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Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ----- Spring 2004

Problem Set 6

Due Tuesday, April 27, in lecture

• Problem 1 [30 points total]

  Consider an ideal Bose gas in three dimensions. In lecture we considered how the specific heat at constant volume, C_V behaved as one approached the BEC transition temperature. Here we will investigate how the specific heat at constant pressure, C_p, behaves. This problem is Pathria problem 7.4. Note that C_V = T(S/T)_V,N, and C_p = T(S/T)_p,N, i.e. both are derivates taken keeping the number of particles N constant (i.e. not keeping fugacity z constant!).

  a) Start by showing that for an ideal Bose gas,

  

  where z is the fugacity, and g_x is the "standard" function defined in connection with Bose gases (Pathria Appendix D).

  Hint: look back in the lecture notes at how we computed the corresponding derivative at constant density n=N / V, i.e. (z/T)_v (where v=1/n is the specific volume), in our calcuation of C_V, and apply a similar trick to this case. [10 pts]

  b) Next compute the total entropy by using S = -(/T)_N,µ. Then, compute the entropy per particle S/N, and show that you get Pathria Eq.(7.1.44a). You only have to do the case T>T_c. [10 pts]

  c) Finally show that,

  

  From the above, show that as T approaches the condensation temperature T_c from above, both and C_p diverge as 1/(T-T_c).

  Hint: to show the first identity above, consider the specific heats per particle C/N. [10 pts]

• Problem 2 [35 points total]

  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). [7 points]

  (b) Do the same for d=2 dimensions (hard disks). [7 points]

  (c) Compute the isothermal compressibility, _T, in terms of the virial expansion to 2nd order, i.e. through B₂ and B₃. [7 points]

  (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 .) [7 points]

  (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. [7 points]