Physical Chemistry , 1st ed.

Compare this equation with another equation that can be derived from ther-

modynamics:





V

E



T

1



(1



/

p

T)

 p (17.30)

where we have used Eto represent the internal energy. These two equations are

strikingly similar. They imply that is related to 1/T.is not equalto 1/T,be-

cause a proportionality factor would cancel from equation 17.30.is certainly

proportional to 1/T:



T

1



As usual in equations like this, a proportionality can be written as an equality

by introducing the appropriate proportionality constant. But rather than

putting this constant in the numerator, the convention is to put it in the de-

nominator. Giving the proportionality constant the symbol k, we have



k

1

T

 (17.31)

The constant kis called Boltzmann’s constantand has a value of 1.381  10 ^23 J/K.

The expression for the partition coefficient qbecomes

q

i

gie    i/kT (17.32)

All previous equations with can be modified accordingly.

Example 17.3

Consider the diagram in Figure 17.5 with the four energy levels. Assuming

that the energy levels are fourfold degenerate (that is,gi4) and that the en-

ergy levels have values of 0.00, 1.00, 2.00, and 3.00  10 ^21 J, what is the

value of the partition function at 25°C 298 K? What is the value for qif

the energy levels are 0.00, 1.00, 2.00, and 3.00  10 ^19 J?

Solution

The summation from equation 17.32 can be set up as

q 4 exp

 4 exp

 4 exp

 4 exp

q 4  1  4 0.784  4 0.615  4 0.482

q11.524

where exp is the same exponential function as ebut allows long exponents to

appear in a more readable form. All of the units cancel, and qis just a num-

ber. For the larger values for the energy levels, it can be shown that

3.00  10 ^21 J



(1.381  10 ^23 J/K)(298 K)

2.00 ^10 ^21 J

(1.381  10 ^23 J/K)(298 K)

1.00  10 ^21 J



(1.381  10 ^23 J/K)(298 K)

0.00 ^10 ^21 J

(1.381  10 ^23 J/K)(298 K)

17.4 The Most Probable Distribution: Maxwell-Boltzmann Distribution 599