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C.2 Consider a gas of bosons in equilibrium with a reservoir of energy and particles at temperature T and chemical potential pl. a) Show that the grand canonical partition function of a single quantum state is given by 1 2 = 1-open 1-exp KBT [5] where ez is the energy of the quantum state, and kp is the Boltzmann constant. b) The density of modes g(v) for radiation in a cavity of volume V is given by 87 g(v)dv = s výdv, where v is the frequency. Using the result from part a), show that the Helmholtz free energy of the radiation is given by P=k97 Lº dv g(v) in (1 – esp ( hv КВТ ] 0 where h is the Planck constant. c) Use the thermodynamic relation ar P=- av (張)。 т to show that the radiation pressure is given by 3 P= 875 -kot 45 kpT ch HINT: 6.° ) * dx a2log (1-e-*1 = - = - TA 45 [6] d) Calculate the compressibility of the radiation ӘР K = -V av ", T [3] and explain the result. e) In a model star, the radiation field and an ideal monatomic gas co-exist in thermal equilibrium inside a cavity of volume V. The number density of the gas is 3 x 1019 atoms/cm². At what temperature are the gas pressure and the radiation pressure equal? [7] ]