Question

At a given temperature, the difference between the specific heats of a diatomic ideal gas and a monatomic gas is partly due to rotational energy of the diatomic molecules. A quantum rigid rotator has energy levels Erot (1) with degeneration given by ħ2 Erot(1) = 1(1+1) g() = 21+1, 1 = 0, 1, 2,... g(0) 21 where I is the moment of inertia. (a) Find the canonical partition function of a gas of N non-interacting diatomic molecules. (b) Evaluate the specific heat of this gas at high temperatures and at low temperatures in which the rotational motion of the molecules has a significant contribution.

Answer 1

Inswer lll.) 21 l=0,1,2.. 91 lao Z e 87 lo A quantum rigid rotator has energy levels Exoties with degenracy grey given by. Evot (l) = h gld) = alt t. (a) Canonical partition function ! First, the partition function for a Single molecule is - B to? eſe+) Z = E (2+1) e E die for high temprature, Sum can be converted to an integral! - B toaletes (26+1) e lite ( I+1) de: dt 1:BO K&T Z de t 8 it Z = J 91 dt -Bh? t z = I 12 Z = 2 I KT ta for N-non interacting molecules ZNE 2 IKBT

(6) N= op taking lagmithmic enZNE N en en com ) Bth2 Energy given by denza OP N. B h2 -2 B2th ² 9) 2 2 N'S NKBT 79 Specific Heat given by- Cua (duet - At High tempratune Cu = d (NKBT) ODOC NKB dt At low temprature TO rog sirth to low Cu=o] because & <tok , then system is confined to ground state and Z=1 = ent=oire. u=o & Cuco so j (92+1 - ßt² ell+1) 60 enz de for T»TE z = O tas always for TECTE