Physical Chemistry Third Edition

1052 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

the maximum value ofW({N}) can be rounded to the closest integer to give the desired

value. The maximum value subject to constraints is found byLagrange’s method of

undeterminedmultipliers.^4 If we wish to find a relative maximum or minimum of a

functionff(x,y,z) subject to two constraints we first express the constraints by

equations of the form

u(x,y,z) 0 (25.2-15a)

and

w(x,y,z) 0 (25.2-15b)

We then find the point at which the following simultaneous equations are satisfied:

∂

∂x

[f+αu+βw] 0 (25.2-16a)

∂

∂y

[f+αu+βw] 0 (25.2-16b)

∂

∂z

[f+αu+βw] 0 (25.2-16c)

The parametersαandβare calledundetermined multipliers. They can depend on other

variables, but not onx,y,orz. If there are more than two constraints, there is an

undetermined multiplier for each constraint.

It is easier to find the maximum of ln(W) than a maximum inW, becauseWequals

a product of many factors, whereas ln(W) equals a sum of many terms, which is easier

to differentiate than a product. The logarithm is a monotonic function of its argument,

so that the largest value of ln(W) occurs together with the largest value ofW. To find

the maximum of ln(W) subject to the constraints of Eqs. (25.2-14a) and (25.2-14b),

there is a set of simultaneous equations, one for eachNi:

∂

∂Ni

⎡

⎣ln(W)+α

(∑

j

Nj−N

)

−β

(∑

j

Njεj−E

)

⎤

⎦0(i1, 2, 3,...)

(25.2-17)

The method is not changed by using the symbol−βinstead ofβfor one of the multi-

pliers.

The logarithm ofWis given by

ln[W({N})]

∑

j

[Njln(gj)−ln(Nj!)] (25.2-18)

We must find an approximation for ln(Nj!) that can be differentiated. This isStirling’s

approximation:

N!≈(2πN)^1 /^2 NNe−N (25.2-19)

(^4) H. Anton,Calculus, 3rd ed., Wiley, New York, 1988, p. 1032ff; R. G. Mortimer,Mathematics for
Physical Chemistry, 3rd ed., Elsevier Academic Press, San Diego, 2005, p. 228ff.