Problem Set 5

Due Tuesday, April 23, in lecture

Problem 1 [15 points]
Consider a non-interacting ideal gas of non-relativistic bosons (i.e. e(p) = p²/2m) in d dimensions, where d is any number. Show that there is no Bose Einstein condensation for any d less than or equal to two.
Problem 2 [35 points total]
Consider a gas with hard core inter-particle interactions, i.e.

u(r) = infinity
0 for r<r_o
for r>r_o

(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, kappa_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, kappa_T, and the coefficient of thermal expansion alpha.) [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 kappa_T and C_p-C_V vary with dimension d for hard core interactions. Comment on any trends you see. [7 points]

Last update: Monday, August 20, 2007 at 12:10:03 PM.