare the isentropic and isothermal dependences of $\phi\left(V_{j}\right)$ on pressure.
$\left[\frac{\partial \phi\left(V_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}=\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{T}}-\left[\frac{\partial \mathrm{S}(\mathrm{aq})}{\partial \mathrm{p}}\right]_{\mathrm{T}} \,\left[\frac{\partial \mathrm{T}}{\partial \mathrm{S}(\mathrm{aq})}\right]_{\mathrm{p}} \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}$
Noting signs [cf. equations (d) and (e)] and the definition of $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$,
$\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}=-\phi\left(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}\right)-\left[-\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right] \,\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}$
$\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}=\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}} \,\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}$
We turn our attention to $\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}$. We recall that
$\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
$\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left(1 / \mathrm{n}_{\mathrm{j}}\right) \, \mathrm{V}(\mathrm{aq})-\left[\left(\mathrm{n}_{1} / \mathrm{n}_{\mathrm{j}}\right) \, \mathrm{V}_{1}^{*}(\ell)\right]$
Hence
$\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}=\frac{1}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}-\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right]_{\mathrm{p}}$
We combine equations (za) and (zd). Hence
\begin{aligned} &-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}= \ &\phi\left(\mathrm{K}_{\left.\mathrm{T}_{\mathrm{j}}\right)}\right. \ &-\left[\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right]_{\mathrm{T}} \,\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{1}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}-\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right] \end{aligned}
Or,
\begin{aligned} -\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}]_{\mathrm{S}(\mathrm{aq})}}\right]=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \ &-\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \, \frac{1}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}^{2} \ &+\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}\left[\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right] \end{aligned}
But
$\alpha(\mathrm{aq}) \, \mathrm{V}(\mathrm{aq})=[\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T}]_{\mathrm{p}}$
And
$\alpha_{1}^{*}(\ell) \, V_{1}^{*}(\ell)=\left[\partial V_{1}^{*}(\ell) / \partial \mathrm{T}\right]_{\mathrm{p}}$
We introduce the latter two equations into equation (zf).
\begin{aligned} {\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} } &=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \ &-\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \, \frac{1}{\mathrm{n}_{\mathrm{j}}} \,[\mathrm{V}(\mathrm{aq}) \, \alpha(\mathrm{aq})]^{2} \ &+\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\alpha(\mathrm{aq}) \, \mathrm{V}(\mathrm{aq}) \, \alpha_{1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)\right] \end{aligned}
But $\sigma(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})$ and $\sigma_{1}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)$. Also $\mathrm{M}_{1} / \mathrm{V}_{1}^{*}(\ell)=\rho_{1}^{*}(\ell)$. Hence,
\begin{aligned} -\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} &=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \ -& {\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \, \frac{1}{\mathrm{n}_{\mathrm{j}}} \, \mathrm{V}(\mathrm{aq}) \,[\alpha(\mathrm{aq})]^{2} } \ &+\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \,\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\alpha(\mathrm{aq}) \, \alpha_{1}^{*}(\ell) \, \mathrm{M}_{1} / \rho_{1}^{*}(\ell)\right] \end{aligned}
Also, $\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}=\mathrm{m}_{\mathrm{j}}$. And [2] $\mathrm{V}(\mathrm{aq}) / \mathrm{n}_{\mathrm{j}}=\left[1 / \mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})\right]+\left[\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})\right]$
Hence we obtain a relation between the two compressions of the apparent molar volumes
\begin{aligned} -\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} &=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \ -\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\right] \,[\alpha(\mathrm{aq})]^{2} \ &+\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \, \alpha(\mathrm{aq}) \, \alpha_{1}^{*}(\ell) \end{aligned}
Footnotes
[1] Unit check on equation (l). $\left.\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}-\left\{\left[\mathrm{K}^{-1}\right]^{2} \,[\mathrm{K}]\right\} /\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1}\right\}$
$\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}-\left[\mathrm{J} \mathrm{m}^{-3}\right]^{-1}$ But $\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{J} \mathrm{m}^{-3}\right]$
[2] From
\begin{aligned} &V(a q)=\frac{w}{\rho(a q)}=\frac{n_{1} \, M_{1}}{\rho(a q)}+\frac{n_{j} \, M_{j}}{\rho(a q)} \ &\frac{V(a q)}{n_{j}}=\frac{n_{1} \, M_{1}}{n_{j} \, \rho(a q)}+\frac{M_{j}}{\rho(a q)} \ &\frac{V(a q)}{n_{j}}=\frac{1}{m_{j} \, \rho(a q)}+\frac{M_{j}}{\rho(a q)} \end{aligned}
1.7.17: Compressions- Isothermal- Solutes- Partial Molar Compressions
A given aqueous solution at temperature $\mathrm{T}$ and near ambient pressure $\mathrm{p}$ contains a solute $j$ at molality $\mathrm{m}_{j}$. The chemical potential $\mu_{j}(\mathrm{aq})$ is related to the molality $\mathrm{m}_{j}$ using equation (a).
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)$
Then
$\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}$
By definition, the partial molar isothermal compression of solute $j$,[1]
$K_{T_{j}}(a q)=-\left(\frac{\partial V_{j}(a q)}{\partial p}\right)_{T}$
Then
$\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\mathrm{K}_{\mathrm{TJ}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T} \,\left[\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}^{2}\right]_{\mathrm{T}}$
Thus by definition,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})$
Hence the difference between $\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}(\mathrm{aq})$ and $\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})$ depends on the second differential of $\ln \left(\gamma_{j}\right)$ with respect to pressure.
Footnotes
[1] The formal definition of $\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})$ is given by equation (a).
$\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{K}_{\mathrm{T}}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}$
However,
$\mathrm{K}_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{n}(\mathrm{j})}$
Then,
$\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial\left(\partial \mathrm{V} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}(\mathrm{j})}$
Or,
$\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
In other words, equation (c) shows that $\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})$ is a Lewisian partial molar property
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The volume of a given solution prepared at fixed temperature and fixed pressure using $\mathrm{n}_{1}$ moles of water and $\mathrm{n}_{j}$ moles of solute $j$ is given by equation (a).
$\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})$
If the solution is prepared using $1 \mathrm{~kg}$ of water($\ell$),
$\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})$
$\mathrm{M}_{1}$ is the molar mass of the solvent, water($\ell$); $\mathrm{V}_{1}(\mathrm{aq})$ and $\mathrm{V}_{j}(\mathrm{aq})$ are the partial molar volumes of water and solute $j$ respectively in the solution. As we change $\mathrm{m}_{j}$ (for a fixed mass of solvent) so both $\mathrm{V}_{1}(\mathrm{aq})$ and $\mathrm{V}_{j}(\mathrm{aq})$ change. An important procedure rewrites equation (b) in the following form where $\mathrm{V}_{1}^{*}(\ell)$ is the molar volume of pure solvent at the same $\mathrm{T}$ and $\mathrm{p}$. Thus,
$\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
$\phi\left(\mathrm{V}_{\mathrm{j}}\right)$ is the apparent molar volume of the solute $j$. The system, an aqueous solution, is displaced by a change in pressure (at fixed $\mathrm{T}$) along a path where the affinity for spontaneous change is zero. In other words the system is subjected to an equilibrium displacement. The isothermal differential dependence of volume $\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)$ is given by equation (d).
$\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right.}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{M}_{1}^{-1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
By definition, the apparent molar (isothermal) compression of the solute, [1-3]
$\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
Similarly for equilibrium molar compression of the pure solvent,
$\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
By definition,
$\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=-\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
Hence,
$\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)$
Moreover recalling that $\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})$,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})$
These equations combined with those yielding $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ from measured $\kappa_{\mathrm{T} 1}^{*}(\ell)$ and $\kappa_{\mathrm{T}}(\mathrm{aq})$ signal an attractive approach to the study of solvent-solute interactions via $\mathrm{K}_{\mathrm{T} j}^{\infty}(\mathrm{aq})$. In this context Gurney [4] identified a cosphere of solvent around a solute molecule where the organization differs from that in the bulk solvent at some distance from a given solute molecule $j$. For example, the limiting partial molar volume of solute $j$ can be understood as the sum of two terms, $\mathrm{V}$(intrinsic) and $\mathrm{V}$(cosphere). Then $\mathrm{V}$(cosphere) is an indicator of the role of solvent-solute interaction, hydration in aqueous solution. Thus,
$\left.V_{j}^{\infty}(a q)=V_{j} \text { (int rinsic }\right)+V_{j}(\cos p h e r e)$
Hence,
$\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}} \text { (int rinsic) }+\mathrm{K}_{\mathrm{T}_{j}}(\cos \text {phere})$
The argument is advanced that $\mathrm{K}_{\mathrm{T}j}$(intrinsic) for simple ions such as halide ions and alkali metal ions is zero. $\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})$ is an indicator of the hydration of a given solute in aqueous solution. $\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})$ is obtained from the dependence of $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ on, for example, concentration $\mathrm{c}_{j}$ using equation(l) for neutral solutes and equation (m) for salts, the latter being based on the DHLL.
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})+\mathrm{a}_{\mathrm{KT}} \,\left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)$
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})+\mathrm{b}_{\mathrm{KT}} \,\left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)^{1 / 2}$
One might have expected an extensive scientific literature reporting $\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})$ for a wide range of solutes. Unfortunately measurement of isothermal compressions of liquids is difficult at least to the precision required for the estimation of $\mathrm{K}_{\mathrm{Tj}}^{\infty}(\mathrm{aq})$. Indeed direct measurement of the volume change of a liquid when compressed at constant temperature is difficult because the isothermal condition is difficult to satisfy. Two procedures have been adopted to over come this problem. In both cases isentropic compressibilities calculated from densities and speeds of sound have been used.
In one set of procedures, isentropic compressibilities, densities and isobaric heat capacities are used to calculate isothermal compressions for a given solution, molality $\mathrm{m}_{j}$. For example, Bernal and Van Hook [5] use the Desnoyers-Philip Equation to evaluate $\phi\left(K_{T_{j}}\right)^{\infty}$ for glucose, sucrose and fructose [5] in aqueous solutions at $348 \mathrm{~K}$. An alternative procedure equates $\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}$ with the experimentally accessible limiting apparent isentropic compression, ∞ φ(K ) Sj . In another approach, the starting point is equation (a) which is differentiated with respect to pressure at constant temperature to yield equation (n).
$\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{Tl}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})$
Equation (n) is divided by volume $\mathrm{V}(\mathrm{aq})$. Hence
\begin{aligned} \mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \,\left[1 / \mathrm{V}_{1}(\mathrm{aq})\right] \, \mathrm{K}_{1}(\mathrm{aq}) \ &+\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \,\left[1 / \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{j}}(\mathrm{aq}) \end{aligned}
We use $\phi_{1}\left[=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \text { and } \phi_{\mathrm{j}}\left[=\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right]$ to express volume fractions.
$\kappa_{\mathrm{T}}(\mathrm{aq})=\phi_{1} \,\left[1 / \mathrm{V}_{1}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})+\phi_{\mathrm{j}} \,\left[1 / \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})$
The latter equation is not tremendously helpful. Although $\kappa_{\mathrm{T}}(\mathrm{aq})$ can be measured, the right hand side involves six terms about which we have no information ‘a priori’ and which depend on the composition of the solution.
Footnotes
[1] F. T. Gucker, Chem. Rev.,1933,14,127.
[2] F. T. Gucker, J. Am. Chem. Soc.,1933, 55,2709.
[3] Units; $\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right] ; \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]$; $\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right]$
For the solution $\kappa_{\mathrm{T}}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{Pa}^{-1}\right]$
For the solvent $\kappa_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\left[\mathrm{Pa}^{-1}\right]$
Isothermal compressions have units of ‘volume per unit of pressure’ whereas compressibilities have units of ‘reciprocal pressure’. $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ is an apparent molar isothermal compression on the grounds that the units of this quantity are $\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]$. Some reports use the term ‘apparent molar isothermal compressibility’ which should be avoided because in the present context this term corresponds to a different property; see J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998,94,2385.
[4] R.W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.
[5] P. D. Bernal and W. A. Van Hook, J. Chem. Thermodyn., 1986,18,955.
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A given solution (at fixed $\mathrm{T}$ and $\mathrm{p}$) is prepared using $1 \mathrm{~kg}$ of solvent water and $\mathrm{m}_{j}$ moles of solute $j$. The compression of this solution $\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)$ is given by equations (a) and (b).
$\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$
$\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})$
where,
$\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
and
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
Both $\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})$ and $\phi\left(\mathrm{K}_{\mathrm{Tj}^{\mathrm{j}}}\right)$ are Lewisian variables. With reference to partial molar volumes,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
Hence
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})$
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}$ is the limiting (infinite dilution) apparent molar compression of solute--$j$. For a given solution $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ is calculated using one of the following equations together with the isothermal compressions of solution and solvent [1-3].
Molality Scale
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\kappa_{\mathrm{T}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
$\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \ +\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}} \,[\rho(\mathrm{aq})]^{-1} \end{gathered}$
Concentration Scale
$\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{TJ}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \ +\kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell) \end{gathered}$
Also
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{Tl}}^{*}(\ell)\right]+\kappa_{\mathrm{T} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
The latter four equations are thermodynamically correct, no assumption being made in their derivation.
In 1933, Gucker reviewed the direct determination of compressibilities of solutions leading to apparent molar compressions of solutes in aqueous solution calculated using equation (k) [4-6].
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\rho(\mathrm{aq})} \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}}}+\mathrm{M}_{\mathrm{j}}\right]-\frac{\kappa_{1}^{*}(\ell)}{\rho_{1}^{*}(\ell)} \, \frac{1}{\mathrm{~m}_{\mathrm{j}}}$
Compressibilties of solutions were directly determined by measuring the sensitivity of $\mathrm{V}(\mathrm{aq})$ to an increase in pressure. Gucker showed that for aqueous salt solutions, $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ is negative and a linear function of $\left(\mathrm{c}_{\mathrm{j}}\right)^{1 / 2}$. Moreover the limiting value, $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}$ is an additive property of $\phi\left(\mathrm{K}_{\mathrm{T}}-1 \mathrm{ion}\right)^{\infty}$
A useful approximation is that for dilute solutions at constant $\mathrm{T}$ and $\mathrm{p}$ containing a neutral solute $j$, $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ is linear function of molaity $\mathrm{m}_{j}$.
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{e} \, \mathrm{m}_{\mathrm{j}}+\mathrm{f}$
Hence,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=\mathrm{K}_{\mathrm{Tj}}^{\infty}(\mathrm{aq})=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}$
We identify ‘f’ in equation (l) as the limiting isothermal apparent molar isothermal compression of solute $j$ in solution (at equilibrium).
Footnotes
[1] $\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right] ; \mathrm{K}_{\mathrm{Tl}}^{*}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]$; $\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right]$
For the solution, $\kappa_{\mathrm{T}}=\mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{Pa}^{-1}\right]$
For the solvent, $\kappa_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\left[\mathrm{Pa}^{-1}\right]$
[2] Isothermal compressions have units ‘volume per unit of pressure’ whereas compressibilities have units of ‘reciprocal pressure’. $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ is an apparent molar isothermal compression on the grounds that the units of this property are $\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]$. Some reports use the term ‘apparent molar isothermal compressibility’.
[3] For an aqueous solution at fixed temperature and pressure prepared using $\mathrm{n}_{1}$ moles of water and $\mathrm{n}_{j}$ moles of solute $j$,
$\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)$
Hence with respect to an equilibrium displacement (i.e. at $\mathrm{A} = 0$) at defined temperature,
$(\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{p})_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}}+\mathrm{n}_{\mathrm{j}} \,\left(\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right)_{\mathrm{T}}$
Hence
$(\partial \mathrm{V}(\mathrm{aq}) / \partial \rho)_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}}-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$
For the solution the (equilibrium) isothermal compressibility,
$\kappa_{\mathrm{T}}(\mathrm{aq})=-\frac{1}{\mathrm{~V}(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
Similarly for the pure solvent,
$\kappa_{\mathrm{T} 1}^{*}(\ell)=-\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
Hence from equation (c),
$\mathrm{V}(\mathrm{aq}) \, \mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)$
We use equation (as) for $\mathrm{V}(\mathrm{aq})$ in conjunction with equation (f).
\begin{aligned} \kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]=\ \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \end{aligned}
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{T} \mathrm{j}}\right)= \ &{\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)\right] / \mathrm{n}_{\mathrm{j}}-\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{n}_{\mathrm{j}}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})} \end{aligned}
But, $\mathrm{V}_{1}^{*}(\ell)=\mathrm{M}_{1} / \rho_{1}^{*}(\ell)$ Then,
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\kappa_{\mathrm{T}}(\mathrm{aq}) \, \frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{T}}(\mathrm{aq})$
Molality \begin{aligned} &\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{\mathrm{l}} \, \mathrm{M}_{\mathrm{l}} \ &\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}-\frac{\kappa_{\mathrm{T} 1}^{*}(\ell)}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq}) \end{aligned}
Or,
$\phi\left(\mathrm{K}_{\mathrm{TJ}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})$
Using again equation (a) to substitute for $\phi\left(\mathrm{V}_{\mathrm{j}}\right)$,
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, & {\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right] } \ &+\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\frac{\mathrm{V}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\right] \end{aligned}
With $\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})$,
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right] \ &+\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\frac{1}{\mathrm{c}_{\mathrm{j}}}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}\right] \end{aligned}
But $\frac{1}{c_{j}}=\frac{M_{j}}{\rho(a q)}+\frac{1}{m_{j} \, \rho(a q)}$ Hence,
\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{Tl}}^{*}(\ell)\right] \ &+\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}+\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho(\mathrm{aq})}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{\mathrm{1}}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}\right] \end{aligned}
$\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \ +\frac{\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})} \end{gathered}$
We start with equation (f).
$\mathrm{V}(\mathrm{aq}) \, \mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$
Mass of solution,
$V(a q) \, \rho(a q)=n_{1} \, M_{1}+n_{j} \, M_{j}$
Or,
$\mathrm{n}_{1}=\left[\mathrm{V}(\mathrm{aq}) \, \rho(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right] / \mathrm{M}_{1}$
We combine equations (f) and (q).
\begin{aligned} &\mathrm{V}(\mathrm{aq}) \, \kappa_{\mathrm{T}}(\mathrm{aq})= \ &\quad \mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \,\left[\frac{\mathrm{V}(\mathrm{aq}) \, \rho(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\mathrm{M}_{1}}\right]+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \end{aligned}
With $c_{j}=n_{j} / V(a q)$
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\mathrm{c}_{\mathrm{j}}}-\frac{\mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})}{\mathrm{c}_{\mathrm{j}} \, \mathrm{M}_{1}}+\frac{\mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}}}{\mathrm{M}_{1}}$
Hence,
$\begin{array}{r} \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \ +\kappa_{\mathrm{Tl}}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell) \end{array}$
Again from equation (j)
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})$
Hence,
\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)= \ &{\left[\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \rho_{1}^{*}(\ell)\right] \,\left[\rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]} \ &\quad+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq}) \end{aligned}
Or
$\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell)$
[4] F. T. Gucker, J.Am.Chem.Soc.,1933,55,2709.
[5] F. T. Gucker, Chem. Rev.,1933,13,111.
[6] From equation (h),
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}-\frac{\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}+\frac{\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}$
$\phi\left(\mathrm{K}_{\mathrm{T}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\rho(\mathrm{aq})} \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}}}+\mathrm{M}_{\mathrm{j}}\right]-\frac{\kappa_{\mathrm{Tl}}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{\star}(\ell)}$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.19%3A_Compressions-_Isothermal-_Solutions-_Apparent_Molar-_Determination.txt
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In 1933 Gucker [1-3] reviewed attempts to measure the apparent isothermal molar compressions of salts in aqueous solution, these attempts dating back to the earliest reliable measurements by Rontgen and Schneider [4] in 1886 and 1887. Gucker showed [2] that for several aqueous salt solutions the apparent isothermal molar compression, $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ is a linear function of the square root of the salt concentration.
$\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}+\mathrm{a} \, \mathrm{c}_{\mathrm{j}}^{1 / 2}$
This general equation holds for $\mathrm{CaCl}_{2}(\mathrm{aq})$ at 60 Celsius. In general terms, $\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)$ for salts is negative becoming less negative as the salt concentration increases. Gibson described an interesting approach which characterises salt solutions in terms of effective pressures, $\mathrm{p}_{\mathrm{c}}$ exerted by the salt on the solvent [5]. This effective pressure is expressed as a linear function of the product of salt and solvent concentrations. The constant of proportionality is characteristic of the salt. Leyendekkers based an analysis using the Tammann-Tait-Gibson (TTG) model, on the assertion that solutes, salts and organic solutes, exert an excess pressure on water in aqueous solution [6,7]. The TTG approach described by Leyendekkers is intuitively attractive but the analysis is based on an extra - thermodynamic assumption [8]. Calculation of an excess pressure requires an estimate of the volume of solute molecules, $\phi_{j}$ in solution. If this property is independent of solute molality $\mathrm{m}_{j}$, the dependence of the volume of a solution (in $1 \mathrm{~kg}$ of water) on solute molality is described by the dependence of the ‘partial molar volume' of water. The difference between $\left[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{m}_{\mathrm{j}} \, \phi(\mathrm{Vj})\right]$ and $\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)$ is understood in terms of an effective pressure on the solvent. The assumptions underlying this calculation are not trivial. Furthermore from a thermodynamic viewpoint, the pressure is the same in every volume element of a solution [8].
Footnotes
[1] F. T. Gucker, J. Am. Chem.Soc.,1933,55,2709.
[2] F. T. Gucker, Chem.Rev.,1933,13,111.
[3] F. T. Gucker, F. W. Lamb, G. A. Marsh and R. M. Haag, J.Am. Chem. Soc.,1950,72,310.
[4] W. C. Rontgen and J. Schneider, Wied. Ann., 1886,29,165;1887,31,36.
[5] R. E. Gibson, J.Am.Chem.Soc.,1934,56,4.;1935,57,284.
[6] J. V. Leyendekkers, J. Chem. Soc. Faraday Trans.1,1981,77,1529; 1982,78,357; 1988,84, 397,1653.
[7] J. V. Leyendekkers, Aust. J. Chem.,1981,34,1785.
[8] M. J. Blandamer, J. Burgess and A. Hakin, J. Chem. Soc. Faraday Trans.1,1986, 82,3681.
1.7.21: Compressions- Isothermal- Binary Aqueous Mixtures
A given binary aqueous mixture is prepared using $\mathrm{n}_{1}$ moles of water ($\ell$) and $\mathrm{n}_{2}$ moles of liquid 2. The volume of the mixture, $\mathrm{V}(\mathrm{mix})$ is given by equation (a) (at fixed temperature and pressure).
$\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\mathrm{mix})$
The mixture is perturbed by a change in pressure at fixed temperature along an equilibrium pathway where the affinity for spontaneous change remains at zero.
$\left(\frac{\partial \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\frac{\partial \mathrm{V}_{1}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{n}_{2} \,\left(\frac{\partial \mathrm{V}_{2}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
The isothermal compression of the mixture $[\partial \mathrm{V}(\mathrm{mix}) / \partial \mathrm{p}]_{\mathrm{T}}$ is an extensive property. The partial differentials $\left[\partial V_{1}(\operatorname{mix}) / \partial p\right]_{T}$ and $\left[\partial V_{2}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}}$ are intensive properties. There is merit in defining an intensive molar compression using equation (c).
$\mathrm{K}_{\mathrm{Tm}}=\frac{\mathrm{K}_{\mathrm{T}}(\operatorname{mix})}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)}=-\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)^{-1} \,[\partial \mathrm{V}(\operatorname{mix}) / \partial \mathrm{p}]_{\mathrm{T}}$
By definition,
$\mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})=-\left[\partial \mathrm{V}_{1}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}}$
And
$\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})=-\left[\partial \mathrm{V}_{2}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}}$
Hence
$\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})$
For an ideal binary mixture,
$\mathrm{V}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)$
$\mathrm{V}_{1}^{*}(\ell)$ and $\mathrm{V}_{2}^{*}(\ell)$ are the molar volumes of the pure liquids at the same $\mathrm{T}$ and $\mathrm{p}$. Therefore, following the argument outlined above,
$\mathrm{K}_{\mathrm{Tm}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)$
By definition,
$\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})-\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})$
Hence the excess molar compression is given by equation (j).
$\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}(\operatorname{mix})=\mathrm{x}_{1} \,\left[\mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})-\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right]$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.07%3A_Compressions/1.7.20%3A_Compressions-_Isothermal-_Salt_Solutions.txt
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The isothermal compressibility of a given binary liquid mixture having ideal thermodynamic properties is related to the isothermal compressions of the liquid components using equation (a) [1].
$\mathrm{K}_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]}{\mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}^{*}(\ell)-\mathrm{V}_{1}^{*}(\ell)\right]}$
The excess compression for a given binary liquid mixture is defined by equation (b).
$\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix})-\mathrm{K}_{\mathrm{Tn}}(\mathrm{mix} ; \mathrm{id})$
Or,
$\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix})-\left[\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right]$
The isothermal compressibilities of ideal and real binary liquid mixtures are defined by equations (d) and (e) respectively.
$\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=-\frac{1}{\mathrm{~V}(\operatorname{mix} ; \mathrm{id})} \,\left(\frac{\partial \mathrm{V}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
$\kappa_{\mathrm{T}}(\operatorname{mix})=-\frac{1}{\mathrm{~V}(\operatorname{mix})} \,\left(\frac{\partial \mathrm{V}(\operatorname{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
For a given binary liquid mixture we can define an excess compressibility using equation (f).
$\kappa_{\mathrm{T}}^{\mathrm{E}}=\kappa_{\mathrm{T}}(\operatorname{mix})-\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})$
Then
\begin{aligned} &\kappa_{\mathrm{T}}^{\mathrm{E}}(\mathrm{mix})=-\frac{1}{\mathrm{~V}(\mathrm{mix})} \,\left(\frac{\partial \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \ &+\frac{1}{\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]} \,\left(\frac{\partial\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]}{\partial \mathrm{p}}\right)_{\mathrm{T}} \end{aligned}
A similar equation was used by Moelwyn-Hughes and Thorpe [3]. They introduced the concept of a compressibility of the excess volume.
$\Delta \kappa_{\mathrm{T}}(\operatorname{mix})=-\frac{1}{\Delta \mathrm{V}(\operatorname{mix})} \,\left(\frac{\partial \Delta \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
In publications by Prigogine and by Moelwyn-Hughes and Thorpe the analysis was taken a step further to facilitate analysis of experimental results. However approximations were made in both treatments. An exact formulation was given by Missen [4] in terms of volume fractions of both components in the corresponding having ideal thermodynamic properties, $\phi_{1}(\text { mix;id })$ and $\phi_{2}(\text { mix;id })$. Hence,
$\kappa_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=-\frac{1}{\mathrm{~V}_{\mathrm{m}}(\operatorname{mix})} \,\left[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} \, \kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})\right]$
A partial compressibility was defined by Moelwyn-Hughes [5]. For liquid 1 in a binary liquid mixture at defined $\mathrm{T}$ and $\mathrm{p}$, the partial compressibility is defined by equation (j).
$\kappa_{T_{1}}(\operatorname{mix})=-\frac{1}{V_{1}(\operatorname{mix})} \,\left(\frac{\partial V_{1}(\operatorname{mix})}{\partial p}\right)_{T}$
Similarly for component 2,
$\kappa_{\mathrm{T} 2}(\operatorname{mix})=-\frac{1}{\mathrm{~V}_{2}(\operatorname{mix})} \,\left(\frac{\partial \mathrm{V}_{2}(\operatorname{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
The excess compressibility of a given binary liquid mixture $\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})$ was defined in equation (f). Hence,
\begin{aligned} \kappa_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=& \phi_{1}(\operatorname{mix}) \, \kappa_{\mathrm{T} 1}^{\mathrm{E}}(\operatorname{mix})+\phi_{2}(\operatorname{mix}) \, \kappa_{\mathrm{T} 2}^{\mathrm{E}}(\operatorname{mix}) \ &+\left[\phi_{1}(\operatorname{mix})-\phi_{1}(\operatorname{mix} ; \mathrm{id})\right] \, \kappa_{\mathrm{T} 1}^{*}(\ell)+\left[\phi_{2}(\operatorname{mix})-\phi_{2}(\operatorname{mix} ; \mathrm{id})\right] \, \kappa_{\mathrm{T} 2}^{*}(\ell) \end{aligned}
It may be noted that ‘true’ partial properties can also be defined for the isothermal compressibility [6]. Then the properties introduced in equations (j) and (k) would be termed specific partial isothermal compressions [6].
