Statistical Physics, Second Revised and Enlarged Edition

74 Diatomicgases

in a state of highrotationalenergyofadiatomicmoleculethebondwillstretchalittle
from the inertia of the atoms, and this will influence the bond strength and hence the
vibrationalfrequency ofthemolecule. However, sucheffects are small in practice
andmaybe(gratefully)neglected.
From the point of statistical physics, this independence has a great simplifying
influence. It means that the partitionfunction Zfactorizes.
Thepartition function is defined (equation ( 6 .2)) as the sum over all states of the
Boltzmann factors of every state. Writingβfor− 1 /kkkBT,and using (7.1) we obtain

Z=

∑

allstates

exp[β(εtrans+εrot+εvib+εelec)]

=

∑

trans

exp[β(εtrans)]×

∑

rot

exp[β(εrot)]

×

∑

vib

exp[β(εvib)]×

∑

elec

exp[β(εelec)] (7.2)

=ZZZtrans×ZZZrot×ZZZvib×ZZZelec (7.3)

Thesimplicityof(7.2)isthat eachfullstate ofthemolecule,bythe assumption of
independence, can be specified by its separate quantum numbers (i.e. state labels) for
translation, rotation, vibration andelectronic excitation. Hence the partitionfunction
factorizes asin (7.3)intoindependent componentparts.
SinceZfactorizes,lnZhas a number of additive terms. As a result there are inde-
pendent additive termsinthethermodynamicfunctions suchasF(equation ( 6 .1 6 )),U
(equation ( 6 .12)) and henceCV.It is particularlyinstructive to note how this works
out for the free energyF,since it becomes clear how to handle the lnN!termof
( 6 .1 6 b).
Substitutingthe form (7.3) for the diatomicgas into thegeneral expression ( 6 .1 6 b)
for any MB gas, we obtain

F=−NkkkBTlnZ+kkkBTlnN!

=−NkkkBTlnZZZtrans+kkkBTlnN!−NkkkBTlnZZZrot

−NkkkBTlnZZZvib−NkkkBTlnZZZelec (7.4)

=FFFtrans+FFFrot+FFFvib+FFFelec (7.5)

The free energy of the gas is decomposed as anticipated into various parts. The
translationalpartFFFtransisdefinedbythefirst two terms of(7.4), whichincludesZZZtrans
together withthelnN! term. Note that thisisidenticalto the totalfree energyofagas
of structureless (i.e. monatomic) molecules as worked out in Chapter 6. Hence F for
thediatomic gasis equalto Ffor the monatomicgasplusadditive contributionsfrom
theinternaldegrees offreedom, thefinalthree terms of(7.4). The extra contribution of