I have an equation from chemistry: $$\frac{\delta (\frac{\Delta G}{T})}{\delta T}=-\frac{\Delta H}{T^2}$$ This is known as the Gibbs-Helmholtz equation.
Now according to Wikipedia, if I integrate this, I would get: $$\frac{\Delta G_{T_2}}{T_2}-\frac{\Delta G_{T_1}}{T_1}=\Delta H(\frac{1}{T_2} -\frac{1}{T_1})$$ Now, this makes sense, however, I watched another video on the same thing on YouTube and that gives a slightly different answer. It uses $\Delta H_{T_1}$ as in, the $\Delta H$ at $T_1$ only and not $\Delta H$ in general which I suppose considers both $T_1$ and $T_2$, i.e., $\Delta H=\Delta H_{T_2}-\Delta H_{T_1}$. But I am not quite sure. Can someone tell me which source is correct?
If anyone thinks it will be helpful, $\Delta G$ is the Gibbs free energy, $T$ is the temperature, $\Delta H$ is the change in enthalpy, and naturally, $T_1$ is the initial temperature and $T_2$ is the final temperature.
This would be correct if $\Delta H$ was a constant, which is not true.
Assume a simple polynomial dependency such as $$\Delta H=\sum_{i=0}^n a_i\, T^i$$ and, for the antiderivative, you will have $$\frac{\Delta G} T=-\int a_i\, T^{i-2}\,dT$$ that is to say $$\frac{\Delta G} T=\frac{a_0}T-a_1 \log(T)-\sum_{i=2}^n \frac{a_i}i T^{i-1}$$ and here you see coming the famous $T\log(T)$ term.
Now, use the bounds.