Helmholtz Free Energy

If the system is in contact with a reservoir, F will be a minimum when the two systems are in equilibrium. To see this, consider an infintesimal transfer of energy from the system to the reservoir at constant temperature. Then
 

But, by definition, , so we see that dU = t ds. Thus, dF = 0, which is the condition of an extremum. To show this is a minimum, recall that since the total energy of the combined system is U = U_R + U_S, the entropy of the combined system is
 

Now recall that the system is in its most probable configuration at equilibrium. This means that the entropy of the combined system is maximized. This can only be true of F_S is a minimum at equilibrium.
    Consider an infinitesimal change in F
 

From the thermodynamic identity found earlier, we see that dU - t ds = -p dV, so this becomes
 

but in general,
 

so we get the identifications
 
 

and

(6.2)

Maxwell Relations

(6.3)

This is the first of what are known as Maxwell relations. We will derive more later in the course.
    Since we have stated that the partition function is extremely important and is used to derive many of the macroscopic properties of the system, we would like to recast the Helmholtz free energy as a function of Z. Start with the definition of F
 

From (6.2) we saw that  so this becomes a differential equation,
 

Dividing through by t, we see that this is equivalent to
 
 

(6.4)

Recall that U is the average energy of the system, <e_S>, and that after defining the partition function we showed that
 

Substituting this for U, we get
 

or
 

We can evaluate the volume dependent function by noting that as t ® 0, the entropy must reduce to ln g₀, where g₀ are the states at the lowest energy e₀. In this limit the energy of all the states reduces to e₀ and _. Thus, . So the entropy becomes
 

But this can be ln g₀ only if A(V) = 0. Thus
 
 

We can rearrage (6.5) to get Z as a function of F. This yields
 

Substituting this into the definition of the probability of the system being in any quantum state associated with energy e_s, we get
 
 

(6.6)