Thermodynamics and Statistical Physics. Preliminary Ph.D. Qualifying Exam. Summer 2009

Transcription

1 Thermodynamics and Statistical Physics Preliminary Ph.D. Qualifying Exam Summer 2009 (Choose 4 of the following 6 problems) Statistical Physics Part 1a) (1 point) Give a definition of the partition function z for a statistical system. 1b) (7 points) Consider a quantum mechanical system with one ground state and one excited state with energy,, above the ground state. The two states are not degenerate, and there is no other excited state. Obtain an expression for the heat capacity, c V, in terms of, the Boltzmann constant, k B, and the absolute temperature, T. 1c) (2 points) Sketch the dependence of c V on temperature and discuss the limiting behavior for high and low temperatures.

2 2) A rubber band at absolute temperature, T, is fastened at one end to a peg, and supports from its other end a weight, W. In a simple microscopic model of a rubber band, it consists of a linked polymer chain of N segments each of length, a, and each oriented either parallel or antiparallel to the vertical direction. A parallel segment has energy p = -Wa, and an antiparallel segment has energy ap = Wa. a) (7 points) Find an expression for the mean length L of the rubber band as a function of W, T, a, and the Boltzmann constant k B b) (3 points) Sketch the mean length L as a function of temperature. Does the mean length of this model rubber band, increase or decrease with increasing temperature? Is this temperature behavior also true for a real rubber band? (Note: Neglect any kinetic energies or weights of the segments, or any interaction energy between them.)

3 3) You are given a system of two identical particles which may occupy two energy levels n = n n = 0, 1. The lowest energy state, 0, is double degenerate. The system is in thermal equilibrium at temperature T. For each of the following cases determine the partition function and the energy of the system. a) (3 points) The particles obey Fermi statistics (only one particle allowed per state) b) (3 points) The particles obey Bose statistics (no limitation on particles per state) c) (3 points) The (now distinguishable) particles obey Boltzmann statistics. d) (1 point) Discuss the conditions under which Fermions or Bosons may be treated as Boltzmann particles.

4 Thermal Physics Part 4) An aluminum rod 50.0 cm in length and with a cross-sectional area of 2.50 cm 2 is inserted into a thermally insulated vessel containing liquid helium at 4.20 K. The rod is initially at 300 K. Aluminum has thermal conductivity of 31.0 J/s cm K at 4.2 K; ignore its temperature variation. Aluminum has a specific heat of cal/g C and density of 2.70 g/cm 3. The density of liquid helium is g/cm 3. The latent heat of vaporization for helium is J/kg. a) (5 points) If half of the rod is inserted into the helium, how many liters of helium boil off by the time the inserted half cools to 4.20 K? (Assume the upper half does not yet cool.) b) (5 points) If the upper end of the rod is maintained at 300 K, what is the approximate boil-off rate of liquid helium after the lower half has reached 4.20 K.

5 5) In a cylinder of an automobile engine, just after combustion, the gas is confined to a volume of 50.0 cm 3 and has an initial pressure of Pa. The piston moves outward to a final volume of 300 cm 3, and the gas expands without energy loss by heat (i.e. adiabatically). Assume expansion is reversible. a) (3 points) If γ = 1.40 for the gas, where γ is the ratio of molar specific heats, what is the final pressure? b) (1 point) How close is this γ to the γ for a monatomic ideal gas? c) (4 points) How much work is done by the gas in expanding? d) (2 points) On the air-standard diesel cycle figure below, the expansion described in this problem takes place from point C to D. Obtain an expression of efficiency for this cycle in terms of γ and the temperature at each of the four points (A through D).

6 6) The Fundamental Equation of Thermodynamics, in terms of internal energy, is: du = TdS PdV Eq. 1 a) (1 point) What is the equivalent equation in terms of Helmholtz energy? b) (5 points) Use the equation in part a) and the properties of partial derivatives to S derive the Maxwell relation that has: V as one of its two terms. T c) (4 points) The Thermodynamics Equation of State is a useful expression that relates the isothermal change in internal energy with respect to volume changes to an expression that only contains relationships between temperature, volume, and pressure. Obtain such an expression from Eq. 1 and the Maxwell relation derived in b).