EMF of cell Cd(s) — Electrochemistry Chemistry Question

💡 Solution & Explanation

Step 1 - Understand the Cell Representation and Cell Reactions The given electrochemical cell is represented as: $$\ce{Cd(s) \mid CdCl2.5H2O(sat.) \parallel AgCl(s) \mid Ag(s)}$$ To analyze the cell, we write the respective electrode half-reactions: * **Anode (Oxidation Half-Reaction):** $$\ce{Cd(s) -> Cd^{2+}(aq) + 2e^-}$$ * **Cathode (Reduction Half-Reaction):** $$\ce{2AgCl(s) + 2e^- -> 2Ag(s) + 2Cl^-(aq)}$$ By combining these two half-reactions, we obtain the balanced overall cell reaction: $$\ce{Cd(s) + 2AgCl(s) -> Cd^{2+}(aq) + 2Ag(s) + 2Cl^-(aq)}$$ From this balanced equation, the number of moles of electrons transferred in the cell reaction ($n$-factor) is: $$n = 2$$ Step 2 - Calculate the Standard Gibbs Free Energy Change ($\Delta G^\circ$) at both Temperatures The standard Gibbs free energy change ($\Delta G^\circ$) is related to the standard cell potential ($E^\circ_{\text{cell}}$) by the fundamental thermodynamic equation: $$\Delta G^\circ = -n F E^\circ_{\text{cell}}$$ Where: * $n = 2$ * $F = 96,500\text{ C mol}^{-1}$ (Faraday's constant) Let us perform the calculations at both specified temperatures: 1. **At $T = 0^\circ\text{C}$ ($273.15\text{ K}$):** The standard EMF is given as $E^\circ_{\text{cell}} = +0.70\text{ V}$. $$\Delta G^\circ(0^\circ\text{C}) = -2 \times 96,500\text{ C mol}^{-1} \times 0.70\text{ V}$$ $$\Delta G^\circ(0^\circ\text{C}) = -135,100\text{ J mol}^{-1} = \boxed{-135.1\text{ kJ mol}^{-1}}$$ This matches the value in **Option (B)**. 2. **At $T = 50^\circ\text{C}$ ($323.15\text{ K}$):** The standard EMF is given as $E^\circ_{\text{cell}} = +0.60\text{ V}$. $$\Delta G^\circ(50^\circ\text{C}) = -2 \times 96,500\text{ C mol}^{-1} \times 0.60\text{ V}$$ $$\Delta G^\circ(50^\circ\text{C}) = -115,800\text{ J mol}^{-1} = \boxed{-115.8\text{ kJ mol}^{-1}}$$ This matches the value in **Option (A)** (or Option (D) in the printed scrambled list). Step 3 - Calculate the Temperature Coefficient and Standard Entropy Change ($\Delta S^\circ$) The change in standard entropy ($\Delta S^\circ$) for the cell reaction is related to the temperature coefficient of the EMF, $\left(\frac{\partial E^\circ_{\text{cell}}}{\partial T}\right)_P$, by: $$\Delta S^\circ = n F \left(\frac{\partial E^\circ_{\text{cell}}}{\partial T}\right)_P$$ Assuming a linear variation of EMF with temperature over the given range, we calculate the temperature coefficient: $$\left(\frac{\partial E^\circ_{\text{cell}}}{\partial T}\right)_P = \frac{E^\circ_{\text{cell}}(T_2) - E^\circ_{\text{cell}}(T_1)}{T_2 - T_1}$$ $$\left(\frac{\partial E^\circ_{\text{cell}}}{\partial T}\right)_P = \frac{0.60\text{ V} - 0.70\text{ V}}{323.15\text{ K} - 273.15\text{ K}} = \frac{-0.10\text{ V}}{50\text{ K}} = -2 \times 10^{-3}\text{ V K}^{-1}$$ Substituting this value into our entropy equation: $$\Delta S^\circ = 2 \times 96,500\text{ C mol}^{-1} \times \left(-2 \times 10^{-3}\text{ V K}^{-1}\right)$$ $$\Delta S^\circ = \boxed{-386\text{ J K}^{-1}\text{ mol}^{-1}}$$ Step 4 - Calculate the Standard Enthalpy Change ($\Delta H^\circ$) Using the Gibbs-Helmholtz thermodynamic relation, the standard enthalpy change ($\Delta H^\circ$) of the cell process is: $$\Delta H^\circ = \Delta G^\circ + T \Delta S^\circ$$ Let us substitute our calculated parameters at $0^\circ\text{C}$ ($273.15\text{ K}$): $$\Delta H^\circ = -135.1\text{ kJ mol}^{-1} + 273.15\text{ K} \times \left(-0.386\text{ kJ K}^{-1}\text{ mol}^{-1}\right)$$ $$\Delta H^\circ = -135.1\text{ kJ mol}^{-1} - 105.44\text{ kJ mol}^{-1} = \boxed{-240.54\text{ kJ mol}^{-1}}$$ *(Note: Performing this calculation using values at $50^\circ\text{C}$ ($323.15\text{ K}$) yields the identical value of $-240.54\text{ kJ mol}^{-1}$ because $\Delta H^\circ$ and $\Delta S^\circ$ are temperature independent).* Step 5 - Evaluate and Justify the Options * **Option (A) is correct:** Our calculation in Step 2 confirms that the standard Gibbs free energy change of the reaction at $50^\circ\text{C}$ is exactly $-115.8\text{ kJ}$. * **Option (B) is correct:** Our calculation in Step 2 confirms that the standard Gibbs free energy change of the reaction at $0^\circ\text{C}$ is exactly $-135.1\text{ kJ}$. * **Option (C) is incorrect:** This option represents the misplaced question statement from printing errors rather than a valid thermodynamic property option. * **Option (D) is incorrect:** This is a duplicate representation in the printed list or represents an incorrect value when comparing thermodynamic constants. $$\text{Correct Options: } \boxed{\text{A, B}}$$