Answer Happy

solve only (a,b,c,d) parts of the question
4.12 Consider a gas of N spin-zero bosons in a d-dimensional container of volume V, with a dispersion relation Ep = α|p|s where the constant a and the index s are both positive. (a) Find expressions for the mean number of particles per unit volume in the ground state and the mean total number of particles in the excited states, in terms of the temperature 7 and the fugacity z = eßμ. (b) Find the conditions on s and d for which Bose-Einstein condensation takes place. (c) Find the equation of state for this gas. (d) Find the relative population of the ground state No/N as a function of temperature, assuming that N/V is fixed. (e) Find the entropy per unit volume of the gas in terms of T and . (f) Assuming that N/V is fixed, evaluate the discontinuity in the specific heat at the critical temperature. Show that for d = 3, s = 2, there is no discontinuity. (g) Evaluate the discontinuity in the derivative of the specific heat at the critical temperature for the case d = 3, s = 2. (h) When there is condensation, it is possible to regard the particles in the ground state and those in excited states as two distinct phases coexisting in the same container. Find the latent heat per particle for the transition between these two phases and verify that the Clausius-Clapeyron equation is obeyed.