It is also possible to formulate a set of equations incorporating rational activity coefficients for the two components of the binary liquid mixture. We start with the equation for the partial molar volume of component 1.
$V_{1}(\operatorname{mix})=V_{1}^{*}(\ell)+R \, T \,\left(\frac{\partial \ln \left(f_{1}\right)}{\partial p}\right)_{T}$
$K_{T 1}(\operatorname{mix})=K_{T 1}^{*}(\ell)-R \, T \,\left(\frac{\partial^{2} \ln \left(f_{1}\right)}{\partial p^{2}}\right)_{T}$
Similarly
$\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})=\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}$
Therefore
$\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})=\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})-\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}+\mathrm{x}_{2} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}\right]$
The two liquid components are characterised by their molar excess properties.
$\mathrm{K}_{\mathrm{T} 1}^{\mathrm{E}}(\operatorname{mix})=-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}$
and
$\mathrm{K}_{\mathrm{T} 2}^{\mathrm{E}}(\operatorname{mix})=-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}$
Therefore
$\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}+\mathrm{x}_{2} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}\right]$
Also
$\mathrm{K}_{\mathrm{T} 1}^{\mathrm{E}}=-\left(\frac{\partial \mathrm{V}_{1}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \text { and } \mathrm{K}_{\mathrm{T} 2}^{\mathrm{E}}=-\left(\frac{\partial \mathrm{V}_{2}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
In other words
$\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}$
Isothermal compressions of liquid mixtures can be directly measured [7]. Hamann and Smith [8] report measurements using binary liquid mixtures at $303 \mathrm{~K}$ and two pressures. Hamann and Smith define excess isothermal molar compressions $\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)$ in terms of volume fraction weighted isothermal compressions of the pure liquids. The volume fractions are defined as follows.
$\phi_{1}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) /\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]$
$\phi_{2}=x_{2} \, V_{2}^{*}(\ell) /\left[x_{1} \, V_{1}^{*}(\ell)+x_{2} \, V_{2}^{*}(\ell)\right]$
Then
$\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix})-\left[\phi_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\phi_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right]$
For most binary aqueous mixtures $\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)$ is negative, plots of $\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)$ against $\phi_{2}$ being smooth curves. The minima in aqueous mixtures containing $\mathrm{THF}$ and propanone the minima are near 0.4 and 0.6 respectively [0].
Footnotes
[1] For a binary liquid mixture having ideal thermodynamic properties,
$\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)$
Then
$\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)$
But
$\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})}{\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}$
Then,
$\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]}{\mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}^{*}(\ell)-\mathrm{V}_{1}^{*}(\ell)\right]}$
[2] I. Prigogine, The Molecular Theory of Solutions, North Holland, Amsterdam, 1957, p.18.
[3] E. A. Moelwyn-Hughes and P. L. Thorpe, Proc. R. Soc. London, Ser. A,1964,278A, 574.
[4] R. W. Missen, Ind. Eng. Chem. Fundam., 1969,8,81.
[5] E. A. Moelwyn-Hughes, Physical Chemistry, Pergamon, London, 2nd. Edn., 1965, .817
[6] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998,94,2385.
[7] J. E. Stutchbury, Aust. J. Chem.,1971,24,2431.
[8] S. D. Hamann and F. Smith, Aust. J. Chem.,1971,24,2431.
[9] For a detailed report on the properties of liquid mixtures see G. M. Schneider, Pure Appl. Chem.,1983,55,479 ; and references therein.
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By definition, the Gibbs energy,
$\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}$
Enthalpy,
$\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}$
Combination of equations (a) and (b) yields an important equation relating Gibbs energy $\mathrm{G}$ and enthalpy $\mathrm{H}$.
$\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}$
Just as we can never know the thermodynamic energy of a system, so we can never know the enthalpy. Consequently analysis of enthalpies is more complicated than analysis of volumetric properties, bearing in mind that the density of a solution (liquid) can be accurately measured. Differences are therefore emphasised in the context of enthalpies.
A differential change in Gibbs energy at constant temperature is related to the changes in enthalpy $\mathrm{dH}$ and entropy, $\mathrm{dS}$.
$\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}$
For an isothermal process from state I to state II, the change in Gibbs energy $\Delta \mathrm{G}$ is given by equation (e).
$\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \, \Delta \mathrm{S}$
Equation (e) signals how enthalpy and entropy changes determine the change in Gibbs energy.
A closed system at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ is prepared using $\mathrm{n}_{1}$ moles of solvent (water) and $\mathrm{n}_{j}$ moles of solute-$j$. The system is at equilibrium such that the composition/organisation is represented by $\xi^{\mathrm{eq}}$ and the affinity for spontaneous change is zero. Using an over-defined representation we define the system as follows.
$\mathrm{G}^{\mathrm{eq}}=\mathrm{G}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]$
Under such circumstances the Gibbs energy $\mathrm{G}$ is a minimum $\mathrm{G}^{\mathrm{eq}}$ when plotted as a function of $\xi$. The enthalpy of this system can be defined using a similar equation.
$\mathrm{H}^{\mathrm{eq}}=\mathrm{H}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]$
It is unlikely that $\mathrm{H}^{\mathrm{eq}}$ corresponds to a minimum in the plot of enthalpy $\mathrm{H}$ against $\xi$. Indeed the same comment applies to the entropy $\mathrm{S}^{\mathrm{eq}}$;
$\mathrm{S}^{\mathrm{eq}}=\mathrm{S}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]$
The plots showing the product $\mathrm{T} \, \mathrm{S}$ and $\mathrm{H}$ against $\xi$ may not show extrema though taken together they produce a minimum in $\mathrm{G}$ at $\xi^{\mathrm{eq}}$.
$\mathrm{G}^{e q}=\mathrm{H}^{e q}-\mathrm{T} \, \mathrm{S}^{e q}$
1.8.02: Enthalpy
There is considerable merit in identifying an extensive property of a closed system called the enthalpy, $\mathrm{H}$. The enthalpy of a closed system is a state variable and defined by equation (a).
$\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}$
We identify a given state by the symbol I having enthalpy $\mathrm{H}[\mathrm{I}]$, energy $\mathrm{U}[\mathrm{I}]$ and volume $\mathrm{V}[\mathrm{I}]$ at pressure $\mathrm{p}$.
$\mathrm{H}[\mathrm{I}]=\mathrm{U}[\mathrm{I}]+\mathrm{p} \, \mathrm{V}[\mathrm{I}]$
This system is displaced to a neighbouring state such that the differential change in enthalpy is $\mathrm{dH}$. Using equation (a),
$\mathrm{dH}=\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}$
But according to the first law of thermodynamics, the differential change in thermodynamic energy $\mathrm{dU}$ is given by ‘$q-p \, d V$’ where $\mathrm{q}$ is the heat accompanying the change. Then,
$\mathrm{dH}=\mathrm{q}-\mathrm{p} \, \mathrm{dV}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}$
or,
$\mathrm{dH}=\mathrm{q}+\mathrm{V} \, \mathrm{dp}$
At constant pressure,
$\mathrm{dH}=\mathrm{q}$
For a change from state I to state II the change in enthalpy is given by equation (g).
$\Delta \mathrm{H}=\int_{\mathrm{I}}^{\mathrm{II}} \mathrm{dH}=\mathrm{H}(\mathrm{II})-\mathrm{H}(\mathrm{I})=\mathrm{q}$
In equation (g) we replace the integral of dH by the difference $\mathrm{H}(\mathrm{II}) - \mathrm{H}(\mathrm{I})$ because enthalpy is a state variable and so $\Delta \mathrm{H}$ is independent of the path between the two states and hence so is $\mathrm{q}$. In liquid solutions, the recorded heat is also independent of the rate of change in chemical composition between state I and state II.
1.8.03: Enthalpy- Thermodynamic Potential
The enthalpy $\mathrm{H}$ of a closed system is related by definition to the thermodynamic energy $\mathrm{U}$; $\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}$. But
$\mathrm{dH}=\mathrm{q}+\mathrm{V} \, \mathrm{dp}$
From the second law of thermodynamics,
$\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0$
Then
$\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq 0$
Thus all spontaneous processes at constant entropy and pressure (i.e. isentropic and isobaric) lower the enthalpy of a closed system. This conclusion finds application in acoustics where the changes in a system perturbed by a travelling sound wave are discussed in terms of changes in enthalpy at constant entropy and pressure. Confining our attention to systems either at equilibrium (i.e. $\mathrm{A} = 0$) or at fixed $\xi$, two key relationships follow from equation (c).
$\mathrm{T}=(\partial \mathrm{H} / \partial \mathrm{S})_{\mathrm{p}}$
and
$\mathrm{V}=(\partial \mathrm{H} / \partial \mathrm{p})_{\mathrm{S}}$
In these terms the extensive variable, volume, is given by the
1. isentropic differential dependence of enthalpy on pressure, and
2. isothermal differential dependence of Gibbs energy on pressure.
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The enthalpy of a solution containing n1 moles of water and nj moles of solute, chemical substance j, is defined by the independent variables, $\mathrm{T}$, $\mathrm{p}$, $\mathrm{n}_{1}$ and $\mathrm{n}_{j}$.
$\mathrm{H}=\mathrm{H}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]$
where [1],
$\mathrm{H}=\mathrm{n}_{1} \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq})$
Here $\mathrm{H}_{1}(\mathrm{aq})$ and $\mathrm{H}_{j}(\mathrm{aq})$ are the partial molar enthalpies of water and solute $j$ in the solution.
$\mathrm{H}_{1}(\mathrm{aq})=\left(\partial \mathrm{H} / \partial \mathrm{n}_{1}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}$
$\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\left(\partial \mathrm{H} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)}$
For a solution prepared using $1 \mathrm{~kg}$ of solvent, water and $\mathrm{m}_{j}$ moles of solute $j$ [2],
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq})$
The chemical potential of the solvent in an aqueous solution is related to the molality of solute $j$, $\mathrm{m}_{j}$ using equation (f) where $\phi$ is the practical osmotic coefficient, a property of the solvent.
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
The chemical potential and partial molar enthalpy are linked using the Gibbs-Helmholtz equation such that at fixed pressure, $\mathrm{d}\left(\mu_{1}(\mathrm{aq}) / \mathrm{T}\right) / \mathrm{dT}=-\mathrm{H}_{1}(\mathrm{aq}) / \mathrm{T}^{2}$. Hence [3]
$\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\mathrm{d} \phi / \mathrm{dT})_{\mathrm{p}}$
By definition the practical osmotic coefficient is unity for ideal solutions at all $\mathrm{T}$ and $\mathrm{p}$. Then the partial molar enthalpy of the solvent in an ideal solution,
$\mathrm{H}_{1}(\mathrm{aq}, \mathrm{id})=\mathrm{H}_{1}^{*}(\lambda)$
The definition of $\phi$ requires that $\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{aq})$ equals $\mathrm{H}_{1}^{*}(\lambda)$. We express the difference between the partial molar enthalpies of the solvent in real and ideal solutions using a relative (partial) molar enthalpy, $\mathrm{L}_{1}(\mathrm{aq})$.
$\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda)$
In equation (i), we encounter another difference in order to take account of the fact that we cannot measure absolute enthalpies of solutions and solvents.
The chemical potential of the solute $j$ (at fixed $\mathrm{T}$ and $\mathrm{p}$, which is close to ambient pressure) is related to the molality $\mathrm{m}_{j}$ using equation (j).
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
From the Gibbs-Helmholtz Equation,
$\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\mathrm{d} \ln \gamma_{\mathrm{j}} / \mathrm{dT}\right)_{\mathrm{p}}$
But activity coefficient $\gamma_{j}$ is defined such that $\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0$ at all $\mathrm{T}$ and $\mathrm{p}$. Moreover for an ideal solution, $\gamma_{j} = 1.0$. Hence,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
In other words, with increasing dilution $\mathrm{H}_{j}(\mathrm{aq})$ approaches a limiting partial molar enthalpy $\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$ which equals the partial molar enthalpy of the solute in an ideal solution. We identify a relative (partial) molar enthalpy of solute $j$, $\mathrm{L}_{\mathrm{j}}(\mathrm{aq})$.
$\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
Hence, at fixed $\mathrm{T}$ and $\mathrm{p}$
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{L}_{\mathrm{j}}(\mathrm{aq})=0$
Therefore for simple solutes in solution in the limit of infinite dilution the relative partial molar enthalpy of solute $j$ is zero [4].
Footnotes
[1] $[\mathrm{J}]=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]$
[2] $\left[\mathrm{J} \mathrm{kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]$
[3] Note the advantage of expressing the composition in terms of molalities rather than in concentrations for which we would have to take account of the dependence of volume on temperature.
[4] An interesting comparison is the molar enthalpy of water($\lambda$) and the limiting molar enthalpy of solute water in a solvent such as methanol. We define a transfer quantity, $\Delta_{\mathrm{tr}} \mathrm{H}^{0}$ [ $= \mathrm{H}^{\infty}$ ($\mathrm{H}_{2}\mathrm{O}$ as solute in a defined solvent) $-\mathrm{H}_{1}^{*}\left(\lambda \mathrm{H}_{2} \mathrm{O}\right)$], characterizing the difference in molar enthalpy of liquid water and the limiting partial molar enthalpy of solute water at ambient pressure and $298.15 \mathrm{~K}$_. $\Delta_{\mathrm{tr}} \mathrm{H}^{0}$ is $0.85$, $4.05$ and $10.11 \mathrm{~kJ mol}^{-1}$ in $\mathrm{CH}_{3}\mathrm{OH}(\lambda)$, $\mathrm{C}_{7}\mathrm{H}_{15}\mathrm{OH}(\lambda)$ and $\mathrm{C}_{2}\mathrm{H}_{4}(\mathrm{O.CO.C}_{3}\mathrm{H}_{7})_{2} (\lambda)$ respectively [5].
[5] S.-O. Nilsson, J. Chem. Thermodyn., 1986, 18, 1115.
1.8.05: Enthalpies- Solutions- Equilibrium and Frozen Partial Molar Enthalpies
A given system at fixed $\mathrm{T}$ and $\mathrm{p}$ is at thermodynamic equilibrium. The enthalpy of the system is perturbed by adding $\delta \mathrm{n}_{j}$ moles of chemical substance $j$. We imagine two possible limiting changes to the system. In one limit the enthalpy of the system changes to a neighbouring state where the extent of chemical reaction remains constant; i.e. at fixed $\xi$. In another limit the enthalpy of the system changes to a neighbouring state where the affinity for spontaneous change $\mathrm{A}$ remains constant. The two differential changes in enthalpy are related.
$\left(\frac{\partial H}{\partial n_{j}}\right)_{A}=\left(\frac{\partial H}{\partial n_{j}}\right)_{\xi}-\left(\frac{\partial \mathrm{A}}{\partial n_{j}}\right)_{\xi} \,\left(\frac{\partial \xi}{\partial A}\right)_{n_{j}} \,\left(\frac{\partial H}{\partial \xi}\right)_{n_{j}}$
We identify the state being perturbed as the equilibrium state where $\mathrm{A} = 0$ and the composition-organisation is represented by $\xi^{\mathrm{eq}$. We identify two quantities describing the impact of adding $\delta \mathrm{n}_{j}$ moles of chemical substance $j$.
Equilibrium partial molar enthalpy,
$\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{A}=0}$
Frozen partial molar enthalpy,
$\mathrm{H}_{\mathrm{j}}\left(\xi^{\mathrm{eq}}\right)=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \xi^{\mathrm{eq}}}$
Because the triple product term on the r.h.s. of equation (a) is not zero at equilibrium (i.e. at $\mathrm{A} = \text { zero}$ and $\xi = \xi^{\mathrm{eq}}$), then $\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)$ is not equal to. By convention, the term ‘ partial molar enthalpy is taken to mean $\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)$.
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A given aqueous solution at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ (close to ambient pressure $\mathrm{p}^{0}$) is prepared using $\mathrm{n}_{1}$ moles of water and $\mathrm{n}_{j}$ moles of a solute, chemical substance-$j$. The enthalpy of this solution $\mathrm{H}(\mathrm{aq})$ is given by equation (a) where $\mathrm{H}_{1}(\mathrm{aq})$ and $\mathrm{H}_{j}(\mathrm{aq})$ are the (equilibrium) partial molar enthalpies of solvent and solute respectively
$\mathrm{H}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq})$
$\mathrm{H}_{1}(\mathrm{aq})=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}$
$\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)}$
Then [1,2],
$\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}$
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)$
Similarly for the solute, chemical substance $j$ (assuming ambient pressure $\mathrm{p}$ is close to the standard pressure) [3,4],
$\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}$
$\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$ is the limiting (infinite dilution) partial molar enthalpy of solute $j$. The enthalpy of a solution prepared using \
(\mathrm{n}_{1}\) moles of water and $\mathrm{n}_{j}$ moles of solute is given by equation (g).
\begin{aligned} \mathrm{H}(\mathrm{aq})=\mathrm{n}_{1} \,\left[\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] \ &\left.+\mathrm{n}_{\mathrm{j}} \, \mid \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned}
For a solution in $1 \mathrm{~kg}$ of water,
\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=&\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned}
We re-arrange equation (h).
\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda) \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned}
Equation (i) is interesting because inside the brackets [….] we have the limiting partial molar enthalpy of the solute and two terms which describe the extent to which the enthalpic properties of the solution differ from those of the corresponding ideal solution. We find it advantageous to describe the property in the brackets […] as the apparent molar enthalpy of the solution, $\phi\left(\mathrm{H}_{\mathrm{j}}\right)$. By definition,
$\phi\left(\mathrm{H}_{\mathrm{j}}\right)=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\mathrm{d} \ln \gamma_{\mathrm{j}} / \mathrm{dT}\right)_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \,(\mathrm{d} \phi / \mathrm{dT})_{\mathrm{p}}$
For a solution prepared using $1 \mathrm{~kg}$ of solvent water.
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)$
But at all $\mathrm{T}$ and $\mathrm{p}$,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0 ; \ln \left(\gamma_{\mathrm{j}}\right)=0 ; \phi=1.0$
Hence,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right)\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}=[\partial \phi / \partial \mathrm{T}]_{\mathrm{p}}=0$
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
We recognize a crucial complication in the treatment of the enthalpies of solutions. Unlike volumetric properties of solutions, we cannot measure the enthalpy of a solution. In other words we need to examine differences. Based on equation (k) we form an equation for the enthalpy of the corresponding solution having thermodynamic properties which are ideal.
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}$
The difference between the two enthalpies is given by equation (p)
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]$
Interesting descriptions of the enthalpies of solutions containing simple solutes are based on the concept of excess thermodynamic properties and pairwise solute-solute interaction parameters. Equation (k) describes the enthalpy of a solution prepared using $1 \mathrm{~kg}$ of water whereas equation (o) describes the enthalpy of the corresponding solution where the thermodynamic properties are ideal. The excess enthalpy $\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)$ is given by equation (q).
$\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{h}_{\mathrm{iji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3} \ldots \ldots$
Footnotes
[1] For the solvent in solutions ( at constant pressure),
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
But
$-\frac{\mathrm{H}_{1}(\mathrm{aq})}{\mathrm{T}^{2}}=\frac{\partial\left[\mu_{1}(\mathrm{aq}) / \mathrm{T}\right]}{\partial T}$
Then $-\frac{\mathrm{H}_{1}(\mathrm{aq})}{\mathrm{T}^{2}}=-\frac{\mathrm{H}_{1}^{*}(\lambda)}{\mathrm{T}^{2}}-\mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}$
[2] $\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2} \,[\mathrm{K}]^{-1} =\left[\mathrm{J} \mathrm{mol}^{-1}\right]$
[3] From $\mu_{j}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$ Then, $-\frac{\mathrm{H}_{\mathrm{j}}(\mathrm{aq})}{\mathrm{T}^{2}}=-\frac{\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})}{\mathrm{T}^{2}}+\mathrm{R} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}$
[4] $\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2} \,[\mathrm{K}]^{-1}=\left[\mathrm{J} \mathrm{mol}{ }^{-1}\right]$
[5] See for example,
1. amides(aq) and peptides(aq); A. H. Sijpkes, A. A. C. Oudhuis, G. Somsen and T. H. Lilley, J. Chem. Thermodyn.,1989,21,343.
2. non-electrolytes in DMSO and H2O; E. M. Arnett and D. R. McKelvey, J. Am. Chem.Soc.,1966,88,2598.
3. alkanes(aq); S. Cabani, G. Conti, V. Mollica and L. Bernazzani, J. Chem. Soc. Faraday Trans.,1991,87,2433.
4. hydrocarbons in polar solvents; C. V. Krishnan and H. L. Friedman, J. Phys. Chem.,1971,75,3598.
5. alkanes in organic solvents; R. Fuchs and W. K. Stephenson, Can. J.Chem.,1985,63,349.
6. organic solutes in alkanes and water. W. Riebesehl, E. Tomlinson and H. J. M. Grumbauer, J.Phys.Chem.,1984,88,4775.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.08%3A_Enthalpy/1.8.06%3A_Enthalpies-_Neural_Solutes.txt
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A given (old) aqueous solution is prepared using $\mathrm{n}_{1}$(old) moles of water($\lambda$) and $\mathrm{n}_{j}$ moles of a simple neutral solute at fixed $\mathrm{T}$ and $\mathrm{p}$. The enthalpy $\mathrm{H}(\mathrm{aq} ; \mathrm{old})$ of this solution is expressed in terms of the molar enthalpy of water($\lambda$), $\mathrm{H}_{1}^{*}(\lambda)$ and the apparent molar enthalpy of the solute $\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)$.
$\mathrm{H}(\mathrm{aq} ; \text { old })=\mathrm{n}_{1}(\text { old }) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)$
We use the description ‘old’ because we envisage preparing a ‘new’ solution by adding $\mathrm{n}_{1}$(added) moles of water, enthalpy $\mathrm{n}_{1}(\text { added }) \, \mathrm{H}_{1}^{*}(\lambda)$.
$\mathrm{H}(\text { added })=\mathrm{n}_{1}(\text { added }) \, \mathrm{H}_{1}^{*}(\lambda)$
The enthalpy of the resultant solution is $\mathrm{H}(\mathrm{aq} ; \text { new })$;
$\mathrm{H}(\text { aq; new })=\left[\mathrm{n}_{1}(\mathrm{old})+\mathrm{n}_{1}(\text { added })\right] \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)$
In effect the ‘old’ solution has been diluted.
$\left.\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=\mathrm{H}(\mathrm{aq} ; \text { new })-\mathrm{H}(\text { aq } ; \text { old })-\left[\mathrm{n}_{1} \text { (added }\right) \, \mathrm{H}_{1}^{*}(\lambda)\right]$
Hence,
$\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}} \text { (new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} \text { (old }\right)\right]$
An isobaric calorimeter measures heat $\mathrm{q}$ characterising the dilution.
\begin{aligned} \mathrm{q} / \mathrm{n}_{\mathrm{j}} &=\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{old}) \rightarrow \mathrm{m}_{\mathrm{j}}(\text { new })\right] \ &=\Delta \mathrm{H}(\text { old } \rightarrow \text { new }) / \mathrm{n}_{\mathrm{j}}=\phi\left(\mathrm{H}_{\mathrm{j}}(\mathrm{j} ; \text { new })\right)-\phi\left(\mathrm{H}_{\mathrm{j}}(\mathrm{j} ; \mathrm{old})\right) \end{aligned}
We imagine a series of experiments in which the molality of solute at the start of the experiment is $\mathrm{m}_{j}$(I). Following dilution the molality is $\mathrm{m}_{j}$(II).
$\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{I}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]=\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{II}\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{I}\right)$
In a calorimetric experiment we record heat $\mathrm{q}$ accompanying a second dilution. Hence,
$\Delta_{\mathrm{dll}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{II}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{III})\right]=\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \text { III }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \text { II }\right)$
In a third dilution we have that
$\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{III}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{IV})\right]=\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{IV}\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \text { III }\right)$
In this experiment the molality of the solution in the sample cell is gradually falling. Combination of the results described by equations (g), (h) and (i) yields the set, $\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{I}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right], \Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}} \text { (II) } \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{III})\right], \Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{III}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{IV})\right] \ldots$ This set is expanded with further dilutions until by extrapolation we obtain for solution $\Delta_{\text {dil }} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{I}) \rightarrow \text { infinite dilution }\right]$. We obtain the enthalpies of dilution for all dilutions in a given set of experiments; i.e. for dilution for solutions II, III, IV…
Alternatively a given solution is diluted by increasing amounts of solvent; e.g. adding ethanol($\lambda$) to a solution of urea in ethanol [1].
In the analysis of enthalpies of solutions simplification of the algebra is achieved by defining a number of L-variables, signalling differences in enthalpies. The relative enthalpy L describes the difference between the enthalpies of real and ideal solutions. For a solution prepared using $\mathrm{w}_{1} \mathrm{~kg}$ of solvent (e.g. water),
$\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg} ; \mathrm{id}\right)$
By definition for the solvent,
$\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda)$
For the solute $j$,
$\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
$\mathrm{L}_{1}(\mathrm{aq})$ and $\mathrm{L}_{j}(\mathrm{aq})$ are the relative partial molar enthalpies of solvent and solute respectively. Similarly in terms of apparent properties,
$\phi\left(\mathrm{L}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}$
$\mathrm{L}=\mathrm{n}_{1} \, \mathrm{L}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{L}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)$
$\phe(\mathrm{L}_{j})$ is the apparent relative molar enthalpy of solute $j$ in solution at molality $\mathrm{m}_{j}$, describing the difference between the apparent molar enthalpies of solute $j$ in real and ideal solutions. In other words we have a direct probe of the role of solute-solute interactions in solution. Both $\mathrm{L}$ and $\phi(\mathrm{L}_{j})$ are (by definition) zero for solutions where the thermodynamic properties are ideal.
This galaxy of variables is clarified if we return to a calorimetric experiment where a solution is diluted. An (old) aqueous solution is prepared using $\mathrm{n}_{1}$ moles of water($\lambda$) and $\mathrm{n}_{j}$ moles of solute producing a solution having enthalpy $\mathrm{H}(\mathrm{aq} ; \mathrm{old})$.
$\mathrm{H}(\mathrm{aq} ; \text { old })=\mathrm{n}_{1}(\text { old }) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)$
To this solution we add (at the same $\mathrm{T}$ and $\mathrm{p}$) $\mathrm{n}_{1}$(added) moles of water($\lambda$).
$\mathrm{H}(\text { added })=\mathrm{n}_{1}(\text { added }) \, \mathrm{H}_{1}^{*}(\lambda)$
in the limit that $\mathrm{n}_{1}$(added) is sufficiently large that the molality $\mathrm{m}_{j}$ of the ‘new’ solution is negligibly small, then
$\operatorname{limit}\left(\text { new } ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}$
$\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)$
By definition,
$\Delta_{\text {dil }} \mathrm{H}=\Delta \mathrm{H}(\text { old } \rightarrow \text { new }) / \mathrm{n}_{j}$
$\Delta_{\mathrm{dil}} \mathrm{H}=-\phi\left(\mathrm{L}_{\mathrm{j}}\right)$
Consistent with our definitions of heat $\mathrm{q}$ and enthalpy change, a positive $\Delta_{\mathrm{dil}}\mathrm{H}$ indicates that dilution is endothermic.
We have not commented on the dependence of either $\phi\left(\mathrm{L}_{\mathrm{j}}\right)$ or $\phi\left(\mathrm{H}_{\mathrm{j}}\right)$ on molality of solute. In order to say something about these variables we need explicit equations for these dependences on composition of solution.
An important approach to the description of the properties of solutions uses excess thermodynamic functions. The quantity $\mathrm{L}(\mathrm{aq})$ defined in equation (n) refers to a solutions prepared using $\mathrm{n}_{1}$ moles of solvent and $\mathrm{n}_{j}$ moles of solute, contrasting the properties of real and ideal solutions. The excess enthalpy $\mathrm{H}^{\mathrm{E}}$ refers to the corresponding solutions prepared using $1 \mathrm{~kg}$ of water and $\mathrm{m}_{j}$ moles of solute $j$.
