# Law of Mass Action

Guldberg and Waage (1884) put forward the law of mass action.

It states that, “At constant temperature, the rate of a chemical reaction is directly proportional to the active mass of the reactant present at that time. If more than one reactant takes part in the reaction then the rate of the reaction is proportional to the product of the active masses of the reactants.”

Active mass means the concentration of the substance expressed in mole per litre or the pressure of a reacting gas in atmosphere. It is usually denoted oy putting the symbol of the substance in square bracket, e. g. , [X]. or $C_x$. $[X] = \dfrac{\text{Mass of X} (G)/ \text{Molecular mass of X g mole}^{-1}}{\text{Volume in litre}}$

Now we consider a general reversible reaction: $aA + bB \leftrightharpoons cC + Dd$

The rate of forward reaction = $k_1[A]^a [B]^b$

The rate of backward reaction = $k_2[C]^c [D]^d$

Where $K_1 \text{and} K_3$ are the respective rate constants.

At equilibrium: $K_1[A]^a [B]^b$ = $K_2[C]^c [D]^d$ $or \hspace{3mm} k = \dfrac{K_1}{K_2} = \dfrac{[C]^c [D]^d}{[A]^a [B]^b}$

Where ‘K’ is equilibrium constant

As stated above the active mass is expressed in terms of mole/ litre (for solutions) or partial pressure (for gases). Similarly, equilibrium constant K is expressed as $K_c or K_p$ for solutions or gases, respectively.

For the above reaction, KP and KC are related as follows: $K_p = \dfrac{[P_C]^c [P_D]^d}{[P_A]^a [P_B]^b}$

Since, PV = nRT $Or \hspace{3mm} P = \dfrac{n}{V}RT = CRT \left(\dfrac{n}{V} = \text{Concentration} \right)$ $K_P = \dfrac{[C]^c [RT]^c \times [D]^d [RT]^d}{[A]^a [RT]^a \times [B]^b [RT]^b}$ $= \dfrac{[C]^c [D]^d}{[A]^a [B]^b [RT]^{(c + d)- (a + b)}}$

Let, $\Delta = (c + d)- a(a + b) \\[3mm] K_p = K_c(RT)^{\Delta n}$

Where R = gas constant, T = absolute temperature and $\Delta n$ = [No. of moles of products - No. of moles of reactants].

Realation between kp and kc with their units

Value of $\Delta n : 0$

Relation $: K_p = K_c$

Units of $K_c$: No units

Units of $K_p$: No units

Value of $\Delta n$ : > 0 or (+) Ve

Relation $: K_p > K_c$

Units of $K_c$: $(mol L^{-1})^{\Delta n}$

Units of $K_p$: $(atm)^{\Delta n}$

Value of $\Delta n$ : < 0 or (-) ve

Relation $: K_p =

Units of $K_c$: $(L mol^{-1})^{\Delta n}$

Units of $K_p$: $(atm)^{-\Delta n}$

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