Physics

Physics, 29.06.2019 23:20, Nitziaruiz1

Consider a cavity of volume v filled with photons (a) show that the logarithm of photon's grand partition function q is (b) show that the equation of state of the photon gas can be specified by where 3 (c) determine the average number of photons (n) in the cavity and show that the mean energy per photon is approximately the following information may be (x) = with the riemann zeta-function ((v)2n. here are some frequently used values: (2) = π2/6, ζ(3) 1.202, e(4) = π4/90

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Physics, 10.07.2019 22:20, breannamiller0822

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.

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Physics, 10.07.2019 22:20, les6965

The grand-canonical ensemble: general relations in this problem, we want to derive general relations in the grand-canonical ensemble. for this, you can assume that all the functions involved are analytic. the grand-canonical partition function 2(t, v,) is given by (1) bpv 2(17,v ) -σνz(t, v)-epv n-0 a) (3p) in the lecture, we showed that o0(t, v, μ) θμ 1 aln2(t,v, u n) b (2) t,v it,v where 2(t, v,u)=-kbt in2 (t, v,u) is the grand-canonical potential. show in an that the variance of the number of particles is given by analogous manner v a2p (n2)(n) v/n. prove the following relations between the b) (2p) we now introduce the volume per particle v derivatives (4) n out,n av an t,v n ov t,v t.n can be written in terms the volume per particle the free energy f(t, v, n) 5 f(t,v, n) nf(t, u), where the free energy per particle f only depends on v and n implicitly. the chemical potential p and the pressure p can be obtained from the free energy p(t,v,n) avtn (6) (t,v,n) = n t,v c) (3p) show that ap =1 aut.v au t.n d) (3p) use your previous results to show that n (8) kt t,v 1 av is the isothermal compressibility. convince yourself that this means that where kt v bp t.n (n2) (n) a n and thus n2) (n)2/(n) o 1/vn -12

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Physics, 10.07.2019 22:30, jennychrin95

Black-body radiation as a photon gas consider a radiation cavity of volume v at temperature t. bose treats the cavity radiation as an ideal gas of photons in thermal equilibrium. a photon gas is an ideal boson gas with zero chemical potential because their number is not conserved. the energy of a photon is proportional to the frequency of the radiation: e = hw and the density of states in w space is: d(w) = w2 t (a) write down the energy density spectrum u(w, t) as a function of w and t (or b). (b) show that the total energy density w(t) scales as u (t) = ato and find b. (c) let wmax be the angular frequency at which the energy density u(w, t) reaches its maximum, shows that wmat t.

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Physics, 21.08.2019 04:00, lizredrose5

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