Basic question about an expression for Gibbs free energy

Question

I'm getting into thermodynamics, and I have a basic question about the following expression for Gibbs free energy:

$G = \sum \mu_i N_i$

Is it correct that this equality only holds for constant temperature and pressure, as follows from integration over N of the expression for $dG$:

$dG = -SdT + VdP + \sum \mu_i dN_i$

Answer 1

Both Chet Miller and Buck Thorn have provided good answers, and explained that

$$G = \sum \mu_i N_i \tag{1}\label{1}$$

is indeed a function of $T$ and $P$. One way to show that is as follows.

As you say, we can derive (1) from the Fundamental Theorem of Thermodynamics for a closed system in which composition can change:

$$ dG = V dP -S dT + \sum_i \mu_i dN_i \tag{2}\label{2}$$

Now take the differential of (1):

$$dG = \sum \mu_i d N_i + \sum N_i d\mu_i $$

Equate that differential with the Fundamental Theorem (2), to give

$$\begin{align} dG = \sum \mu_i d N_i + \sum N_i d\mu_i &= V dP -S dT + \sum_i \mu_i dN_i \\ \sum N_i d\mu_i &= V dP -S dT \end{align}$$

which shows that a change in either $T$ or $P$ at constant composition necessarily results in a change in some chemical potential $\mu_i$. Hence, $\mu_i$ is a function of both $T$ and $P$, as is $G$ in equation (1).

For a system composed of a pure substance,

$$\begin{align} N d\mu &= V dP -S dT \\ d\mu &= \overline{V} dP -\overline{S} dT \end{align}$$

Answer 2

In the equation

$$G = \sum \mu_i N_i$$

$G$ is the absolute value of the Gibbs free energy at some point in state space. While not explicitly stated, the values of the chemical potentials $\mu_i$ are in general functions of composition and other state variables (e.g. T and p), rendering $G$ a function of these same state variables.

On the other hand, the equation

$$dG = -SdT + VdP + \sum \mu_i dN_i$$

describes an infinitesimal difference in the free energy between two (infinitesimally close) points in that space.

Neither equation puts constraints on T and p, that is, both are general.