Question

Which of the following expressions correctly relates the limiting molar conductivity (\(\Lambda_m^\circ\)) of aluminum sulfate, \(\text{Al}_2(\text{SO}_4)_3\), to its individual ionic components according to Kohlrausch's Law?

• A. \(\Lambda_m^\circ = \lambda^\circ(\text{Al}^{3+}) + \lambda^\circ(\text{SO}_4^{2-}) \)
• B. \(\Lambda_m^\circ = 2\lambda^\circ(\text{Al}^{3+}) + 3\lambda^\circ(\text{SO}_4^{2-}) \)
• C. \(\Lambda_m^\circ = 3\lambda^\circ(\text{Al}^{3+}) + 2\lambda^\circ(\text{SO}_4^{2-}) \)
• D. \(\Lambda_m^\circ = \frac{1}{2}\lambda^\circ(\text{Al}^{3+}) + \frac{1}{3}\lambda^\circ(\text{SO}_4^{2-}) \)

Hint:

Always scale the individual ionic values by the balance coefficients for molar conductivity. For equivalent conductivity, these stoichiometric coefficients are normalized.

Answer 1

The Correct Option is B

Concept: Kohlrausch's law of independent migration of ions states that the total limiting molar conductivity of an electrolyte equals the sum of the individual contributions from its cations and anions: \[ \Lambda_m^\circ = \nu_+\lambda^\circ_+ + \nu_-\lambda^\circ_- \] Where \(\nu_+\) and \(\nu_-\) denote the stoichiometric coefficients of the positive and negative ions.

Step 1: Analyze the complete dissociation process.
One formula unit of aluminum sulfate dissociates completely in water according to this equilibrium: \[ \text{Al}_2(\text{SO}_4)_3(s) \xrightarrow{\text{H}_2\text{O}} 2\text{Al}^{3+}(aq) + 3\text{SO}_4^{2-}(aq) \] Here, \(\nu_+ = 2\) and \(\nu_- = 3\). Substituting these values yields: \[ \Lambda_m^\circ = 2\lambda^\circ(\text{Al}^{3+}) + 3\lambda^\circ(\text{SO}_4^{2-}) \]