166 Dealingwith interactions
Theendresultisthatintheclassicallimitwesimplyreplace sums over states
by integrals over the appropriate phase space. It is worth again stressing that in the
prequantum era ofBoltzmann,it was a monumentalachievement to maketheabove
averagingpostulate.
As one example of the classical approach, consider again the calculation of the one-
particle partitionfunctionZfor an ideal monatomic gas. In Chapter 6 we used ideas of
k-space todothe sum over states as anintegralink(see ( 6 .4)) togive finallythe result
(see(6.6))Z=V( 2 πMkkkBT)^3 /^2 /h^3 .The classical partition function is obtained as
asimpleintegraloftheBoltzmannfactor(exp−ε/kkkBT)over the one-particlephase
space
ZZZclass=∫
exp[−p^2 /( 2 MkkkBT)]ddpxddpyddpzdxdydzSince the energy does not depend on position, the space part of the integral simply
gives afactorVandthe momentum partis treatedthe same wayas was thekintegral
in Chapter 6. The result is thatZZZclass=V( 2 πMkkkBT)^3 /^2 , i.e. the same as before, only
without the constant factorh^3 , which we now know is needed to convert phase space
to quantum states.
1 4.4.2 Imperfect gases
We are now readytolookatimperfectgases. One can realizeimmediatelythat the
difference to be taken into account is that (unlike in the previous paragraph) the energy
is nowdependent on molecular position as wellas momentum. Therefore we must
workinthe conceptualframeworkofthe canonicalensemble, whichinvolves the
assembly partition functionZZA(see (12. 5 )), worked out from all possible energies
ofthewhole assemblyofNgas particles. As explainedin section 12.2,ifwefirst
workoutZZA,then we can compactlyandconvenientlycompute thermodynamic
quantities. Here we are particularly concerned with the pressure, in order to quantify
the corrections to the equation ofstatefor animperfectgas wheninteractions are
switched on. What we shall expect to obtain is a result of the form
P
kkkBT=n+B 2 n^2 +B 3 n^3 ··· (14.4)wheren=N/Visthe numberdensityofthegas. Equation (14.4)iscalledthevirial
expansionof the equation of state, and theBcoefficients are functions of temperature
only. Thefirst term alone givesPV=NkkkBT,theidealgas equation ofstate.
The attempttocalculateZZAproceedsasfollows. Suppose that thereis a non-zero
intermolecular potential energy(r 1 ,r 2 ...rN)between theNmolecules. For a gas,
one can safely assume that this potentialenergyisderivedfrom a sum over allpairs
ofmolecules ofthepairpotentials,φ(rij)beingthe potentialenergyofapairof
moleculesadistancerij=ri−rrjapart.
The partitionfunctionZZAisdefinedas the sum over allpossible assembly states
ofthequantities exp(−E/kkkBT)whereEisthe energyofthe assemblystate. Thusit