GATE 2024 CY – Question 58

Physical Chemistry · Equilibrium: Laws of thermodynamics. Standard states. Thermochemistry. Thermodynamic functions and their relationships: Gibbs-Helmholtz and Maxwell relations, Gibbs-Duhem equation, Van’t Hoff equation. Criteria of spontaneity and equilibrium. Absolute entropy. Partial molar quantities. Thermodynamics of mixing. Chemical potential. Fugacity, Activity and Activity coefficient. Chemical equilibria. Dependence of equilibrium constant on temperature and pressure. · 2 marks · Numerical answer

$\Delta S^\circ$ (in $\mathrm{J\,mol^{-1}K^{-1}}$) for the given reaction at 298 K is ________. (Given: $\log K_1=10.6$, where $K_1$ is the equilibrium constant. $\Delta H^\circ=-54\ \mathrm{kJ\,mol^{-1}}$ and $R=8.314\ \mathrm{J\,mol^{-1}K^{-1}}$) (rounded off to two decimal places)

Copper aqua complex reacts with ethylenediamine to displace two water ligands.

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Correct answer: 21.61 to 21.83

Explanation

**Thermodynamic relations at 298 K:**
$$\Delta G^\circ=-RT\ln K,\qquad\Delta G^\circ=\Delta H^\circ-T\Delta S^\circ .$$

Combining,
$$\Delta S^\circ=\frac{\Delta H^\circ-\Delta G^\circ}{T}=\frac{\Delta H^\circ}{T}+R\ln K .$$

**Numbers.**
- $\ln K=2.303\times\log K=2.3026\times10.6=24.41$
- $\dfrac{\Delta H^\circ}{T}=\dfrac{-54\,000}{298}=-181.21\ \text{J mol}^{-1}\text{K}^{-1}$
- $R\ln K=8.314\times24.41=202.93\ \text{J mol}^{-1}\text{K}^{-1}$

$$\Delta S^\circ=-181.21+202.93=\mathbf{21.72\ J\,mol^{-1}K^{-1}}.$$

(The positive entropy change reflects the release of two water molecules when one ethylenediamine chelate forms.)