Problem Set 1

Problem Set 1

DQ 1 -- Due Tuesday, February 9, 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).
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?

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Problems -- Due Thursday, February 11, by 5pm.

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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]