Problem Set 2

Problem Set 2

DQ 2 -- Due Tuesday, February 1, by 5pm

If one allows an ideal 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? Specifically, you should find that p and V are related algebraically, with p ∼ 1/V^γ. What is the value of the exponent γ for each case? Don't just give a general result, give the numerical values of γ for the ideal gas. You may use your results from problem 1 of Problem Set 1.

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Problems -- Due Thursday, February 3, by 5pm.

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Problem 1 [10 points total]
Taking 2nd derivatives of the appropriate thermodynamic potentials for the ideal gas, as found in Problem Set 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.
Problem 2 [10 points total]
Prove the following relationship between the specific heat at constant pressure, C_p, and the coefficient of thermal expansion α.
(∂C_p/∂p)_T = −TV[α² + (∂α/∂T)_p]
Problem 3 [15 points total]
In a particular engine a gas is compressed in the initial stroke of the piston. Measurements of the instantaneous temperature, carried out during the compression, reveal that the temperature increases according to the relation:
T = (V/V_o)^ηT_o
where T_o and V_o are the initial temperature and volume and η is a constant. The gas is compressed to the volume V₁ (where V₁ < V_o). Assume that the gas is a monatomic ideal gas of N atoms, and assume the process is quasi-static (i.e. the system is always instantaneously in equilibrium).
a) Calculate the mechanical work done on the gas. [5 points]
b) Calculate the change in the total energy of the gas. [5 points]
c) Calculate the heat transfer Q to the gas. For what value of η is Q = 0? Show that this corresponds to the case of adiabatic compression (see this week's discussion question, above). [5 points]
[Hint: you may use the facts you know about an ideal gas, i.e. pV = Nk_BT, and E = (3/2)Nk_BT.]