Statistical thermodynamics may be used to find the radiation pressure p for cavity (or black body) radiation in terms of the energy per unit volume u. (a) an ideal quantum gas comprises non-interacting identical particles with discrete quantum states labelled 1, 2, partition function is given by z (t, v,n)- > exp(-b(n,& , + п,& , +} пп. (i) define the symbols n1, 81, 82, (iiexplain why, for photons, the partition function may be expressed as: in zo (t, v)- exp(-be, [4 marks (b) for photons, the number of states with angular frequency in the range o to o + do is given by vo'do f (o)do п'с3 use the results of part (a), the definition of helmholtz free energy f and an appropriate meaning for x to show that v . x* n(1-е") dx. f = [6 marks] (c) (i write the ftr in terms of f, with t and v as the independent variables, and hence show that pressure p and entropy s are given respectively by: p = and s av от (ii) by considering the relationship between f and ln z, and between the mean internal energy e and ln z, show that а(bf) ap (iii) find simple expressions for p, s and e. explain why these can be obtained without evaluating the integral in the expression for f in part (b). (it may be to know that the integral is negative.) (iv) hence show that p [8 marks (d) the radiation present in the very early universe has expanded adiabatically and cooled down. use the expression for s from part (c) to find the relation between the temperature t and the radius r of the universe.