Problem Set 1

Problem Set 1

Due Thursday, February 13, uploaded to Blackboard by 11:59pm

Problem 1 [25 points total]
Consider the classical ideal gas. In Notes 1-3, we found that the total entropy could be written as:
S(E, V, N) = (N/N_o)S_o + Nk_B ln [ (E/E_o)^3/2(V/V_o)(N/N_o)^-5/2 ]
where E is the total internal energy, V is the total volume, and N is the number of particles. E_o, V_o, N_o, S_o, and k_B 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 Φ(T, V, µ), by the method of Legendre transforms. [10 points]
(b) Find the familiar equation of state, pV = Nk_BT, 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]
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).
What can you say about this new potential X? What does X imply about the variables T, p, and µ? Have you seen this before?
Problem 3 [10 points total]
Taking 2nd derivatives of the appropriate thermodynamic potentials for the ideal gas, as found in problem 1, compute the specific heats C_V and C_p, the compressibilities κ_T and κ_S, 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 Notes 1-8.