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

Problem Set 4

Due Tuesday, April 16, in lecture

• Problem 1 [5 points]

  The Stefan-Boltzmann law states that a black body at temperature T radiates power per unit surface area equal to sigma*T⁴ where sigma is a universal constant independent of the material properties of the body. Assuming that the sun and the earth are black bodies, and that the earth is in thermal equilibrium with the sun [i.e. energy absorbed = energy emitted] calclate the temperature of the earth in terms of the temperature of the sun. Look up the parameters you need in order to compute a number for this estimate of earth's temperature.

• Problem 2 [15 points]

  N Fermions A of spin 1/2 are introduced into a large volume V at temperature T. Two Fermions may combine to create a Boson with spin 0 via the interaction,

  A + A <--> A₂

  Creation of the molecule A₂ costs energy e_o > 0. At equilibrium, the system will contain N_F Fermions and N_B Bosons. Provide expressions from which the ratio N_B/N_F can be calculated, and perform the calculation explicitly for T=0. What would this (T=0) ratio be, if the particles were classical (i.e. quantum statistics can be neglected). Explain the difference.

• Problem 3 [15 points]

  a) Find the chemical potential µ for an ideal (non-relativistic) Fermi gas at low temperature, to second order in T, at fixed pressure p. Note, this is a different expansion from that at fixed density N/V. (Hint: what is E/V at fixed p?)

  b) The Gibbs free energy is related to the chemical potential by G(T,p,N) = Nµ(T,p). The entropy can be derived from the Gibbs free energy, and hence from the chemical potential, by using,

  S = - (dG/dT)_p,N = - N(dµ/dT)_p

  Using your result from part (a), find the entropy of the Fermi gas to lowest order in T. Does your result agree with that given in Pathria §8.1 Eq.(41), as derived there from the Helmholtz free energy?

• Problem 4 [15 points]

  Consider an ideal Bose gas composed of molecules with an internal degree of freedom. Assume that this internal degree of freedom can have only one of two energy values, the ground state e_o = 0, and an excited state, e₁. Determine the Bose Einstein condensation temperature of the gas as a function of e₁. Show in particular that for e₁/k_BT >> 1,

  T_c/T_co = 1- (2/3)(1/zeta(3/2))e ^-e₁/k_BT

  where T_co is the transition temperature when e₁ is infinite, and zeta is the Riemann zeta function.