The change of Gibbs free energy (for a single phase or constitute system) is

$$dG = -SdT + VdP +{\mu}dn$$

By using the fact that $T$ and $P$ are intensive properties and $n$(mole) is an extensive property,

$G = n\mu$ is derived in my lecture note.

But I am not sure about that, because, with the definition of $dG$, $G$ is the function of $(T,P,n)$.

Where are the dependency of $T$ and $P$?

This interpretation is only valid for constant temperature and pressure situations?

like, $G = G_{0}(T,P) + n\mu$?


$G$ depends on $T$ and $P$ through the chemical potential:

$$G(T,P,n) = \mu(T,P,n) \cdot n$$

Recall that $\mu = \left(\frac{\partial G}{\partial n}\right)_{T,P}$. If $G$ is a function of some set of variables, then its derivatives (and in particular, $\mu$) are functions of the same set of variables. Note also in this case that $\left(\frac{\partial \mu}{\partial n}\right)_{T,P} = 0$, in accordance with the assumptions used to derive $G = \mu n$ (though these assumptions do not universally hold for all thermodynamical systems).


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