Proof of the chemical potential formula My textbook as well as Wikipedia both state the formula $\mu_A = \mu_A^0 + RT\ln\{A\}$ which gives the chemical potential $\mu_A$ of a chemical species $A$ as function of the standard chemical potential $\mu_A^0$ and the activity (relative to standard conditions) $\{A\}$.

This formula can also be used to derive the Gibbs free energy formula as $\Delta_rG_{T, p} = \Delta_rG_{T, p}^0 + RT \ln Q_r$ where $Q_r$ is the reaction coefficient.

Both formulas appear to be widely used in chemistry, but I have not found a decent explanation for why they are correct. The texts I have seen did not provide sources for the claim and omitted explanation.

Is this an empirical formula, or is it a consequence of statistical mechanics? I would like to find a source with a clear proof or explanation for the origin of this equation. I know that some will define the activity precisely using this equation, but then the question becomes why the activity so-defined corresponds to concentration for ideal solutions. That is, why in an ideal solution, we have $\mu_A = \mu_A^0 + RT\ln c_A$ where $c_A$ is concentration of $A$ relative to standard concentration.
The Sackur-Tetrode equation can be written as $$S^{Trans}=R\ln \left( \frac{e}{C\Lambda^3} \right)$$ where $C$ is the concentration and $\Lambda$ the thermal wavelength. You can rewrite this as $$S^{Trans}=R\ln \left( \frac{e}{\Lambda^3 C^\circ\frac{C}{C^\circ}} \right) = R\ln \left( \frac{e}{C^\circ\Lambda^3} \right)-R\ln\left( \frac{C}{C^\circ} \right) = S^{\circ,Trans}-R\ln\left( \frac{C}{C^\circ} \right)$$ where $C^\circ$ is an arbitrary standard concentration. When you combine it with enthalpy and the other entropy terms you get $$G = G^{\circ}+RT\ln\left( \frac{C}{C^\circ} \right) $$