h3

Consider 1 cm3 of neon gas at a pressure of 1 bar and temperature of 300 K,

containing 2.4—1019 atoms. By integrating (17.63) from 0 up to kB T we ¬nd

that there are 6.7 — 1025 quantum states with energies up to kB T . Thus the

probability that any particular quantum state is occupied is much smaller

than one, and the probability that any state is doubly occupied is negligible.

17.5 Fermi“Dirac, Bose“Einstein and Boltzmann statistics 469

Therefore there will be no distinction between fermions and bosons and the

system will behave as in the classical or Boltzmann limit. In this limit the

occupancies nk 1 and hence

»e’βµk 1. (17.64)

In the Boltzmann limit, including the lowest order deviation from the limit,

the occupancies and grand partition function are given by (upper sign: FD;

lower sign: BE)

nk ≈ »e’βµk “ »2 e’2βµk + · · · , (17.65)

e’βµk “ 1 »2 e’2βµk + · · · .

βpV = ln Ξ ≈ » (17.66)

k

2

k

Since N = k nk , the ideal gas law pV = N kB T is recovered in the Boltz-

mann limit. The ¬rst-order deviation from ideal-gas behavior can be ex-

pressed as a second virial coe¬cient B2 (T ):

N2

p N ’1

≈ + ···,

+ NA B2 (T ) (17.67)

kB T V V

h3 Λ3

’1

=± = ± 5/2 ,

NA B2 (T )FD (17.68)

BE

2(4πmkB T )3/2 2

where Λ is the de Broglie thermal wavelength

h

def

Λ=√ . (17.69)

2πmkB T

Avogadro™s number comes in because B2 is de¬ned per mole and not per

molecule (see (16.59)). This equation is proved as follows: ¬rst show that

’1

NA B2 = ±q2 V /(2q 2 ), where q = k exp(’βµk ) and q2 = k exp(’2βµk ),

and then solve q and q2 by approximating the sum by an integral:

2 k2

h2 n2

’βµk

q= e , µk = = . (17.70)

2ma2

2m

k

Here, n2 = n2 + n2 + n2 with nx , ny , nz ∈ {0, ±1, ±2, . . .} (see (17.61)). Use

x y z

has been made of the periodicity requiring that k = (2π/a)n. Since the

occupation numbers are high (n is large), the sum can be approximated by

∞ ∞ ∞

βh2 2 a3 V

q≈ exp ’ dn = 3 (2πmkB T )3/2 = 3 . (17.71)

n

2ma2 h Λ

’∞ ’∞ ’∞

470 Review of statistical mechanics

Note that this q is the single particle canonical translational partition funct-

ion of a particle in a periodic box.14 Also note that the quantum deviation

from the Boltzmann limit,15 due to particle symmetry, is of the order h3 .

In the Boltzmann limit, the grand partition function Ξ and the single-

particle canonical partition function q are related by

ln Ξ = »q (17.72)

and thus

qN

Ξ = e»q = »N . (17.73)

N!

N

Since Ξ = N »N QN (see (17.40)), it follows that the N -particle canonical

partition function QN for non-interacting particles equals

qN

QN = . (17.74)

N!

The N ! means that any state obtained by interchanging particles should not

be counted as a new microstate, as we expect from the indistinguishability

of identical quantum particles. It is a result of quantum symmetry that

persists in the classical limit. It™s omission would lead to thermodynamic

functions that are neither intensive nor extensive (the Gibbs™ paradox) as

the following will show.

Consider a gas of N non-interacting atoms in a periodic box of volume V ,

with translational single-atom partition function (17.71)

V

q= (17.75)

Λ3

Using (17.74) and the Stirling approximation (see footnote on page 455) for

N !, the Helmholtz free energy is given by

qN

A = ’kB T ln QN = ’kB T ln N ’N

Ne

q q

= ’N kB T ln ’ N kB T = ’N kB T ln ’ pV. (17.76)

N N

From this follows the absolute thermodynamic potential of the gas

G A + pV q

= ’RT ln

μ= = (17.77)

n n N

14 The same result is obtained if the box is not periodic, but closed with in¬nitely high walls. The

wave functions must than vanish at the boundaries and thus be composed of sine waves with

wave lengths that are whole fractions of twice the box length. This leads to 8— higher density

of points in n-space, of which only one octant (positive n) is valid, and thus to the same value

of the integral.

15 Any quantum correction to classical behavior should contain h; the classical limit is often

viewed as the limit for h ’ 0, which is nonsensical for a physical constant, but useful.

17.5 Fermi“Dirac, Bose“Einstein and Boltzmann statistics 471

Λ 3 p0 p

= RT ln + RT ln 0 , (17.78)

kB T p

where p0 is (any) standard pressure. We recover the linear dependence of μ

of the logarithm of the pressure. Without the N ! we would have found μ to

be proportional to the logarithm of the pressure divided by the number of

particles: a nonsensical result.

The single-particle partition function q is still fully quantum-mechanical.

It consists of a product of the translational partition function qtrans (com-

puted above) and the internal partition function, which “ in good approxi-

mation “ consists of a product of the rotational partition function qrot and

the internal vibrational partition function qvib , all for the electronic ground

state. If there are low-lying excited electronic states that could be occupied

at the temperatures considered, the internal partition function consists of a

sum of vibro-rotational partition functions, if applicable multiplied by the

degeneracy of the electronic state, for each of the relevant electronic states.

The rotational partition function for a linear molecule with moment of

inertia I (rotating in 2 dimension) equals

2 J(J+ 1)

(2J + 1) exp ’

qrot = (17.79)

2IkB T

J

2

T ˜ 1 ˜

= 1+ + + ... , (17.80)

σ˜ 3T 15 T