PX284 - I1 - definition

the chemical potential, $μ$ , is defined as the change in energy on adding a particle to the system
from the first law of thermodynamics

d U = T d S - p d V - μ d N

μ = (\frac{\partial U}{\partial N})_{S}, V

μ = {(\frac{\partial F}{\partial N})}_{V, T} = {(\frac{\partial G}{\partial N})}_{p, T}

note: $G (p, T, N)$ is extensive, ie: scales with the system size
therefore, not only is $μ = (\frac{\partial G}{\partial N})_{p, T}$ , moreover, it is the gibbs free energy per particle: $μ = \frac{G}{N}$