Kohlrausch’s Law

Kohlrausch’s law, also known as Kohlrausch’s law of independent ionic migration, was established by the German physicist Friedrich Kohlrausch in 1875-1879. This law is in connection with the molar conductivity of an electrolyte solution (λ_m), and it helps to give an accurate value for the molar conductivity at rather dilute conditions.

The λ_m of a solution at a given concentration is the conductance of the volume of solution containing one mole of electrolyte kept between two electrodes with the unit area of cross-section and distance of unit length.

It is observed that λ_m increases as the concentration of electrolyte decrease. This is because λ_m is the conductance of that volume solution containing one mole of an electrolyte. As dilution increases, the volume containing one mole of electrolyte increases, thus increasing λ_m.

Using this law, it is possible to calculate the value of λ_m for infinite dilution of a given electrolyte, called limiting molar conductivity and represented by λ₀, if we know the value of the same for its constituents. 

“Law of Independent Migration of Ions”

There are a couple of important concepts required to establish Kohlrausch’s Law:

The Motion of Ions is Random

We tend to think of the movement of ions towards electrodes when an electric field is applied to be orderly and neat. This is far from the truth. 

When there is no field applied, the molecules of water and electrolyte ions tend to be in a state of Brownian motion. When an electric field is applied to the solution, there is an occasional jump towards the direction of said field. But, this is a very tiny effect; multiple such jumps are required to advance beyond a single molecule.

The Ions Migrate Independently

An ionic compound consists of least one cation and one anion. It can be seen that the limiting equivalent conductivities of salts with a common ion differ by a constant. These differences are the differences in the conductivities of the ions that are different between the salts.

Statement and Mathematics

Kohlrausch’s law states that at the limit of infinite dilution, each ionic species makes a contribution to the conductivity of the solution that depends only on the nature of that particular ion, and is independent of the other ions present.

This can be expressed mathematically as:

λ₀ = Σλ₀⁺ + Σλ₀^–

where: 
λ₀ is the limiting molar conductivity,
Σλ₀⁺ is the sum of limiting molar conductivities of all the cations present in the electrolyte
Σλ₀^– is the sum of limiting molar conductivities of all the anions present in the electrolyte.

This provides the values of molar conductivities of electrolytes in cases where extrapolation does not work.

Applications of Kohlrausch’s Law

The main advantages of this law are:

1. Kohlrausch law helps us in the determination of limiting molar conductivities for any electrolyte. Weak electrolytes have lower molar conductivities and lower degrees of dissociation at higher concentrations. The graph plotted between molar conductivity and c^1/2 (where c is the concentration) is not a straight line for weak electrolytes. The molar conductivity of weak electrolytes increases steeply at lower concentrations. Therefore, λ₀ cannot be obtained by extrapolation of molar conductivity to zero concentration.
2. Kohlrausch law also helps us in determining the value of dissociation constant α from the value of molar conductivity and limiting molar conductivity for a weak electrolyte at a given concentration. This is given by the equation,
   
   α = λ_m / λ₀
   where
   λ_m is the molar conductivity at any concentration
   λ₀ is the limiting molar conductivity