**Problem : **

*G* and *H* are functions of which three variables each?

Simply by writing out the identity associated with each, we can extract
the variables from the differentials. We see that *G* is a function of
*τ*, *p* and *N*, and that *H* is a function of *σ*, *p* and
*N*.

**Problem : **

Suppose that we wanted to define an energy *A* that was a function of
*σ*, *V* and *μ*. Give *A* in terms of *U* and appropriate other
variables, and give the differential identity for *A*.

Let *A* = *U* - *μN*. Then *dA* = *dU* - *μ*, *dN* - *N*, *dμ*, or
*dA* = *τ*, *dσ* - *p*, *dV* - *N*, *dμ*.

**Problem : **

State the definitions of *H*, *G*, and *F*. You have to memorize them!

*H* = *U* + *pV*, *F* = *U* - *τσ*, *G* = *U* = *pV* - *τσ*.

**Problem : **

A given system is to be expanded at constant temperature and constant
number of particles. We might say it undergoes an "isothermal
expansion". Find the energy that most simply describes how the energy
changes in this process, and write the simplified differential.

We want to find the energy that has *τ* and *N* as differentials, so
we choose *F*, the Helmholtz Free Energy. Then *dF* = - *p*, *dV*. We can
then easily see how the energy change relates to the pressure.

**Problem : **

Explain a process in which the enthalpy remains constant.

If a system remains at constant entropy, pressure and number, then no
matter what happens to, say, the temperature, the enthalpy will not
change.