Kohlrausch Law

Kohlrausch law states that at infinite dilution, when dissociation is complete, each ion makes a definite contribution towards equivalent conductance of the electrolyte irrespective of the nature of the ion with which it is associated and the value of equivalent conductance at infinite dilution for any electrolyte is the sum of the contribution of its constituent ions (cations and anions). Thus, we can say it states that ‘conductivity of ions of an electrolyte at infinite dilution is constant and it does not depend on nature of co-ions.’

\[\lambda_{eq}^{\infty} = \lambda_{c}^{\infty} + \lambda_{a}^{\infty}\]

\[\lambda_{eq}^{\infty}\] = Molar conductivity at infinite dilution 

\[\lambda_{C}^{\infty}\]= Conductivity of cation at infinite dilution 

\[ \lambda_{a}^{\infty}\]= Conductivity of anion at infinite dilution 

When the concentration of the electrolyte is almost zero, at that point, molar conductivity is called limiting molar conductivity. 

The molar conductivity of the solution can be defined as the volume of the solution that is conducting that also contains one mole of electrolyte when kept between two electrodes with a unit area of cross-section and one unit length of distance. With the decrease in the concentration, the molar conductivity increases. The increase in the molar conductivity is due to the increase in the volume that comprises one mole of electrolytes. The molar conductivity is known as limiting molar conductivity, Ëm°, when the concentration of the electrolyte approaches zero. 

Explanation of Kohlrausch Law

Any random electrolyte is the general case of this law which can be denoted as \[ A_{x}B_{y}\]. 

Thus mathematically, the limiting molar conductivity of \[ A_{x}B_{y}\] can be represented as:

\[ \lambda _{AxBy}^{\infty} = 2 \lambda_{A + B}^{\infty} + \lambda_{B - x}^{\infty}\]

Where \[\lambda^{\infty}\] is the limiting molar conductivity of the electrolyte chosen. 

When a cation is the same in both the electrolytes, then the difference in the molar conductivity of the two electrolytes does not depend upon the cation and is only dependent on the change that happens in their anions. The statement mentioned is also true if the anions are the same and the cations are different.  

For instance, if there are two pairs of electrolytes with the same cation A and D in each pair, then the difference between their limiting molar conductivities is not affected by A or D. This can be mathematically represented as,

\[ \lambda _{AB}^{\infty} - \lambda _{AC}^{\infty} = \lambda _{DB}^{\infty} - \lambda _{DC}^{\infty}\]

Where

\[ \lambda _{AB}^{\infty}\] is limiting molar conductivity of AB,

\[\lambda _{AC}^{\infty}\] is limiting molar conductivity of AC,

\[ \lambda _{DB}^{\infty}\] is limiting molar conductivity of DB, and

 \[ \lambda _{DC}^{\infty}\] is limiting molar conductivity of DC.

Uses of Kohlrausch’s Law 

• Kohlrausch’s law is used to calculate molar conductivity at infinite dilution for the weak electrolytes. It’s very difficult or impossible to calculate the molar conductivity of weak electrolytes at infinite dilution. as the conductance of these types of solutions is very low and dissociation of these electrolytes is not completed at high dilutions as well. For example, acetic acid is a weak electrolyte and its molar conductivity at infinite dilution can be calculated by Kohlrausch’s Law. It can be represented as follows: 

\[\mu^{\infty}\] = Molar conductance at infinite dilution

\[\mu^{\infty} = \mu^{\infty} = m \lambda _{+}^{\infty} + m \lambda _{-}^{\infty}\]

• m and n are a number of ions formed. 

• For example – molar conductance of aluminium sulphate at infinite dilution can be written as follows – 

• The formula of aluminium sulphate is \[Al_{2}(SO_{4})_{3}.\]

• So, molar conductance at infinite dilution = \[ \mu _{Al_{2}}^{\infty } (SO_{4})_{8} = 2 \lambda _{Al^{s}}^{\infty } + 3\lambda _{SO{_{4}}^{2-}}^{\infty }\] 

• The knowledge of molar conductivities at infinite dilution of the strong electrolyte like HCl, \[ (CH_{3}COONa)]\, and NaCl and the molar conductivity of acetic acid at infinite dilution can be obtained as follows:

\[ \lambda_{m(HCl)}^{\infty } = \lambda _{H^{+}}^{\infty }  + \lambda _{Cl^{-}}^{\infty } \]

\[\lambda_{m(NaCl)}^{\infty } = \lambda _{Na^{+}}^{\infty }  + \lambda _{Cl^{-}}^{\infty }\]

\[\lambda_{m(CH_{3}COONa)}^{\infty } = \lambda _{CH_{3}Coo^{-}}^{\infty }  + \lambda _{Na^{+}}^{\infty }\]

\[\lambda_{m(CH_{3}COOH)}^{\infty } = \lambda _{CH_{3}Coo^{-}}^{\infty }  + \lambda _{H^{+}}^{\infty }\]

\[\lambda_{m(CH_{3}COOH)}^{\infty } = [\lambda _{CH_{3}Coo^{-}}^{\infty } + \lambda _{Na^{+}}^{\infty }]  - [ \lambda _{Na^{+}}^{\infty } + \lambda _{Cl^{-}}^{\infty }] [ \lambda _{H^{+}}^{\infty } + \lambda _{Cl^{-}}^{\infty }\]

• Determination of the degree of dissociation of the weak electrolytes is given by the following equation:

\[ \alpha = \frac{\lambda _{m}}{ \lambda _{m}^{o}} \]

• Determining the dissociation constant (K) of weak electrolytes

\[ K_{a} = \frac{\Lambda _{m}^{2} .C}{\Lambda _{m}^{2}\left [ 1 - \frac{\Lambda _{m}}{\Lambda _{m}^{o}} \right ] } \]

\[ \Rightarrow K_{a} = \frac{\Lambda _{m}^{2}.C}{\Lambda _{m}^{o2}\left ( \frac{\Lambda _{m}^{o} - \Lambda _{m}}{\Lambda _{m}} \right )}\]

\[ \Rightarrow K_{a} = \frac{\Lambda _{m}^{2}C}{\Lambda _{m}^{o}(\Lambda _{m}^{o} - \Lambda _{m})} \]

• Thus, the dissociation constant for weak electrolytes at a specific concentration of the solution can be calculated by using the following formula-

\[ K_{c} = \frac {Ca ^{2}} {1 - \alpha} \]

Where

K = dissociation constant,

C = concentration of the solution, and 

α = degree of dissociation. 

• Kohlrausch’s law is used for the calculation of solubility of moderately soluble salt. Some salts that dissolve in very small quantities in water are called moderately or sparingly soluble salts. For example – silver chloride, barium sulphate, lead sulphate, etc. 

• Acid dissociation constant \[K_{a}\] can also be calculated by Kohlrausch’s law. 

• When the concentration of the electrolyte is almost zero, at that point, molar conductivity is called limiting molar conductivity. By Kohlrausch’s law, we can determine limiting molar conductivity for an electrolyte.