Question

II. ENTROPY AND DISTINGUISHABILITY (Adapted from Blundell & Blundell) The Sackur-Tetrode equation for the entropy of an ideal gas is given by where ρ-N/V is the (number) density of the gas, and th-hN2tmkBT is the thermal wavelength a) [5 marks] Show that S is an extensive quantity. b) [5 marks] Show that the entropy of a gas of distinguishable particles is instead given by relationship Swhere Fis the Helm or where F is the Helmholtz free energy. c) [5 marks] Show that the quantity in b) is not extensive.

Answer 1

Sackur - Tetrode equation S = NK_s [5/2 - In (5 lambda)] = K K_B [5/2 - In (N/v middot h^3/(2m pi k_B T)^3/2)] = N k_3 In [(V/N) (5/N)^5/2 (4 pi m/3 h^2)^3/2] + 5/2 N k_B where E = 3/2 N k_B T Let as now increase volume v, no.of particle N and energy E by alpha, then, s' = alpha N k_B In [(alpha v/alpha N) (alpha E/alpha N)^3/2 (4 pi m/3 h^2)^3/2] + 5/2 alpha N k_B = alpha [N k_B In {V/N) (5/N)^3/2 (4 pi m/3 h^2)^3/2] + 5/2 alpha N k_B = alpha 5 Additive law is satisfied so, it is extensive quantity. The number of quantum states, delta T = 1/h^3N doubleintegral pi^N_i = 1 dq_i dp_i The energy of the system squareroot 2 = sigma pi^2/2m therefore delta T = v^N/h^3 N integral pi^N_i = 1 dp_i The integral over momentum space can be evaluated subject to the constraint (5 - delta squareroot 2) lessthanorequalto