# Why is $\Delta_r G^0$ temperature dependent in the van 't Hoff equation?

One can derive for a equilibrium reaction between ideal gases

$$\sum_i \nu_i R_i \leftrightarrow \sum_i \nu_i P_i$$ the expression $$\Delta_r G^0=-RTln(K) \quad \textrm{where} \quad K=\frac{\Pi_{\textrm{products}} p_i^{\nu_i}}{\Pi_{\textrm{reactants}} p_i^{\nu_i}}$$ under the assumption that $$P,T$$=const. and that G is minimized. I know how to do this. But in my derivation the resulting $$\Delta_r G^0=\sum_i \nu_i \mu_i^0$$ is independent of the temperature $$T$$ since it is the Gibbs energy at standard conditions.

Yet in this wikipedia article it is stated, that this Gibbs energy is dependent on the temperature $$T$$. In particular

$$\Delta_r G^0=\Delta_r H^0-T \Delta_r S^0$$ Shouldn't it be $$\Delta_r G^0=\Delta_r H^0-T_0 \Delta_r S^0$$ Where $$T_0$$ is standard temperature? I know this is a very specific question and I didn't give my derivation, but maybe someone stumbled upon this issue before. The literature isn't helping me, as many of the texts are very sloppy with the maths.

• I have resolved the issue. As this is not treated in any literature I red, I will add an explanation as an answer, once I got more spare time. In the mean time: One should not make the mistake and think $\mu(p,T)=\mu_0+RT\ln(p/p_0)$ with $\mu_0$ being independent of $T$. Indeed it is $\mu_0(T)$. – TheoreticalMinimum Dec 6 '19 at 18:42