My question is the following :
I have a system at (T,V,N) which is composed of two phases : (T1,V1,N1) and (T2,V2,N2).
Initially I wanted to proove that $ \mu_1 = \mu_2 $, but I had troubles.
To proove it i use the fact that $F=F_1+F_2$ must be minimised at equilibrium because we are in (T,V,N).
So we have :
I can write : $V=V_1+V_2$, $N=N_1+N_2$ and s N and V are fixed, $dV_2=-dV_1$, $dN_2=-dN_1$.
Then I wanted to say "well, $N_1$, $V_1$, $T$ are independant variables so I have to cancel the terms in frond of $dN_1$, $dV_1$ and I would have $P_1=P_2$ and $\mu_1=\mu_2$. But there is these terms in $dT_1$ and $dT_2$ that I don't know how to replace by $dT$...