Physical Chemistry , 1st ed.

where we have used the fact that (^) iNiN. Since e^ is a constant (eis a con-
stant and is a constant), we factor that term out to get
Ne^ 
i
gie i (17.18)
Notice what equation 17.18 gives us. The total number of particles Nmust satisfy
an expression in terms of the degeneracies of the energy levels,gi, and an expo-
nential expression that is related to the energy of the ith quantum state. It is also
dependent on some exponential e^ and the constant , whose forms we don’t
know yet. But equation 17.18 suggests that there will be a relationship between
the two constants and and the energy and number of particles in the system.
The expression (^) igie iis going to be a common one in statistical ther-
modynamics, so it is useful to give it a symbol. We define qas
q (^) 
i
gie i (17.19)
This quantity qis called the partition function.It plays a central role in statis-
tical thermodynamics. Because we defined our system as a canonical ensem-
ble,qis commonly called the canonical ensemble partition function.
Even though we don’t know the absolute number of particles Niin energy
state (^) i, we can determine what fractionof the total particles are in that energy
state (and then, if we know the total number of particles, we can calculate the
absolute number Ni.) We do this by using the expressions in equations 17.17
and 17.18. The fraction is given by the expression Ni/N, which according to
those equations is


N

N

igi

e

e

(^) e


q
 i

The exponential e^ cancels:


N

N

i^1

q

gie i (17.20)

Consider this expression. For any given distribution (and certainly for the most

probable distribution) of a canonical ensemble,qis a constant that depends on

the temperatures, numbers of particles, and volumes of the microstates.

Degeneracy of the ith energy state is also a constant for a given substance, and

eand are also constants. Therefore, the only variable so far is (^) i, the energy
of the quantum state. The population of any energy level is a negative expo-
nential function of the value of the energy level above the ground state, a func-
tion that looks like Figure 17.9, that is, the population of the energy levels de-
creases exponentially with increasing energy. This type of population
distribution is called the Maxwell-Boltzmann distribution(sometimes more
concisely called the Boltzmann distribution). Notice that the term is not
present in equation 17.20. The implication here is that is not much of a con-
cern to us. However, the constant remains, and determining the value of
is an important step in the development of statistical thermodynamics.
The partition function qis still a part of equation 17.20, however. We can
eliminate qby determining the ratio of the population of the ith energy level
to the population of the kthenergy level:





1

q

gie i





1

q

gke k

N

N

i





N

N

k

gie^ e  i

e^ 

i

gie   i

596 CHAPTER 17 Statistical Thermodynamics: Introduction

x

1

ex

Figure 17.9 This is the general shape of a neg-
ative exponential, which is the heart of the
Boltzmann distribution. Relating this to equation
17.20, it implies that the higher in energy a state
is, the less it will be populated (as long as degen-
eracy is not considered).