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.

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    $\begingroup$ 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 $\mu_0(T)$. $\endgroup$ – TheoreticalMinimum Dec 6 '19 at 18:42

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