# The relation between Chemical potential and Gibbs free energy ($n\mu = G$) is global?

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).