The Chemistry Maths Book, Second Edition

612 Chapter 21Probability and statistics

numbers (treated as continuous variables) subject to the constraints that the total

number of particles nis constant and that the total energy Eis constant:

(21.29)

We first take the logarithm of Wand use Stirling’s approximation (21.27) to simplify

the expression:

(21.30)

The maximum value ofln 1 Wsubject to the constraints is then obtained by the

method of Lagrange multipliers; that is, we find the maximum value of the auxiliary

function

(21.31)

where αand βare the multipliers (see Section 9.4). For a stationary value (maximum)

with respect ton

i

,

Therefore, because , we haveln 1 n

i

1 = 1 α 1 + 1 βε

i

so that the most

probable distribution of particles is given by the occupation numbers

(21.32)

With appropriate interpretations of the quantities αand β, this set of numbers is

called the Boltzmann distribution. Thus,β 1 = 1 − 12 kT, so that and

The fractional number of particles in state iis then

where is called the partition function.

qe

i

k

kT

i

=

=

−

∑

1

ε

n

nq

e

i

kT

i

=

−

1

ε

i

k

i

i

k

kT

nne e

i

==

−

∑∑

==

11

α

ε

nee

i

kT

i

=

−

α

ε

ne

i

i

=

αβε+

∂

∂

=−

n

Wn

i

i

ln ln

∂

∂

=

∂

∂

++ =

φ

αβε

nn

W

ii

i

ln 0

φαβε=+ +

==

∑∑

lnWnn

i

k

i

i

k

ii

11

lnWn nnnn nnnln ln ( ln ) ( ln

i

k

i

i

k

ii

=!− != −− −

==

∑∑

11

ii

)

nn En

i

k

i

i

k

ii

== , = =

==

∑∑

11

constant ε constant