I have the following equation,
$$ K_r=\left ( \frac{P}{RT} \right )^{v}exp \left \{ \sum_{s}\left [ (\beta_{s,r}-\alpha_{s,r}) \left \langle \frac{h_s}{RT}-\frac{s_s}{R}\right \rangle \right ] \right \} $$
where $$ v=\sum_{s}\left(\beta_{s,r}-\alpha_{s,r}\right) $$ $$ \frac{h_s}{RT}=-\frac{a_1}{T^2}+a_2\frac{ln(T)}{T}+a_3+a_4\frac{T}{2}+a_5\frac{T^2}{3}+a_6\frac{T^3}{4}+a_7\frac{T^4}{5}+\frac{a_8}{T} $$ $$ \frac{s_s}{R}=-\frac{a_1}{2T^2}-\frac{a_2}{T}+a_3ln(T)+a_4T+a_5\frac{T^2}{2}+a_6\frac{T^3}{3}+a_7\frac{T^4}{4}+a_9 $$
R, P, $a_1$ to $a_9$, $\beta$ and $\alpha$ are constants.
I am interested in obtaining $$ \frac{\mathrm{d} K_r}{\mathrm{d} T} $$ But I don't know how to proceed. how do I derive a summation inside an exponential?
Thanks for your help.
Hint
You can make your life much easier taking first the logarithms of both sides.
The idea of using $\log(K_r)$ instead of $K_r$ comes from different places. First, from definition since $$\Delta G=\Delta H- T \Delta S=-RT \log(K_r)$$ But, to me, the most important is the numerical conditioning of the problem : very poor when using the $K_r$'s (even if they vary by only one or two orders of magnitude), very good everywhere when using the $\log(K_r)$'s. All the above apply to chemical and/or physical equilibrium.
I "wasted" more than $50$ years in this area. If you are interested, I shall be delighted to continue discussions around these topics. Cheers.