Q24 / 120 · VITEEE 2007 · Chemistry

Enthalpy is equal to

  • A
    $-\mathrm{T}^{2}\left[\frac{\delta(\Delta \mathrm{G})}{\delta \mathrm{T}}\right]_{\mathrm{V}}$
  • B
    $-\mathrm{T}^{2}\left[\frac{\delta(\mathrm{G} / \mathrm{T})}{\delta \mathrm{T}}\right]_{\mathrm{P}}$
  • C
    $\mathrm{T}^{2}\left[\frac{\delta(\mathrm{G} / \mathrm{T})}{\delta \mathrm{T}}\right]_{\mathrm{V}}$
  • Correct answer
    D
    $-\mathrm{T}^{2}\left[\frac{\delta(\Delta \mathrm{G})}{\delta \mathrm{T}}\right]_{\mathrm{P}}$

Solution

Gibbs Helmholtz Equation $\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \Delta \mathrm{S}$ ....(1) differentiate this equation w.r.t. temperature at constant pressure $\begin{array}{l} \left(\frac{\partial \Delta \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{P}}=\left(\frac{\partial \mathrm{G}_{\mathrm{y}}}{\partial \mathrm{T}}\right)_{\mathrm{P}}-\left(\frac{\partial \mathrm{G}_{\mathrm{x}}}{\partial \mathrm{T}}\right)_{\mathrm{P}} ...(2) \\ =-\mathrm{S}_{\mathrm{y}}-\left(-\mathrm{S}_{\mathrm{x}}\right) \\ =-\left(\mathrm{S}_{\mathrm{y}}-\mathrm{S}_{\mathrm{x}}\right)=-\Delta \mathrm{S} ...(3) \end{array}$ where $\Delta \mathrm{S}$ change in entropy on combining equation (1) \& (3) we get $\Delta \mathrm{G}=\Delta \mathrm{H}+\mathrm{T}\left(\frac{\partial(\Delta \mathrm{G})}{\partial \mathrm{T}}\right)_{\mathrm{P}}$ ...(4) Equation (4) is on alternative form of Gibbs Helmholtz equation Dividing equ. (4) by $\mathrm{T}^{2}$, we get $\frac{\Delta \mathrm{G}}{\mathrm{T}^{2}}=\frac{\Delta \mathrm{H}}{\mathrm{T}^{2}}+\frac{1}{\mathrm{~T}}\left[\frac{\partial(\Delta \mathrm{G})}{\partial \mathrm{T}}\right]_{\mathrm{P}}$ on rearrangement, we get $\begin{array}{l} {\left[\frac{\partial(\Delta \mathrm{G})}{\partial \mathrm{T}}\right]_{\mathrm{P}}=-\frac{\Delta \mathrm{H}}{\mathrm{T}^{2}}} \\ \therefore \Delta \mathrm{H}=-\mathrm{T}^{2}\left[\frac{\partial(\Delta \mathrm{G})}{\partial \mathrm{T}}\right]_{\mathrm{P}} \end{array}$

Hence, the correct answer is option: 'D'.

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