Which of the following is a correct equation to show change in mo… — Solution | JEE Chemistry

MCQMediumJEE 2024Electrolytic Conductance & Kohlrausch's Law

Which of the following is a correct equation to show change in molar conductivity with respect to concentration for a weak electrolyte, if the symbols carry their usual meaning?

A
$\Lambda_m^2 C - K_a \Lambda_m^{\circ 2} + K_a \Lambda_m \Lambda_m^\circ = 0$
B
$\Lambda_m - \Lambda_m^\circ + AC^{1/2} = 0$
C
$\Lambda_m - \Lambda_m^\circ - AC^{1/2} = 0$
D
$\Lambda_m^2 C + K_a \Lambda_m^\circ - K_a \Lambda_m \Lambda_m^\circ = 0$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: A weak electrolyte with molar conductivity $\Lambda_m$ , limiting molar conductivity $\Lambda_m^\circ$ , concentration $C$ , and dissociation constant $K_a$ .

Find: The correct equation showing the change of molar conductivity with concentration.

For a weak electrolyte, the degree of dissociation is

\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}

Derivation Using Ostwald's Dilution Law

Using Ostwald's dilution law,

K_a = \frac{C\alpha^2}{1-\alpha}

Substitute

\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}

to get

K_a = \frac{C\left(\frac{\Lambda_m}{\Lambda_m^\circ}\right)^2}{1-\frac{\Lambda_m}{\Lambda_m^\circ}}

On rearranging,

\Lambda_m^2 C - K_a \Lambda_m^{\circ 2} + K_a \Lambda_m \Lambda_m^\circ = 0

Therefore, the correct option is A.

The first solution text contains an intermediate approximation and wording inconsistency, but the second approach gives the correctly rearranged relation that matches option A.

Common mistakes

Using the strong electrolyte relation $\Lambda_m = \Lambda_m^\circ - AC^{1/2}$ is incorrect here because that applies to strong electrolytes, not weak electrolytes. For a weak electrolyte, use degree of dissociation and Ostwald's dilution law instead.
Writing $K_a = C\alpha^2$ and omitting the factor $1-\alpha$ is wrong for the exact weak electrolyte relation. The correct expression is $K_a = \frac{C\alpha^2}{1-\alpha}$ before rearrangement.
Confusing $\Lambda_m$ with $\Lambda_m^\circ$ leads to an incorrect final equation. $\Lambda_m^\circ$ is the limiting molar conductivity at infinite dilution, while $\Lambda_m$ is the value at concentration $C$ .

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