3. (a) Show that Ber + a = ln(1 Fīr) - Inñr. where & is the energy of (single particle) state r, 7, is the mean occupati

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3 A Show That Ber A Ln 1 Fir Innr Where Is The Energy Of Single Particle State R 7 Is The Mean Occupati 1 (50.18 KiB) Viewed 24 times

3. (a) Show that Ber + a = ln(1 Fīr) - Inñr. where & is the energy of (single particle) state r, 7, is the mean occupation number, and the upper (lower) signs are for FD (BE) statistics, respectively. (b) Show that In 2 = 7 In(17 nr), where Z is the grand partition function. (e) i. From a knowledge of the partition function Z derived in class, write an expression for the entropy S of an ideal FD gas. Express your answer solely in terms of ñr, the mean number of particles in stater. ii. Write a similar expression for the entropy S of a BE gas. iii. What do these expressions for S become in the classical limit when ñ <1? Use the above two results and S = k(In 2 + N + BĒ), which we found in class. (d) Show that the answer to (c)iii is what you would expect for an ideal gas with "fudged clas- sical" statistics (i.e., classical statistics with the correction factor for indistinguishability thrown in). (Start by finding the entropy of a single particle, which has probabilities P, = n/N of being in stater. Then find the entropy of all Ñ indistinguishable particles.)