Physical Chemistry Third Edition

1056 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

The most probable distribution can now be written

Nj

N

z

gje−βεj (25.3-4a)

The probability of finding a randomly selected molecule in molecule energy leveljis

equal to the fraction of molecules that are in this level:

pj

Nj

N



1

z

gje−βεj (probability of levelj) (25.3-4b)

The name “partition function” was chosen becausezis related to the way in which

the molecules are partitioned among the possible molecular states. The German name

for a partition function isZustandsumme, literally translated as “state sum” or “sum over

states.” This name is more descriptive than “partition function” and is occasionally used

in the English language, with or without translation. The German name is the reason

for using the letterzto denote the molecular partition function, but the letterqis also

used by some authors.

Each term in Eq. (25.3-3) is proportional to the number of states in that level, so we

can write the molecular partition function as a sum over states instead of a sum over

levels:

z

∑

i

e−βεi (sum over molecule states) (25.3-5)

The molecular partition function can be interpreted as a sum of probabilities of states,

with the probability of a state of zero energy set equal to unity. Its value can be thought

of as an effective total number of states accessible to a molecule under the given

conditions.

The Parameterβ

In order to determine what the parameterβrepresents, we substitute the expression for

Njgiven in Eq. (25.3-4a) into the constraint of Eq. (25.2-14b):

1

z

∑

j

εjgje−βεj〈ε〉

E

N



U

N

(25.3-6)

where we have again equated the system energy eigenvalueEto the thermodynamic

energyU. Equation (25.3-6) can be written in another way, using a mathematical trick

that identifies a derivative and interchanges the order of summing and differentiating.

U

N



1

z

∑

j

εjgje−βεj−

1

z

∑

j

∂

∂β

(

gje−βεj

)

−

1

z

∂

∂β

⎡

⎣

∑

j

gje−βεj

⎤

⎦−^1

z

∂z

∂β

−

(

∂ln(z)

∂β

)

V

(25.3-7)

The energy eigenvalues depend on the volume, soVmust be held fixed in the differ-

entiation. The interchange of summing and differentiating an infinite series is valid if

the series is uniformly convergent. This means that the series converges at least at a

given rate for all values of the variables involved. We assume this to be the case.