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.
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.
State the definitions of H, G, and F. You have to memorize them!
H = U + pV, F = U - τσ, G = U = pV - τσ.
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.
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.