PHY 418: Statistical Mechanics I
Prof. S. Teitel: stte@pas.rochester.edu ---- Spring 2022
DQ 1 -- Due Tuesday, January 25, by 5pm
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).Post your response on the Discussion Board at this link: DQ1What can you say about this new potential X? What does X imply about the variables T, p, and µ? Have you seen this before?
Upload your solutions to Blackboard at this link: PS1
Consider the classical ideal gas. In Notes 1-3, 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 Φ(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]