$\mathrm{H}^{\mathrm{E}}=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{id}\right)$
thus,
$\mathrm{H}^{\mathrm{E}}= \left[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\right]-\left[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]$
Or,
$\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]$
Therefore
$\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)$
Again the development of equation (x) reflects our continuing interest in differences with respect to enthalpies. Nevertheless the key isobaric calorimetric equation requires that the measured ratio ($\mathrm{q} / \mathrm{~n}_{j}$) for the process solvent + solute forming an ideal solution (at fixed $\mathrm{T}$ and $\mathrm{p}$) equals the standard enthalpy of solution for pure substance $j$, $\Delta_{s \ln } \mathrm{H}^{0}$. For neutral solutes, the dependence of partial molar enthalpy of solute $\mathrm{H}_{j}(\mathrm{aq})$ on solute molality mj is small such that the recorded ($\mathrm{q} / \mathrm{~n}_{j}$) for real solutions can often be equated to the corresponding limiting enthalpy of solution, $\Delta_{s \ln } \mathrm{H}^{0}$ because in an ideal solution the standard partial molar enthalpy of a solute equals the partial molar enthalpy of the solute at infinite dilution. For solute $j$,
$\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{0}=\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~s})$
Significantly modern calorimeters are sufficiently sensitive to measure heat $\mathrm{q}$ when a known but small amount of substance $j$ is dissolved in a known amount of solvent. In many cases the dependence of $\Delta_{\sin } \mathrm{H}$ ion solute molality is, for small neutral solutes, negligibly small such that $\Delta_{\sin } \mathrm{H}$ is assumed to equal $\Delta_{\sin } \mathrm{H}^{0}$ [2].
Heats of solution can be analysed in terms of group contributions to the enthalpy of solution for a given series of solutes [3]. Moreover the dependence of $\Delta_{\sin } \mathrm{H}^{\infty}$ for a given solute on temperature yields the corresponding limiting isobaric heat capacity of solution, $\Delta_{s \ln } C_{p}^{\infty}$ [4]. In fact by measuring $\Delta_{\sin } \mathrm{H}^{\infty}$ for solutes in two solvents, the derived property is the standard enthalpy of transfer [5].
\begin{aligned} \Delta_{s \ln } \mathrm{H}_{\mathrm{j}}^{\infty} &\text { solvent } \mathrm{B} \rightarrow \text { solvent } \mathrm{A}) \ &=\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{\infty}(\text { solvent } \mathrm{A})-\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{\infty}(\text { solvent } \mathrm{B}) \end{aligned}
Such a study identified a quite striking extremum for limiting partial molar enthalpies of solution for $\mathrm{NaBH}_{4}$ in water + 2-methylpropan-2-ol mixtures at low alcohol mole fractions and $298.2 \mathrm{~K}$ and hence a quite striking reversal of sign in limiting partial molar isobaric heat capacities for $\mathrm{NaBPh}_{4}$ in this binary aqueous mixture [5]; see also data for dialkyl sulfonates in alcohol + water mixtures [6] and tri-n-alkyl phosphates in water + DMF binary mixtures [7]. Indeed an extensive literature describes the enthalpies of solution for neutral solutes and, where the results concern one solute in two or more solvents, the corresponding enthalpy of transfer; cf. equation (z) [8].
Where the results describe a series of closely related neutral solutes, it is often possible to estimate contributions from individual groups (e.g. $\mathrm{CH}_{2}$ and $\mathrm{OH}$) to a given limiting enthalpy of transfer [9].
In many reports, the results of calorimetric experiments show clear evidence of a dependence of partial molar enthalpy of a given solute on molality of the solution. One of the first reports of such a dependence for neutral solutes was published in 1940 [10]. Hence a direct signal is obtained of enthalpic solute-solute interactions in solution.
An aqueous solution is prepared using water($\mathrm{w}_{1} = 1 \mathrm{~kg}$) and $\mathrm{m}_{j}$ moles of solute $j$ at defined $\mathrm{T}$ and $\mathrm{p}$.
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq})$
In the event that the thermodynamic properties of this solutions are ideal the enthalpy of the solution is given by equation (zb).
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
The excess enthalpy $\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)$ is given by equation (zc).
$\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)-\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
$\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)$ can be expressed as a power series in molality $\mathrm{m}_{j}$.
$\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}$
A given solution contains solute $j$ such that an isobaric calorimeter is used to measure the heat of dilution. We obtain the enthalpy per mole of solute on going from molality $\mathrm{m}_{j}$(initial) to $\mathrm{m}_{j}$(final), $\Delta_{\mathrm{dil}} \mathrm{H}$.
$\Delta_{\mathrm{dil}} \mathrm{H}=\mathrm{H}^{\mathrm{E}}\left(\mathrm{m}_{\mathrm{j}} \text { - final }\right) / \mathrm{m}_{\mathrm{j}}(\text { final })-\mathrm{H}^{\mathrm{E}}\left(\mathrm{m}_{\mathrm{j}}-\text { initial }\right) / \mathrm{m}_{\mathrm{j}} \text { (initial) }$
Equations (zd) and (ze) yield an equation for measured $\Delta_{\mathrm{dil}} \mathrm{H}$ in terms of enthalpic solute-solute pairwise and triplet interaction parameters.
\begin{aligned} \Delta_{\mathrm{dil}} \mathrm{H}=\mathrm{h}_{\mathrm{ij}} \,\left[\mathrm{m}_{\mathrm{j}}(\text { final })\right.&\left.-\mathrm{m}_{\mathrm{j}}(\text { initial })\right] / \mathrm{m}^{0} \ &+\mathrm{h}_{\mathrm{jij}} \,\left[\left\{\mathrm{m}_{\mathrm{j}}(\text { final })\right\}^{2}-\left\{\mathrm{m}_{\mathrm{j}}(\text { initial })\right\}^{2}\right] /\left(\mathrm{m}^{0}\right)^{2} \end{aligned}
In most cases, authors concentrate attention on pairwise interaction parameters [11] between identical (homotactic) and different (heterotactic) solute molecules in a given solution [12]. The concept of solute-solute pairwise (and higher order) interaction parameters allows quite detailed patterns to emerge from enthalpies of dilution of neutral solutes in salt solution [13].
Footnotes
[1] E.g. adding ethanol($\lambda$) to a solution of urea in ethanol($\lambda$) ; D. Hamilton and R. H. Stokes, J. Solution Chem.,1972,1,223.
[2]
1. A. Roux and G. Somsen, J. Chem. Soc. Faraday Trans. 1, 1982, 78, 3397;ureas(aq) and amides(aq).
2. W. Zielenkiewicz, J. Thermal Anal.,1995,45,615; 1988, 33,7.
3. D. Hallen, S.-O. Nilsson, W. Rothschild and I. Wadso, J. Chem. Thermodyn., 1986,18,429; n-alkanols in $\mathrm{H}_{2}\mathrm{O}$ and $\mathrm{D}_{2}\mathrm{O}$.
4. S.-O. Nilsson, J. Chem.Thermodyn.,1986,18,1115; solute water in alcohols($\lambda$) and esters($\lambda$).
[3]
1. G. Della Gatta, G. Barone and V. Elia, J. Solution Chem., 1986, 15, 157; n-alkamides(aq).
2. F. Franks and B. Watson, Trans. Faraday Soc.,1969,65,2339; amines(aq).
3. S. Cabani, G. Conti and L. Lepori, Trans. Faraday Soc., 1971, 67, 1943.; cyclic ethers(aq).
4. S. Cabani, G. Conti and L. Lepori, Trans. Faraday Soc., 1971, 67, 1933; cyclic amines.
5. K. P. Murphy and S. J. Gill, Thermochim. Acta, 1989, 139,279; diketopiperazine(aq).
6. J. M. Corkhill, J. F. Goodman and J. R. Tate, Trans. Faraday Soc., 1969, 65, 1742
[4]
1. S.-O. Nilsson and I. Wadso, J. Chem. Thermodyn.,1986,18,673; esters(aq).
2. S. J. Gill, N. F. Nichols and I. Wadso, J. Chem. Thermodyn., 1975, 7, 175.
[5] E. M. Arnett and D. R. McKelvey, J. Am. Chem. Soc., 1966,88, 5031
[6] C. V Krishnan and H. L. Friedman, J Solution Chem.,1973,2,37.
[7]
1. C. de Visser, H. J. M. Grunbauer and G. Somsen, Z. Phys. Chem. Neue Folge, 1975,97,69.
2. S. Lindenbaum, D. Stevenson and J. H. Rytting, J. Solution Chem., 1975, 4,893.
[8]
1. S. Cabani, G. Conti, V. Moliica and L. Bernazzani, J. Chem. Soc. Faraday Trans.,1991, 87,2433; neutral solutes in aq. soln and in solution in octanol.
2. W. Riebesehl, E. Tomlinson and H. M. Grunbauer, J. Phys. Chem., 1984, 88, 4775; neutral solutes in aq. solution and in solution in 2,2,4-trimethylpentane.
[9]
1. C. V. Krishnan and H. L. Friedman, J. Phys. Chem.,1971, 75, 3598; alcohols in non-aqueous solvents.
2. E. M. Arnett and D. R. McKelvey, J. Am. Chem. Soc., 1966, 88, 2598.
3. C. V. Krishnan and H. L. Friedman, J. Phys. Chem.,1969, 73, 1572; Solutes in water, propylene carbonate and DMSO.
4. G. Castronuovo, R.P.Dario and V.Elia, Thermochim Acta,1991,181,305.
5. A. H. Sijpkes, G. Somsen and S. G. Blankenborg, J. Chem. Soc. Faraday Trans.,1990, 86,3737.
6. R. Fuchs and W.K Stephenson, Can. J. Chem.,1985, 63,349; alkanes in organic solvents.
7. W. Riebeschl and E. Tomlinson, J. Phys. Chem.,1984,88,4770; organic solutes in 2,2,4-trimethylpentane.
8. S. Cabani, G. Conti, V. Mollica and L. Bernazzani, J. Chem. Soc. Faraday Trans.,1991,87,2433.
9. D.Hamilton and R.H.Stokes, J. Solution Chem.,1972,1,223.; dilution of urea in six solvents.
[10] F. T. Gucker and H. B. Pickard, J. Am.Chem.Soc.,1940,62,1464.
[11]
1. J. E. Reading, P. A. Carlisle, G. R. Hedwig and I. D. Watson, J. Solution Chem., 1989,18,131; Me-subst amino acids(aq).
2. M. Bloemendal and G. Somsen, J. Chem. Eng. Data,1987,32,274; amides(DMF); see also J. Solution Chem.,1988,17,1067.
[12]
1. K. Nelander, G. Olofsson, G. M. Blackburn, H. E. Kent and T. H. Lilley, Thermochim. Acta, 1984,78,303.
2. B. Andersson and G. Olofsson, J. Solution Chem.,1988,17,169.
3. F. Franks, M. A. J. Quickenden, D. S. Reid and B. Watson, Trans. Faraday Soc.,1970,66,582.
[13]
1. R. B. Cassel and R. H. Wood, J. Phys. Chem.,1974,78,2460.
2. P. J. Cheek, M. A. Gallardo-Jimenez and T. H. Lilley, J. Chem. Soc Faraday Trans.,1,1988,84,3435; formamide(aq).
3. H. Piekarski and W. Koierski, Thermochim. Acta, 1990, 164, 323;(DMF(aq).
4. G. Barone, P. Cacase, G. Castronuovo and V. Elia, Carbohydrate Research, 1981,91,101;oligosaccharides(aq)
5. M. J. Blandamer, M. D. Butt and P. M. Cullis, Thermochim. Acta, 1992, 211,49; urea(aq).
6. G. Perron and J. E. Desnoyers, J.Chem.Thermodyn.,1981,13,1105.
7. F. Franks, M. Pedley and D. S. Reid, J. Chem. Soc. Faraday Trans., 1, 1976,72,359.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.08%3A_Enthalpy/1.8.07%3A_Enthalpies-_Solutions-_Dilution-_Simple_Solutes.txt
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The excess Gibbs energy $\mathrm{G}^{\mathrm{E}}$ for a dilute aqueous solution containing a simple solute $j$ prepared using $1 \mathrm{~kg}$ of solvent, water is given by equation (a).
$\mathrm{G}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]$
In terms of Gibbs energies pairwise solute-solute interaction parameters,
$\mathrm{G}^{\mathrm{E}}=\mathrm{g}_{\mathrm{jj}} \,\left[\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right]^{2}$
The excess enthalpy[1]
$\mathrm{H}^{\mathrm{E}}=\mathrm{h}_{\mathrm{ij}} \,\left[\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right]^{2}$
where [cf. Gibbs –Helmholtz Equation],
$\mathrm{h}_{\mathrm{ij}}=-\mathrm{T}^{2} \,\left\{\partial\left[\mathrm{g}_{\mathrm{jj}} / \mathrm{T}\right] / \partial \mathrm{T}\right\}_{\mathrm{p}}$
Here $\mathrm{h}_{\mathrm{jj}}$ is the pairwise solute-solute enthalpic interaction parameter. For the solvent,
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}-\mathrm{M}_{1} \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}$
Using the Gibbs-Helmholtz Equation,
$\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)-\mathrm{M}_{1} \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}$
For the solute,
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+2 \, \mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}$
Then using the Gibbs-Helmholtz Equation [2]
$\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}$
Alternatively we may express the enthalpy of the solution in terms of the apparent molar enthalpy of the solute, $\phi\left(\mathrm{H}_{\mathrm{j}}\right)$.
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)$
For the ideal solution,
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}$
where $\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$. Then
$\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]$
Hence using equation (c),
$\phi\left(\mathrm{H}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}+\mathrm{h}_{\mathrm{ji}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}$
We use these equations in the analysis of a calorimetric data where a given solution is diluted. The solution is prepared using $\mathrm{n}_{1}$ moles of solvent (water) and $\mathrm{n}_{j}$ moles of a simple solute $j$. Then
$\mathrm{H}(\mathrm{I} ; \mathrm{aq})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{I} ; \mathrm{aq}\right)$
A new solution is prepared by adding (in the calorimeter) $\Delta \mathrm{n}_{1}$ moles of solvent, Then
$\mathrm{H}(\mathrm{II} ; \mathrm{aq})=\left(\mathrm{n}_{1}+\Delta \mathrm{n}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{II} ; \mathrm{aq}\right)$
Thus the molality of solute $j$ changes from $\mathrm{m}_{j}$(I) $\left[=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right]$ to $\mathrm{m}_{j}$(II) $\left[=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\Delta \mathrm{n}_{1}\right) \, \mathrm{M}_{1}\right]$. Therefore,
$\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{I} ; \mathrm{aq}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}+\left[\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right]$
And
$\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{II} ; \mathrm{aq}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}+\left[\mathrm{h}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\Delta \mathrm{n}_{1}\right) \, \mathrm{M}_{1}\right]$
In fact we record the heat $\mathrm{q}$ (at constant pressure) when $\Delta \mathrm{n}_{1}$ moles of solvent are added to solution I to form solution II. Thus,
$\mathrm{q}=\mathrm{H}(\mathrm{II} ; \mathrm{aq})-\mathrm{H}(\mathrm{I} ; \mathrm{aq})-\Delta \mathrm{n} \, \mathrm{H}_{1}^{*}(\lambda)$
Footnotes
[1] From equation (a) and (b) $\mathrm{G}^{\mathrm{E}}=\left[\mathrm{J} \mathrm{kg}^{-1}\right]$ From equation (c) $\mathrm{H}^{\mathrm{E}}=\left[\mathrm{J} \mathrm{kg}^{-1}\right]$
[2] A check on the equations with reference to solution prepared using $1 \mathrm{~kg}$ of solvent.
\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{H}_{1}^{*}(\lambda)-\mathrm{M}_{1} \, \mathrm{h}_{\mathrm{ji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\right] \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\right] \end{aligned}
Or,
\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*} &(\lambda)-\mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \ &+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \end{aligned}
Then
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{i}} \, \mathrm{H}_{\mathrm{i}}^{\infty}(\mathrm{aq})+\mathrm{H}^{\mathrm{E}}$
1.8.09: Enthalpies- Salt Solutions- Apparent Molar- Partial Molar and Relative Ent
Description of the enthalpies of salt solutions is similar to that given for neutral solutes except that account is taken of the fact that one mole of a given salt can with complete dissociation produce $v$ moles of ions. The chemical potential of the solvent in an aqueous salt solution (at constant temperature and ambient pressure) is given by equation (a).
$\mu_{1}(\mathrm{aq})=\mu_{1}^{\star}(\lambda)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
Here $\phi$ is the practical osmotic coefficient where $\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0$ at all $\mathrm{T}$ and $\mathrm{p}$. Using the Gibbs-Helmholtz Equation,
$\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)+\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}$
Also
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)$
By definition,
$\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda)$
The chemical potential of a salt $j$ in aqueous solution is given by equation (e).
$\mu_{j}(a q)=\mu_{j}^{0}(a q)+v \, R \, T \, \ln \left(Q \, m_{j} \, \gamma_{\pm} / m^{0}\right)$
where, at all $\mathrm{T}$ and $\mathrm{p}$,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1.0$
Using the Gibbs-Helmholtz Equation,
$\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}$
For a salt solution having ideal thermodynamic properties,
$\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
By definition, the relative partial molar enthalpy of the salt,
$\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
In the limit of infinite dilution the relative partial molar enthalpy of a salt is zero. Thus
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{L}_{\mathrm{j}}(\mathrm{aq})=0$
For a solution prepared using $\mathrm{w}_{1} \mathrm{~kg}$ of water($\lambda$),
\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right) &=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{1} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}} \ +\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \end{aligned}
But
$\mathrm{n}_{1} \, \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{1} \, \mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1}$
\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}} \ &+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \end{aligned}
\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=& \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \,\left\{\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right.\ &\left.+\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\} \end{aligned}
The term in the brackets {….} defines the apparent molar enthalpy of salt $j$, $\phi\left(\mathrm{H}_{\mathrm{j}}\right)$.
$\phi\left(\mathrm{H}_{\mathrm{j}}\right)=\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}+\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}$
Using equation (o),
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)$
In other words we have grouped all the parameters describing the properties of the salt in a real solution under a single term, $\phi(\mathrm{H}_{j})$. For a solution prepared using $1 \mathrm{~kg}$ of water,
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)$
At all $\mathrm{T}$ and $\mathrm{p}$,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1 ; \ln \left(\gamma_{\pm}\right)=0 ; \phi=1$
Hence,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right)\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}=[\partial \phi / \partial \mathrm{T}]_{\mathrm{p}}=0$
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}}\right)=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
1.8.10: Enthalpy of Solutions- Salts
The chemical substance $\mathrm{NaCl}(\mathrm{s})$ is a hard crystalline solid with a high melting point, $1074 \mathrm{K}$. All the more remarkable therefore is the observation when a few crystals are dropped into water($\ell$) the crystals disintegrate into ions with no dramatic change in the temperature of the water. One concludes that the intensity of interactions between ions in the crystal is comparable to that between ions and water molecules in the aqueous solution. Not surprisingly therefore enthalpies of solutions have been extensively investigated.
The enthalpy of an aqueous solution prepared at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ using $\mathrm{n}_{1}$ moles of water and $\mathrm{n}_{j}$ moles of salt is given by equation (a) where $\phi\left(\mathrm{H}_{j}\right)$ is the apparent molar enthalpy of salt $j$ in solution.
$\mathrm{H}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)$
Before the solution was prepared the enthalpy of the system, $\mathrm{H}(\mathrm{no}-\mathrm{mix})$ is given by equation (b) where $\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~s})$ is the molar enthalpy of solid salt $j$.
$\mathrm{H}(\mathrm{no}-\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~s})$
Using an isobaric calorimeter, heat $\mathrm{q}$ is recorded for the solution process.
$\mathrm{q}=\mathrm{H}(\mathrm{aq})-\mathrm{H}(\mathrm{no}-\mathrm{mix})=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~s})\right]$
Or
$\phi\left(H_{j}\right)-H_{j}^{*}(s)=q / n_{j}$
In many studies [1] using sensitive calorimeters ($\mathrm{q}/\mathrm{n}_{j}$) can be recorded for the production of quite dilute solutions such that $\phi\left(\mathrm{H}_{j}\right)$ is effectively equal to $\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$. In other cases $\Delta_{s \ln } \mathrm{H}(\mathrm{s} \rightarrow \mathrm{aq})$ is found to depend on the molality of the resultant solution. One procedure [2] fits the measured enthalpy of solution to a quadratic in the molality of salt.
$\Delta_{s \ln } \mathrm{H}(\mathrm{s} \rightarrow \mathrm{aq})=\Delta_{\mathrm{sin}} \mathrm{H}^{0}(\mathrm{~s} \rightarrow \mathrm{aq})+\mathrm{A} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{B} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}$
In other cases the Debye-Huckel limiting law is used as a basis for extrapolating $\Delta_{\mathrm{s} \ln } \mathrm{H}(\mathrm{s} \rightarrow \mathrm{aq})$ to the required infinite dilution value [3].
Footnotes
[1]
1. R. K. Mohanty, T. S. Sarma, S. Subramanian and J.C. Ahluwalia, Trans. Faraday Soc.,1971,67,305.
2. N Van Meurs, Th. W Warmerdam and G. Somsen, Fluid Phase Equilib., 1989,49,263.
3. M. E. Estep, D. D. Macdonald and J. B. Hyne, J. Solution Chem.,1977,6,129.
[2] C. V. Krishnan and H. L. Friedman, J. Phys.Chem.,1970,74,3900 .
[3] e.g. Om. N. Bhatnagar and C. M. Criss, J. Phys Chem.,1969,73,174.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.08%3A_Enthalpy/1.8.08%3A_Enthalpies-_Solutions-_Simple_Solutes-_Interaction_Parameters.txt
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One mole of salt in solution can, with complete dissociation, produce $ν$ moles of ions. Hence for a given solution prepared using $\mathrm{n}_{1}$ moles of water($\ell$) and $\mathrm{n}_{j}$ moles of salt, the enthalpy $\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)$ is given by equation (a).
\begin{aligned} &\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)= \ &\begin{aligned} \mathrm{n}_{1} \, & {\left[\mathrm{H}_{1}^{*}(\ell)+\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] } \ &+\mathrm{n}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\partial \ln \gamma_{\pm} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned} \end{aligned}
With a little re-arrangement,
\begin{aligned} &\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)= \ &\quad \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell) \ &\quad+\mathrm{n}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\pm} / \partial \mathrm{T}\right)_{\mathrm{p}}+\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] \end{aligned}
The terms within the brackets [….] define the apparent molar enthalpy of salt $j$ in aqueous solution, $\phi\left(\mathrm{H}_{j}\right)$.
$\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)$
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
By definition
$\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg} ; \mathrm{id}\right)$
$\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\ell)$
$\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})$
$\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{L}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)$
Thus,
$\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg} ; \mathrm{id}\right)=0$
Equation (e) forms the basis of comments on changes in enthalpy when a salt solution is diluted by adding $\Delta \mathrm{n}_{1}$ moles of water($\ell$). Hence
\begin{aligned} \Delta_{\text {dil }} \mathrm{H}=\left[\left(\mathrm{n}_{1}\right.\right.&\left.\left.+\Delta \mathrm{n}_{1}\right) \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)\right] \ &-\left[\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}-\text { initial }\right)\right]-\Delta \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell) \end{aligned}
Or,
$\Delta_{\text {dil }} \mathrm{H}=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)-\phi\left(\mathrm{H}_{\mathrm{j}}-\text { initial }\right)\right]$
If in a given experiment where ‘$\mathrm{n}_{j} = 1 \mathrm{~mol}$’ and $\Delta \mathrm{n}_{1}$ is large such that $\phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)$ equals $\phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)^{\infty}$, equation (k) is re-written as shown in equation (l). Then,
$\Delta_{\text {dil }} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)=-\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]$
Or,
$\Delta_{\mathrm{dil}} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)=-\phi\left(\mathrm{L}_{\mathrm{j}}\right)$
If for such a dilution, heat passes from the surroundings into the system , $\Delta_{\text {dil }} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)$ is positive and $\phi\left(L_{j}\right)$ is negative. Thus direct calorimetric measurement of $\Delta_{\mathrm{dil}} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)$ yields the relative apparent molar enthalpy of the salt in solution at molality $\mathrm{m}_{j}$.
However we need to comment in more detail on the analysis of heats of dilution for salt solutions. We envisage a situation where a calorimeter records the heat associated with dilution of a given salt solution from an initial molality $\mathrm{m}_{\mathrm{i}}$ to a final molality $\mathrm{m}_{\mathrm{f}}$. A data set often includes pairs of $\mathrm{m}_{\mathrm{i}}-\mathrm{m}_{\mathrm{f}}$ values together with the accompanying enthalpy change, $\Delta \mathrm{H}(\text { old } \rightarrow \text { new })$ which yields the difference in apparent molar enthalpies of the two salt solutions, cf. equation (k). Thus
$\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\right]$
Or,
$\left[\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\right]=\Delta \mathrm{H}(\text { old } \rightarrow \text { new }) / \mathrm{n}_{\mathrm{j}}$
We note that the molalities of the ‘new’ and ‘old’ solutions differ and therefore the contributions of ion-ion interactions to the apparent molar enthalpies differ. In the event that sufficient solvent is added that $\mathrm{m}_{\mathrm{f}}$ is effectively zero, then $\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)$ is the infinitely dilute property $\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}$.
The excess enthalpy $\mathrm{H}^{\mathrm{E}}$ is given by equation (p).
$\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]$
For salt solutions $\mathrm{H}^{\mathrm{E}}$ is not negligible as a consequence of intense ion-ion interaction. However in order to calculate $\mathrm{H}^{\mathrm{E}}$ and hence obtain an indication of the strength of these interactions we return to equation (m) and note that experiment yields the difference between $\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)$ and $\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)$. Since there are no ion-ion interactions at infinite dilution, the difference $\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\left\{\text { i.e. } \phi\left(\mathrm{L}_{\mathrm{j}}\right)\right\}$ is obtained as a function of $\mathrm{m}_{j}$(old).
A key component of the difference $\left[\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\right]$ is charge-charge interaction in the real solutions which is calculated using, for example, the Debye-Huckel equations. These equations start out with a relation between $\ln \left(\gamma_{\pm}\right)$ where $\gamma_{\pm}$ is the mean ionic activity coefficient and I the ionic strength (or, in a simple solution, molality $\mathrm{m}_{j}$). These equations are differentiated with respect to temperature (at fixed pressure) requiring therefore the corresponding dependences of molar volume $\mathrm{V}_{1}^{*}(\ell)$ and relative permittivity $\varepsilon_{\mathrm{r}}^{*}(\ell)$ of the solvent. Not surprisingly a large chemical literature describes a range of procedures for analysing the calorimetric results. In most cases the starting point is the Debye-Huckel Limiting Law.
For $\mathrm{Bu}_{4}\mathrm{N}^{+}\mathrm{Br}^{-}(\mathrm{aq})$, the dependence of $\phi\left(\mathrm{L}_{\mathrm{j}}\right)$ on $\mathrm{m}_{j}$ was expressed [1] using equation (q). $\mathrm{S}_{\mathrm{H}}$ was taken from the compilation published by Helgeson and Kirkham [2].
$\phi\left(L_{j}\right)=S_{H} \,\left(m_{j} / m^{0}\right)^{1 / 2}+\sum B_{i} \,\left(m_{j} / m^{0}\right)^{(i+1) / 2}$
For $\left(\mathrm{HOC}_{2}\mathrm{H}_{4}\right)_{4}\mathrm{N}^{+}\mathrm{Br}^{-}(\mathrm{aq})$, an extended Debye –Huckel equation was used having the following form [3].
$\begin{gathered} \phi\left(\mathrm{L}_{\mathrm{j}}\right)=\mathrm{S}_{\mathrm{H}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} \,\left[\frac{1}{1+\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}}-\frac{\sigma \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}}{3}\right] \ +\mathrm{B} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{C} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3 / 2} \end{gathered}$
The dependence of $\phi\left(\mathrm{L}_{\mathrm{j}}\right)$ on $\mathrm{m}_{j}$ for 1,1’-dimethyl-4,4’-dipyridinium dichloride(aq; $298 \mathrm{~K}$) was expressed [3] using a simple polynomial in $\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}$.
