Question

1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz free energy: A = -kTinQ. Using the fundamental thermo- dynamic relation of Helmheltz free energy i.e. dA = -SAT - PDV + pdN), express P, and p in terms of Q. (b) The canonical partition function for N non-interacting, indistinguishable parti- cles in volume V at temperature T is given by Q(N,V,T) = where where q(V.T) is the partition function of single particle. Express P and ii of this non-interacting system in terms of q, N, V and T. (c) The partition function q in (b) for an atom is given by 12 m3/2 g(V, 7) = (B)"V, where m is the mass of the particle and h is Planck's constant. Show that the pressure expression you derived in (b) leads to the ideal gas law. (d) Using the q given in (c), calculate the chemical potential of 1 mol Ar (ideal gas) in V = 1L and T = 298 K.

Answer 1

a) A =-KTMS from the fondamental egn of thermodynamus, TA = – S&T-partuan -*=-(20) IN = -[-KT INC] IN = KT (am), N. bad [b = KT (amy haven when boter u= A va ahor tross 2 ja 1200 = a (-ktm 0) Nb 2027 =-KT lamy P lan Tv For tripoorn (1) 9(NVT) = A Sing = nmq-MN! - Ning (NmN-N) - An Ning - WINNIN (wing storting samahu

av 7 P= NT fangan Nowing = Ning + N-winn. elm o Sin = av (Ning+w-winnen & [Nina], to = lng (n) than = 0+Nama IN - [p = NET (sing ) IN Coling av = 2 (avimą +N-NINJA = 2v meter + IV-B (WIN), = my trenger ti-n cering 1. = Ing+N (2) IN TX-X -Inne 21 = m ()+Na hindav Ú= -47 ( amige her =-kt In E43 -wkil olunan fue = -K1 m(*) — Nk1 Change her

si LEHT q (T) = (241m)% v sinq = m (-21m1% +- Inv. (oh), How p- NKT (2mg) = NKT + for = Nki t hin the equation for identgan. (d) 9 (V 1) = (ummor mg =mfort mr ; f = f(T) ts> Hince, u = - KT M V = - KT ln (2 ) ¥ Ekim 4 21an kp7jxx -0 Now, ks = e = = -38 x 17°sk T = 298 K = 6.626 X10 34 JS. v = TL N = 1 mil. m = 39.948 amu 639.948 6-023X 7023 = 6-63% 10-239 Putting these valuesm ego o we get, bebasaxt pa z o X80tby M=-1152x10-19 Haz t 97