Answer a total of any THREE out of the four questions. Put the solution to each problem in a separate blue book and put the number of the problem and your name on the front of each book. If you su...
Answer a total of any THREE out of the four questions. Put the solution to each problem in a separate blue book and put the number of the problem and your name on the front of each book. If you submit solutions to more than three problems, only the first three problems as listed on the exam will be graded. Some possibly useful information: Sterling's asymptotic series: N In N-N + 1 ln(2nN] In N! N → oo, as 2 dr r exp(-ar2) - /-exp(-) ) with Re(a) > 0 dz exp(-αχ2 ßr) + Consider a classical relativistic ideal gas with a fixed number N > 1 of free, massless, identical 1 omentum p the energy is Ep clpl. particles occupying a volume V. For a massless particle of m (a) Derive expressions for the partition function Z(T, V, N), the thermal energy U, and the en- (b) Determine the constant pressure and constant volume heat capacities cp and cv. (c) Derive expressions for the entropy S and the Helmholtz free energy F-U-TS, and explicitly thalpy H = U + PV verify the relation S(T,V, N) =-(an where V and N are held fixed. 2. Consider a classical ideal nonrelativistic gas of identical atoms in thermal equilibrium at a tem- -. The perature T, with the gas being confined by a neutral atom trap with potential V(r) number of atoms, N, is fixed and large. (a) Determine the partition function Z(N,T) and the entropy S(N,T) of the system
Answer a total of any THREE out of the four questions. Put the solution to each problem in a separate blue book and put the number of the problem and your name on the front of each book. If you submit solutions to more than three problems, only the first three problems as listed on the exam will be graded. Some possibly useful information: Sterling's asymptotic series: N In N-N + 1 ln(2nN] In N! N → oo, as 2 dr r exp(-ar2) - /-exp(-) ) with Re(a) > 0 dz exp(-αχ2 ßr) + Consider a classical relativistic ideal gas with a fixed number N > 1 of free, massless, identical 1 omentum p the energy is Ep clpl. particles occupying a volume V. For a massless particle of m (a) Derive expressions for the partition function Z(T, V, N), the thermal energy U, and the en- (b) Determine the constant pressure and constant volume heat capacities cp and cv. (c) Derive expressions for the entropy S and the Helmholtz free energy F-U-TS, and explicitly thalpy H = U + PV verify the relation S(T,V, N) =-(an where V and N are held fixed. 2. Consider a classical ideal nonrelativistic gas of identical atoms in thermal equilibrium at a tem- -. The perature T, with the gas being confined by a neutral atom trap with potential V(r) number of atoms, N, is fixed and large. (a) Determine the partition function Z(N,T) and the entropy S(N,T) of the system