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• 1: Van der Waals fluid
  Consider a van der Waals gas, and assume that its energy equation of state is \(E = \frac32 NkT – a/V\). Calculate its (a) constant-volume heat capacity and (b) isothermal compressibility; comment on whether the compressibility is higher or lower than that of an ideal gas at the same temperature and density, and justify your result in terms of the physical meaning of the parameters a and b. 

• 2: Thermodynamics of a plastic rod
  In a certain temperature range, the tension force F applied to a stretched plastic rod is related to its length by F = aT ² (L–L₀), where L₀ is the rod's unstretched length and a a positive constant. When L = L₀, the heat capacity of the rod at constant length is C_L = bT, where b is a constant.
  (a) Write down the fundamental thermodynamic identity for this system, relating dE, dS and dL.
  (b) The entropy S of the rod is a function of T and L. Compute (∂S/∂L)_T.
  (c) Assuming you know S(T₀, L₀), find S(T, L) at any other values of T and L. (Hint: Calculate first the change of entropy with temperature at the length L₀, where the heat capacity is known.)
  (d) If one starts at T = T_i and L = L_i, and stretches the thermally insulated rod quasistatically until it attains a length L_f, what is the final temperature T_f? Is T_f larger or smaller than T_i?

• 3: Small oscillations in a gas cylinder
  A cylinder with insulating walls is divided into two equal compartments by means of an insulating piston of mass M, which can slide without friction. The cylinder has cross section area A, and the compartments are of length L. Each compartment contains n moles of a classical ideal gas with \(C_p^~/C_V^~= \gamma\) at a temperature \(T_0^~\).
  (a) Suppose the piston is adiabatically displaced a small distance \(x \ll L\). Calculate the pressures \(P_1^~\) and \(P_2^~\), and the temperatures \(T_1^~\) and \(T_2^~\) in the two chambers, to first order in x.
  (b) Find the frequency of small adiabatic oscillations of the piston about its equilibrium position.
  

Aside from discussions I may have had with other students in the class about how to solve these problems, the solutions to this homework set I am submitting are entirely my own. While I was writing these solutions I did not look at anyone else's solutions, and I was not told by anyone what to write.

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