Home

Contact
Info

Course
Info

Calendar

Homework

Lecture
Notes

Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ----- Spring 2005

Problem Set 7

Due Thursday, April 14, in lecture

• Problem 1 [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 _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 2 [15 points]

  A linear molecule of N identical atoms has a vibrational spectrum given by the angular frequencies,

  _m=_osin(m/2N), for m = 1,2, ..., N-1 (i.e. these are the frequencies of the normal modes of vibration).

  a) Show, by explicit calculation, that the vibrational contribution to the specific heat of this molecule at very high temperature, T-> , is

      C = (N-1)k_B                

  b) Show that for lower temperatures, such that 1/N << k_BT/_o << 1,

      C ~ T                          

  c) How does C vary with temperature at very low temperatures, such that  k_BT/_o << 1/N?    

• Problem 3 [25 points]

  Consider a gas with hard core inter-particle interactions, i.e.

  u(r)

  =

  infinity      0

  for r<ro for r>ro

  (a) Compute the 2nd and 3rd virial coefficients, B₂ and B₃ for d=1 dimension (hard rods).

  (b) Do the same for d=2 dimensions (hard disks).

  (c) Compute the isothermal compressibility, _T, in terms of the virial expansion to 2nd order, i.e. through B₂ and B₃.

  (d) Compute the specific heat at constant pressure, C_p, in terms of the virial expansion to 2nd order, i.e. through B₂ and B₃. (Hint: to compute C_p you might wish to use the formula that relates it to C_V, _T, and the coefficient of thermal expansion .)

  (e) Using your results in parts (a) and (b), and using Pathria Eqs. (9.3.15) and (9.3.20) to get B₂ and B₃ for hard spheres in d=3 dimensions, make a table showing how _T and C_p-C_V vary with dimension d for hard core interactions. Comment on any trends you see.