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, B_{2} and B_{3} 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 B_{2} and B_{3}. [7 points]
(d) Compute the specific heat at constant pressure, C_{p}, in terms of the virial expansion to 2nd order, i.e. through B_{2} and B_{3}. (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 B_{2} and B_{3} 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]