PHYS-4420THERMODYNAMICSSTATISTICAL.doc

PHYS-4420 THERMODYNAMICS STATISTICAL MECHANICS SPRING 2001 FINAL EXAMINATION Tuesday, May 1, 2001 Your grade will be sent to you by e-mail by midnight, Wednesday, May 2, 2001 NAME: _______________________________________ e-mail address:___________________ There are seven pages to this examination. Check to see that you have them all. To receive credit for a problem, you must show your work, or explain how you arrived at your answer. 1. (20%) Consider a collection of N identical, distinguishable harmonic oscillators, all of frequency ?. The energies that one these oscillators can take on (measured relative to the ground state) are ?n = nh?. Where n is an integer that can take on values from 0 to ?. a) (5%) Find the partition function ?, for one of these oscillators. (Hint: for x 1) ? = _______________ b) (5%) Find the partition function Z, for the collection of oscillators. (Since the oscillators are distinguishable, no N! is needed in the denominator.) Z = _______________ c) (5%) Find the Helmholtz function F, for this collection of oscillators. (Hint: F = – kT ln Z) F = _______________ d) (5%) Find the entropy S, for this collection of oscillators. Hint: . This will not give a particularly neat expression. However, you can check your result as follows: as T goes to zero, S goes to zero, and as T gets very large, S goes to . S = __________________________________ 2. (50%) The figure shows a p-V diagram, for n moles of a monatomic ideal gas, dreamed up by an entrepreneur who wants to improve on the Carnot cycle. The cycle begins at a where the pressure is p0, the volume is V0, and the temperature in T0. It proceeds along the isobaric path ab to b, where the volume is 2V0. Then it goes along the isothermal bc to c where the volume is 4V0. Next it goes along the isobaric cd to d where the volume is 2V0, and finally along the isothermal da back to a. (Remember: pV = nRT) a) (4%) Calculate t