The Pitzer equations describing the properties of salt solutions also provide a basis for examining the enthalpies of dilution of, for example [4], $\mathrm{NaCl}(\mathrm{aq})$. An interesting group of papers [5] compares relative apparent molar enthalpies of salts in $\mathrm{D}_{2}\mathrm{O}$ and $\mathrm{H}_{2}\mathrm{O}$; i.e. $\phi\left(\mathrm{L}_{\mathrm{j}} ; \mathrm{D}_{2} \mathrm{O}\right)-\phi\left(\mathrm{L}_{\mathrm{j}} ; \mathrm{H}_{2} \mathrm{O}\right)$. The compositions of the salt solutions are expressed in aquamolalities; i.e. $\mathrm{m}_{j}$ moles of salt in $55.1$ moles of solvent. The difference is expressed as a quadratic in aqueous molality using Kerwin’s equation.
$\phi\left(\mathrm{L}_{\mathrm{j}} ; \mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{D}_{2} \mathrm{O}\right)=\mathrm{k}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{k}_{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}$
Further examples are listed in reference [6].
Footnotes
[1] J. E. Mayrath and R. H. Wood, J. Chem. Thermodyn., 1983,15,625; and references therein.
[2] H. C. Helgeson and D. H Kirkham, Am. J. Sci.,1974,274,1199.
[3] G. Perron and J. E. Desnoyers, J. Solution Chem.,1972,1,537.
[4] R. H. Busey, H. F. Holmes and R. E. Mesmer, J.Chem.Thermodyn., 1984,16, 343.
[5]
1. A. S. Levine and R. H. Wood, J.Phys.Chem.,1973,77,2390.
2. Y.-C. Wu and H.L.Friedman, J.Phys.Chem.,1966,70,166.
3. J. E. Desnoyers, R. Francescon, P. Picker and C. Jolicoeur, Can. J. Chem., 1971, 49,3460
[6]
1. S. Lindenbaum, J.Chem.Thermodyn., 1971, 3,625; J. Phys. Chem., 1971,75,3733; $\mathrm{Na}^{+}$ and $\mathrm{Bu}_{4}\mathrm{N}^{+}$ salts of carboxylic acids(aq).
2. R. H. Wood and F. Belkin, J. Chem. Eng. Data, 1973, 18,184; $\left(\mathrm{HOC}_{2}\mathrm{H}_{4})_{4}\mathrm{N}^{+}\mathrm{Br}(\mathrm{aq})$.
3. D. D. Ensor, H. L. Anderson and T. G. Conally, J. Phys. Chem.,1974,78,77.
4. D. D. Ensor and H. L. Anderson, J. Chem. Eng. Data, 1973, 18,205; $\mathrm{NaCl}(\mathrm{aq})$.
5. G. E. Boyd, J. W. Chase and F. Vaslow, J. Phys. Chem., 1967, 71, 573; $\mathrm{R}_{4} \mathrm{~N}^{+} \mathrm{X}(\mathrm{aq})$.
6. S. Lindenbaum, J. Phys.Chem.,1969,73,4734; $\left[\mathrm{Bu}_{3} \mathrm{~N}-\left(\mathrm{CH}_{2}\right)_{8}-\mathrm{NBu}_{3}\right] \mathrm{X}_{2}$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.08%3A_Enthalpy/1.8.11%3A_Enthalpies-_Salt_Solutions-_Dilution.txt
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It is generally assumed that the Born Equation yields a difference in Gibbs energies rather than Helmholtz energies and so one can use the Gibbs-Helmholtz Equation for the dependence on temperature at fixed pressure to yield the Born-Bjerrum Equation, assuming that ($\mathrm{dr}_{\mathrm{j}} / \mathrm{dT}$) is zero.
\begin{aligned} &\Delta(\mathrm{pfg} \rightarrow \mathrm{s} \ln ) \mathrm{H}_{\mathrm{j}}\left(\mathrm{c}_{\mathrm{j}}=1 \mathrm{moldm} \mathrm{dm}^{-3} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}\right)= \ &\quad-\left[\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}\right] \,\left[1-\left(1 / \varepsilon_{\mathrm{r}}\right)-\left(\mathrm{T} / \varepsilon_{\mathrm{r}}\right) \,\left(\partial \ln \varepsilon_{\mathrm{r}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned}
In fact an early calorimetric study showed that in terms of predicting the enthalpies of solution for salts, the Born equation is inadequate, often predicting the wrong sign. [1,2]
Differentiation of equation (a) with respect to temperature yields an equation for the partial molar isobaric heat capacity of ion $j$ in a solution having ideal thermodynamic properties.
\begin{aligned} &C_{p j}\left(\operatorname{sln} ; c_{j}=1 \mathrm{~mol} \mathrm{dm}{ }^{-3} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}\right) \ &=-\left[\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}\right] \,\left[\partial\left\{\left(1 / \varepsilon_{\mathrm{r}}\right)+\left(\mathrm{T} / \varepsilon_{\mathrm{r}}\right) \, \partial \ln \varepsilon_{\mathrm{r}} / \partial \mathrm{T}\right\} / \partial \mathrm{T}\right] \end{aligned}
Footnotes
[1] F. A. Askew, E. Bullock, H. T. Smith, R. K. Tinkler, O. Gatty and J. H. Wolfenden, J. Chem. Soc., 1934, 1368.
[2] For estimation of single in enthalpies see M. Booij and G. Somsen, Electrochim Acta,1983,28,1883.
1.8.13: Enthalpies- Liquid Mixtures
For an ideal binary liquid mixture the Gibbs energy at temperature T is given by equation (a).
$\mathrm{G}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{n}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right]$
From the Gibbs-Helmholtz equation,
$\mathrm{H}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{H}_{2}^{*}(\ell)$
Hence for an ideal binary liquid mixture,
$\mathrm{H}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{H}_{1}^{*}(\ell) \text { and } \mathrm{H}_{2}(\operatorname{mix} ; \mathrm{id})=\mathrm{H}_{2}^{*}(\ell)$
The molar enthalpy of a real binary liquid mixture is given by equation (d).
$\mathrm{H}_{\mathrm{m}}=\mathrm{x}_{1} \, \mathrm{H}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{H}_{2}(\operatorname{mix})$
Therefore the molar enthalpy of mixing for a real binary liquid mixture is given by equation (e).
$\Delta_{\text {mix }} H_{m}=x_{1} \,\left[H_{1}(\operatorname{mix})-H_{1}^{*}(\ell)\right]+x_{2} \,\left[H_{2}(\operatorname{mix})-H_{2}^{*}(\ell]\right.$
Significantly equations (b) and (e) show that the molar enthalpy of mixing of an ideal binary liquid mixture, $\Delta_{\text {mix }} H_{m}(\mathrm{id})$ is zero. The latter condition offers an important point of reference for isobaric calorimetry. [1] If we discover that the mixing of two liquids (at constant pressure) is not zero, the measured molar heat of mixing [$=\Delta_{\text {mix }} \mathrm{H}_{\mathrm{m}}$] is an immediate indicator of the extent to which the properties of a given mixture are not ideal.
Nevertheless it is important to set down a link between the measured enthalpies of mixing with the activity coefficients of two liquid components. To this end we start with the equation for the chemical potentials of liquid component 1 in a liquid mixture at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ (which is close to ambient); equation (f).
$\mu_{1}(\mathrm{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)$
where
$\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1 \text { at all } T \text { and } p \text {. }$
The Gibbs - Helmholtz equation yields an equation for the partial molar enthalpy of component 1 in the liquid mixture. Thus
$\mathrm{H}_{1}(\operatorname{mix})=\mathrm{H}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}$
Similarly,
$\mathrm{H}_{2}(\operatorname{mix})=\mathrm{H}_{2}^{*}(\ell)-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}$
Hence,
\begin{aligned} &\mathrm{H}_{\mathrm{m}}(\operatorname{mix})= \ &\quad \mathrm{H}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})-\mathrm{R} \, \mathrm{T}^{2} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\} \end{aligned}
We also obtain equations for the excess molar enthalpies of the two components (at defined $\mathrm{T}$ and $\mathrm{p}$).
$\mathrm{H}_{1}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}$
and
$\mathrm{H}_{2}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}$
The excess molar enthalpy,
$\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}(\text { mix })=-\mathrm{R} \, \mathrm{T}^{2} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\}$
At fixed pressure, the differential dependence of $\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}(\operatorname{mix})$ on temperature yields the corresponding excess isobaric heat capacity of mixing.
Footnotes
[1] J. B. Ott and C. J. Wormald, Experimental Thermodynamics, IUPAC Chemical Data Series, No. 39, ed. K. N. Marsh and P. A. G. O’Hara, Blackwell, Oxford, 1994, chapter 8.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.08%3A_Enthalpy/1.8.12%3A_Enthalpies-_Born-Bjerrum_Equation-_Salt_Solutions.txt
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A closed system (in addition to the thermodynamic energy $\mathrm{U}$) is characterised by two functions of state.
1. Temperature, $\mathrm{T}$: an intensive variable.
2. Entropy, $\mathrm{S}$: an extensive variable.
The concept of entropy is particularly valuable in commenting on the direction of spontaneous chemical reaction [1].
The Second Law of Thermodynamics states that for spontaneous chemical reaction in a closed system [2].
$\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi$
where
$\mathrm{A} \, \mathrm{d} \xi \geq 0$
These two equations comprise the Second Law [3]. The product of the affinity for spontaneous chemical reaction and the extent of chemical reaction (i.e. accompanying change in composition) can never be negative. This is the thermodynamic `selection rule' for which there are absolutely no exceptions. Chemists base their analysis of chemical processes on the certainty of this rule (or axiom). The key point is the sense of direction of spontaneous change which emerges [4].
In the event that the affinity for spontaneous change is zero, no change in chemical composition occurs in a closed system; i.e. $\mathrm{d}\xi$ is zero and the rate of change $\mathrm{d}\xi / \mathrm{dt}$ is zero. The system and surroundings are in equilibrium. Hence
$\mathrm{T} \, \mathrm{dS}=\mathrm{q} \quad(\text { at } \mathrm{A}=0)$
The latter equation has a particular set of applications. We imagine a closed system for which the affinity for spontaneous change is zero. We perturb the system by a change in pressure such that there is a corresponding change in composition-organisation in the system. However as we change the pressure along a certain pathway, we assert that the affinity for spontaneous change is always zero. Then between states I and II, at constant temperature $\mathrm{T}$
$\mathrm{T} \, \mathrm{S}(\mathrm{II})-\mathrm{T} \, \mathrm{S}(\mathrm{I})=\mathrm{T} \, \int_{\text {statel }}^{\text {state II }} \mathrm{dS}=\mathrm{q}$
The pathway between these two states is called reversible or an equilibrium transformation. In fact the change in pressure must be carried out infinitely slowly because we must allow the chemical composition/molecular organisation to hold to the condition that there is no affinity for spontaneous change.
All processes in the real world (i.e. all natural processes) are irreversible; there is a defined direction for spontaneous changes [5].
Footnotes
[1] Many authors offer an explanation of the property, entropy. One view is that to attempt an explanation of the "meaning" of entropy is a complete waste of time (M. L. McGlashan, J. Chem. Educ., 1966,43, 226). A wide-ranging discussion is given by P.L. Huyskens and G.G. Siegel,Bull. Soc.Chem.Belg., 1988,97, 809, 815 and 823.
E.A. Guggenheim [Thermodynamics, North-Holland, Amsterdam, 1950]. This monograph is often cited for the following bold statement (page 11): "There exists a function $\mathrm{S}$ of the state of a system called the entropy of the system .....".
H. Margenau [The Nature of Physical Reality, McGraw-Hill, New York, 1950] states 'Entropy is as definite and clear a thing as other thermodynamic quantities'.
The common view in introductory chemistry textbooks for many years has been that entropy is a measurement of randomness and/or disorder. However this view is unhelpful if not meaningless [E. T. Jaynes, Am. J.Phys.,1965,33,391; F. L. Lambert, J. Chem. Educ.,1999,76,1385; 2002,79,187.] indeed a myth [W Brostow, Science 1972,178,211.] and an educational disaster [M. Sozlibir, J.K.Bennett, J. Chem. Educ.,2007,84,1204.]
The generally accepted view [F. L. Lambert, J.Chem.Educ.,2002,79,1241] is that an entropy increase results from the energy of molecular motion becoming more dispersed or ’spread out’; e.g. in the two classic examples of a system being warmed by hotter surroundings or, isothermally, when a system’s molecules have greater volume for their energetic movement the energy of molecular motion becoming more dispersed or ‘spread out’; e.g. in the two classic examples of as system being warmed by hotter surroundings or, isothermally, when a system’s molecules have greater volume for their energetic movement. The concept is exceptionally valuable because entropy increase can be seen by chemists as simply involving the energy associated with mobile molecules spreading out more in three-dimensional space, whether a new total system of ‘less hot plus once-cooler’ or isothermally in a larger volume. This simple view is equivalent to the dispersal of energy in phase space. In quantum mechanical terms, ’energy dispersal’ means that a system will come to equilibrium in a final state that is optimal because it affords a maximal number of accessible energy arrangements. Even though the system can be in only one arrangement at one instant, its energy is truly dispersed because at the next instant it can be in a different arrangement: this amounts to a ‘temporal dance’ over a very small fraction of the hyper-astronomical number of microstates predicted by the Boltzmann relation. The account given here is based on a written comments in correspondence from F. L. Lambert.
[2] From equation (a), $\mathrm{T}. \left.\mathrm{dS}=[\mathrm{K}]-\left[\mathrm{JK}^{-1}\right]=[]\right]+\left[\mathrm{J} \mathrm{} \mathrm{mol}^{-1}\right]=[\mathrm{J}]$
[3] Equations (a) and (b) can be re-expressed in terms of the contribution to the change in entropy $\mathrm{dS}$ by a process (e.g. chemical reaction) within the system $\mathrm{d}_{\mathrm{i}}\mathrm{S}$. Then
$\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{T} \, \mathrm{d}_{\mathrm{s}} \mathrm{S}$
where
$\mathrm{d}_{\imath} \mathrm{S}>0$
Equation (B) is the Second Law in that $\mathrm{d}_{\mathrm{i}}\mathrm{S}$ cannot be negative. For a reversible process $\mathrm{d}_{\mathrm{i}}\mathrm{S}=0$. But for all processes in the real world, $\mathrm{d}_{\mathrm{i}}\mathrm{S}$ is positive. In other words all spontaneous processes occur in the direction whereby there is a positive contribution from $\mathrm{d}_{\mathrm{i}}\mathrm{S}$ to the change in entropy $\mathrm{dS}$.
[4] This concept of spontaneous change, coupled with the idea that changes occur in a predefined direction is linked with the idea that time is "one-sided". (a) I. Prigogine, From Being to Becoming, Freeman, San Francisco, 1980, page 6. (b) see also G. Nicolis and I. Prigogine, Self-Organization in Non-Equilibrium Systems, Wiley, New York, 1977.
[5] Equation (a) forms the basis of an oft-quoted comment. For an isolated system, $\mathrm{q}$ is zero. Then $\mathrm{T} \, \mathrm{dS}=\mathrm{A} \, \mathrm{d} \xi$ where $\mathrm{A} \, \mathrm{d} \xi \geq 0$
So for all spontaneous processes in an isolated system, $\mathrm{dS} >0$. This is the basis of the statement that the entropy of the universe is increasing if the universe can be treated as an isolated system. But these comments stray from immediate interests of chemists.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.09%3A_Entropy/1.9.01%3A_Entropy_-_Second_Law_of_Thermodynamics.txt
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The volume of a given closed system at equilibrium prepared using $\mathrm{n}_{1}$ moles of solvent (water) and $\mathrm{n}_{j}$ moles of solute-$j$ is defined by the set of independent variables shown in equation (a).
$\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \xi^{\mathrm{eq}}\right]$
The same set of independent variables defines the entropy $\mathrm{S}$.
$\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \xi^{\mathrm{eq}}\right]$
We envisage that the system is displaced by a change in pressure along a path where the system remains at equilibrium (i.e. $\mathrm{A} = 0$) and the volume remains the same as defined by equation (a). In a plot of entropy against $\mathrm{p}$, the gradient of the plot at the point defined by the independent variables, $\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \boldsymbol{\xi}^{\mathrm{eq}}\right]$ is given by equation (c).
Isochoric
$\left(\frac{\partial S}{\partial p}\right)_{V, A=0}$
The set of derivatives is completed by the following partial derivatives.
Isothermal
$\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}$
Isobaric
$\left(\frac{\partial S}{\partial T}\right)_{p, A=0}$
1.9.03: Entropy and Spontaneous Reaction
It is often stated that the entropy of a system is a maximum at equilibrium. This is not generally true and is certainly not the case for closed systems at either (a) fixed $\mathrm{T}$ and $\mathrm{p}$, or (b) fixed $\mathrm{T}$ and $\mathrm{V}$.
We rewrite the Master Equation in the following way:
$\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dU}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero }$
Temperature $\mathrm{T}$ is positive and non-zero. At constant energy and constant volume (i.e. isoenergetic and isochoric), spontaneous processes are accompanied by an increase in entropy. This statement is important in statistical thermodynamics where the condition, ‘constant $\mathrm{U}$ and constant $\mathrm{V}$’ is important.
The following equation defines the enthalpy $\mathrm{H}$ of a closed system.
$\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}$
Then
$\mathrm{dU}=\mathrm{dH}-\mathrm{p} \, \mathrm{dV}-\mathrm{V} \, \mathrm{dp}$
From equation (a),
$\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero }$
Hence,
$\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero }$
Temperature $\mathrm{T}$ is always positive. Hence at constant enthalpy and pressure (i.e. iso-enthalpic and isobaric) all spontaneous processes produce an increase in entropy.
We have identified two sets of conditions under which an increase in entropy accompanies a spontaneous process. If we follow through a similar argument with respect to the Gibbs energy, the outcome is not straightforward. By definition,
$\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}$
Then
$\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT}$
Or,
$\mathrm{S}=-\mathrm{dG} / \mathrm{dT}+\mathrm{dH} / \mathrm{dT}-\mathrm{T} \, \mathrm{dS} / \mathrm{dT}$
But from equation (e)
$\mathrm{dH} / \mathrm{dT}=\mathrm{T} \, \mathrm{dS} / \mathrm{dT}+(\mathrm{V} / \mathrm{T}) \, \mathrm{dp} / \mathrm{dT}-(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi / \mathrm{dT}$
Hence,
$\mathrm{S}=-(\mathrm{dG} / \mathrm{dT})+\mathrm{V} \,(\mathrm{dp} / \mathrm{dT})-\mathrm{A} \,(\mathrm{d} \xi / \mathrm{dT}) \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero }$
Clearly no definite conclusions can be drawn about changes in entropy S under isobaric - isothermal conditions. We stress these points because again it is often tempting to link, misguidedly, entropies to the degree of ‘muddled-up-ness’. This is the basis of many explanations of entropy. For example, neither the volume nor energy of a deck of cards change on shuffling. Whether what actually happens on shuffling a new well-ordered deck of cards clarifies the meaning of entropy seems doubtful.
1.9.04: Entropy- Dependence on Temperature
Using a calculus operation, the isochoric dependence of entropy of temperature is related to the corresponding isobaric dependence. Thus
$\left(\frac{\partial S}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial T}\right)_{p}-\left(\frac{\partial S}{\partial p}\right)_{T} \,\left(\frac{\partial p}{\partial V}\right)_{T} \,\left(\frac{\partial V}{\partial T}\right)_{p}$
But
$\left(\frac{\partial S}{\partial p}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{p}$
Hence,
$\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}} \,\left[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right]^{2}$
Or,
$\left(\frac{\partial S}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial T}\right)_{p}-\frac{\left(E_{p}\right)^{2}}{K_{T}}$
The final term in equation (c) contains the variable $\mathrm{p}-\mathrm{V}-\mathrm{T}$.
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A key equation relates the chemical potential and partial molar entropy of solute-$j$. For a given solute $j$ in an aqueous solution,
$\mathrm{S}_{\mathrm{j}}(\mathrm{aq})=-\left[\partial \mu_{\mathrm{j}}(\mathrm{aq}) / \partial T\right]_{\mathrm{p}}$
In order to appreciate the importance of equation (a) we initially confine our attention to the properties of a solution whose thermodynamic properties are ideal. A given aqueous solution contains solute $j$ at temperature $\mathrm{T}$ and ambient pressure (which is close to the standard pressure).
$\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Using equation (a),
$\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
$\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ is the partial molar entropy of solute $j$ in an ideal aqueous solution having unit molality [1,2]. Therefore for both real and ideal solutions [3],
$\operatorname{limit}\left(m_{j} \rightarrow 0\right) S_{j}(a q ; T ; p)=+\infty$
In other words the limiting partial molar entropy for solute $j$ is infinite. Interestingly if the aqueous solution contains two solutes $j$ and $\mathrm{k}$, then the following condition holds for solutions at temperature $\mathrm{T}$ and pressure $\mathrm{p}$.
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0 ; \mathrm{m}_{\mathrm{k}} \rightarrow 0\right)\left[\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{S}_{\mathrm{k}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\right]=\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{S}_{\mathrm{k}}^{0}(\mathrm{aq})$
Similarly
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0 ; \mathrm{m}_{\mathrm{k}} \rightarrow 0\right)\left[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{\mathrm{k}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\right]=\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{k}}^{0}(\mathrm{aq})$
The partial molar entropy of solute $j$ in a real solution is given by equation (g).
\begin{aligned} &S_{j}(a q ; T ; p)= \ &S_{j}^{0}(a q ; T ; p)-R \, \ln \left(m_{j} / m^{0}\right)-R \, \ln \left(\gamma_{j}\right)-R \, T \,\left[\frac{\partial \ln \left(\gamma_{j}\right)}{\partial T}\right]_{p} \end{aligned}
A given aqueous solution having thermodynamic properties which are ideal contains a solute $j$, molality $\mathrm{m}_{j}$. The partial molar entropy of the solvent is given by equation (h).
$\mathrm{S}_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
Hence,
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{S}_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})$
When an ideal solution is diluted the partial molar entropy of the solvent approaches that of the pure solvent.
A given aqueous solution is prepared using $1 \mathrm{~kg}$ of solvent and $\mathrm{m}_{j}$ moles of solute $j$ at temperature $\mathrm{T}$ and pressure $\mathrm{p}$, the latter being close to the standard pressure. The entropy of the solution is given by equation (j).
$\mathrm{S}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{S}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}(\mathrm{aq})$
In the event that the thermodynamic properties of the solution are ideal the entropy of the solution is given by equation (k).
\begin{aligned} &\mathrm{S}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)= \ &\quad \mathrm{M}_{1}^{-1} \,\left[\mathrm{S}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}
Interestingly,
\begin{aligned} &\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{S}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)= \ &\mathrm{M}_{1}^{-1} \, \mathrm{S}_{1}^{*}(\ell)+(0) \,\left[\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{R} \, \ln \left(0 / \mathrm{m}^{0}\right)\right] \end{aligned}
But
$\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{m}_{\mathrm{j}} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)=0$
In other words the entropy for an ideal solution in the limit of infinite dilution is given by the entropy of the pure solvent. For a real solution,
$\mathrm{S}_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\phi \, \mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}$
Hence,
\begin{aligned} &\mathrm{S}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right) \ &=\mathrm{M}_{1}^{-1} \,\left[\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\phi \, \mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right] \ &\quad+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right] \end{aligned}
The difference $\left[\mathrm{S}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)-\mathrm{S}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)\right]$ yields the excess entropy, $\mathrm{S}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)$.
Footnotes
[1] For a salt solution, the standard partial molar entropy of the salt is given by the sum of standard partial molar entropies of the ions. For a 1:1 salt, $S_{j}^{0}(a q)=S_{+}^{0}(a q)+S_{-}^{0}(a q)$
[2] Y. Marcus and A. Loewenschuss, Annu. Rep. Prog. Chem., Ser. C, Phys. Chem., 1984, 81, chapter 4.
[3] For comments on the entropy of dilution of salt solutions see (a classic paper), H. S. Frank and A. L. Robinson, J. Chem. Phys.,1940,8,933.
[4] For comments on partial molar entropies of apolar solutes in aqueous solutions see, H. S. Frank and F. Franks, J. Chem. Phys.,1968,48,4746.
1.9.06: Entropies- Liquid Mixtures
The chemical potential of liquid component 1 in a binary liquid mixture (at temperature $\mathrm{T}$ and pressure $\mathrm{p}$, close to the standard pressure $\mathrm{p}^{o}$) is related to the mole fraction $\mathrm{x}_{1}$ using equation (a).
$\mu_{1}(\operatorname{mix} ; \mathrm{id})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)$
But
$\mathrm{S}_{1}(\operatorname{mix})=-\left[\partial \mu_{1}(\operatorname{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}}$
Then,
$\mathrm{S}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)\) Hence the molar entropy of mixing of an ideal binary liquid mixture (at defined $\mathrm{T}$ and $\mathrm{p}$) is given by equation (d). \[\Delta_{\text {mix }} S_{m}(\text { id })=-R \,\left[x_{1} \, \ln \left(x_{1}\right)+x_{2} \, \ln \left(x_{2}\right)\right]$
The chemical potential of component 1 in a real binary liquid mixture (at temperature $\mathrm{T}$ and pressure $\mathrm{p}$, near the standard pressure) is given by equation (e).
$\mu_{1}(\operatorname{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)$
Then
$\mathrm{S}_{1}(\mathrm{mix})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)-\mathrm{R} \, \ln \left(\mathrm{f}_{1}\right)-\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}$
$S_{1}(\operatorname{mix})=S_{1}(\operatorname{mix} ; \text { id })-R \, \ln \left(f_{1}\right)-R \, T \,\left[\partial \ln \left(f_{1}\right) / \partial T\right]_{p}$
Similarly,
$\mathrm{S}_{2}(\mathrm{mix})=\mathrm{S}_{2}(\mathrm{mix} ; \mathrm{id})-\mathrm{R} \, \ln \left(\mathrm{f}_{2}\right)-\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}$
The extent to which the partial molar entropies for each liquid component in a given liquid mixture differs from that in the corresponding ideal mixture depends on the rational activity coefficient and its dependence on temperature. Hence we define excess partial molar entropies for both liquid components.
$S_{1}^{E}=-R \, \ln \left(f_{1}\right)-R \, T \,\left[\partial \ln \left(f_{1}\right) / \partial T\right]_{p}$
and
$S_{2}^{E}=-R \, \ln \left(f_{2}\right)-R \, T \,\left[\partial \ln \left(f_{2}\right) / \partial T\right]_{p}$
For the binary mixture,
\begin{aligned} S_{m}^{E}=-R\left\{x_{1} \, \ln \left(f_{1}\right)\right.&+x_{1} \,\left[\partial \ln \left(f_{1}\right) / \partial T\right]_{p} \ &\left.+x_{2} \, \ln \left(f_{2}\right)+x_{2} \,\left[\partial \ln \left(f_{2}\right) / \partial T\right]_{p}\right\} \end{aligned}
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.09%3A_Entropy/1.9.05%3A_Entropies-_Solutions-_Limiting_Partial_Molar_Entropies.txt
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The Gibbs energy is an extensive state variable defined by the following equation.
$\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}$
Instead of Gibbs energy the terms Gibbs free energy and Gibbs function are often used. Physicists prefer the term Gibbs function [1]. The term ‘free energy’ is not encouraged. Everyday experience tells us that no energy is ‘free’.
Footnote
[1] Nevertheless the French term ‘enthalpie libre’ ( i.e. free enthalpy) for $\mathrm{G}$ has merit. Enthalpy is defined by $\mathrm{H} = \mathrm{ U} + \mathrm{p V}$. Then $\mathrm{G} = \mathrm{ H} – \mathrm{ TS}$. The product $\mathrm{TS}$ is the linked energy in a system from which no work can be produced. Hence the available or ‘free’ part of the enthalpy is the Gibbs energy.
