Statistical Physics, Second Revised and Enlarged Edition

76 Diatomicgases

7 .2.3 Vibrational contribution

Similarly, the contribution from vibrational motion has been calculated earlier in
thisbook. Eachdiatomicmoleculeis accurately representedby a one-dimensional
harmonic oscillator, andfurthermore themolecules are veryweakly interacting.
Therefore,the vibrations of ourNgas molecules can be described as an assem-
blyofNidenticalandweaklyinteractingharmonic oscillators. Thisisthe problem
discussedfully in section 3.2. Theheat capacitycontributionCV,vibisgiven precisely
by (3.1 5 ) and is illustrated in Fig. 3.10.
Theheat capacityCV,vibiscalculatedto risefrom zero toNkkkBas the tempera-
ture is raised, the change occurringaround the scale temperatureθ.As discussed in
section 3.2,kkkBθis the energy level spacing of an oscillator, equal tohνwhereνisthe
classicalfrequency. Its value, therefore, will dependon thebondstrengthbetween
the two atoms of the molecule and on their masses. For typical commongasesθturns
out to have a value in the 2000–6000 K region (roughly 2000 K for O 2 , 3000 K for
N 2 and 6000 K for H 2 where the masses arelighter). At room temperature, therefore,
vibration remains substantiallyunexcited so thatCV,vib≈0 for all diatomicgases.
At elevated temperatures the onset of vibration is seen, and the heat capacity rises.
However,inpractice,itoftenhappens that,beforeCV,vibbecomes verylarge, the
diatomic gas dissociates into 2Nmonatomic gas molecules, the vibrational energy
having overcome thebonding energy.

7 .2.4 Rotational contribution

Having effectively disposed of electronic excitation and vibration, both theoretically

andexperimentally, we now turn to the rotationalcontribution. Thiswill befounda

more substantial topic!

The rotation of a linear molecule is modelled, bearing in mind the strictures of

section 7.1,bythe motion ofarigidrotator. The rotatorhas afixedmoment ofinertia

Iaround an axis perpendicular to its own axis; spinning motion around its own axis is

neglected. The solution for the quantum states of the rotator will be quoted here, but

theyshouldlookfamiliartoastudentwhohasstudiedanyangularmomentumproblem

in quantum mechanics. Basically the requirement for the wavefunction to be single

valuedupon a 2πrotationleads to quantization ofan angular momentum component

inunits of.Theallowedvalues of(angular momentum)^2 becomel(l+ 1 )^2 ,with

l=0, 1 , 2 ,....And hence the allowed energy levels are given by

εl=l(l+ 1 )^2 / 2 I

≡l(l+ 1 )kkkB (7.7a)

where the temperaturesodefinedrepresents a characteristic scale temperaturefor

rotation. These levels are degenerate, there being

gl=( 2 l+ 1 ) (7.7b)