Derive the van’t Hoff equation given in problem 16.51, using Eqs.16.12 and 16.15. Note: the d(g /T) at constant P for each component can be expressed using the relations in Eqs. 16.18 and 16.19.

Question

Derive the van’t Hoff equation given in problem 16.51, using Eqs.16.12 and 16.15. Note: the [latex]d (\overline{ g } / T )[/latex] at constant P for each component can be expressed using the relations in Eqs. 16.18 and 16.19.

Accepted Answer

[latex] ext { Eq. 16.12: } \Delta G ^0= v _{ C } \overline{ g }_{ C }^0+ v _{ D } \overline{ g }_{ D }^0- v _{ A } \overline{ g }_{ A }^0- v _{ B } \overline{ g }_{ B }^0[/latex]

[latex] ext { Eq. 16.15: } \ln K =\Delta G ^0 / \overline{ R } T \quad ext { Eq. 16.19: } \quad \Delta G ^0=\Delta H ^{\circ}- T \Delta S ^0[/latex]

[latex]\begin{aligned}\frac{d \ln K}{d T} & =-\frac{d}{d T}\left(\frac{\Delta G^0}{\bar{R} T} ight)=-\frac{1}{\bar{R} T} \frac{d \Delta G^0}{d T}+\frac{\Delta G^0}{\bar{R} T^2}=\frac{1}{\bar{R} T^2}\left[\Delta G^0-T \frac{d \Delta G^0}{d T} ight] \& =\frac{1}{\bar{R} T^2}\left[\Delta G^0+T \Delta S^0 ight] \quad ext { used Eq.16.19 } \frac{d \bar{g}}{d T}=-\bar{s} \& =\frac{1}{\bar{R} T^2} \Delta H^0\end{aligned}[/latex]

Question 16.CSGP.103: Derive the van’t Hoff equation given in problem 16.51, using......

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