Which out 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
- \( \Lambda_m^2 C - K_a \Lambda_m + K_a \Lambda_m^{\circ 2} = 0 \)
- \( \Lambda_m - \Lambda_m^{\circ} + AC \frac{1}{2} = 0 \)
- \( \Lambda_m - \Lambda_m^{\circ} - AC \frac{1}{2} = 0 \)
- \( \Lambda_m^2 C + K_a \Lambda_m^{\circ 2} - K_a \Lambda_m^{\circ} = 0 \)
The Correct Option is A
Approach Solution - 1
To determine which equation correctly describes the change in molar conductivity with respect to concentration for a weak electrolyte, we need to consider the dissociation equilibrium and conductivity of weak electrolytes.
Molar conductivity (\(\Lambda_m\)) of an electrolyte is given by:
\(\Lambda_m = \frac{\kappa}{C}\)
where \(\kappa\) is the conductivity and \(C\) is the concentration.
The molar conductivity of a weak electrolyte at any concentration differs from its limiting molar conductivity (\(\Lambda_m^\circ\)) at infinite dilution. For weak electrolytes, as the concentration decreases, \(\Lambda_m\) approaches \(\Lambda_m^\circ\).
The relationship between molar conductivity and concentration for weak electrolytes is rather complex, and various models approximate it. A commonly used relationship involves the degree of dissociation, \(\alpha\), linked with the equilibrium constant (\(K_a\)). For a weak electrolyte:
\(\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}\)
The equilibrium constant (\(K_a\)) can also be expressed as:
\(K_a = C\alpha^2 = C\left(\frac{\Lambda_m}{\Lambda_m^\circ}\right)^2\)
Rearranging gives us an expression reflective of the equation:
\(\Lambda_m^2 C - K_a \Lambda_m + K_a \Lambda_m^{\circ 2} = 0\)
This equation represents a quadratic relationship between molar conductivity, concentration, and the dissociation constant of a weak electrolyte.
Let's evaluate the given options:
- \(\Lambda_m^2 C - K_a \Lambda_m + K_a \Lambda_m^{\circ 2} = 0\): This equation correctly represents the relationship as derived above, making it the correct option.
- \(\Lambda_m - \Lambda_m^{\circ} + AC \frac{1}{2} = 0\): Incorrect. This equation does not represent the typical quadratic form expected for weak electrolytes.
- \(\Lambda_m - \Lambda_m^{\circ} - AC \frac{1}{2} = 0\): Incorrect. Similar reasoning to the previous one, it does not align with expected relationships.
- \(\Lambda_m^2 C + K_a \Lambda_m^{\circ 2} - K_a \Lambda_m^{\circ} = 0\): Incorrect. The terms rearranged do not satisfy the known relationship for weak electrolytes.
Therefore, the correct answer is option \(\Lambda_m^2 C - K_a \Lambda_m + K_a \Lambda_m^{\circ 2} = 0\).
Approach Solution -2
The relationship between molar conductivity $\Lambda_m$, molar conductivity at infinite dilution $\Lambda_m^\circ$, and concentration $C$ for a weak electrolyte can be derived from the dissociation equilibrium. The correct equation involves the dissociation constant $K_a$ and accounts for the variation of $\Lambda_m$ with concentration. For weak electrolytes, the molar conductivity $\Lambda_m$ is related to the degree of dissociation $\alpha$ as:
\[ \alpha = \frac{\Lambda_m}{\Lambda_m^\circ}. \]
The dissociation constant $K_a$ is expressed as:
\[ K_a = \frac{C\alpha^2}{1 - \alpha}. \]
Substituting $\alpha =\frac{\Lambda_m}{\Lambda_m^\circ}$ into the equation:
\[ K_a = \frac{C \left(\frac{\Lambda_m}{\Lambda_m^\circ}\right)^2}{1 - \frac{\Lambda_m}{\Lambda_m^\circ}}. \]
Simplifying and rearranging, the equation becomes:
\[ \Lambda_m^2 C - K_a \Lambda_m^{\circ 2} + K_a \Lambda_m \Lambda_m^\circ = 0. \]
This is the equation that correctly represents the relationship between molar conductivity, concentration, and dissociation constant for a weak electrolyte.
Final Answer: (1)
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