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Spring 2005 PHY 203: Introduction to Statistical Mechanics Assignment #2 Due at the beginning of class on Wednesday, 2/09/05 Problems: 1. The following describes a method used to measure the specific heat ratio γ ≡ cP /cV of a gas (assumed ideal). The gas is confined within a vertical cylinder and supports a freely moving piston of mass m. Gravity (acceleration g) acts on the piston, but its effect on the gas is negligible. The cross-sectional area of the piston (and the cylinder) is A. Atmospheric pressure is P0 , and when the system is in mechanical and thermal equilibrium, the volume of the gas is V0 . The piston is now displaced slightly from its equilibrium position and is found to oscillate with frequency ν. (The oscillations eventually die out, but not before an accurate measure can be made.) The oscillations are slow enough that the gas can be assumed to be in thermal equilibrium at all times, but fast enough that there is no appreciable exchange of heat with the piston or cylinder. Express γ in terms of ν and the system parameters. 2. (a) Show that for two large systems in thermal contact, the number Ω0 (E (0) , E1 ) can be expressed as a Gaussian in the variable E1 . Determine the root-mean-square deviation of E1 from the mean value Ē1 in terms of the temperature and the specific heats of the systems. (b) Make an explicit evaluation of the root-mean-square deviation of E1 in the special case when the system A1 and A2 are ideal classic gases. 3. Study the statistical mechanics of an extreme relativistic gas characterized by the single-particle energy states: ε(nx , ny , nz ) = h q 2 nx + n2y + n2z 2L Show that the ratio CP /CV is in this case 4/3 instead of 5/3. Work along the lines of Pathria section 1.4, but do not try to calculate Ω(E) explicitly. Just calculate γ. 4. A mole of argon and a mole of helium are contained in vessels of equal volume. If argon is at 300 K, what should the temperature of helium be so that the two have the same entropy? 1