# What is the proof of Kohlrausch's Law of Independent Migration of Ions? [closed]

What is the proof of Kohlrausch's Law of Independent Migration of Ions? How did Kohlrausch calculate the limiting molar conductance of a weak electrolyte? I am asking this in context of electrochemistry and limiting molar conductance of a weak electrolyte.

There is a simple demonstration of Kohlrausch's law. Start with four electrolytes with common ions, say AX, AY, BX and BY. The law says that the difference in limiting molar conductivities of two salts with a common ion is independent of that ion, for instance: $$\Lambda_m^\circ (\ce{AX})-\Lambda_m^\circ (\ce{AY})=\Lambda_m^\circ (\ce{BX})-\Lambda_m^\circ (\ce{BY})$$ The reason for this is that A contributes the same amount to the conductivities of solutions of AX or AY, and similarly B contributes the same amount to the conductivities of solutions of BX or BY (provided molar concentrations are the same of course).

If you want a more fundamental explanation you can use Coulomb's law to compute the energy of a pair of charges at a distance r from each other. Apply it to the pair of counterions formed when a simple 1:1 salt dissociates in solution:

$$U = \frac{e^2}{4\pi \epsilon\epsilon_0 r}$$

where $$\epsilon$$ is the dielectric constant of the medium (assumed constant, approx 80 for water).

Consider now how the energy of the interaction changes with distance. At $$\pu{r= 1 Å}$$ the energy is $$\pu{17.4 kJmol-1}$$. Repeat the computation for other distances and tabulate the results:

$$\begin{array}{c|c|c}\hline r(Å) & U (\pu{kJmol-1}) & \pu{C(M)}\\\hline 1 & 17.4 & 481\\ 10 & 1.74 &0.481\\ 100 & 0.174& \pu{4.81e-4}\\ 1000 & 0.0174&\pu{4.81e-7} \\\hline \end{array}$$

Compare the energies to kT, a measure of the thermal energy available to the molecules causing random motion in the solution. If kT is greater than the interaction energy, then the ions are unlikely to be close to each other. At a temperature of 300K, $$RT=\pu{2.5 kJmol-1}$$. This means that if two ions are at a distance of more than a few nm they are unlikely to have any significant effect on each other's position. In other words, they will act independently. This is embodied by the corollary of Kohlrausch's law that conductivity is linearly dependent on concentration at very low concentrations, because in that regime ions act independently.

You can also estimate how the mean distance between ions changes with concentration (using some rough back-of-the envelope calculations), this is shown in the last column of the table. It suggests that if you keep to ~mM concentrations of simple salts, the behavior of the electrolyte should approximate the zero concentration (limiting) prediction of Kohlrausch's law.

As for the second question, the limiting molar conductivity $$\Lambda_m^\circ$$ is a value you obtain by extrapolating experimental data (molar conductivity versus concentration) to the limit of infinite dilution (zero concentration). For a weak 1:1 electrolyte you can perform a linear fit using Ostwald's dilution law

$$\frac{1}{\Lambda_m} = \frac{1}{\Lambda_m^\circ} + \frac{\Lambda_m c}{K_a (\Lambda_m^\circ )^2}$$

and determine the intercept.

In general $$\Lambda_m^\circ$$ can be calculated by fitting the experimental data (molar conductivity versus concentration) to a polynomial in concentration and computing the value expected at zero concentration (the intercept).

• What is 'k' and 'T' in 'kT = 2.5 kilojoule per mole'? Commented Apr 15, 2021 at 13:27
• That's Boltzmann's constant... Commented Apr 15, 2021 at 14:01
• @BuckThorn Then it should be RT. Aside of the value wrong by N_A, kT has dimension of energy. (As is known R=k . N_A.) Commented Jul 23, 2022 at 4:49
• And for some unknown reasons, energy in the post uses exponent -1. ( $kT= \pu{2.5 kJ-1 mol-1}$) Commented Jul 23, 2022 at 4:56
• @Poutnik thanks for bringing that to my attention! Commented Jul 23, 2022 at 5:56