**Helmholtz Free Energy**

Define the **Helmholtz free energy** as

*F* = *U* ‑ t s (6.1)

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

*dF* = *dU* ‑ t ds

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*

*dF* = *dU* ‑ t *d*s ‑ s *d*t

From the thermodynamic identity found earlier, we see that *dU* ‑ t *d*s = –*p**dV*, so this becomes

*dF* = ‑*p**dV* ‑ s *d*t

but in general,

so we get the identifications

and (6.2)

*Maxwell Relations*

*Maxwell Relations*

Now consider the second derivatives and. We know that they must be equal to each other. Substituting the equalities in (6.2), we get the relation

(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*

*F* = *U* ‑ t s

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, S>, and that after defining the partition function we showed that

Substituting this for *U*, we get

or

*F* = ‑t ln *Z* + t *A*(*V*)

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

But this can be ln *g*_{0} only if *A*(*V*) = 0. Thus

*F* = ‑t ln *Z* (6.5)

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

*Z* = e^{‑F/}^{t}

Substituting this into the definition of the probability of the system being in any quantum state associated with energy e_{s}, we get