1.10.02: Gibbs Energy- Thermodynamic Potential
The Gibbs energy of a system,
$\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}$
For a closed single phase system, changes in thermodynamic energy $\mathrm{dU}$ and Gibbs energy $\mathrm{dG}$ are related by the following equation.
$\mathrm{dG}=\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT}$
The change in thermodynamic energy $\mathrm{dU}$ is related to the affinity for spontaneous change using the Master Equation.
$\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0$
We use equation (b) by substituting for $\mathrm{dU}$ in equation (a). Hence,
$\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0$
Chemists carry out most of their experiments under the twin conditions, constant pressure (usually ambient) and constant temperature (often near room temperature). Hence we can see why the latter equation is so important. At fixed $\mathrm{T}$ and $\mathrm{p}$,
$\mathrm{dG}=-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0$
Hence under common laboratory conditions the direction of spontaneous change (e.g. chemical reaction) is in the direction for which $\mathrm{G}$ decreases. The spontaneous ‘flow’ of a chemical reaction (at fixed $\mathrm{T}$ and $\mathrm{p}$) is down the plot of $\mathrm{G}$ against extent of reaction, $\xi$; high to low $\mathrm{G}$. This statement opens the door to the quantitative study of chemical reactions. Thus from equation (e),
$\mathrm{A}=-\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}$
The Gibbs energy decreases until the affinity for spontaneous change is zero; i.e. equilibrium. Then,
$\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{t}, \mathrm{p}}^{\mathrm{eq}}=0$
At equilibrium the Gibbs energy is a minimum [1]. In general terms, a thermodynamic potential is an extensive property of a closed system which reaches an extremum at equilibrium under specified conditions. For processes in closed systems at fixed $\mathrm{T}$ and $\mathrm{p}$, the thermodynamic potential is $\mathrm{G}$. Thus $\mathrm{T}$ and $\mathrm{p}$ are the natural variables for $\mathrm{G}$.
Experience shows that for a given system there is one unique composition which corresponds to the minimum in Gibbs energy (at fixed $\mathrm{T}$ and $\mathrm{p}$). In fact chemistry would be a very difficult subject (and it is difficult as it is) if there were many minima such that it was just a matter of chance which minimum a system ended up in following spontaneous chemical reaction.
The conclusions advanced above refer to the Gibbs energy of a closed system; i.e. a macroscopic property. We cannot at this stage draw conclusions about the properties of the chemical substances making up the system. At the molecular level a whole range of processes may be taking place; chemical reaction, diffusion, molecular collisions. We cannot comment on these using equation (g). It may be that one or more of these processes contributes towards an increase in Gibbs energy. However these processes operate in such a way that the fluctuations in Gibbs energies in small domains are opposed, holding the overall system at a minimum in $\mathrm{G}$.
The Gibbs energy is a contrived property. It is not the ‘energy’ of the system. Nevertheless we can begin to ‘understand’ this property by returning to equation (d). Consider a system at equilibrium and at constant temperature ; i.e. $\mathrm{A}$ = 0\) and $\mathrm{dT} =0$. Then
$V=\left(\frac{\partial G}{\partial p}\right)_{T, A=0}$
The familiar property, volume, is the differential dependence of Gibbs energy on pressure at constant temperature and at equilibrium. If we can assume that the coffee mug on this desk is at equilibrium, although I do not know (and can never know) its Gibbs energy, I know that the volume offers a direct measure of the dependence of its Gibbs energy on pressure. Indeed the link between a property which can be readily measured (e.g. volume or density) offers chemists a pathway into the Gibbs energy and a detailed thermodynamic analysis.
Footnote
[1] G. Willis and D. Ball, J.Chem.Educ.,1984,61,173.
1.10.03: Gibbs Energies- Solutions- Solvent and Solute
A given solution (at temperature $\mathrm{T}$ and pressure $\mathrm{p}$, where the latter is close to the standard pressure) is prepared using $1 \mathrm{~kg}$ of water and $\mathrm{m}_{\mathrm{j}}$ moles of a simple solute. We consider the differential dependence of the excess Gibbs energy for the solution $\mathrm{G}^{\mathrm{E}}$ on molality $\mathrm{m}_{\mathrm{j}}$.
$\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]$
Hence, at fixed $\mathrm{T}$ and $\mathrm{p}$,
\begin{aligned} (1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right]=\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]-\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}\right] \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}\right] \end{aligned}
But according to the Gibbs-Duhem equation,
$-\phi-\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}\right]+1+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}\right]=0$
Hence, we obtain an equation for $\ln \left(\gamma_{j}\right)$ as a function of the differential dependence of $\mathrm{G}^{\mathrm{E}}$ on $\mathrm{m}_{\mathrm{j}}$.[1]
$\ln \left(\gamma_{\mathrm{j}}\right)=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right]$
If we substitute for $\ln \left(\gamma_{j}\right)$ in the equation for $\mathrm{G}^{\mathrm{E}}$, an equation for $\phi$ in terms of $\mathrm{G}^{\mathrm{E}}$ is obtained.
$1-\phi=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}-\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right]$
A more elegant derivation of equation (e) starts out with the equation (a) for the excess Gibbs energy written in the following form.
$\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right] / \mathrm{R} \, \mathrm{T}=1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)$
Then at fixed $\mathrm{T}$ and $\mathrm{p}$,
$(1 / \mathrm{R} \, \mathrm{T}) \,\left\{\mathrm{d}\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right] / \mathrm{dm}_{\mathrm{j}}\right\}=-\left(\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}\right)+\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}$
But according to the Gibbs-Duhem equation,
$-\left(\mathrm{d} \phi / d m_{\mathrm{j}}\right)+\left(\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{m}_{\mathrm{j}}\right)=(\phi-1) / \mathrm{m}_{\mathrm{j}}$
Then,
$1-\phi=-(1 / \mathrm{R} \, \mathrm{T}) \,\left\{\mathrm{d}\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right] / \mathrm{dm}_{\mathrm{j}}\right\} \, \mathrm{m}_{\mathrm{j}}$
Or,
$1-\phi=-(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right] \, \mathrm{m}_{\mathrm{j}}$
The latter equation does not however require that $(1-\phi)$ is a linear function of $\mathrm{m}_{\mathrm{j}}$. The actual form of this dependence has to be obtained by experiment.
Footnotes
[1] $\ln \left(\gamma_{\mathrm{j}}\right)=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]^{-1} \,\left[\mathrm{K}^{-1} \,\left[\mathrm{J} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{-1}=[1]\right.$
[2] $(1-\phi)=\left[\frac{1}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}\right] \,\left[\frac{\left[\mathrm{J} \mathrm{kg}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right.}+\frac{\left[\mathrm{J} \mathrm{kg}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]}\right]=[1]$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.01%3A_Gibbs_Energy.txt
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The Gibbs energy of a closed system at temperature $\mathrm{T}$ is related to the enthalpy $\mathrm{H}$ using equation (a) [1].
$\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}$
The differential change in Gibbs energy at constant temperature is related to the change in enthalpy $\mathrm{dH}$ using equation (b).
$\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}$
For a process taking place in a closed system involving a change from state I to state II, the change in Gibbs energy is given by equation (c).
$\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \, \Delta \mathrm{S}$
The latter equation signals how changes in enthalpy and entropy determine the change in Gibbs energy. A given closed system at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ is prepared using $\mathrm{n}_{1}$ moles of solvent (water) and $\mathrm{n}_{\mathrm{j}}$ moles of solute $j$. The system is at equilibrium such that the composition/organisation is represented by $\xi^{\mathrm{eq}$ and the affinity for spontaneous change is zero. We summarise this state of affairs as follows.
$\mathrm{G}^{e q}=\mathrm{G}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]$
In a plot of $\mathrm{G}$ against $\xi$ (at fixed $\mathrm{T}$ and $\mathrm{p}$) the Gibbs energy is a minimum at $\xi^{\mathrm{eq}}$ [2]. The enthalpy $\mathrm{H}^{\mathrm{eq}}$ of the equilibrium state can be represented in a similar fashion.
$\mathrm{H}^{e q}=\mathrm{H}^{e q}\left[\mathrm{~T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]$
However it is unlikely that $\mathrm{H}^{\mathrm{eq}}$ at $\xi^{\mathrm{eq}}$ corresponds to a minimum in enthalpy $\mathrm{H}$ when $\mathrm{H}$ is plotted as a function of $\xi$. A similar comment applies to the entropy $\mathrm{S}^{\mathrm{eq}}$;
$\mathrm{S}^{\mathrm{eq}}=\mathrm{S}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]$
However taken together $\mathrm{H}^{\mathrm{eq}}$ and $\mathrm{S}^{\mathrm{eq}}$ produce the minimum in $\mathrm{G}$ at $\mathrm{G}^{\mathrm{eq}}$.
$\mathrm{G}^{\mathrm{eq}}=\mathrm{H}^{\mathrm{eq}}-\mathrm{T} \, \mathrm{S}^{\mathrm{eq}}$
In summary; at thermodynamic equilibrium
1. $\mathrm{A}$ is zero,
2. $\mathrm{G}$ is a minimum and
3. the rate of change of composition/organisation $\mathrm{d}\xi/\mathrm{dt}$ is zero.
The latter condition emerges from the conclusion that this rate is zero if there is no affinity for change. For a given system at defined $\mathrm{T}$ and $\mathrm{p}$, the state for which $\mathrm{G}$ is a minimum is unique [3]. Indeed if this was not the case, chemistry would be a very difficult subject. In a given spontaneous chemical reaction proceeds until the composition/organisation reaches $\xi^{\mathrm{eq}}$. In other words the Gibbs energy is the important thermodynamic potential, certainly forming the basis of treatments of chemical reactions in closed systems at fixed $\mathrm{T}$ and $\mathrm{p}$ [4]. However a word of caution is in order. The Gibbs energy of a system differs from the thermodynamic energy $\mathrm{U}$. In fact the Gibbs energy is a somewhat contrived property but aimed at a description of closed systems at fixed $\mathrm{T}$ and $\mathrm{p}$. Nevertheless the Gibbs energy can be given practical significance. We consider a system at equilibrium (i.e. $\mathrm{A} = 0$) at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ where in this state (state I) the Gibbs energy is $\mathrm{G}[\mathrm{I}]$. The system is displaced by a change in pressure to a neighbouring equilibrium state (at constant $\mathrm{T}$). The equilibrium isothermal dependence of Gibbs energy $\mathrm{G}[\mathrm{I}]$ on pressure equals the volume of the system, $\mathrm{V}[\mathrm{I}]$ [5].
$V=\left[\frac{\partial G}{\partial p}\right]_{T, A=0}$
In other words we may not know the Gibbs energy of a system (in fact never know) at least we know that the pressure dependence is the volume which we can readily measure. The isobaric dependence of $\mathrm{G}[\mathrm{I}]$ on temperature for an equilibrium displacement yields the entropy.
$\mathrm{S}=-\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}=0}$
Four key points can be made.
1. The equilibrium state for a system at constant $\mathrm{T}$ and $\mathrm{p}$ corresponds to a minimum in Gibbs energy.
2. The minimum in Gibbs energy of a given system is unique.
3. In the non-equilibrium, there is no direct relationship between the gradient $(\partial \mathrm{G} / \partial \xi)$ and the rate of spontaneous change, $(\partial \xi / \partial t)$.
4. The equilibrium state is stable; $(\partial \mathrm{A} / \partial \xi)<0$ at $\xi^{\mathrm{eq}}$ [6].
Footnotes
[1]
\begin{aligned} &\mathrm{G}=[\mathrm{J}] ; \mathrm{T} \, \mathrm{S}=[\mathrm{K}] \,\left[\mathrm{J} \mathrm{K}^{-1}\right]=[\mathrm{J}] ; \mathrm{p} \, \mathrm{V}=\left[\mathrm{N} \mathrm{m}^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{N} \mathrm{m}]=[\mathrm{J}] \ &\mathrm{A} \, \xi=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \,[\mathrm{mol}]=[\mathrm{J}] \end{aligned}
[2] See for example
1. G. Willis and D. Ball, J. Chem. Educ.,1984, 61, 173, and
2. P. L. Corio, J. Phys. Chem.,1983, 87, 2416.
[3] This point is discussed in the monograph, F. Van Zeggeren and S. H. Story, The Computation of Chemical Equilibria, Cambridge University Press, 1970.
[4] The dependence of rate of reaction on composition is described using the law of mass action and rate constants. The law of mass action is in these terms, extrathermodynamic , meaning that the law does not follow from the first and second laws.
[5] For completeness we consider the case where in equilibrium state [I] at $\xi^{\mathrm{eq}}[\mathrm{I}]$, the system is displaced by a change to a neighbouring state having the same composition/organisation, $\xi[\mathrm{I}]$; i.e. the system is ‘frozen’. The isothermal dependence of $\mathrm{G}[\mathrm{I}]$ on pressure at constant composition equals the volume. Thus, $\mathrm{V}[\mathrm{I}]=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi[\mathrm{I}]}$ Similarly, $\mathrm{S}[\mathrm{I}]=-\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi[\mathrm{I}]}$
The identities of $\mathrm{S}$ and $\mathrm{V}$ at constant ‘$\mathrm{A}=0$’ and at $\xi^{\mathrm{eq}}[\mathrm{I}]$ arise from the fact that $\mathrm{V}$ and $\mathrm{S}$ are strong state variables.
[6] Consider a simple cup in which we have placed (delicately) a steel ball near the top edge.
1. We release the ball, the ball rolls down and the ball gathers speed.
2. At the bottom of the cup the speed of the ball is a maximum (i.e. maximum in kinetic energy) where the potential energy is a minimum.
3. The ball continues through the bottom of the cup and rises to the opposite edge.
4. The ball slows down and then changes direction, falling back to the bottom of the cup.
5. If the surface of the cup is perfectly smooth ( i.e. no energy lost to friction), the ball oscillates about the bottom of the well of the cup. This mechanical model is not correct for chemical reactions in closed systems. In fact the rate of the chemical reaction decreases as a system approaches the minimum in Gibbs energy.
1.10.05: Gibbs Energies- Raoult's Law
We consider a closed system containing a (homogeneous) mixture of two volatile liquids. The closed system is connected to a pressure measuring device which records that at temperature $\mathrm{T}$ the pressure inside the closed system is $\mathrm{p}(\text{tot})$. The composition of the liquid mixture is assayed; the mole fractions of the two components of the liquid are $\mathrm{x}_{1}$ and $\mathrm{x}_{2}$ (where $\mathrm{x}_{2} = 1 – \mathrm{x}_{1}$). Thus the system contains two components so that in terms of the Phase Rule, $\mathrm{C} = 2$. There are two phases, vapour and liquid, so $\mathrm{P}$ equals $2$. Thus in terms of the Rule, $\mathrm{P} + \mathrm{~F} = \mathrm{~C} + 2$, we have fixed the composition and the temperature using up the two degrees of freedom. Hence the pressure $\mathrm{p}(\text{tot})$ is fixed.
The foundation of thermodynamics is experiment. So, in considering the properties of water in dilute aqueous solutions, we take account of the observation that the equilibrium vapour pressure $\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq})$ of water in equilibrium with water in an aqueous solution (at fixed temperature) is approximately a linear function of the mole fraction of water in the solution; equation (a).
$\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq}) \cong \mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1}$
Thus as mole fraction $\mathrm{x}_{1}$ approaches unity (the composition of the solution approaches pure water where $\mathrm{x}_{1}$ is unity), the equilibrium vapour pressure $\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq})$ approaches the vapour pressure of pure liquid water $\mathrm{p}_{1}^{*}(\ell)$ at the same temperature. At this stage we introduce the concept of an ideal solution. We assert that for an ideal solution the approximation (a) is an equation. Thus
$\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})=\mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1}$
In other words, $\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})$ is a linear function of mole fraction composition of the solution [1]. We have linked the (equilibrium) vapour pressure of the solvent to the composition of the solution. Returning to the results of experiments, we invariably find that as real solution becomes more dilute (i.e. as $\mathrm{x}_{1}$ approaches unity) $\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq})$ for real solutions approaches $\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})$ for the corresponding ideal solution at the same temperature. Therefore, we rewrite equation (b) as an equation for real solutions by introducing a new quantity called the (rational) activity coefficient, $\mathrm{f}_{1}$. Then,
$\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq}) \cong \mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1} \, \mathrm{f}_{1}$
where, by definition,
$\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 0\right) \mathrm{f}_{1}=1.0$
Although equations (c) and (d) have simple forms, rational activity coefficients carry a heavy load in terms of information. Thus for a given solution $\mathrm{f}_{1}$ describes the extent to which interactions involving solvent water in the real solution differ from those in the corresponding ideal solution. The challenge of expressing this information in molecular terms is formidable.
Footnotes
[1] Note that, $\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})-\mathrm{p}_{1}^{*}(\ell)<0$. Thus adding a solute lowers the vapour pressure of the solvent. However the total vapour pressure of a binary liquid mixture can be either increased or decreased by adding a small amount of solute, the change being characteristic of the solute; G. Bertrand and C. Treiner, J. Solution Chem.,1984,13, 43.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.04%3A_Gibbs_Energies-_Equilibrium_and_Spontaneous_Change.txt
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A given solution is prepared using $1 \mathrm{~kg}$ of water($\ell$) and $\mathrm{m}_{\mathrm{j}}$ moles of solute $j$. The chemical potential of the solvent water is related to $\mathrm{m}_{\mathrm{j}}$ using equation (a) where pressure $\mathrm{p}$ is close to the standard pressure, $\mathrm{p}^{0}$
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}$
Here $\mu_{1}^{*}(\ell)$ is the chemical potential of solvent water at the same $\mathrm{T}$ and $\mathrm{p}$; $\phi$ is the practical osmotic coefficient which is unity for a solution having thermodynamic properties which are ideal; $\mathrm{M}_{1}$ is the molar mass of water. The chemical potential of the solute $\mathrm{j}$, $\mu_{j}(\mathrm{aq})$ is related to the molality $\mathrm{m}_{\mathrm{j}}$ using equation (b).
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
The chemical potentials of solute and solvent are linked by the Gibbs-Duhem equation which for aqueous solutions (at fixed $\mathrm{T}$ and $\mathrm{p}$) containing $1 \mathrm{~kg}$ of water takes the following form.
$\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0$
We draw equations (a) and (b) together in an equation for the Gibbs energy of a solution prepared using $1 \mathrm{~kg}$ of solvent water. Then [1],
$\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})$
Or,
\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right] \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}
If the thermodynamic properties of the solution are ideal, both $\phi$ and $\gamma_{\mathrm{j}}$ are unity.
\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right] \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}
The difference between $\mathrm{G}(\mathrm{aq})$ and $\mathrm{G}(\mathrm{aq} ; \mathrm{id})$ is the excess Gibbs energy $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})$ [2];
$\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]$
A related quantity is the excess molar Gibbs energy $\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{aq}) \quad\left\{=\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right\}\] . Then [3] $\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]$ The dependence of \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})$ on $\mathrm{m}_{\mathrm{j}}$ emerges from equation (g).
$(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=\mathrm{d}(1-\phi)+\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)$
But, from the Gibbs-Duhem equation,
$\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=-\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right]$
Then [4]
$(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=-\left[(1-\phi) / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}}$
Equation (k) relates $(1-\phi)$ to the dependence of $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})$ on molality $\mathrm{m}_{\mathrm{j}}$. The relationship between $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})$ and $\gamma_{\mathrm{j}}$ is given by equation (l) [5].
$\ln \left(\gamma_{\mathrm{j}}\right)=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{dG}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{dm}_{\mathrm{j}}\right]$
At this point we make a key extrathermodynamic assumption. We assert that (at fixed temperature and pressure) the excess Gibbs energy $\mathrm{G}^{\mathrm{E}}$ is related to molality $\mathrm{m}_{\mathrm{j}}$ of neutral solute $\mathrm{j}$ using equation (m). Thus
$\mathrm{G}^{\mathrm{E}}=\mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{g}_{\mathrm{ij}}+\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3} \ldots \ldots$
Here $g_{j j}, g_{j j j} \ldots$ are the coefficients in a virial type of equation. Thus $g_{j j}$ measures the contribution of pairwise solute-solute interactions to $\mathrm{G}^{\mathrm{E}} (\mathrm{aq})$; $g_{j j j}$ is a triplet interaction term. For quite dilute solutions the dependence of $\mathrm{G}^{\mathrm{E}} (\mathrm{aq})$ on $\mathrm{m}_{\mathrm{j}}$ is effectively described by the pairwise term, $g_{j j}$.
$\mathrm{G}^{\mathrm{E}}=\mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}$
Here $g_{j j}$ is expressed in $\left[\mathrm{J kg}^{-1}\right]$, being the (Gibbs) energy of interaction in a solution containing $1 \mathrm{~kg}$ of water. Pairwise solute-solute Gibbs energy interaction parameters are characteristic of solute $j$, temperature and pressure.
At this stage we have not defined either the sign or magnitude of $g_{j j}$. Clearly if pairwise solute-solute interactions are attractive/cohesive, both $g_{j j}$ and $\mathrm{G}^{\mathrm{E}}$ are negative. In the next stage of the analysis we use equation (l) to obtain an equation for $\ln \left(\gamma_{j}\right)$ in terms of $g_{j j}$ and molality $\mathrm{m}_{\mathrm{j}}$. Thus [6]
$\ln \left(\gamma_{\mathrm{j}}\right)=[2 / \mathrm{R} \, \mathrm{T}] \, \mathrm{g}_{\mathrm{jj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}$
Hence equation (o) requires that (at fixed $\mathrm{T}$ and $\mathrm{p}$) if $g_{j j}$ is negative $\ln \left(\gamma_{\mathrm{j}}\right)$ decreases with increase in $\mathrm{m}_{\mathrm{j}}$ whereby $\mu_{j}(\mathrm{aq})<\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})$. We anticipate that the sign and magnitude of $g_{j j}$ reflect the hydration characteristics of the two solute molecules because these characteristics determine the impact of cosphere overlap on the properties of the solution.
We turn to the properties of the solvent. Thus [7]
$(1-\phi)=-(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}$
Or,
$\phi=1+(1 / R \, T) \, g_{j} \,\left(m^{0}\right)^{-2} \, m_{j}$
From the equation for $\left[\mu_{1}(\mathrm{aq})-\mu_{1}(\mathrm{aq} ; \mathrm{id})\right]$, the difference in chemical potentials of solvent water in real and ideal solutions, it follows that negative $g_{j j}$ requires that $\mu_{1}(\mathrm{aq})>\mu_{1}(\mathrm{aq} ; \mathrm{id})$, the solvent in the real solution being at a higher chemical potential than in the corresponding ideal solution. In other words the non-ideal properties of the solvent are also related to the pairwise interaction parameter $g_{j j}$ and $\mathrm{m}_{\mathrm{j}}$.
As a check on the procedures described above we draw the equations together to recover the original equation for $\mathrm{G}^{\mathrm{E}} (\mathrm{aq})$.
\begin{aligned} &\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right] \ &=\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \,\left[-(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{jj}} \,\left(1 / \mathrm{m}^{0}\right)^{2} \, \mathrm{m}_{\mathrm{j}}+(2 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}} \,\left(1 / \mathrm{m}^{0}\right)^{2}\right] \ &=\mathrm{g}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \end{aligned}
Thus for dilute solutions both $\phi$ and $\operatorname{ln}\left(\gamma_{\mathrm{j}}\right)$ are linear functions of $\mathrm{m}_{\mathrm{j}}$. Equation (n) forms the basis for understanding the properties of dilute aqueous solutions where the solutes are non-ionic [8]. The underlying theme is the idea that solute -solute interactions in these solutions can be understood in terms of cosphere- cosphere interactions [9,10]. Description of the properties of real solutions based on equation (a) is closely related to descriptions of dilute solutions developed for metallurgical systems [11,12]. Similarly procedures are discussed using site-site pair correlation functions for molecular interaction energies [13] and using quasi-chemical models [14].
Footnotes
[1] $\mathrm{G}\left(\mathrm{aq} ; \mathrm{W}_{1}=1 \mathrm{~kg}\right]=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{kg}^{-1}\right]$
[2] $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right] \,[\mathrm{I}]=\left[\mathrm{J} \mathrm{kg}^{-1}\right]$
[3] $\mathrm{G}_{\mathrm{m}}{ }^{\mathrm{E}}(\mathrm{aq})=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,[\mathrm{I}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]$
[4]
$\begin{gathered} (1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=\mathrm{d}(1-\phi)-\left(\mathrm{l} / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right] \ (1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=\mathrm{d}(1-\phi)-\mathrm{d}(1-\phi)-\left[(1-\phi) / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}} \ (1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{j}\right]=-\left[(1-\phi) / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}} \end{gathered}$
[5] From equation (g), $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)\right]$
$(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{dm}_{\mathrm{j}}=(1-\phi)-\mathrm{m}_{\mathrm{j}} \,\left(\mathrm{d \phi} / \mathrm{dm}_{\mathrm{j}}\right)+\ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}$
But from the Gibbs-Duhem equation, $\mathrm{d}\left[\mathrm{m}_{\mathrm{j}} .(1-\phi)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)=0$
Or,
$-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+(1-\phi) \, \mathrm{dm}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{j}\right)=0$
Or,
$-\mathrm{m}_{\mathrm{j} \,} \,\left(\mathrm{d} \phi / \mathrm{dm} \mathrm{m}_{\mathrm{j}}\right)+(1-\phi)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}=0$
[6]
\begin{aligned} &\ell \mathrm{n}\left(\gamma_{j}\right)=(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{g}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\right] / \mathrm{dm}_{\mathrm{j}} \ &\ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)=(2 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}} \end{aligned}
Or, $[1]=[1] \,\left[\mathrm { J } \mathrm { mol } ^ { - 1 } \mathrm { K } ^ { - 1 } \, \left[\mathrm{K}^{-1} \,\left[\mathrm{J} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{-2} \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\right.\right.$
[7] From, $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]$ But $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{g}_{i \mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}$
Then, $\mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} / \mathrm{R} \, \mathrm{T}=\mathrm{m}_{\mathrm{j} \,}(1-\phi)+(2 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}$
Then $(1-\phi)=-(1 / R \, T) \, g_{i j} \,\left(m^{0}\right)^{-2} \, m_{j}$
[8] M. J. Blandamer, J. Burgess, J. B. F. N. Engberts and W. Blokzijl, Ann. Rep. Progr. Chem., Sect. C, Phys. Chem., C, 1990,87,45.
[9] R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.
[10] See for example, W. G. McMillan Jr. and J. E. Mayer, J. Chem. Phys., 1945, 13, 276.
[11] L. S. Darken, Trans.Metallurg. Soc.AIME, 1967, 239, 80.
[12]
1. R. Schuhmann, Metallurg. Trans.B, 1985, 16, 807.
2. A. D. Pelton and C. W. Bale, Metallurg. Trans.,A,1986,17,1211.
3. S. Srikanth, K. J. Jacob and K. P. Abraham,Steel Research, 89,6.
[13] R. P. Currier and J. P. O'Connell, Fluid Phase Equilib., 1987,33, 245.
[141 J. Abusleme and J. H. Vera, Can. J. Chem. Eng., 1985, 63,845.
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The practical osmotic coefficient can be calculated knowing the dependence of $\gamma_{\mathrm{j}}$ on molality of solute $j$. Of course at this stage we do not know the form of the dependence of $\gamma_{\mathrm{j}}$ on $\mathrm{m}_{\mathrm{j}}$. In fact $\gamma_{\mathrm{j}}$ also depends on the solute, temperature and pressure. But for a given system (at fixed $\mathrm{T}$ and $\mathrm{p}$) we might express $\phi$ as a series expansion of the molality $\mathrm{m}_{\mathrm{j}}$. Thus,
$\phi=1+a_{1} \, m_{j}+a_{2} \, m_{j}^{2}+a_{3} \, m_{j}^{3}+\ldots \ldots$
Interestingly this assumed dependence is equivalent to a series expansion in mole fraction of solute $\mathrm{x}_{\mathrm{j}}$ for $1 \mathrm{nf}_{1}$, where $\mathrm{f}_{1}$ is the (rational) activity coefficient for the solvent [1,2].
