6

The theory of simple gases

In the last chapter we have laid the foundations for quantum statistical mechanics. Let us summarisethe most important results here. A quantum state of a collection of identical particles is a fully anti-symmetric (for fermions) or symmetric (bosons) many-particle state. Such a state can be constructedfrom single-particle states by a Slater determinant (in thecase of fermions) or a symmetrised linearcombination of products of single-particle states (bosons). In this chapter we shall work out further thecase of non-interacting particles for which the partition functions usually factorise, thereby renderinga full analytic solution feasible.

6.1 An ideal gas in other quantum-mechanical ensembles – occupation numbers

Quantum states for ideal gases arefully characterised by specifyinghow many particles there are ineach available state. From this it follows that, if we have a set of single-particle quantum states|i〉,the many-particle state is specified by the numbersni of particles in each such state. If we haveNparticles, we must have

∑i

ni = N.

If the single-particle states are eigenstates of the singleparticle Hamiltonian with energiesεi , we canevaluate the energy of the system to be

E = ∑i

niεi .

The number of single-particle products in such a state is

N!n1! ·n2! · · · .

In any sum over all the eigenstates, each set(n1,n2, . . .) should be countedonly once.In Maxwell-Boltzmann counting, we sum over all possible configurations fordistinguishablepar-

ticles and then divide byN! Therefore, the effective weight with which we take the configuration(n1,n2, . . .) into account is

g(n1,n2, . . .) =1

n1! ·n2! · · ·instead of the correct factorg= 1 which is taken into account in Bose-Einstein counting. Thetwo areequivalent only if each state contains at most one particle,which occurs at high enough temperatureand large enouh volume.

The canonical partition function can be evaluated as

Q(N,T) = ∑{ni}

′g(n1,n2, . . .)e

−β ∑i εini

37

38

where∑′{ni} denotes a sum over all configurations(n1,n2, . . .) with ∑i ni = N. Because of this last

restriction, it is not easy to evaluate this partition function. Note that for Bose-Einstein (BE) andFermi-Dirac (FD) statistics, the weight factorg = 1, and that for Maxwell-Boltzman statisticsg =1/(n1! ·n2! · · · ).

In order to proceed, we look at the grand canonical partitionfunction, where the restriction∑i ni =N does not come into play:

Z (µ ,T) = ∑{ni}

g(n1,n2, . . .)eβ ∑i(µ−εi)ni .

The nice property of this partition function is that it factorises into a product of sums overni . In thecase of BE statistics:

Z (µ ,T) =∞

∑n1=0

eβ(µ−ε1)n1∞

∑n2=0

eβ(µ−ε2)n2 · · · .

Each of the factors is a geometric series which can be evaluated analytically:

∞

∑n=0

eβ(µ−ε)n =1

1−eβ(µ−ε)

Note, however, that in order for the grand canonical partition function to be well-defined, it is neces-sary thatµ < ε0, whereε0 is the ground state energy.

For FD statistics, the situation is even simpler: each of theni only assumes the values 0 or 1.

1

∑n=0

eβ(µ−ε)n = 1+eβ(µ−ε).

For Maxwell-Boltzmann counting, with 1/(n1! ·n2! · · · ), the factors are identified as the power seriesexpansions of the exponential function:

∞

∑n=0

1n!

eβ(µ−ε)n = exp[

eβ(µ−ε)]

.

It is also possible to evaluate the average occupations of the levels. For Bose-Einstein statistics,we obtain:

〈ni〉 =∑∞

n1=0eβ(µ−ε1)n1 · · ·∑∞ni=0nieβ(µ−εi)ni · · ·

∑∞n1=0eβ(µ−ε1)n1 · · ·∑∞

ni=0eβ(µ−εi)ni · · · .

All factors in the numerator and the denominator are identical, except for thei-th factor, which yields:

〈ni〉 =∑∞

ni=0nieβ(µ−εi)ni

∑∞ni=0eβ(µ−εi)ni

This can be evaluated as

〈ni〉 =∂

∂β (µ − εi)ln

[

1

1−eβ(µ−εi)

]

=1

eβ(εi−µ)−1.

This is the famous Bose–Einstein distribution function.For Fermi-Dirac statistics, we obtain

〈ni〉 =∂

∂β (µ − εi)ln[

1+eβ(µ−εi)]

=1

eβ(εi−µ) +1.

39

〈ni〉

0

0.5

1

1.5

2

2.5

3

-2 -1 0 1 2 3

BE

MB

FD

(ε −µ)/kBT

Figure 6.1: Bose–Einstein, Fermi–Dirac and Maxwell–Boltzmann distribution functions.

Finally, for Maxwell-Boltzmann counting, we have, not really surprisingly:

〈ni〉 =∂

∂β (µ − εi)ln[

exp(

eβ(µ−εi))]

= eβ(µ−εi).

In figure 6.1 we show the different distribution functions.

6.2 Examples: gaseous systems composed of molecules with internal motion

Consider a gas consisting of molecules with internal degrees of freedom. These can include electronicor nuclear spin, and vibrational or rotational motions of the nuclei. We neglect the interaction betweendifferent molecules, which is justified in the gas phase whentheir mutual separations are on averagevery large. We furthermore suppose that the thermal wavelength is much smaller than the system size,so that Boltzmann counting is justified.

In the usual way, we may factorise the partition function into partition functions of the individualmolecules:

Q(N,T,V) =1N!

[Q(1,T,V)]N

where the single-molecule partition function has the form:

Q(1,T,V) = V

(

2πmkBTh2

)3/2

j(T).

To obtain this expression, it is necessary to split up the degrees of freedom: we consider the centreof mass coordinates separately from the internal degrees offreedom. The centre of mass coordinatesof the molecules yield the free, ideal gas partition function, whereas the internal degrees of freedomgenerate the internal, molecular partition functionj(T):

j(T) = ∑i

gie−εi/(kBT).

The factorgi is the multiplicity (degeneracy) of the statei. We do not have to include the countingfactor 1/n1! sincen1 ≤ 1 in the regime considered.