Expressing ΔH and ΔS using heat capacities and then using these to express ΔG

Essentially, I think I derived the correct expressions for ΔH and ΔS, but for some reason I can't combine them into the correct form of ΔG.


Workings below:


[![enter image description here](https://i.stack.imgur.com/PzPJM.jpg)](https://i.stack.imgur.com/PzPJM.jpg)


[![enter image description here](https://i.stack.imgur.com/46aYN.png)](https://i.stack.imgur.com/46aYN.png)




$$\Delta G(T\_1) = \Delta H(T\_1) - T\_1 \Delta S(T\_1)$$


$$\Delta G(T\_2) = \Delta H(T\_2) - T\_2 \Delta S(T\_2)$$


$$\Delta G(T\_2) - \Delta G(T\_1) = \Delta H(T\_2) - \Delta H(T\_1) + T\_1 \Delta S(T\_1) - T\_2 \Delta S(T\_2) = \\
\int\_{T\_1}^{T\_2}{\Delta C\_\mathrm{p}(T)\mathrm{d}T} + T\_1 \Delta S(T\_1) - T\_2 \left(\Delta S(T\_1)+\int\_{T\_1}^{T\_2}{\frac{\Delta C\_\mathrm{p}(T)}{T}\mathrm{d}T}\right) = \\ \int\_{T\_1}^{T\_2}{\Delta C\_\mathrm{p}(T)\mathrm{d}T} + (T\_1 - T\_2)\Delta S(T\_1) - T\_2 \int\_{T\_1}^{T\_2}{\frac{\Delta C\_\mathrm{p}(T)}{T}\mathrm{d}T}$$


For $\Delta C\_\mathrm{p}$ constant with $T$:


$$\Delta G(T\_2) - \Delta G(T\_1) = \Delta C\_\mathrm{p}(T\_2 - T\_1 ) + (T\_1 - T\_2)\Delta S(T\_1) - T\_2 \Delta C\_\mathrm{p} \ln{\frac{T\_2}{T\_1}} = \\ (\Delta C\_\mathrm{p} - \Delta S(T\_1))(T\_2 - T\_1 ) - T\_2 \Delta C\_\mathrm{p} \ln{\frac{T\_2}{T\_1}}$$


 0