Question

For a salt XY, which is a strong electrolyte, the plot of \(\Lambda_m\) versus \(\sqrt{c}\) has a slope of \(-90.0\text{ S cm}^2\text{ mol}^{-3/2}\text{ L}^{1/2}\) at \(298\text{ K}\). At \(0.01\text{ M}\) concentration of XY, the value of \(\Lambda_m\) is \(145.0\text{ S cm}^2\text{ mol}^{-1}\). The limiting molar conductivity of \(\text{Y}^-\) ion (\(\lambda^\circ_{\text{Y}^-}\), in \(\text{S cm}^2\text{ mol}^{-1}\)) at \(298\text{ K}\) will be (Given: \(\lambda^\circ_{\text{X}^+} = 74.0\text{ S cm}^2\text{ mol}^{-1}\))

• A. \(76.0\)
• B. \(80.0\)
• C. \(100.0\)
• D. \(90.0\)

Hint:

Be extra precise when calculating the square root of concentration decimals. For example, $\sqrt{0.01} = 0.1$, not $0.01$. Once you successfully solve for the total limiting conductance $\Lambda_m^\circ$, simply subtract the value of the known ion to get the unknown ion instantly.

Answer 1

The Correct Option is B

Concept: For strong electrolytes, the variation of molar conductivity (\(\Lambda_m\)) with concentration (\(c\)) follows a linear relationship at low concentrations. This physical behavior is accurately modeled by the Debye-Huckel-Onsager equation: \[ \Lambda_m = \Lambda_m^\circ - A\sqrt{c} \] Where:

• \(\Lambda_m\) is the molar conductivity at a given concentration \(c\).

• \(\Lambda_m^\circ\) is the limiting molar conductivity (the molar conductivity at infinite dilution).

• \(A\) is a constant that depends on the valence of the electrolyte, the nature of the solvent, and the absolute temperature.

• The slope of the plot of \(\Lambda_m\) versus \(\sqrt{c}\) is equal to \(-A\).
Additionally, according to Kohlrausch's Law of Independent Migration of Ions, the limiting molar conductivity of a total electrolyte can be expressed as the sum of the individual limiting molar conductivities of its component cations and anions. For a 1:1 salt like XY, which dissociates completely via \(\text{XY} \rightarrow \text{X}^+ + \text{Y}^-\), the relation is: \[ \Lambda_m^\circ(\text{XY}) = \lambda^\circ_{\text{X}^+} + \lambda^\circ_{\text{Y}^-} \] Let us carefully execute the calculation across structured steps:

Step 1: Extract and interpret information from the given data.
We are provided with the following parameters:

• Slope of the \(\Lambda_m\) vs \(\sqrt{c}\) plot = \(-90.0\text{ S cm}^2\text{ mol}^{-3/2}\text{ L}^{1/2}\). Comparing this to the equation \(\Lambda_m = \Lambda_m^\circ - A\sqrt{c}\), we have: \[ \text{Slope} = -A = -90.0 \quad \Rightarrow \quad A = 90.0 \]

• Molarity / Concentration of the solution, \(c = 0.01\text{ M}\)

• Molar conductivity at this concentration, \(\Lambda_m = 145.0\text{ S cm}^2\text{ mol}^{-1}\)

• Limiting molar conductivity of the cation, \(\lambda^\circ_{\text{X}^+} = 74.0\text{ S cm}^2\text{ mol}^{-1}\)

Step 2: Calculate the overall limiting molar conductivity \(\Lambda_m^\circ\) of the salt XY.
Substitute our values directly into the Debye-Huckel-Onsager equation: \[ 145.0 = \Lambda_m^\circ - 90.0 \times \sqrt{0.01} \] Since \(\sqrt{0.01} = \sqrt{\frac{1}{100}} = \frac{1}{10} = 0.1\), the equation simplifies to: \[ 145.0 = \Lambda_m^\circ - 90.0 \times 0.1 \] \[ 145.0 = \Lambda_m^\circ - 9.0 \] Isolating \(\Lambda_m^\circ\) by moving \(9.0\) to the left side: \[ \Lambda_m^\circ = 145.0 + 9.0 = 154.0\text{ S cm}^2\text{ mol}^{-1} \]

Step 3: Apply Kohlrausch's law to solve for the unknown ionic conductivity \(\lambda^\circ_{\text{Y}^-}\).
Using the additive property for the 1:1 electrolyte XY: \[ \Lambda_m^\circ(\text{XY}) = \lambda^\circ_{\text{X}^+} + \lambda^\circ_{\text{Y}^-} \] Substitute the computed value of \(\Lambda_m^\circ = 154.0\) and the given value of \(\lambda^\circ_{\text{X}^+} = 74.0\): \[ 154.0 = 74.0 + \lambda^\circ_{\text{Y}^-} \] Subtract \(74.0\) from both sides to find the limiting molar conductivity of the anion \(\text{Y}^-\): \[ \lambda^\circ_{\text{Y}^-} = 154.0 - 74.0 \] \[ \lambda^\circ_{\text{Y}^-} = 80.0\text{ S cm}^2\text{ mol}^{-1} \] The limiting molar conductivity of the \(\text{Y}^-\) ion is \(80.0\text{ S cm}^2\text{ mol}^{-1}\), which corresponds to option (2).