Question

Consider an electrochemical cell: $A\left(s\right) \left|A^{n+}\left(aq, \, 2\, M\right)\right|\left|B^{2n+}\left(aq,\,1\,M\right)\right|B\left(s\right).$ The value of $ΔH^\theta$ for the cell reaction is twice that of $ΔG^\theta$ at $300\, K$. If the emf of the cell is zero, the $ΔS^\theta$ (in $JK^{−1}\, mol^{−1}$) of the cell reaction per mole of $B$ formed at $300\, K$ is ____.
(Given: $ln(2)$ = $0.7, R$ (universal gas constant) = $8.3\, JK^{−1}\, mol^{−1}$. $H, S$ and $G$ are enthalpy, entropy and Gibbs energy, respectively.)

Answer 1

$A(s)\left|A^{\ln }(a q .2 M)\right|\left|B^{+2 a}(a q, 1 M)\right| B(s)$
$\Delta H^{\circ}=2 \Delta G_{0}^{0} $
$ E_{\text {cell }}=0$
Cell $Rx A \rightarrow A^{+n}+n e^{-} \times 2$
$B^{+2 n}+2 n e^{-} \rightarrow B(s)$
$2 A(s)+B_{1 M} \,{}^{+2 n}(a q) \rightarrow 2 A_{2 M}\,{}^{+n}(a q)+B(s)$
$\Delta G=\Delta G^{\circ}+R T \ln \frac{\left[A^{+n}\right]^{2}}{\left[B^{+2 n}\right]}$
$\Delta G^{\circ}=-R T \ln \frac{\left[A^{+n}\right]^{2}}{\left[B^{+2 n}\right]}$
$=-R T \cdot \ln \frac{2^{2}}{1}=-R T \cdot \ln 4$
$\Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ}$
$\Delta G^{\circ}=2 \Delta G^{\circ} T \Delta S^{\circ}$
$\Delta S^{\circ}=\frac{\Delta G^{\circ}}{T}=\frac{R T \ln 4}{T}$
$=-8.3 \times 2 \times 0.7=11.62\, J / K .mol$