$\operatorname{lnf}_{1}=\mathrm{b}_{1} \, \mathrm{x}_{\mathrm{j}}^{2}+\mathrm{b}_{1} \, \mathrm{x}_{\mathrm{j}}^{3}+\mathrm{b}_{3} \, \mathrm{x}_{\mathrm{j}}^{4}+\ldots \ldots$
Here $\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3} \ldots$ depend on the solute (for given $\mathrm{T}$ and $\mathrm{p}$). The link between the two equations can be expressed as follows.
$\mathrm{b}_{1}=-\left[(1 / 2)+\mathrm{M}_{1}^{-1} \, \mathrm{a}_{1}\right]$
$\mathrm{b}_{2}=-\left[(2 / 3)+2 \, \mathrm{M}_{1}^{-1} \, \mathrm{a}_{1}+\mathrm{M}_{1}^{-2} \, \mathrm{a}_{2}\right]$
$\mathrm{b}_{3}=-\left[(3 / 4)+3 \, \mathrm{M}_{1}^{-1} \, \mathrm{a}_{1}+3 \, \mathrm{M}_{1}^{-2} \, \mathrm{a}_{2}+\mathrm{a}_{3} \, \mathrm{M}_{1}^{-3}\right]$
Footnotes
[1] J. J. Kozak, W. S. Knight and W. Kauzmann, J. Chem. Phys., 1968,48, 675.
[2] By definition, for a solution j in solvent, chemical substance 1,
$\mathrm{x}_{\mathrm{j}}=\mathrm{m}_{\mathrm{j}} /\left(\mathrm{M}_{\mathrm{l}}^{-1}+\mathrm{m}_{\mathrm{j}}\right)$
where $\mathrm{M}_{1}$ is the molar mass of solvent expressed in $\mathrm{kg mol}^{-1}$. Hence molality of solute $j$,
$\mathrm{m}_{\mathrm{j}}=\mathrm{x}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{l}}^{-1} \,\left(1-\mathrm{x}_{\mathrm{j}}\right)^{-1}$
We expand $\left(1-x_{j}\right)^{-1}$ based on the premise that $0<\mathrm{x}_{\mathrm{j}}<<1.0$ for dilute solutions. Then,
$\mathrm{m}_{\mathrm{j}}=\mathrm{x}_{\mathrm{j}} \mathrm{M}_{1}^{-1} \,\left[1+\mathrm{x}_{\mathrm{j}}+\mathrm{x}_{\mathrm{j}}^{2}+\mathrm{x}_{\mathrm{j}}^{3}+\ldots \ldots\right]$
or, m x j jj jj M x x x = ⋅+ + ++ − 1 1 234 [ .....] (d)
Here we carry all terms up to and including the fourth power of xj. But from the two methods for relating µ1(aq) to the composition of a solution, 1nx f M m 11 1 j ( ) ⋅ =−⋅ ⋅ φ (e)
Then, 1 1 11 1 1 2 2 3 3 nx f M m m m m j jjj ( ) [ .....] ⋅ =− ⋅ ⋅ + ⋅ + ⋅ + ⋅ + a a a (f)
or, 1 11 1 11 2 2 3 3 4 nf n M m m m m j jj j j x a a a =− − − ⋅ + ⋅ + ⋅ + ⋅ + ( ) [ .....] (g)
But for dilute solutions, −− =+ + + 11 2 3 4 2 34 nx x x x j jj j j ( ) ( /) ( /) ( /) x (h)
Using equation (c) for mj as a function of xj in the context of equation (f), we obtain an equation for ln(f1). 1 234 1 2 34 nf x x x jj j j ( ) ( /) ( /) ( /) x =+ + + −− − − xxxx jj jj 234 1 4 j 1 1 1 3 j 1 1 1 2 j 1 1 − M ⋅ x ⋅ a − 2 ⋅ M ⋅ x ⋅ a − 3⋅ M ⋅ x ⋅ a − − − − ⋅ ⋅ −⋅ ⋅ ⋅ − − M xa M xa 1 j j 2 3 2 1 2 4 2 3 − ⋅⋅ − M xa 1 j 3 4 3 (i)
Hence, 1n(f1) = { [( / ) ( )] } − +⋅ ⋅ − 1 2 1 1 1 2 a M x j } { [(2 / 3) (2 a M ) (a M )] x3 j 2 2 1 1 1 1 + − + ⋅ ⋅ + ⋅ ⋅ − − +− + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − −− { [ / ) ( ) ( ) ( )] } 34 3 3 1 1 1 2 1 2 3 1 3 4 aM aM aM x j (j)
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.07%3A_Gibbs_Energies-_Solutions-_Parameters_Phi_and_ln%28gamma%29.txt
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In real solutions, solute molecules are not infinitely far apart. With increase in solute concentration, the mean separation of solute molecules decreases [1]. Deviations in the properties of real solutions of neutral solutes from ideal can be understood in term of contact, overlap and interaction between cospheres of solvent surrounding solute molecules [2]. Two limiting cases can be identified. In one case overlap occurs between cospheres for which the organisation of solvent molecules are compatible, leading to attractive interaction between two solute molecules; a stabilising effect. In the opposite case the organisation of solvent in the cospheres is incompatible leading to repulsion between the solute molecules; i.e a destabilising effect. These ideas can be formulated quantitatively leading to an understanding of the factors controlling the properties of solutes in aqueous solution.
A given solution is prepared using $1 \mathrm{~kg}$ of water($\ell$) and $\mathrm{m}_{j}$ moles of solute $j$. The chemical potential of the solvent water is related to $\mathrm{m}_{j}$ using equation (a).
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}$
Here $\mu_{1}^{*}(\ell)$ is the chemical potential of solvent water at the same $\mathrm{T}$ and $\mathrm{p}$; $\phi$ is the practical osmotic coefficient which is unity for a solution having thermodynamic properties which are ideal; $\mathrm{M}_{1}$ is the molar mass of water. The difference $\left[\mu_{1}(\mathrm{aq})-\mu_{1}^{*}(\ell)\right]$ equals $\left[-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right]$. Hence for an ideal solution where $\phi$ is unity addition of a solute lowers the chemical potential of the solvent; i.e. stabilises the solvent.
The chemical potential of the solute $\mu_{\mathrm{j}}(\mathrm{aq})$ is related to the molality $\mathrm{m}_{j}$ using equation (b).
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
We rewrite equation (b) in an extended form.
$\mu_{j}(a q)=\mu_{j}^{0}(a q)+R \, T \, \ln \left(m_{j} / m^{0}\right)+R \, T \, \ln \left(\gamma_{j}\right)$
Or,
$\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Hence $\gamma_{j}$ measures the extent to which the chemical potential of solute $j$ in the real solution differs from that in an ideal solution. If $\gamma_{j} > 1$ [and hence $\ln \left(\gamma_{j}\right)>0$] $\mu_{j}(\mathrm{aq})<\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})$, solute-solute interactions destabilise the solute. If $\gamma_{\mathrm{j}}<1$ [and hence $\ln \left(\gamma_{\mathrm{j}}\right)<0$] $\mu_{\mathrm{j}}(\mathrm{aq})>\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})$, and so these interaction stabilise the solute. [$\mathrm{NB} \gamma_{j}$ cannot be negative.]
The chemical potentials of solute and solvent are linked by the Gibbs – Duhem equation which for aqueous solutions (at fixed $\mathrm{T}$ and $\mathrm{p}$) containing $1 \mathrm{~kg}$ of water takes the following form.
$\left(1 / M_{1}\right) \, d \mu_{1}(a q)+m_{j} \, d \mu_{j}(a q)=0$
Then,
\begin{aligned} \left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right] \ &+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}
Or,
$-\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)=0$
The latter equation links changes in the osmotic coefficient $\phi$ and $\gamma_{j}$. In other words, a perturbation which affects the solvent feeds back on to the properties the solute. This is Gibbs-Duhem communication. In the present context equation (g) explains why interaction between cospheres feeds back to the properties of the solute. Consequently the Gibbs-Duhem equation is used to switch between equations describing $\phi$ and $\gamma_{j}$. Thus from equation (g),[3]
$\mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=0$
Equation (h) can be written in two forms depending which direction we wish to proceed. Thus from equation (h)[4],
$(1-\phi)=-\left(1 / m_{j}\right) \, \int_{m(j)=0}^{m(j)} m_{j} \, d \ln \left(\gamma_{j}\right)$
In other words we have an equation for $\phi$ and in terms of $\gamma_{j}$. Alternatively [5] we can express $\gamma_{j}$ in terms of $\phi$ and its dependence on $\mathrm{m}_{j}$.
$\left.\int_{m(j)=0}^{m(j)} d \ln \left(\gamma_{j}\right)=\int_{m(j)=0}^{m(j)} d(1-\phi)+\int\left[(1-\phi) / m_{j}\right)\right] \, d m_{j}$
Then
$\ln \left(\gamma_{\mathrm{j}}\right)=(\phi-1)+\int_{\mathrm{m}(\mathrm{j})=0}^{\mathrm{m}(\mathrm{j})}(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)$
Hence $\gamma_{j}$ can be calculated from knowing $\phi$ and its dependence on dependence on $\mathrm{m}_{j}$.
Footnotes
[1] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edn.,1959, chapter 1.
[2] J. J. Kozak, W. S. Knight and W. Kauzmann, J. Chem. Physics,1968,48,675.
[3] \begin{aligned} &-\phi \, d m_{j}-m_{j} \, d \phi+m_{j} \, d m_{j} / m_{j}+m_{j} \, \ln \left(\gamma_{j}\right)=0\ &\text { Or, } \mathrm{dm}_{\mathrm{j}}-\phi \, d \mathrm{~m}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+\mathrm{m}_{\mathrm{j}} \, \ln \left(\gamma_{\mathrm{j}}\right)=0 \end{aligned}
[4] $\begin{array}{r} \int_{\mathrm{m}(\mathrm{j} j=0}^{\mathrm{m}(\mathrm{j})} \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right]=-\int_{\mathrm{m}(\mathrm{j})=0}^{\mathrm{m}(\mathrm{j})} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \ \mathrm{m}_{\mathrm{j}} \,(1-\phi)=-\int_{\mathrm{m}(\mathrm{j})=0}^{\mathrm{m}(\mathrm{j})} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \end{array}$
[5] From equation (h), $\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(\phi-1)\right]$
Or,
$\left.\int_{m(j)=0}^{m(j)} d \ln \left(\gamma_{j}\right)=\int_{m(j)=0}^{m(j)} d(\phi-1)+\int\left[(\phi-1) / m_{j}\right)\right] \, d m_{j}$
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The chemical potential of solute $j$ in aqueous solution, molality $\mathrm{m}_{j}$, at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ (which is close to ambient) is given by equation (a).
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
In developing an understanding the of factors which contribute to $\mu_{\mathrm{j}}(\mathrm{aq})$, a model for solutions developed by Gurney is often helpful [1].
A co-sphere is identified around each solute molecule $j$ where the organization of solvent molecules differs from that in the bulk solvent at the same $\mathrm{T}$ and $\mathrm{p}$. In a solution where the thermodynamic properties of the solute $j$ are ideal, there are no solute-solute interactions such that the activity coefficient $\gamma_{j}$ is unity. In real solutions the fact that $\gamma_{j} \neq 1$ can be understood in terms of co-sphere---co-sphere interactions together for salt solutions strong charge-charge interactions.
The model [2] identifies two zones. Zone A describes solvent molecules close to the solute molecule, the number of such solvent molecules being the primary hydration number. Zone B describes the solvent molecules outside Zone A. Their organization differs from that in the bulk solvent as a consequence of the presence of solute molecule (or, ion) $j$. Zone C lies beyond zone B where the organization of solvent is effectively the same as that in pure solvent at the same $\mathrm{T}$ and $\mathrm{p}$. There is merit in not being too pedantic concerning the definitions of zones A, B and C.
For real solutions co-sphere----co-sphere interactions are accounted for using for example the term $\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}$ in the equation describing the partial molar volume $\mathrm{V}_{\mathrm{j}}(\mathrm{aq})$ for solute $j$ in a real solution.
Footnotes
[1] R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.
[2] H. S. Frank and W.-Y. Wen, Discuss. Faraday Trans.,1957,24,756.
1.10.10: Gibbs Energies- Solutions- Cosphere-Cosphere Interactions
For neutral solutes in aqueous solutions, the solvent plays a key role in determining the form and magnitude of solute-solute interactions. One description of these interactions uses the Gurney model for solute cospheres.[1] Cosphere-cosphere interaction, involving the solvent, feeds back to the properties of the solute; i.e. Gibbs - Duhem communication. Two limiting cases are identified.
1. The hydration characteristics (i.e. solute-solvent interaction) of the two solute molecules, $\alpha$ and $\beta$ are quite different such that in the region of overlap their structural influence on water-water interactions are incompatible/antagonistic. Thus the pairwise Gibbs energy interaction parameter $\mathrm{g}_{j j}$ is positive; $\gamma_{j} > 1$ and $\ln \left(\gamma_{\mathrm{j}}\right)>0$. The intuitive idea is that incompatability is synonymous with repulsion (almost human behaviour-- a dangerous anthropomorphic argument).
2. In the second case, the hydration cospheres of the two solute molecules α and β are similar to the extent that in the region of cosphere overlap there is an enhancement of water-water interactions and hence attraction. Thus the pairwise Gibbs energy interaction parameter $\mathrm{g}_{j j}$ is negative; $\gamma_{j} < 1$ and $\ln \left(\gamma_{j}\right)<0$. Where solute molecules α and β are apolar this attraction is called hydrophobic bonding/attraction.
Footnotes
[1] See for example, R. P. Currier and J. P. O’Connell, Fluid Phase Equilibria,1987,33,245.
1.10.11: Gibbs Energies- Solutions- Solute-Solute Interactions- Pairwise
Analysis of the thermodynamic properties of aqueous solutions was taken a step further by Savage and Wood who envisage two solute molecules A and B in aqueous solution [1]. The total pairwise interaction between these molecules is described in terms of pairwise group-group interaction parameters. Then, for example, the pairwise enthalpic solute-solute interaction parameter $\mathrm{H}_{\mathrm{AB}$ is written as the sum of products, $\mathrm{n}_{\mathrm{i}}^{\mathrm{A}} \, \mathrm{n}_{\mathrm{j}}^{\mathrm{B}} \, \mathrm{h}_{\mathrm{ij}}$ where $\mathrm{n}_{\mathrm{i}}^{\mathrm{A}}$ is the number of A-groups in solute molecule $\mathrm{i}$ and $\mathrm{n}_{\mathrm{j}}^{\mathrm{B}}$ is the number of B-groups in solute molecule $\mathrm{j}$ where $\mathrm{h}_{\mathrm{ij}}$ is a pairwise enthalpic group interaction parameter. A similar analysis is carried out for interaction Gibbs energies leading to pairwise Gibbs energy parameters $\mathrm{g}_{\mathrm{ij}}$. So, for example, $\mathrm{g}(\mathrm{OH}-\mathrm{OH})$ is negative characteristic of a hydrophilic-hydrophilic interaction. Whereas $\mathrm{g}\left(\mathrm{OH}-\mathrm{CH}_{2}\right)$ is positive indicating 'repulsion' within hydrophobic-hydrophilic pairs. Interestingly $\mathrm{g}\left(\mathrm{CH}_{2}-\mathrm{CH}_{2}\right)$ is negative" which is indicative of a hydrophobic-hydrophobic attraction (cf. hydrophobic bonding); the corresponding enthalpic pairwise parameter is positive. Thus it is tempting to speculate that hydrophobic attraction is entropy driven [2]; for further comments see references [3-13].
The general approach is readily extended to a consideration of pairwise interactions between added solutes and both initial and transitions states for given chemical reactions in aqueous solution [14-18].
Footnotes
[1] J.J. Savage and R. H. Wood, J. Solution Chem.,1976,5,733.
[2] J.J. Spitzer, S. K. Suri and R. H. Wood, J. Solution Chem.,1985,14,5; and references therein.
[3] S. K. Suri and R. H. Wood, J. Solution Chem.,1986,15,705.
[4] S. K. Suri, J.J.Spitzer, R. H. Wood, E.G.Abel and P.T. Thompson, J. Solution Chem.,1986,14,781.
[5] A. L. Harris, P. T. Thompson and R. H. Wood, J. Solution Chem.,1980, 9,305.
[6] For the role of solute stereochemistry see F. Franks and M. D. Pedley, J. Chem. Soc. Faraday Trans. 1, 1983,79,2249.
[7] Amides in N,N-dimethvl formamide: M. Bloemendal and G. Somsen, J. Solution Chem.,1983,12,83.
[8] Solutions in DMF with a modification of the role of the solvent; M. Bloemendal and G. Somsen, J. Solution Chem.,1987,16,367
[9] Interaction between amides and urea in aqueous solution; P. J. Cheek and T. H. Lilley, J. Chem. Soc. Faraday Trans.1, 1988,84,1927.
[10]
1. Diols(aq); S. Andini, G. Castronuovo, V. Ella and L. Fasano, J. Chem. Soc. Faraday Trans.. 1990. 86, 3567.
2. Amino acids and peptides; T H. Lilley and R. P. Scott, J. Chem. Soc. Faraday Trans. 1, 1976,72,359.
3. Monosaccharides(aq); G. Barone, G. Castronuovo, D. Doucas, V.Elia and G. A. Mattia, J.Phys.Chem.,1983,87,1931.
[11] Urea and polyols(aq); G. Barone, V. Elia and E. Rizzo, J. Solution Chem.,1982,11,687.
[12] Volumes and heat capacities of aromatic solutes(aq); S. Cabani, P. Gianni,V. Mollica and L. Lepori, J. Solution Chem., 1981,10,563.
[13] Small peptides(aq); enthalpies; O. V. Kulikov, A. Zielenkiewicz, W. Zielenkiewicz. V. G. Badelin and A.Krestov , J. Solution Chem., 1993,22,59.
[14] M. J. Blandamer, J. Burgess, I . M. Horn, J. B. F. N. Engberts and P. Warrick Jr., Colloids and Surfaces, 1990,48,139.
[15] M. J. Blandamer, J. Burgess, J. B. F. N. Engberts and W. Blokzijl, Annu. Rep. Prog. Chem., Sect C, Phys. Chem.,1990,87,45.
[16] W. Blokzijl, J. B. F. N. Engberts, J. Jager and M. J. Blandamer, J. Phys. Chem.,1987, 91,6022.
[17] M. J. Blandamer, J. Burgess and J. B. F. N. Engberts, Chem. Soc Rev.,1985,14,237.
[18] M. J. Blandamer and J. Burgess, Pure Appl. Chem.,1982, 54,2285.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.09%3A_Gibbs_Energies-_Solutes-_Cospheres.txt
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A given aqueous solution containing two neutral solutes, $\mathrm{i}$ and $\mathrm{j}$, (e.g. urea and sucrose) was prepared using $1 \mathrm{~kg}$ of water at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ ( which was close to ambient). The molalities of the two solutes were $\mathrm{m}_{\mathrm{i}}$ and $\mathrm{m}_{\mathrm{j}}$ The chemical potential of the solvent in the mixed aqueous solution is given by equation (a) where for an ideal solution the practical osmotic coefficient $\phi$ is unity. Then,
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)$
The chemical potentials of the two solutes are related to their molalities using equations (b) and (d).
$\mu_{\mathrm{i}}(\mathrm{aq})=\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)$
where
$\left.\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{i}}=1.0 \quad \text { (at all } \mathrm{T} \text { and } \mathrm{p}\right)$
Similarly,
$\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)$
where
$\left.\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0 \quad \text { (at all T and } \mathrm{p}\right)$
Therefore the Gibbs energy of a solution prepared using $1 \mathrm{~kg}$ of water is given by equation (f).
\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1} / \mathrm{kg}=1.0\right)=\left(1 / \mathrm{M}_{1}\right) \, & {\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] } \ +\mathrm{m}_{\mathrm{i}} \,\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\right.&\left.\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right] \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}
For the corresponding solution having ideal thermodynamic properties
\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1} / \mathrm{kg}=1.0\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \ +\mathrm{m}_{\mathrm{i}} \,\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)\right] \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}
But the excess Gibbs energy is defined by equation (h).
$\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{G}(\mathrm{aq})-\mathrm{G}(\mathrm{aq} ; \mathrm{id})$
Hence
$\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{R} \, \mathrm{T}=\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right) \,(1-\phi)+\mathrm{m}_{\mathrm{j}} \, \ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \ln \left(\gamma_{\mathrm{i}}\right)$
Actually $\phi$, $\gamma_{\mathrm{i}}$ and $\gamma_{\mathrm{j}}$ are linked. They cannot change independently. According to the Gibbs-Duhem equation (at fixed $\mathrm{T}$ and $\mathrm{p}$),
$\left(1 / M_{1}\right) \, d \mu_{1}(a q)+m_{j} \, d \mu_{j}+m_{i} \, d \mu_{i}=0$
We consider the case where the solution is perturbed by a change in molality, $\mathrm{dm}_{\mathrm{j}}$, recognising that the chemical potentials of solvent and solutes change. Thus,
$\left(1 / M_{1}\right) \,\left(d \mu_{1}(a q) / d m_{j}\right)+m_{j} \,\left(d \mu_{j} / d m_{j}\right)+m_{i} \,\left(d \mu_{i} / d m_{j}\right)=0$
Therefore from equation (j),
\begin{aligned} &\left(1 / \mathrm{M}_{1}\right) \, \frac{\mathrm{d}}{\mathrm{dm}_{\mathrm{j}}}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \ &+\mathrm{m}_{\mathrm{i}} \, \frac{\mathrm{d}}{\mathrm{dm}_{\mathrm{j}}}\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right] \ &\quad+\mathrm{m}_{\mathrm{j}} \, \frac{\mathrm{d}}{\mathrm{dm}_{\mathrm{j}}}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}
In the case considered here molality $\mathrm{m}_{\mathrm{i}}$ does not change. Then (at constant, $\mathrm{m}_{\mathrm{i}}$, $\mathrm{T}$, $\mathrm{p}$ and mass of solvent),
$-\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right) \,\left[\frac{\partial \phi}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}+(1-\phi)+\mathrm{m}_{\mathrm{j}} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}+\mathrm{m}_{\mathrm{i}} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}=0$
With the help of the latter equation we explore how the Gibbs energy depends on the molality of solute $j$ at constant $\mathrm{m}_{\mathrm{i}}$. Then from equation (i),
\begin{aligned} \frac{1}{R \, T} \,\left[\frac{\partial G^{\mathrm{E}}(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right] &=-\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right) \,\left[\frac{\partial \phi}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}+(1-\phi) \ &+\mathrm{m}_{\mathrm{j}} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}+\ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})} \end{aligned}
Comparison of equations (m) and (n) shows that the differential dependence of $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})$ on $\mathrm{m}_{\mathrm{j}}$ at constant $\mathrm{m}_{\mathrm{i}}$ is related to $\ln \left(\gamma_{\mathrm{s}}\right)$;
$\ln \left(\gamma_{\mathrm{j}}\right)=\frac{1}{\mathrm{R} \, \mathrm{T}} \,\left[\frac{\partial \mathrm{G}^{\mathrm{E}}(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right]$
If the solution is dilute then $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})$ can be described using pairwise Gibbs energy interaction parameters. Thus
$\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{g}_{\mathrm{ji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+2 \, \mathrm{g}_{\mathrm{ij}} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2}+\mathrm{g}_{\mathrm{ii}} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2}$
Here $\mathrm{g}_{i i}$ and $\mathrm{g}_{j j}$ are homotactic pairwise Gibbs energy interaction parameters whereas $\mathrm{g}_{i j}$ is the corresponding heterotactic parameter. According to equation (p) the differential dependence of $\mathrm{G}^{\mathrm{E}}(\mathrm{aq})$ on molality $\mathrm{m}_{\mathrm{j}}$ at constant $\mathrm{m}_{\mathrm{i}}$ is given by equation (q). Thus,
$\left[\frac{\partial \mathrm{G}^{\mathrm{E}}(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}=2 \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}+2 \, \mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{i}}$
Combination of equations (o) and (q) yields an equation for $\ln \left(\gamma_{j}\right)$ as a function of two pairwise interaction parameters. Then [1],
$\ln \left(\gamma_{\mathrm{j}}\right)=\left[\frac{2}{R \, T}\right] \,\left[\frac{1}{m^{0}}\right]^{2} \,\left[g_{i j} \, m_{j}+g_{i j} \, m_{i}\right]$
In other words $\ln \left(\gamma_{\mathrm{j}}\right)$ is simply related to the molality of the two solutes. In many applications we are concerned with a solution in which $\mathrm{m}_{\mathrm{i}} >> \mathrm{m}_{\mathrm{j}}$ such that the solution contains only a trace of solute $j$. The activity coefficient for solute $j$ is written $\gamma_{\mathrm{j}}^{\mathrm{T}}$. The latter describes the effect of solute-solute interactions on solute $j$. If we set $\mathrm{m}_{\mathrm{j}} \cong 0$, then equation (r) yields an equation for $\gamma_{\mathrm{j}}^{\mathrm{T}}$ in terms of $\mathrm{m}_{\mathrm{i}}$.
$\ln \left(\gamma_{\mathrm{j}}^{\mathrm{T}}\right)=\left[\frac{2}{\mathrm{R} \, \mathrm{T}}\right] \,\left[\frac{1}{\mathrm{~m}^{0}}\right]^{2} \, \mathrm{g}_{\mathrm{ij}} \, \mathrm{m}_{\mathrm{i}}$
Footnote
[1] $\ln \left(\gamma_{\mathrm{j}}\right)=\left[\frac{2}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}\right] \,\left[\frac{1}{\left[\mathrm{~mol} \mathrm{~kg}{ }^{-1}\right]}\right]^{2} \,\left[\mathrm{J} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right]$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.12%3A_Gibbs_Energies-_Solutions-_Two_Neutral_Solutes.txt
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An explanation of the properties of a given solute j in aqueous solutions is in terms of the formation of a hydrate; j.hJ$\mathrm{H}_{2}\mathrm{O}$ where h is the hydration number independent of temperature and pressure[1]. In summary there are two descriptions of the solutions prepared using $\mathrm{n}_{1}$ moles of water and $\mathrm{n}_{j}$ moles of solute $j$. In description A there are $\mathrm{n}_{j}$ moles of solute, chemical substance j, and n1 moles of solvent. In description B there are nj moles of solute j.h$\mathrm{H}_{2}\mathrm{O}$ and $\left(n_{1}-h \, n_{j}\right)$ moles of water [2]. At fixed $\mathrm{T}$ and $\mathrm{p}$ the system is at equilibrium, being therefore at a minimum in Gibbs energy. The Gibbs energy is not dependent on our description of the system [3]; it does not know which description we favour!
We imagine two open dishes in a partially evacuated chamber at constant $\mathrm{T}$. Each dish contains the same amount of a given solution but we label one dish A and the other dish B [4]. Further the Gibbs energies are equal; $\mathrm{G}(\mathrm{A}) = \mathrm{G}(\mathrm{B})$. The vapour pressures are the same so that $\mu_{1}(\mathrm{aq} ; \mathrm{A})=\mu_{1}(\mathrm{aq} ; \mathrm{B})$. For dish A,
$\mathrm{G}(\mathrm{A})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})$
For dish B.
$\mathrm{G}(\mathrm{B})=\left(\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \, \mathrm{h}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{jh}} \, \mu_{\mathrm{jh}}(\mathrm{aq})$
Here $\mu_{j \mathrm{~h}}(\mathrm{aq})$ is the chemical potential of hydrate j.h$\mathrm{H}_{2}\mathrm{O}$ in solution. We notes that $\mathrm{n}_{\mathrm{j}}=\mathrm{n}_{\mathrm{jh}}$. Because $\mathrm{G}(\mathrm{A})=\mathrm{G}(\mathrm{B})$, and the chemical potentials of the solvent are the same, $\mu_{j h}(a q)=\mu_{j}(a q)+h \, \mu_{1}(a q)$. The molality of hydrate j.h$\mathrm{H}_{2}\mathrm{O}$, $\mathrm{m}_{\mathrm{jh}}=\mathrm{n}_{\mathrm{jh}} /\left[\left(\mathrm{n}_{1}-\mathrm{h} \, \mathrm{n}_{\mathrm{jh}}\right) \, \mathrm{M}_{1}\right]$ whereas the molality of solute $j$ $\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}\right]$. Then at fixed $\mathrm{T}$ and $\mathrm{p}$,
\begin{aligned} &\mu_{\mathrm{jh}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{jh}} \, \gamma_{\mathrm{jh}} / \mathrm{m}^{0}\right)= \ &\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{h} \,\left\{\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \end{aligned}
In the $\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0$ at all $\mathrm{T}$ and $\mathrm{p}$; in the $\operatorname{limit}\left(\mathrm{m}_{\mathrm{jh}} \rightarrow 0\right) \gamma_{\mathrm{jh}}=1.0$ at all $\mathrm{T}$ and $\mathrm{p}$. In the same limit, $\phi=1$. Hence assuming $\mathrm{h}$ is independent of $\mathrm{m}_{j}$.
