Physical Chemistry , 1st ed.

In order to maximize ln in terms of the two constraints, we can combine

the equations for ln ,N, and Einto a single linear combination and maxi-

mize that resulting, three-term sum. However, we do not know the relative

magnitudes of the individual terms in the sum. We will therefore multiply two

of the three terms with a weighting factor. We use as the weighting factor for

N, and as the weighting factor for E. This technique is referred to as

Langrange’s method of undetermined multipliersand gives us

ln 



NE

as the expression to minimize. (The negative sign on the last term is for our future

convenience, and will be justified shortly.) Substituting for ,N, and E,we get

Nln Nj Njln N

gj

j





i

Ni

i

Ni i (17.13)

as the expression to minimize.

Since the compactness of a distribution is dictated by the occupation num-

bers, we take the derivative of this expression with respect to the Njvalues (all

jof them) and require that they collectively be zero, as befits the maximum

point in the plot of a function:







Nj

Nln N

j

Njln 

N

gj

j





i

Ni

i

Ni i  0 (17.14)

for each value ofNj. Equation 17.14 thus gives us jexpressions that must

equal zero.

The derivative ofNln Nwith respect to Njis zero, since Nln Nis a con-

stant. Even though Njand Nirepresent different occupation numbers, for a

large enough system there will always be instances in which NiNj. Therefore,

the effect of the derivative in terms ofjis to eliminate all the terms in the re-

maining summations except one, the one in which NiNj. What remains are

iequations of the form

ln 

N

gi

i

 1 

  i 0 i1,2,3,... (17.15)

For simplicity, we redefine the undetermined multiplier as



 1 (17.16)

Rearranging, we get

ln 

N

gi

i



  i

ln 

N

gi

i

  

i



N

gi

ie

  i



N

gi

ie^ e   i

Nigie^ e i (17.17)

If we sum the values of both sides of equation 17.17 over possible values ofi,

we get



i

NiN

i

gie^ e    i

17.4 The Most Probable Distribution: Maxwell-Boltzmann Distribution 595