I am uncertain whether chemical potentials stay constant during chemical equilibrium.
Consider a closed container divided into two parts 1 and 2 filled with ideal gas particles. The barrier between parts 1 and 2 allows particle exchange.
The Gibbs energy for parts 1 and 2 are
The total Gibbs energy for the container is
which has differential
It seems like the equilibrium conditions are
(Not sure if it is necessarily true that $N_1d\mu_1=N_2d\mu_2=0$.)
What I can conclude from here is that in chemical equilibrium, 1 and 2 exchange the same number of particles, ie. 1 gains some particles from 2 but gives back the same amount of particles to 2 so that the net change is zero?
Is my reasoning correct? Are there any cases when chemical potentials stay constant during chemical equilibrium?