Problem 1 [35 points total]
Consider the classical ideal gas. In lecture, we found that the total entropy could be written as:
S(E,V,N) = (N/No)So + NkB ln [ (E/Eo)3/2(V/Vo)(N/No)-5/2 ]
where E is the total internal energy, V is the total volume, and N is the number of particles. Eo, Vo, No, So, and kB are constants.
(a) Starting from the above S(E,V,N), find the Helmholtz free energy A(T,V,N), the Gibbs free energy G(T,p,N), and the Grand Potential Σ(T,V,µ), by the method of Legendre transforms. [10 points]
(b) Find the familiar equation of state, pV = NkBT, by taking an appropriate 1st derivative of an appropriate thermodynamic potential. [5 points]
(c) Find the chemical potential µ, by taking an appropriate first derivative of a thermodynamic potential. By comparing to your result in part (a), show explicitly that the chemical potential is the same as the Gibbs free energy per particle. [5 points]
(d) Find the pressure p, by taking an appropriate first derivative of a thermodynamic potential. By comparing to your result in part (a), show explicitly that the pressure is the same as the negative of the Grand Potential per volume. [5 points]
(e) By computing the appropriate 2nd derivatives of the appropriate thermodynamic potentials, compute the specific heats CV and Cp, the compressibilities κT and κS, and the coefficient of thermal expansion α. Show by direct comparison of these results that the two specific heats, and the two compressibilities, are indeed related to each other by the general formulae derived in lecture. [5 points]
(f) If one allows the gas to expand isothermally (i.e. at constant temperature), how does the volume vary with the pressure? If one allows the gas to expand adiabatically (i.e. at constant entropy), how does the volume now vary with the pressure? You must show how you derive your results. [5 points]
Problem 2 [10 points total]
Consider taking the Legendre transform of the energy, E(S,V,N), with respect to S, V, and N, to get a new thermodynamic potential, X(T,p,µ).
(a) Show that X(T,p,µ) is identically zero. [5 points]
(b) What does this imply about the variables T, p, and µ? Where have you already seen this result before? [5 points]