Proving Minimization of Helmholtz Free Energy at Equilibrium

[Silviu]

Homework Statement

Show that for a reaction occurring at constant T and V, F is minimized at equilibrium.

Homework Equations

$F=U-TS$
$TdS=dU+pdV-\mu dN$

The Attempt at a Solution

$dF=dU-d(TS)=dU-TdS-SdT=dU-dU -pdV+ \mu dN -S dT=-pdV - SdT + \mu dN$. At constant T and V this reduces to $dF = \mu dN$. But I don't know what to do from here. Also in the next problem we have to use the fact that F is minimized at equilibrium to prove a relation between the chemical potentials of certain elements involved in a chemical reaction, so I assume i can't set $dN=0$, as in a chemical reaction N changes and it seems that the fact that F is minimal at equilibrium holds in chemical reactions, too.
How can I solve this? Thank you!

 

[Mapes]

When multiple species exist, the last term must be $\Sigma\mu_i \,dN_i$, no? That's quite different from $\mu\,dN$ in the context of your problem.

Then, you have to prove that $F$ is minimized rather than maximized or at a saddle point. Use the second derivative to do this.