$\mu_{\mathrm{jh}}^{0}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{h} \, \mu_{1}^{*}(\ell)$
We use equation (d) and reorganise equation (c) as an equation for $\gamma_{j}$.
$\ln \left(\gamma_{\mathrm{j}}\right)=\ln \left(\mathrm{m}_{\mathrm{jh}} / \mathrm{m}_{\mathrm{j}}\right)+\mathrm{h} \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\ln \left(\gamma_{\mathrm{jh}}\right)$
We assert that the formation of hydrate by solute $j$ accounts for the fact that the properties of solute $j$ are not ideal. We also assert that the properties of the hydrate are ideal; $\gamma_{\mathrm{jh}}=1$. Moreover, $\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}_{\mathrm{jh}}\right)=1-\left(\mathrm{h} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)$ Then,
$\ln \left(\gamma_{\mathrm{j}}\right)=-\ln \left[1-\left(\mathrm{h} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right)\right]+\mathrm{h} \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
If the solution is dilute , $\phi \cong 1$. Then,
$\ln \left(\gamma_{\mathrm{j}}\right)=2 \, h \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}$
The hydrate model for activity coefficients can be understood in the following fashion. When $\delta n_{j}$ moles of solute are added to a solution molality $\mathrm{m}_{j}$, $\mathrm{h} \, \delta \mathrm{n}_{\mathrm{j}}$ moles of water are removed from ‘solvent’ and transferred to the solute. In these terms each solute molecule responds to this increased competition for solvent by other solute molecules and therefore ‘knows’ that there are other solute molecules in the solution. Any communication between solute molecules in solution is reflected in the extent to which $\gamma_{j}$ differs from unity.
Footnotes
[1] L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 2nd. Edn., 1970,Section 2.13.
[2] $\mathrm{h} \, \mathrm{n}_{\mathrm{j}}$ must be $<\mathrm{n}_{1}$
[3] E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1977, chapter 2.
[4] M. J. Blandamer, J. B. N. Engberts, P. T. Gleeson and J. C. R. Reis, Chem. Soc. Rev., submitted,.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.13%3A_Gibbs_Energies-_Solutions-_Hydrates_in_Aqueous_Solution.txt
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An aqueous solution is prepared using $mathrm{n}_{\mathrm{j}}$ moles of salt $\mathrm{MX}$ and $\mathrm{n}_{1}$ moles of water. The properties of the system are accounted for using one of two possible Descriptions.
Description I
The solute $j$ comprises a 1:1 salt MX molality $\mathrm{m}(\mathrm{MX})\left[=\mathrm{n}(\mathrm{MX}) / \mathrm{w}_{1}\right.$ where $\mathrm{w}_{1}$ is the mass of water]. The single ion chemical potentials, are defined in the following manner
\begin{aligned} &\mu\left(\mathrm{M}^{+}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{x}^{-}\right)} \ &\mu\left(\mathrm{X}^{-}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{M}^{+}\right)} \end{aligned}
Then the total Gibbs energy (at fixed $\mathrm{T}$ and $\mathrm{p}$) is given by equation (b). $\mathrm{G}(\mathrm{aq} ; \mathrm{I})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})$
\begin{aligned} &+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{\#}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \ &+\mathrm{n}_{\mathrm{j}}\left\{\mu^{\#}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{X}^{-}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \end{aligned}
Description II
According to this Description each mole of cation is hydrated by $\mathrm{h}_{\mathrm{m}}\left(\mathrm{H}_{2}\mathrm{O}\right)$ moles of water and each mole of anion is hydrated by $\mathrm{h}_{\mathrm{x}}\left(\mathrm{H}_{2}\mathrm{O}\right)$ moles of water. Hence the single ion chemical potentials are defined as follows.
$\mu\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\left\lfloor\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)\right\rfloor$
at constant $\mathrm{T}$, $\mathrm{p}$, $\mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right)$ and,
$\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right]$
at constant $\mathrm{T}$, $\mathrm{p}$, $\mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right)$ Then,
$\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right]$
$\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right]$
Hence the (equilibrium) Gibbs energy (at defined $\mathrm{T}$ and $\mathrm{p}$) is given by the following equation.
\begin{aligned} &\mathrm{G}(\mathrm{aq})=\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right] \, \mu_{1}(\mathrm{aq}) \ &\quad+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{\prime \prime}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \ &+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{\mathrm{y}}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)\right. \ &\left.\quad+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-} \text {(II) } / \mathrm{m}^{0}\right\}\right] \end{aligned}
But the Gibbs energies defined by equations (b) and (g) are identical (at equilibrium at defined $\mathrm{T}$ and $\mathrm{p}$). Hence [1],
\begin{aligned} &\mu^{\prime \prime}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\prime \prime}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[1-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right)\right]\ &+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{I}) \, \gamma_{-}(\mathrm{I})\right\}\ &=-\left(h_{m}+h_{X}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, R \, T \, M_{1} \, m_{j}\right\}\ &+\mu^{\prime \prime}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{\prime \prime}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{X}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)\ &+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+} \text {(II) } \, \gamma_{-} \text {(II) }\right\} \end{aligned}
We use the latter equation to explore what happens in the limit that $\mathrm{n}_{j}$ approaches zero. Thus, $\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right) \gamma_{+}(\mathrm{I})=1 ; \gamma_{-}(\mathrm{I})=1 ; \gamma_{+}(\mathrm{II})=1 ; \gamma_{-}(\mathrm{II})=1 ; \mathrm{m}_{\mathrm{j}}=0$ Hence,
\begin{aligned} &\mu^{\#}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)= \ &\mu^{\#}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right) \ &-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}{ }^{*}(\ell) \end{aligned}
We obtain an equation linking the ionic chemical potentials. Thus,
$\begin{array}{r} \ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \ -2 \, \ln \left[1-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right)\right] \ +\ln _{+} \gamma_{+}(\mathrm{II})+\ln \gamma_{-}(\mathrm{II}) \end{array}$
Then in dilute solutions,
$\begin{array}{r} \ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \ +\ln \gamma_{+}(\mathrm{II})+\ln \gamma_{-}(\mathrm{II}) \end{array}$
But $\ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \ln \gamma_{\pm}(\mathrm{I})$ Then, $2 \, \ln \gamma_{\pm}(\mathrm{I})=2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+2 \, \ln \gamma_{\pm}(\mathrm{II})$
We identify relationships between single ion activity coefficients in an extra-thermodynamic analysis. Thus from equation (k),
$\ln \gamma_{+}(\mathrm{II})=\ln \gamma_{+}(\mathrm{I})-2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{m}}$
$\ln \gamma_{-}(\mathrm{II})=\ln \gamma_{-}(\mathrm{I})-2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{x}}$
It is noteworthy that in these terms the solution can be ideal using description I where $\gamma_{\pm} = 1.0$ but non-ideal using description II. Nevertheless, these equations show how the activity coefficient of the hydrated ion (description II) is related to the activity coefficient of the simple ion (description I).
Footnote
[1] From equations (b) and (g), (dividing by $\mathrm{n}_{j}$)
\begin{aligned} &\left[\mu^{n}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]\ &+\left[\mu^{\prime \prime}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]=\ &-\left(h_{m}+h_{x}\right) \, \mu_{1}(a q)+\ &+\left[\mu^{\prime \prime}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+} \text {(II) } / \mathrm{m}^{0}\right\}\right]\ &+\left[\mu^{\prime \prime}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \end{aligned}
Then
\begin{aligned} &\text { en }\left[\mu^{*}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]\ &+\left[\mu^{*}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]=\ &-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\}\ &+\left[\mu^{*}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right]\ &+\left[\mu^{\prime \prime}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \end{aligned}
Or,
\begin{aligned} &{\left[\mu^{\prime \prime}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)\right.} \ &+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm { m } ( \mathrm { M } ^ { + } ; \mathrm { I } ) \, \mathrm { m } \left(\mathrm{X}^{-} ;(\mathrm{I}) /\left(\mathrm{M}^{+} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right\}\right.\right. \ &\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{I}) \, \gamma_{-}(\mathrm{I})\right\} \ &=-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \ &+\mu^{*}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}}\left(\mathrm{H}_{2} \mathrm{O}\right) ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{X}}\left(\mathrm{H}_{2} \mathrm{O}\right) ; \mathrm{aq}\right) \ &+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{II}) \, \gamma_{-}(\mathrm{II})\right\} \end{aligned}
Using the definition of $\mu^{\prime \prime}\left(\mathrm{M}^{+} ; \mathrm{I}\right)$ and $\mu^{\prime \prime}\left(\mathrm{X}^{-} ; \mathrm{I}\right)$ and equations (e) and (f) for description (II),
\begin{aligned} &\frac{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)}{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right)}= \ &\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{M}_{1} \, \mathrm{n}_{1}} \, \frac{\mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \, \mathrm{n}_{\mathrm{j}}\right]}{\mathrm{n}_{\mathrm{j}}} \, \frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{M}_{1} \, \mathrm{n}_{1}} \, \frac{\mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \, \mathrm{n}_{\mathrm{j}}\right]}{\mathrm{n}_{\mathrm{j}}} \end{aligned}
Thus,
$\frac{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)}{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right)}=\left[1-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]^{2}$
Therefore,
\begin{aligned} &\mu^{\# \prime}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\# \prime}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \ln \left[1-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right)\right] \ &\quad+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{I}) \, \gamma_{-}(\mathrm{I})\right\} \ &=-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \ &\quad+\mu^{\# *}\left(\mathrm{M}^{+} ; \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{\# \#}\left(\mathrm{X}^{-} ; \mathrm{h}_{\mathrm{X}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{II}) \, \gamma_{-}(\mathrm{II})\right\} \end{aligned}
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.14%3A_Gibbs_Energies-_Salt_Hydrates.txt
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The solid crystalline salt $\mathrm{NaCl}$ comprises a lattice of sodium $\mathrm{Na}^{+}$ and chloride $\mathrm{Cl}^{-}$ ions. The charge number on each sodium ion, $\mathrm{z}_{+}$, is $+ 1$; the charge number on each chloride ion, $\mathrm{z}_{-}$, is $- 1$. The amount of sodium ions $\mathrm{ν}_{+}$ produced by one mole of sodium chloride is 1 mol. The amount of chloride ions $\mathrm{ν}_{-}$ produced by one mole of sodium chloride is 1 mol. The electric charge on the sodium ions in one mole of sodium chloride is $\left(\mathrm{v}_{+} \, \mathrm{z}_{+} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{e} \right) \mathrm{~C}$ where $\mathrm{e}$ is the unit charge and $\mathrm{N}_{\mathrm{A}}$ is the Avogadro constant. The product $\left(\mathrm{N}_{\mathrm{A}} \, \mathrm{e}\right) \quad\left\{=[\mathrm{C}] .[\mathrm{mol}]^{-1}\right\}$ is the Faraday constant $\left{=\left[\mathrm{C} \mathrm{mol}^{-1}\right]\right\}$. Similarly the electric charge on the chloride ions in 1 mol of sodium chloride equals, $\left(\mathrm{v}_{-} \, \mathrm{z}_{-} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{e} \right) \mathrm{~C}$. The total electric charge on one mole of solid sodium chloride equals $\left[\left(\mathrm{v}_{+} \, \mathrm{z}_{+} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{e}\right)+\left(\mathrm{v}_{-} \, \mathrm{z}_{-} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{e}\right)\right] C$ which equals zero.
We make these points in order to highlight the fact that the total electric charge on 1 mol of sodium cations (in for example 53 g of common salt) is enormous, being 96 500 C. Very few laboratories can handle such enormous electric charges. Chemists cope because the electric neutrality condition always operates [1]. When we set down equations describing the properties of salt solutions we ensure that the electric neutrality condition is not violated. However when we turn to the task of developing molecular models for these systems we recognize the magnitude of the forces involved.
Footnotes
[1] One model of Utopia is a society when there are equal number of men and women. It is interesting to note that from the perspective of each male, the Utopian society has a majority of women. Similarly each woman lives in a male dominated society. Life is the same for ions.
1.10.16: Gibbs Energies- Salt Solutions- Born Equation
The Born Equation [1] is based on a BBB model, “brass balls in a bathtub” [2]. The solvent is treated as a dielectric continuum characterised at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ by its relative permittivity $\varepsilon_{r}$. The ions are treated as hard non-polarizable spheres, having radius $r_{j}$. The Born Equation describes the difference in thermodynamic properties of a mole of i-ions forming a perfect gas and a mole of j-ions in an ideal solution at temperature $\mathrm{T}$ and pressure $\mathrm{p}$. The calculation is not straightforward [3-5]. What emerges is the difference in Helmholtz energies (at constant $\mathrm{T}$ and $\mathrm{V}$) for a mole of $j$ ions in incompressible liquid phases. The calculated quantities refer to the energies associated with the electric fields over the limits $r_{j} \leq r \leq \infty$. In these terms the Born Equation describes the electrical part of the change in chemical potential on transferring an ion from the gas phase, permittivity $\varepsilon_{0}$, to a solvent, relative (electric) permittivity $\varepsilon_{r}$. In effect the Born Equation yields parameters characterizing the difference between the properties of one mole of $j$ ions in ideal systems having equal concentrations at fixed $\mathrm{T}$ and $\mathrm{p}$ [6].
$=-N_{A} \,\left(z_{j} \, e\right)^{2} \,\left[1-\left(1 / \varepsilon_{r}\right)\right] / 8 \, \pi \, r_{j} \, \varepsilon_{0}$
Similarly for transfer of one mole of $j$ ions from an ideal solution in solvent $\mathrm{s}_{1}$ to an ideal solution in solvent $\mathrm{s}_{2}$, the transfer chemical potential is given by the Born Equation assuming $r_{j}$ is independent of solvent.
$\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \,\left[\left(1 / \varepsilon_{\mathrm{r}}(\mathrm{s} 2)\right)-\left(1 / \varepsilon_{\mathrm{r}}(\mathrm{s} 1)\right)\right] / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}$
Many attempts have been made to modify the Born Equation in order to attain agreement between theory and measured thermodynamic ionic properties, particularly in the case of aqueous salt solutions. A common concern is the extent to which near-neighbor water molecules form an electrostricted layer [7] around ions in solution, namely a layer of solvent molecules having dielectric properties which differ from those of the pure solvent at the same temperature and pressure [8 - 11]. A common concern in this subject is the definition of the ionic radius for a given ion [12 - 16]. There is no agreement concerning a set of ‘absolute’ ionic radii. As Conway pointed out in 1966, ‘ …theories .. based on the Born equation seem to have reached an asymptotic level of usefulness..’ [17].
Nevertheless correlations involving thermodynamic properties of salt solutions play an important role.
Standard partial molar entropies for alkali metal halides in various solvents are linear functions of the corresponding entropies in aqueous solution [18]. A similar correlation is reported for ionic entropies in mixed aqueous solvents and the corresponding entropies in aqueous solutions [19]. With reference to enthalpies, the analysis also suffers from the fact that the ionic radius is sensitive to temperature [20].
Footnotes
[1] M. Born, Z. Phys., 1920, 1, 45.
[2] H. S. Frank quoted by H. L. Friedman, J. Electrochem. Soc., 1977, 124, 421c.
[3] H. S. Frank, J. Chem. Phys., 1955, 23, 2023.
[4] J. E. Desnoyers and C. Jolicoeur, Modern Aspects of Electrochem, ed. B. Conway and J.O’M. Bockris.1969, 5,1.
[5] J. E. Desnoyers, R. E. Verrall and B. E. Conway, J. Chem. Phys., 1965,43, 243.
[6]
\begin{aligned} & Delta(\mathrm{pfg} \rightarrow \mathrm{s} \ln ) \mu_{\mathrm{j}}=\left[\mathrm{mol}^{-1}\right] \,[\mathrm{C}]^{2} \,\{[1]-[1]\} /[1] \,[1] \,[\mathrm{m}] \,\left[\mathrm{Fm}^{-1}\right]= \ &{[\mathrm{mol}]^{-1} \,\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] /[\mathrm{m}] \,\left[\mathrm{A}^{2} \mathrm{~s}^{4} \mathrm{~kg}^{-1} \mathrm{~m}^{-3}\right]=\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{kg} \mathrm{m}^{2} \mathrm{~s}^{-2}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]} \end{aligned}
[7] M. H. Abraham, E. Matteoli and J. Liszi, J. Chem. Soc. Faraday Trans.I, 1983, 79,2781; and references therein.
[8] D. R. Rosseinsky, Chem. Rev.,1965,65,467; and references therein.
[9] A. A. Rashin and B. Hornig, J. Phys. Chem., 1985, 89,5588.
[10] T. Abe, Bull. Chem. Soc. Jpn.,1991, 64,3035.
[11] S. Goldman and R.G.Bates, J.Am.Chem.Soc.,1972,94,1476.
[12] L. Pauling, Nature of the Chemical Bond, Cornell University Press, Ithaca, 3rd edn., 1960, chapter 8.
[13] M. Bucher and T. L. Porter, J. Phys. Chem.,1986,90,3406; and references therein.
[14] M. Salomon, J.Phys.Chem.,1970,74,2519.
[15] K. H. Stern and E. S. Amis, Chem. Rev.,1959,59,1.
[16] Y. Marcus, Chem. Rev.,1988,88,1475.
[17] B. E. Conway, Annu. Rev. Phys. Chem.,1966,17,481.
[18] C. M. Criss, R. P. Held and E. Luksha, J.Phys.Chem.,1968,72,2970.
[19] F. Franks and D. S .Reid, J. Phys.Chem.,1969,73,3152.
[20] B. Roux, H.-A. Yu and M. Karplus, J. Phys. Chem.,1990, 94,4683.
1.10.17: Gibbs Energies- Salt Solutions- Lattice Models
Lattice models for salt solutions have attracted and continue to attract interest [1]. Ions in a salt solution are regarded as occupying lattice sites, the lattice parameter increasing as a solution is diluted; solvent molecules occupy the interstices of the lattice. This model for salt solutions generates interest because the distribution of ions about a central reference j-ion is therefore known. This theory requires that $\ln \gamma_{\pm}$ is a linear function of $\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{1 / 3}$ for salt-i; the cube-root law. This dependence is observed for reasonably concentrated salt solutions [2]. Unfortunately convincing evidence for lattice structures is not forthcoming. For example, the electrical conductivities of salt solutions cannot be understood in terms of lattice structures.
Footnotes
[1]
1. J. C. Ghosh, J. Chem. Soc., 1918, 449, 627, 707, 790.
2. H. S. Frank and P. T. Thompson, in Structure of Electrolytic Solutions, ed. W. J. Hamer, Wiley, New York, 1959, p.113.
3. J. E. Desnoyers and B. E. Conway, J. Phys. Chem., 1964,68, 2305.
4. L. Bahe, J. Phys. Chem., 1972,76, 1062, 1608.
5. C. W. Murphy, J. Chem. Soc. Faraday Trans. 2, 1982,78, 881.
6. B. N. Ghosh, J. Ind. Chem. Soc., 1983,60, 141, 607; 1981,58, 675; 1984,61, 213.
7. I. Horsak and I. Slama, Collect. Czech. Chem. Commum., 1987,52, 1672.
[2] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edition revised,1965, pp. 226
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.15%3A_Gibbs_Energies-_Salt_Solutions-_Electric_Neutrality.txt
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The electric potential $\phi_{j}$ (at ion-$j$; ion-ion interaction) describes the electric potential at a given reference $j$-ion arising from all other $i$-ions in solution. The contribution to the chemical potential of one mole of $j$-ions is obtained using the Guntleberg charging process. Thus,
$\Delta \mu_{\mathrm{j}}(\text { ion }-\mathrm{j} \text {; ion }-\text { ion int eractions })=\int_{0}^{\mathrm{z}_{\mathrm{j}} \,{ }_{\mathrm{e}}} \varphi_{\mathrm{j}}(\text { at ion }-\mathrm{j} \text {; ion }-\text { ion }) \, \mathrm{d}\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)$
Hence[1],
$\Delta \mu_{\mathrm{j}}(\text { ion }-\mathrm{j} ; \text { ion }-\text { ion int eractions })=-\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \, \mathrm{N}_{\mathrm{A}}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}}$
The charge-charge interactions are the only source of deviations in the properties of a given solution from ideal. Then for ion-$j$,
$\ln \left(\gamma_{\mathrm{j}}\right)=\Delta \mu_{j}(\text { ion }-\mathrm{j}<-\longrightarrow>\text { ion atmos. }) / \mathrm{R} \, \mathrm{T}$
Hence for the ionic activity coefficient $\gamma_{j}$,
$\ln \left(\gamma_{\mathrm{j}}\right)=-\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \, \mathrm{N}_{\mathrm{A}}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}}$
Ionic activity coefficients have no practical significance. We require an equation for the mean ionic activity coefficient for a salt in solution. For one of mole of salt in solution,
$v \, \ln \left(\gamma_{\pm}\right)=v_{+} \, \ln \left(\gamma_{+}\right)+v_{-} \, \ln \left(\gamma_{-}\right)$
or,
$\ln \left(\gamma_{\pm}\right)=\frac{1}{\left(v_{+}+v_{-}\right)}\left[v_{+} \, \ln \left(\gamma_{+}\right)+v_{-} \, \ln \left(\gamma_{-}\right)\right]$
Hence
$\ln \left(\gamma_{\pm}\right)=-\frac{1}{\left(v_{+}+v_{-}\right)} \,\left[v_{+} \, z_{+}^{2}+v_{-} \, z_{-}^{2}\right] \,\left[\frac{e^{2} \, N_{A}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, R \, T} \, \frac{\kappa}{(1+\kappa \, a)}\right]$
Or[2,3],
$\ln \left(\gamma_{\pm}\right)=-\left[\frac{\left|z_{+} \, z_{-}\right| \, e^{2} \, N_{A}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{t} \, R \, T} \, \frac{K}{(1+K \, a)}\right]$
In this connection the distance ‘a’ characterises both cations and anions in the salt. In the Debye-Huckel Limiting Law (DHLL) the term $(1+\kappa \, a)$ is approximated to unity thereby assuming that $(1>>K \, a)$. Then using the term $\mathrm{S}_{\gamma}$, for a 1:1 salt equation (h) is rewritten as follows.
$\ln \left(\gamma_{\pm}\right)=-\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} /\left[1+\mathrm{b} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\right]$
where
$\mathrm{b}=\beta \, \mathrm{a}$
and [4]
$\beta=\left[\frac{2 \, \mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}^{2} \, \rho_{1}^{*}(\ell) \, \mathrm{m}^{0}}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2}$
For aqueous solutions at ambient pressure and $298.15 \mathrm{~K}$, $\beta=3.285 \times 10^{9} \mathrm{~m}^{-1}$ [5].
The quantity $\mathrm{b}$ depends on a distance parameter [6] ‘a’ which characterises salt $j$ and reflects the role of repulsion between ions in determining the chemical potentials of a salt in solution. Hence with increase in distance ‘a’ so the denominator increases and $\ln \left(\gamma_{j}\right)$ is not so strongly negative as predicted by the DHLL. For large ‘a’ and high ionic strengths, the salts are not stabilised to the extent required by the DHLL. The integrated form of the Gibbs-Duhem equation yields an equation for $(\phi-1)$ in terms of molality $\mathrm{m}_{j}$ [7]. Thus,
$\mathrm{m}_{\mathrm{j}} \,(1-\phi)=-\int_{0}^{\mathrm{m}(\mathrm{j})} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\pm}\right)$
Hence, [6]
$(1-\phi)=\left[\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} / 3\right] \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{-1} \, \sigma(\mathrm{x})$
and
$\sigma(x)=\left(3 / x^{3}\right) \,\left[(1+x)-(1+x)^{-1}-2 \, \ln (1+x)\right]$
Then the excess molar Gibbs energy for a solution containing a 1:1 salt is given by equation (o).
$\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{S}_{\gamma} \, \mathrm{m}_{\mathrm{j}}^{3 / 2} \,\left(\mathrm{m}^{0}\right)^{-1} \,\left[\sigma(\mathrm{x}) / 3-(1+\mathrm{x})^{-1}\right]$
Footnotes
[1]
\begin{aligned} \Delta \mu_{j}(---) &=\frac{[\mathrm{A} \mathrm{s}]^{2} \,\left[\mathrm{mol}^{-1}\right]}{[1] \,\left[\mathrm{F} \mathrm{m}^{-1}\right] \,[1]} \, \frac{\left[\mathrm{m}^{-1}\right]}{\left\{[1]+\left[\mathrm{m}^{-1}\right] \,[\mathrm{m}]\right\}} \ &=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{mol}^{-1}\right]}{\left[\mathrm{As} \mathrm{} \mathrm{V}^{-1}\right]}=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{mol}^{-1}\right]}{\left[\mathrm{As} \mathrm{As} \mathrm{} \mathrm{J}^{-1}\right]}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \end{aligned}
[2] Condition of electric neutrality; $v_{+} \, z_{+}=-v_{-} \, z_{-}$ or, $v_{-}=-v_{+} \, z_{+} / z_{-}$
Then,
\begin{aligned} & \frac{1}{\left(v_{+}+v_{-}\right)} \,\left[v_{+} \, z_{+}^{2}+v_{-} \, z_{-}^{2}\right] \ =& {\left[\frac{1}{v_{+}-v_{-} \, z_{+} / z_{-}}\right] \,\left[v_{+} \, z_{+}^{2}-v_{+} \, z_{+} \, z_{-}\right]=\frac{z_{-}}{\left(z_{-}-z_{+}\right)} \,\left[z_{+}^{2}-z_{+} \, z_{-}\right] } \ =&-z_{+} \, z_{-}=\left|z_{+} \, z_{-}\right| \end{aligned}
[3] $\ln \left(\gamma_{\pm}\right)=\frac{[1] \,[\mathrm{A} \mathrm{s}]^{2} \,[\mathrm{mol}]}{[1] \,[1] \,\left[\mathrm{As} \mathrm{} \mathrm{J}^{-1} \mathrm{As}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]} \, \frac{\left[\mathrm{m}^{-1}\right]}{\left\{1+\left[\mathrm{m}^{-1}\right] \,[\mathrm{m}]\right\}}=[1]$
[4]
\begin{aligned} \beta=\left\{[1] \,[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]^{2} \,\left[\mathrm{kg} \mathrm{m}^{-3}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\right\}^{1 / 2} \ /\left\{\left[\mathrm{J}^{-1} \mathrm{C}^{2} \mathrm{~m}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]\right\}^{1 / 2}=\left[\mathrm{m}^{-1}\right] \end{aligned}
[5] For DH parameters for aqueous solutions to high $\mathrm{T}$ and $\mathrm{p}$, see D. J. Bradley and K. S. Pitzer,J. Phys.Chem.,1979,83,1599;1983;87,3798.
[6] Parameter ‘a’ is sometimes called ‘ion size’ . But as S. Glasstone [Introduction to Electrochemistry, D.van Nostrand, New Jersey, 1943, page 145, footnote] points out ‘the exact physical significance cannot be expressed precisely’. Nevertheless an important consideration is the relative sizes of ions and solvent molecules; B. E. Conway and R. E. Verrall, J.Phys.Chem.,1966, 70,1473.
