Calculate Λm° (in Ω^-1 cm^2 mol^-1) for SrCl2 at 25°C, from the following data: Conc. 0.25 M \ — Electrochemistry Chemistry Question

💡 Solution & Explanation

Step 1 - State the Debye-Hückel-Onsager Equation The molar conductivity of a strong electrolyte in a dilute solution varies linearly with the square root of its concentration. This relationship is mathematically described by the **Debye-Hückel-Onsager (DHO) equation**: $$\Lambda_m = \Lambda_m^\infty - A\sqrt{c}$$ Where: * $\Lambda_m$ is the molar conductivity of the electrolyte at a concentration $c$ (in $\text{ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}$). * $\Lambda_m^\infty$ is the limiting molar conductivity (molar conductivity at infinite dilution) of the electrolyte (in $\text{ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}$). * $A$ is a constant that depends on the nature of the electrolyte (specifically its stoichiometry and ion charges), the solvent, and the temperature. * $c$ is the molar concentration of the electrolyte (in $\text{mol L}^{-1}$). Step 2 - Substitute the First Set of Experimental Data We are given the first set of experimental values for strontium chloride ($\ce{SrCl2}$) at $25^\circ\text{C}$: * Concentration ($c_1$) = $0.25\text{ M}$ * Molar conductivity ($\Lambda_{m,1}$) = $260\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}$ First, calculate the square root of the concentration: $$\sqrt{c_1} = \sqrt{0.25\text{ mol L}^{-1}} = 0.5\text{ (mol L}^{-1}\text{)}^{1/2}$$ Substitute these values into the Debye-Hückel-Onsager equation: $$260\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1} = \Lambda_m^\infty - A \times 0.5\text{ (mol L}^{-1}\text{)}^{1/2} \quad \text{--- (Equation 1)}$$ Step 3 - Substitute the Second Set of Experimental Data We are given the second set of experimental values: * Concentration ($c_2$) = $1.0\text{ M}$ * Molar conductivity ($\Lambda_{m,2}$) = $250\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}$ Calculate the square root of the concentration: $$\sqrt{c_2} = \sqrt{1.0\text{ mol L}^{-1}} = 1.0\text{ (mol L}^{-1}\text{)}^{1/2}$$ Substitute these values into the Debye-Hückel-Onsager equation: $$250\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1} = \Lambda_m^\infty - A \times 1.0\text{ (mol L}^{-1}\text{)}^{1/2} \quad \text{--- (Equation 2)}$$ Step 4 - Solve the System of Linear Equations for $\Lambda_m^\infty$ To solve for the limiting molar conductivity ($\Lambda_m^\infty$), we can eliminate the constant $A$. Subtract Equation 2 from Equation 1: $$\left(260 - 250\right)\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1} = \left(\Lambda_m^\infty - 0.5A\right) - \left(\Lambda_m^\infty - 1.0A\right)$$ $$10\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1} = 0.5A$$ Solve for the constant $A$: $$A = \frac{10}{0.5}\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}\text{ (L mol}^{-1}\text{)}^{1/2}$$ $$A = 20\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}\text{ (L mol}^{-1}\text{)}^{1/2}$$ Now, substitute the value of $A = 20\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}\text{ (L mol}^{-1}\text{)}^{1/2}$ back into Equation 2: $$250\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1} = \Lambda_m^\infty - 20 \times 1.0$$ $$250\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1} = \Lambda_m^\infty - 20\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}$$ $$\Lambda_m^\infty = 250\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1} + 20\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}$$ $$\Lambda_m^\infty = \mathbf{270\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}}$$ Thus, the limiting molar conductivity at infinite dilution ($\Lambda_m^\infty$) for $\ce{SrCl2}$ is exactly $270\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}$. Step 5 - Evaluate and Explain the Options * **Option (A) is incorrect:** This value ($280$) is too high and does not satisfy the linear relationship of the Debye-Hückel-Onsager equation with the given data. * **Option (B) is incorrect:** This value ($265$) would be obtained through a calculation error in finding the value of the constant $A$ or the square roots of the concentrations. * **Option (C) is correct:** As mathematically calculated, extrapolation using the Debye-Hückel-Onsager equation yields a limiting molar conductivity of exactly $270\text{ ohm}^{-1}\text{ cm}^2\text{ mol}^{-1}$. * **Option (D) is incorrect:** This value ($275$) is incorrect and does not satisfy the experimental coordinates on the molar conductivity plot. $$\text{Correct Option: } \boxed{\text{C}}$$