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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/(TT_{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 interparticle 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, _{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}, _{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 B_{2} and B_{3} 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]
