Kohlrausch Law and conductivity
For what condition do the equations 1 and 2 apply? I think that 2 is for strong electrolytes at low concentration though im not sure. So what about at high concentrations of strong electrolytes? What about weak electrolytes? 2) When would you use equation 1, when would you use equation 2? Are they equivalent?


Equation 1 defines the molar conductivity:
$$[6]\;\;\mathrm{Λ_{m}} = \frac{\kappa}{c}$$
It is generally applicable. You use it to compute the molar conductivity from experimental data (conductivity and molar concentration of electrolyte). Here $\kappa$ is the conductivity. For an ideal measurement cell and electrolyte it is defined as
$$\kappa = \frac{l}{RA}$$
where l and A describe the geometry of the cell. A more general definition is possible for an arbitrary geometry or sample composition.
Equation 2 is called Kohlrausch's law and sure enough was derived by Kohlrausch based on the behaviour of strong electrolytes, to which it commonly applies at low concentrations:
$$[7]\;\;\mathrm{Λ_{m}} = \mathrm{Λ_{M}^\circ} - k\sqrt{c}$$
You might fit this equation to concentration versus molar conductivity data to determine limiting molar conductivities $\mathrm{Λ_{m}^\circ}$, and from that obtain the limiting molar conductivities of the component ions. The parameter k (also obtained from the fit) may be modeled by applying the Debye-Huckel-Onsager theory.
For weak electrolytes one might invoke Ostwald's dilution law.


In what situation should you use equation [8]?


Equation [8] is the law of independent migration of ions:
$$[8]\;\;\mathrm{Λ_{M}}^0 = \lambda{_{M+}}^0+\lambda{_{A-}}^0$$
where the $\lambda$ are limiting ionic conductivities, properties of individual ionic species in a specific solvent at infinite dilution. As written the equation applies to a solution containing a single 1:1 electrolyte. More generally $\Lambda_{M}^0 = \sum_i \nu_i\lambda_i^0$, where the $\nu$ are stoichiometric coefficient of individual ions in the electrolyte solution. The equation allows prediction of the limiting conductivity from the electrolyte composition provided limiting conductivities have been compiled. You can also use the equation to parse limiting conductivities of individual components from data for different simple electrolytes.


One of my questions is to calculate the concentration of impurities in tap water, using the conductivity measurement. How can I do that when I have more than one unknown in the equation?


Conductivity measurements can be used to estimate the concentration of total dissolved solids in tap water. Typically standard conversions factors are used to convert from conductivity to TDS, the conversion factor depending on the water source, as explained in the wikipedia:

The relationship of TDS and specific conductance of groundwater can be approximated by the following equation:


TDS = keEC


where TDS is expressed in mg/L and EC is the electrical conductivity in microsiemens per centimeter at 25 °C. The correlation factor ke varies between 0.55 and 0.8.2.

There are also online calculators (such as from lenntech) that convert between TDS and conductivity. However if you have performed a water composition analysis you can combine this with tabulated conductivities for different ions, as the lenntech website explains:

The EC determination can be done in different ways. One possibility is the usage of an ion specific conductivity coefficient. This coefficient is listed in charts though for their usage it is necessary to have an exact water analysis because every single ion affects the conductivity. An example for such a chart can be found in the „Handbook of Chemistry and physics” 76th edition, S. 5-90“.

See also ISO document ISO 7888:1985 and links from the EPA.