Standard Gibb’s energy of reaction ( _ r G ^ ) at a certain temperature can be computed…

Standard Gibb’s energy of reaction (\(\Delta_{\mathrm{r}} \mathrm{G}^{\circ}\)) at a certain temperature can be computed as \(\Delta_{r} G^{o}=\Delta_{r} H^{o}-T \cdot \Delta_{r} S^{o}\)and the change in the value of \(\Delta_{\mathrm{r}} \mathrm{H}^{\circ}\)and \(\Delta_{\mathrm{r}} \mathrm{~S}^{0}\)for a reaction with temperature can be computed as follow:
  \[\begin{array}{l} \Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{T}_{2}}^{\circ}-\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{T}_{1}}^{\circ}=\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\circ}\left(\mathrm{T}_{2}-\mathrm{T}_{1}\right) \\ \Delta_{\mathrm{r}} \mathrm{~S}_{\mathrm{T}_{2}}^{\circ}-\Delta_{\mathrm{r}} \mathrm{~S}_{\mathrm{T}_{1}}^{\circ}=\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\circ} \ln \left(\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}\right) \end{array}\]
\[\begin{array}{l} \Delta_{\mathrm{r}} \mathrm{G}^{\circ}=\Delta_{\mathrm{r}} \mathrm{H}^{\circ}-\mathrm{T} \cdot \Delta_{\mathrm{r}} \mathrm{~S}^{\circ}\\ \text { and by } \Delta_{\mathrm{r}} \mathrm{G}^{\circ}=-\mathrm{RT} \ln \mathrm{~K}_{\mathrm{eq}} \text {. } \end{array}\]
Consider the following reaction:
\(\mathrm{CO}(\mathrm{~g})+2 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}(\mathrm{~g})\) 
Given: \(\Delta_{f} \mathrm{H}^{\circ}\left(\mathrm{CH}_{3} \mathrm{OH}, \mathrm{~g}\right)\)= –201 kJ/mol; \(\Delta_{f} \mathrm{H}^{\circ}(\mathrm{CO}, \mathrm{~g})\)= –144 kJ/mol
S^o(CH₃OH, g) = 240 J/K-mol; S^o(H₂, g) = 29 JK^–1mol^–1
S^o(CO, g) = 198 J/mol-K; \(\mathrm{C}_{\mathrm{p}, \mathrm{~m}}^{o}\left(\mathrm{H}_{2}\right)\)=28.8 J/mol-K
\(\mathrm{C}_{\mathrm{p}, \mathrm{~m}}^{o}(\mathrm{CO})\)=29.4 J/mol-K; \(\mathrm{C}_{\mathrm{p}, \mathrm{~m}}^{\mathrm{o}}\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) = 44 J/mol-K
and \(\ln \left(\frac{320}{300}\right)=0.06\), all data at 300 K
\(\Delta_{\mathrm{r}} \mathrm{H}^{\circ}\)at 300 K for the reaction is :