In this problem you will show that the fluctuations in the internal energy are very small if there is a large number of particles. a. Start with U = U.P. 8 and find an expression for U2 in terms of the partition function Z. b. Next compute of = U () in terms of the partition function. Hint: Your answer should be in terms of Z not log Z. c. Compare with your answer in problem (1) and show that 1 o = 32 Cv of Hint: Write everything out in terms of Z, not log Z. d. The relative fluctuation is ou/. Note that both and Cy are extensive quantities and so they are each proportional to N. Use this to show that ou/ ~ 1/N.