[7] From equation (i) when by definition $\mathrm{x}=\mathrm{b} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}$ and $\mathrm{k}=\left|\mathrm{Z}_{+} \, \mathrm{Z}_{-}\right| \, \mathrm{S}_{\gamma} / \mathrm{b}$
Hence,
\begin{aligned} &\ln \left(\gamma_{\pm}\right)=-(b \, k) \,(x / b) /(1+x)=-k \, x /(1+x) \ &\mathrm{d} \ln \left(\gamma_{\pm}\right)=-\mathrm{k} \,\left\{[1 /(1+\mathrm{x})]-\left[\mathrm{x} /(1+\mathrm{x})^{2}\right]\right\} \, \mathrm{dx}=-\mathrm{k} \,\left\{[1+\mathrm{x}-\mathrm{x}] /[1+\mathrm{x}]^{2}\right\} \, \mathrm{dx} \ &\mathrm{d} \ln \left(\gamma_{\pm}\right)=-\mathrm{k} \, \mathrm{dx} /[1+\mathrm{x}]^{2} \end{aligned}
Therefore,
$(1-\phi) \, m_{j}=-\int_{0}^{x} m_{j} \,\left\{-k /(1+x)^{2}\right\} \, d x$
Or, $(1-\phi)=\left(k / x^{2}\right) \, \int_{0}^{x}\left\{x^{2} /(1+x)^{2}\right\} \, d x$
Standard integral:
\begin{aligned} &\int_{0}^{x}\left\{x^{2} /(a \, x+b)^{2}\right\} \, d x= \ &\left\{(a \, x+b) / a^{3}\right\}-\left\{b^{2} / a^{3} \,(a \, x+b)\right\}-\left(2 \, b / a^{3}\right) \, \ln (a \, x+b) \end{aligned}
With $a=b=1$,
$\int_{0}^{x}\left\{x^{2} /(1+x)^{2}\right\} \, d x=(1+x)-[1 /(1+x)]-2 \, \ln (1+x)$
Thus,
$(1-\phi)=(k \, x / 3) \,\left\{\left(3 / x^{3}\right) \,\left[(1+x)-(1+x)^{-1}-2 \, \ln (1+x)\right]\right.$
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.18%3A_Gibbs_Energies-_Salt_Solutions-_Debye-Huckel_Equation.txt
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According to the Debye-Huckel analysis the mean ionic activity coefficient is given by equation (a).
$\ln \left(\gamma_{\pm}\right)=-\left[\frac{\left|z_{+} \, z_{-}\right| \, e^{2} \, N_{A}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{t} \, R \, T} \, \frac{K}{(1+K \, a)}\right]$
In the DHLL the term $(1+\kappa \, a)$ is approximated to unity thereby assuming that $(1>>\kappa \, a)$. By definition
$\mathrm{S}_{\mathrm{\gamma}}=\left[\frac{2 \, \pi \, \mathrm{N}_{\mathrm{A}} \, \mathrm{M}_{1} \, \mathrm{m}^{0}}{\mathrm{~V}_{1}^{*}(\ell)}\right]^{1 / 2} \,\left[\frac{\mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{3 / 2}$
Then[1]
$\ln \left(\gamma_{\pm}\right)=-\left|z_{+} \, z_{-}\right| \, S_{\gamma} \,\left(m_{j} / m^{0}\right)^{1 / 2}$
The practical osmotic coefficient for quite dilute salt solutions is also a linear function of $\left(\mathrm{m}_{j}\right)^{1 / 2}. Indeed \(\phi$ and $\ln \left(\gamma_{\pm}\right)$ are simply related.
$1-\phi=-(1 / 3) \, \ln \left(\gamma_{\pm}\right)$
Footnotes
[1] From,
$\mathrm{S}_{\gamma}=\frac{\mathrm{e}^{3} \,\left[2 \, \mathrm{N}_{\mathrm{A}} \, \rho_{1}^{*}(\ell)\right]^{1 / 2}}{8 \, \pi \,\left[\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}\right]^{3 / 2}}$
Then, $\frac{1}{\pi}=\frac{\pi^{1 / 2}}{\pi^{3 / 2}}$ and $\frac{1}{8}=\frac{1}{4^{3 / 2}}$
1.10.20: Gibbs Energies- Salt Solutions- DHLL- Derived Parameters
Granted that the DHLL forms a starting point for understanding the role of ion-ion interactions in aqueous solutions, we use the DHLL to explain the impact of these interactions on related properties such as volumes and enthalpies. We write the Debye-Huckel coefficient as a function of three variables:
1. the molar volume of the solvent $\mathrm{V}_{1}^{*}(\ell)$,
2. the relative permittivity of the solvent, $\varepsilon_{r}$ and
3. the temperature.
Here we take account of the fact that $\mathrm{V}_{1}^{*}(\ell)$ and $\varepsilon_{r}$ depend on both temperature and pressure.
$S_{\gamma}=\left[\frac{2 \, \pi \, \mathrm{N}_{\mathrm{A}} \, \mathrm{M}_{1} \, \mathrm{m}^{0}}{\mathrm{~V}_{1}^{*}(\ell)}\right]^{1 / 2} \,\left[\frac{\mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{4 \, \pi \, \varepsilon^{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{3 / 2}$
$\mathrm{S}_{\gamma}$ is written in the following form.
$\mathrm{S}_{\gamma}=\mathrm{E} \,\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1 / 2} \,\left(\varepsilon_{\mathrm{r}}\right)^{-3 / 2} \,(\mathrm{T})^{-3 / 2}$
where
$\mathrm{E}=\left[2 \, \pi \, \mathrm{N}_{\mathrm{A}} \, \mathrm{M}_{1} \, \mathrm{m}^{0}\right]^{1 / 2} \,\left[\frac{\mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{4 \, \pi \, \varepsilon^{0} \, \mathrm{R}}\right]^{3 / 2}$
Hence[1]
$\mathrm{S}_{\gamma}=\mathrm{E} \, \mathrm{F}$
where
$\mathrm{F}=\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1 / 2} \,\left(\varepsilon_{\mathrm{r}}\right)^{-3 / 2} \,(\mathrm{T})^{-3 / 2}$
In terms of our interest in the dependence of $\mu_{\mathrm{j}}(\mathrm{aq})$ for salt $j$ on temperature leading to partial molar enthalpies we require $\left[\partial \mathrm{S}_{\gamma} / \partial \mathrm{T}\right]_{\mathrm{p}}$ which is calculated using the dependences of both $\mathrm{V}_{1}^{*}(\ell)$ and $\varepsilon_{r}$ on temperature yielding $(\partial \mathrm{F} / \partial \mathrm{T})_{\mathrm{p}}$. For partial molar isobaric heat capacities we require the second differential $\left(\partial^{2} \mathrm{~F} / \partial \mathrm{T}^{2}\right)_{\mathrm{P}}$. The predicted dependence by DHLL of the partial molar volume $\mathrm{V}_{\mathrm{j}}(\mathrm{aq})$ on salt molality involves the derivative $(\partial \mathrm{F} / \partial \mathrm{p})_{\mathrm{T}}$.
Calculations are considerably helped using a PC in conjunction with equations describing the $\mathrm{T} - \mathrm{p}$ dependences of $\mathrm{V}_{1}^{*}(\ell)$ and $\varepsilon_{r}$.
Footnotes
[1] $\mathrm{E}=\left[[1] \,[1] \,\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\right]^{1 / 2} \,\left[\frac{[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]}{[1] \,[1] \,\left[\mathrm{C}^{2} \mathrm{~J}^{-1} \mathrm{~m}^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right.}\right]^{3 / 2}$
or, $\mathrm{E}=[\mathrm{mol}]^{-1 / 2} \,[\mathrm{m}]^{3 / 2} \,[\mathrm{K}]^{3 / 2}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{1 / 2} \,[\mathrm{K}]^{3 / 2}$
and $\mathrm{F}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1 / 2} \,[1]^{-3 / 2} \,[\mathrm{K}]^{-3 / 2}$
Hence, $\mathrm{S}_{\gamma}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{1 / 2} \,[\mathrm{K}]^{3 / 2} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1 / 2} \,[1]^{-3 / 2} \,[\mathrm{K}]^{-3 / 2}=[1]$
1.10.21: Gibbs Energies- Salt Solutions- DHLL- Empirical Modifications
The success of equations based on the Debye-Huckel equations is often modest and so attempts are made to describe quantitatively the dependences of $\phi$, $\ln \left(\gamma_{\pm}\right)$ and $\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})$ on $\mathrm{m}_{j}$ to higher molalities. In most cases attempts are made to moderate the stabilization of the salt with increasing ionic strength. The obvious procedure centres on incorporating a denominator into the DHLL as illustrated by the Guntleberg equation.
$\ln \left(\gamma_{\pm}\right)=-\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} /\left\{1+\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\right\}$
The Guggenheim Equation starts with equation (a) and adds a further term, linear in ionic strength.
$\ln \left(\gamma_{\pm}\right)=-\left[\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} /\left\{1+\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\right\}\right]+\mathrm{b} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)$
The quantity ‘b’ is characteristic of the salt. Another obvious development uses the same approach in the context of the DHLL. An interesting equation takes the following form for the solution containing a salt $j$.
$\ln \left(\gamma_{\pm}\right)=-\left[\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\right]+\mathrm{B} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Here $\mathrm{B}$ describes the role of ion size and the impact of cosphere-cosphere interactions specific to a particular salt.
In most approaches, the starting point in an equation for $\ln \left(\gamma_{\pm}\right)$ as a function of ionic strength, the equation for the dependence of $\phi$ on ionic strength being obtained using the integral of equation (c). An interesting approach suggested by Bronsted starts out with a virial equation for $1 - \phi$ in terms of molality $\mathrm{m}_{j}$.
$1-\phi=\alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Hence [1]
$\ln \left(\gamma_{\pm}\right)=-3 \, \alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-2 \, \beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)$
Footnote
[1] From
\begin{aligned} &\ln \left(\gamma_{\pm}\right)=(\phi-1)+\int_{0}^{m_{j}}(\phi-1) \, \mathrm{d} \ln m_{j} \ &\ln \left(\gamma_{\pm}\right)=(\phi-1)-\int_{0}^{\mathrm{m}_{\mathrm{j}}}\left[\left\{\alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right\} / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}} \ &\left.\ln \left(\gamma_{\pm}\right)=(\phi-1)-\int_{0}^{m_{j}}\left[\left\{\alpha / m_{j} \, m^{0}\right)^{1 / 2}\right\}+\left\{\beta / m^{0}\right\}\right] \, d_{j} \ &\ln \left(\gamma_{\pm}\right)=(\phi-1)-\left[2 \, \alpha \,\left(m_{j} / m^{0}\right)^{1 / 2}+\beta \,\left(m_{j} / m^{0}\right)\right]_{0}^{m_{j}} \ &\ln \left(\gamma_{\pm}\right)=-\alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)-2 \, \alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \ & \ln \left(\gamma_{\pm}\right)=-3 \, \alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-2 \, \beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \end{aligned}
1.10.22: Gibbs Energies- Salt Solutions- Solvent
A given salt solution contains a 1:1 salt $j$ (e.g. $\mathrm{NaCl}$) in which the salt, chemical substance $j$, completely dissociates into ions. In other words, the total molality of solutes equals $2 \, m_{j}$. By definition the chemical potential of water in this aqueous solution, $\mu_{1}(\mathrm{aq})$ (at fixed temperature and pressure, the latter being ambient and hence close to the standard pressure $\mathrm{p}^{0}$) is given by equation (a).
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
For the corresponding ideal solution, $\phi = 1.0$ at all $\mathrm{T}$ and $\mathrm{p}$, Hence,
$\mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\ell)-2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
Just as for solutions containing neutral solutes, the minus sign in equation (b) means that added salt stabilizes the solvent in an ideal solution; $\mu_{1}(\mathrm{aq} ; \mathrm{id})<\mu_{1}^{*}(\ell)$.
For water in an aqueous salt solution containing salt $j$, molality $\mathrm{m}_{j}$, where each mole of salt forms $v$ moles of ions with complete dissociation, the chemical potential of the solvents is given by equation (c).
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}\left(\ell ; \mathrm{p}^{0}\right)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}(0)}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}$
For the ideal dilute solution, $\phi =1.0$. Here $\mu_{1}^{*}\left(\ell, \mathrm{p}^{0}\right)$ is the standard chemical potential of water at temperature $\mathrm{T}$. Alternatively we may switch the reference chemical potential for the solvent to the pure liquid at the same pressure [1,2].
$\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}$
Footnotes
[1] For relevant Tables see;’ R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd edn.(revised), Butterworths, London, 1965, Appendix 8.
[2] The impact of salts on osmotic coefficients is illustrated by the properties of aqueous solutions containing alkylammonium salts.
1. S. Lindenbaum, J. Phys.Chem.,1971, 75,3733; and references therein.
2. G. E. Boyd, A. Schwartz and S. Lindenbaum, J. Phys.Chem.,1966, 70, 821; and references therein.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.19%3A_Gibbs_Energies-_Salt_Solutions-_Debye-Huckel_Limiting_Law.txt
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A given salt solution contains a single salt $j$ which completely dissociates to form $ν$ moles of ions from one mole of salt. Then the chemical potential of the salt $j$ in aqueous solution at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ is given by equation (a).
$\mu_{j}(a q)=\mu_{j}^{0}(a q)+v \, R \, T \, \ln \left(Q \, m_{j} \, \gamma_{\pm} / m^{0}\right)+\int_{p^{0}}^{p} V_{j}^{\infty}(a q) \, d p$
Here $\mu_{\mathrm{j}}^{0}(\mathrm{aq})$ is the chemical potential of the salt in solution at the same temperature and the standard pressure where molality $\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}$ and mean ionic activity coefficient $\gamma_{\pm} = 1$. The chemical potential of water in aqueous solution at temperature $\mathrm{T}$ and pressure $\mathrm{p}$ is given by equation (b) where $\phi$ is the practical osmotic coefficient [1-3],
$\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}$
If we confine our attention to the properties of solutions at ambient pressure (which is very close to the standard pressure) then we can ignore the integrals in equations (a) and (b). Hence the Gibbs energy of the solution at the same $\mathrm{T}$ and $\mathrm{p}$ prepared using $1 \mathrm{~kg}$ of water is given by equation (c).
\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right] \end{aligned}
As for solutions containing neutral solutes we cannot put a number value to $\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)$. If the properties of this salt solution are in fact ideal (in a thermodynamic sense) then $\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}: \mathrm{id}\right)$ is given by equation (d).
\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=&\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{\mathrm{l}} \, \mathrm{m}_{\mathrm{j}}\right] \ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right] \end{aligned}
Hence in the case where $j = \mathrm{~NaCl}$, $ν = 2$ and $\mathrm{Q} = 1$. In the next stage we use differences between $\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)$ and $\mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)$ to define excess Gibbs energies for a solution prepared using $1 \mathrm{~kg}$ of water. Then
$\mathrm{G}^{\mathrm{E}}=\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)$
For salt $j$,
$\mathrm{G}^{\mathrm{E}}=\mathrm{V} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \left(\gamma_{\pm}\right)\right]$
According to the Gibbs-Duhem for a solution at constant temperature and constant pressure,
$\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0$
Hence for salt $j$,
\begin{aligned} &\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \ &\quad+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}
We are concerned with the dependence of chemical potential on the molality of salt. Thus the amount of solvent and, for the salt, both $\mathrm{Q}$ and $v$ are fixed. Hence
$\mathrm{d}\left[-\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right]=0$
Hence,
$-\phi \, d m_{j}-m_{j} \, d \phi+m_{j} \, d m_{j} / m_{j}+m_{j} \, d \ln \left(\gamma_{t}\right)=0$
Then,
$(\phi-1) \, d m_{j}+m_{j} \, d \phi=m_{j} \, d \ln \left(\gamma_{\pm}\right)$
Hence we obtain an equation for $\ln \left(\gamma_{\pm}\right)$ in terms of the dependence of $(\phi - 1)$ on molality bearing in mind that $\ln \left(\gamma_{\pm}\right)$ equals zero and $\phi$ equals 1 at ‘$\mathrm{m}_{j} = 0$’.
$\ln \left(\gamma_{\pm}\right)=(\phi-1)+\int_{0}^{m_{j}}(\phi-1) \, d \ln \left(m_{j}\right)$
From equation (f), the dependence of $\mathrm{G}^{\mathrm{E}}$ on $\mathrm{m}_{j}$ is given by equation (m).
\begin{aligned} &(1 / \mathrm{V} \, \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}} \ &=\left[1-\phi+\ln \left(\gamma_{\pm}\right)\right]-\mathrm{m}_{\mathrm{j}} \,\left(\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\pm}\right) / \mathrm{dm}_{\mathrm{j}}=0 \end{aligned}
But according to equation (i),
$-\phi-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}+1+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\pm}\right) / \mathrm{dm}_{\mathrm{j}}=0$
Hence[4],
$\ln \left(\gamma_{\pm}\right)=(1 / \mathrm{V} \, \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}$
Footnotes
[1] Compilation of Data for polyvalent electrolytes; R. N. Goldberg, B. R. Staples, R. L. Nuttall and R. Arbuckle, NBS Special Publication 485, 1977.
[2] Thermal Properties of Aqueous Univalent –Univalent Electrolytes, V.B.Parker, NBS, 2, 1965.
[3] J.-L. Fortier and J. E. Desnoyers, J. Solution Chem.,1976,5,297.
[4]
$\ln \left(\gamma_{\pm}\right)=\left[\frac{1}{[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}\right] \,\left[\frac{\mathrm{J} \mathrm{kg}^{-1}}{\mathrm{~mol} \mathrm{~kg}^{-1}}\right]=[1]$
1.10.24: Gibbs Energies- Salt Solutions- Pitzer's Equations
The Debye-Huckel treatment of the properties of salt solutions is based on a linearization of the Botzmann Equation leading to an equation for the radial distribution function, $\mathrm{g}_{i j} (\mathrm{r})$. If a further term is taken into the expansion, the equation for $\mathrm{g}_{i j} (\mathrm{r})$ takes the following form [1].
$g_{i j}(r)=1-q_{i j}+\left(q_{i j}^{2} / 2\right)$
When equation (a) was tested against the results of a careful Monte Carlo calculation the conclusion was drawn that the three-term equation is good approximation [2]. The result is a set of equations for both the practical osmotic coefficient $\phi$ and mean ionic activity coefficient for the salt in a solution having ionic strength $\mathrm{I}$ [3,4]. The theory has been extended to consider the properties of salt solutions at high $\mathrm{T}$ and $\mathrm{p}$ [5,6]. In fact key parameters in Pitzer equations covering extensive ranges of $\mathrm{T}$ and $\mathrm{p}$ have been extensively documented [7]. The Pitzer treatment has been extended to a consideration of the properties of mixed salt solutions [8].
Footnotes
[1] K. S. Pitzer, Acc. Chem.Res.,1977,10,371.
[2] D. N. Card and J. P. Valleau, J. Chem. Phys.,1970,52,6232.
[3] K. S. Pitzer, J.Phys.Chem.,1973,77,268.
[4] Activity and Osmotic Coefficients for
1. 1:1 salts: K. S. Pitzer and G. Mayorga, J.Phys.Chem.,1973,77,2300.
2. For 2:2 salts: K. S. Pitzer and G. Mayorga, J. Solution Chem., 1974, 3,539.
3. For 3:2 salts etc;
1. K. S. Pitzer and L. V. Silvester, J.Phys.Chem.,1978,82,1239.
2. L. F. Silvester and K. S. Pitzer, J. Solution Chem.,1978,7,327.
4. K. S. Pitzer, J. R. Peterson and L. E. Silvester, J. Solution Chem.,1978,7,45.
[5]
1. R. C. Phutela, K. S. Pitzer and P. P. S. Saluja, J. Chem. Eng. Data, 1987, 32,76.
2. H. F. Holmes and R. E. Mesmer, J. Phys.Chem.,1983,87,1242.
[6] R. C. Phutela and K. S. Pitzer, J. Phys Chem.,1986, 90,895.
[7]
1. J. Ananthaswamy and G. Atkinson, J. Chem. Eng. Data,1984,29,81.
2. R. P. Beyer and B. R. Staples, J. Solution Chem.,1986,15,749.
3. P. P. S. Salija, K. S. Pitzer and R. C. Phutela, Can J.Chem.,1986,64,1328.
4. R. T. Pabalan and K. S. Pitzer, J. Chem. Eng. Data, 1988,33,354.
[8]
1. R. C. Phutela and K. S. Pitzer, J. Solution Chem.,1986,15,649.
2. A. Kumar, J. Chem. Eng. Data,1987,32,106.
3. K. S. Pitzer and J. J. Kim, J. Am. Chem.Soc.,1974,96,5701.
4. K. S. Pitzer and J. M. Simonson, J. Phys.Chem,.,1986,90,3005.
5. C. J. Downes and K.S. Pitzer, J. Solution Chem.,1976,5,389.
6. J. C. Peiper and K. S. Pitzer, J.Chem.Thermodyn.,1982,14,613.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.23%3A_Gibbs_Energies-_Salt_Solutions-_Excess_Gibbs_Energies.txt
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The Debye-Huckel equations form the starting point for detailed analyses of the properties of salt solutions. An interesting method examines excess thermodynamic properties of salt solutions and their dependence on salt molality [1]. For an aqueous solution containing a 1:1 salt in 1 kg of solvent, the excess Gibbs energy $\mathrm{G}^{\mathrm{E}}$ is given by equation (a).
$\mathrm{G}^{\mathrm{E}}=2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \gamma_{\pm}\right]$
The corresponding excess molar Gibbs energy is given by equation (b).
$\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=2 \, \mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \gamma_{\pm}\right]$
The corresponding excess molar enthalpy ${\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}$ is obtained from calorimetric data. Hence the molar excess entropy is calculated using equation (c).
$\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}-\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}}$
A common observation is that ${\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}$ and $\mathrm{T}.{\mathrm{S}_{\mathrm{m}}}^{\mathrm{E}}$ are more sensitive to both the molality $\mathrm{m}_{j}$ and the salt than is ${\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}$. For dilute salt solutions ${\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}} < 0$ as a consequence of charge-charge interactions between the ions leading to a stabilisation. Further ${\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}$ and $\mathrm{T}.{\mathrm{S}_{\mathrm{m}}}^{\mathrm{E}}$ are more sensitive to switching the solvent from $\mathrm{H}_{2}\mathrm{O}$ to $\mathrm{D}_{2}\mathrm{O}$ than is ${\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}$.
The dependence of ${\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}$ on salt at fixed $\mathrm{m}_{j}$ shows that in addition to charge-charge interactions there are further interactions which are characteristic of the ions in a given salt. So the suggestion is that even in the absence of charge-charge interactions the properties of the solution would not be ideal by virtue of cosphere - cosphere interactions along the lines suggested by Gurney. But there are two types of solutes in a simple 1:1 salt solution so we must consider in the analysis of these properties at least $\mathrm{g}_{++}$, $\mathrm{g}_{+-}$ and $\mathrm{g}_{--}$ pairwise ion-ion Gibbs energy interaction parameters. Consequently the analysis is not straightforward.
An interesting approach examines patterns in $\ln \left(\gamma_{\pm}\right)$ for a series of 1:1 salts at fixed molality $\mathrm{m}_{j}$, temperature and pressure [2-4]. A most dramatic change is observed for $\mathrm{Pr}_{4}\mathrm{N}^{+}$ salts (aq; $0.2 \mathrm{~mol kg}^{-1}$). Thus $\ln \left(\gamma_{\pm}\right)>\left(\ln \gamma_{\pm}(\mathrm{DHLL}-\text { calc })\right)$ for the fluoride salt ; i.e. a higher chemical potential than calculated simply on the basis of the DHLL-- a destabilisation. But $\ln \left(\gamma_{\pm}\right)>\left(\ln \gamma_{\pm}(\mathrm{DHLL}-\text { calc })\right)$ for the corresponding iodide salt; i.e. lower chemical potential than calculated simply on the grounds of the DHLL--- a stabilisation. The dependence of $\ln \gamma_{\pm}$ for $\mathrm{K}^{+}$, $\mathrm{Rb}^{+}$ and $\mathrm{Cs}^{+}$ on the anion $\mathrm{F}^{-}$, $\mathrm{Cl}^{-}$, $\mathrm{Br}^{-}$ and $\mathrm{I}^{-}$ is much more modest. The pattern signals the important role of hydrophobic-hydrophobic, hydrophilic-hydrophobic and hydrophilic-hydrophilic ion-ion interactions. Indeed there is considerable merit in the approach [4]. The pattern emerges in a comparison of salt effects on rates of hydrolysis in aqueous salt solutions [5]. This conclusion is supported by the observation that the dependence of $\ln \gamma_{\pm}$ on $\left(\mathrm{m}_{\mathrm{j}}\right)^{1 / 2}$ for $\left(\mathrm{HOCH}_{2} \mathrm{CH}_{2}\right)_{4} \mathrm{~N}^{+} \mathrm{Br}^{-}$ deviates from the DHLL pattern in a direction indicating a more hydrophilic character for the cation than in the cases of $\mathrm{Pr}_{4}\mathrm{N}^{+}$ and $\mathrm{Et}_{4}\mathrm{N}^{+}$ [6].
The patterns identified in $\ln \gamma_{\pm}$ signal that the ion-ion pair potential for ions in solution comprises several components. This recognition forms the basis of the treatment developed by Friedman and coworkers. The pair potential $\mathrm{u}_{i j}$ for two ions charge $\mathrm{z}_{\mathrm{i}}.\mathrm{e}$ and $\mathrm{z}_{\mathrm{j}}.\mathrm{e}$ in a solvent having relative permittivity $\varepsilon_{r}$ is expressed in the form shown in equation (d) [6].
$u_{i j}(r)=\left[\frac{\left(z_{i} \, e\right) \,\left(z_{j} \, e\right)}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{r} \, r}\right]+C O R_{i j}+C A V_{i j}+\mathrm{GUR}_{i j}$
The first term takes account of charge-charge interactions; i.e. the Coulombic term, COUL. The term $\mathrm{COR}_{i j}$ is a repulsive core potential , being a function of the sizes of the ions $i$ and $j$. The $\mathrm{CAV}_{i j}$ term takes account of a special effect arising from the interactions between ion + solvent cavities. The impact of cosphere overlap is taken into account by the Gurney potential, $\mathrm{GUR}_{i j}$. The pair potential is used in conjunction with McMillan-Mayer theory [8]. The $\mathrm{GUR}__{i j}$ term includes an adjustable parameter $\mathrm{A}_{i j}$, the change in Helmholtz energy, $\mathrm{F}$ when one mole of solvent in the overlap region returns to the bulk solvent. There are therefore three such terms, $\mathrm{A}_{++}$, $\mathrm{A}_{+-}$, and $\mathrm{A}_{--}$ for the three types of overlap. These terms are related to the corresponding volumetric $\mathrm{V}_{i j}$, entropy $\mathrm{S}_{i j}$ and energy $\mathrm{U}_{i j}$ terms. The analysis is slightly complicated by the fact that the derived thermodynamic functions refer to a solution in osmotic equilibrium with the solvent at the standard pressure; the MM state. Conversion is required to thermodynamic parameters for a solution of the same salt at the same molality at the standard pressure.
Footnotes
[1] Y.-C. Wu and H. L. Friedman, J.Phys.Chem.,1966,70,166.
[2] H. S. Frank, Z. Phys. Chem., 1965,228,364.
[3] H. S. Frank and A. L. Robinson, J.Chem.Phys.,1940,8,933; this paper concerns the dependence of partial molar entropies on composition of salt solutions; the analysis set the stage for subsequent developments in this subject.
[4] J. E. Desnoyers, M. Arel, G. Perron and C. Jolicoeur, J. Phys. Chem., 1969, 73, 3346.
[5] M. J. Blandamer, J. B. F. N. Engberts, J. Burgess, B. Clark and A. W. Hakin, J. Chem. Soc. Chem.Commun.,1985,414; see also apparent molar Cp; J. L. Fortier, P.-A. Leduc and J. E. Desnoyers, J. Solution Chem.,1974,3,323.
[6]
1. W-Y. Wen and S. Saito, J.Phys.Chem.,1965,69,3659.
2. S. Lindenbaum and G.E. Boyd, J. Phys.Chem.,1964,68,911.
[7] P. S. Ramanthan and H. L. Friedman, J.Chem.Phys.,1971,54,1086.
[8] W. G. McMillan and J. E. Mayer, J.Chem.Phys.,1945,13,276.
[9] H. L. Friedman and C. V. Krishnan, Ann. N. Y. Acad. Sci.,1973,204,79.
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textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/1.10%3A_Gibbs_Energies/1.10.25%3A_Gibbs_Energies-_Salt_Solutions-_Cosphere_-_Cosphere_Interactions.txt
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