Question

Question 10 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, ...,r ,....The partition function is given by Z (T,V,N)- > exp(-B(n,&, + п,&, +...)} пп. (i) Define the symbols n1, n2,...,n,...and 81, 82, ..., Er,... (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 helpful 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. [2 marks

Answer 1

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