Question

25/04/19

Please can you show me how to do this part of the question. I nearly got it, but I’m not sure where the 8*pi factor comes from at the front. The energy is epsilon = c*p, which may be difficult to read.

Consider an ideal gas of N identical reletivistic free particles in a box of volune VThe single particle enorgy is proportional to the magnitude of its momentum op with constant c. Show that the ane-body partition function is ch where h is Planck's constant. Yon may find the following intogral useftu nal 100 z"e 'and" in "anti.

Then please can you go on to show the following via finding the helmholtz free energy. I nearly got it, except for the 4 at the front.

Thanks!

Answer 1

Z_N = 1/h^3 intgeral d^3 r_1 d^3 r_2 .... d^3 r_N integral d^3 p_1 d^3 p_2 .... d^3 p_N e^-pe Z_1 = 1/h^3 V integral 4 pi p^2 e^-cp/k_B T dp = 1/h^3 V 4 pi integral^infinity_0 p^2 e^-cp/k_B T dp {beta = 1/k_b T} {integral^infinity_0 x^n e^-ax dx = n!/a^n + 1} so Z_1 = 1/h^3 V 4 pi [2! (k_B T)^3/(c)^2 + 1] Z_1 = 8 rightarrow V [k_B T/ch]^3 Z_1 = 8 pi V (k_B T/ch)^3 \r\n since Z_N = (Z_1)^N/N! = [8 pi V (K_B/vh)^3]^N F = -k_B T ln [Z_N] F = -k_B T ln [(B pi V)^N/N! (k_B T/ch)^3N] = > dF = dV - dT S S = -(partial differential F/ partial differential T) S = k_B [ln [(8 pi V)^N/N! (k_B T/ch)^3N] + T 1 3N times 8 pi V/N! (k_B T/ch)^3N - 1 k_B/ch/8 pi V/N! (k_B T/ch) 3N] S = Nk_B [3 ln [k_B T/ch] +
please like. Thank you. . .