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Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ----- Spring 2002

Midterm Exam

Physics 418            Midterm Exam            Spring 2002

Answer both parts I and II below. Please show all you work. Explain your steps to maximize your chances for partial credit.

Once you have finished the exam, take this problem sheet home, redo the problems, and turn them in at Monday's lecture. When you work on the problems at home you may consult only your class notes and Pathria; do not discuss the problems with any one else, or refer toany other texts. Your performance on this “homework” can be taken into consideration if you have done poorly on the in-class exam.

Problem

Consider a classical ideal gas of indistinguishable non-relativistic point particles of mass m placed in the gravitational field near the surface of the Earth, i.e. a particle at height y has a potential energy U(y)=mgy. Assume the gas is everywhere at a constant temperature T. Because of the potential U(y), the pressure p and density n of the gas will vary with height y.

The goal of this problem is to find how the pressure p(y) and density n(y) vary with height y.

To solve this problem, consider conceptually dividing the gas up into a set of slabs of very small thickness Dy, as shown in the figure below. The dashed line separating the slab at height y from the slab at height y+Dy can be thought of as an imaginary wall that allows the exchange of both heat and particles between the two slabs. Since the two slabs are in equilibrium, they must therefore have the same temperature T and chemical potential µ. By computing the appropriate partition function of the gas in the slab at height y, and using the fact that T and µ. are independent of height y, you should be able to solve the problem!

Part I [50 points]

Solve this problem using the canonical ensemble, by computing the canonical partition function, the Helmholtz free energy, and the chemical potential of the gas in the slab at height y. Then use the fact that the chemical potential is independent of height y to find how the density varies with y. Then find how the pressure varies with y.

Part II [50 points]

Solve this same problem using the grand canonical ensemble, by computing the grand partition function of the slab at height y. This should allow you find how the pressure varies with y. Then find how the density varies with y.