Statistical Physics, Second Revised and Enlarged Edition

78 Diatomicgases

7.3 The heat capacityof hydrogen

As an example of the above results, we consider in greater detail the heat capacity of
hydrogen. From what wehavelearnedsofarinthischapter, we anticipate theheat
capacityofH 2 to resemblethesketchshowninFig. 7.1. Andinbroadterms thisis
correct.
At temperaturesjust above theboiling point ofH 2 ,the measuredvalue ofCVis
3
2 NkkkB.Atthese temperatures, themoleculesdo not rotate! The moment ofinertiaof
H 2 is so smallthat thel=1rotationalstateisfurtherthankkkBTabove the ground
(l= 0 ) rotationalstate. In pictorialterms this means that themolecules remainina
fixed orientation as theymove in thegas; collisions do not impart enough energyto
start rotation. To return to the ideas of Chapter 3, the degree of freedom is frozen out
since we areinthe extreme quantumlimit,i.e.kkkBTε,whereεisthe energy
level spacing.
Incidentally, we may note in passing that it is even more valid to neglect rotation
in a monatomicgas (or axialrotationinthediatomic case). Althoughtherelevant
moment of inertia is not zero, it is extremelysmall since most of the atomic mass is
in the nucleus. Hence the first rotational excited state becomes unattainably high, and
rotationisfrozen out at alltemperatures.
At low temperatures, then, onlythe translational degrees of freedom of thegas are
excited, and for the MB gas (equation (6.9)) each molecule contributes^32 kkkBTto the
totalenergy fromits translationinthreedimensions.
By the time room temperature is reached, the rotational motion has become fully
excited.Another way oflooking at the problemis to note that the rotation ofan axially
symmetricmolecule provides afurther twodegrees offreedom, two since two angles
are needed to specify the direction of the molecule. And at room temperature the
classicallimit,kkkBTε,isvalid. Hence eachmolecule now contributesfromits

Boiling

point

Rotational

temperature Φ

Vibrational

temperature 

Translation

Rotation

Vibration

T (non-linear scale)

7 2 5 2 3 2

CV

NkkkB

Fig. 7. 1 The variation ofheat capacityCVwithtemperaturefor adiatomic gas, showing schematically

the contributions oftranslation,rotation andvibration.