BioPHYSICAL chemistry

(8.14)

The partition function then provides a measure of the occupancy of a
certain energy level relative to the ground state. As an example, consider
the simple harmonic oscillator. It will be shown in Chapter 11 that oscil-
lators have energy levels that are equally separated with a value of hν,
where his Planck’s constant. Defining the ground state to have zero energy
yields the following for the partition function using eqn 8.13:

q= 1 +e−hν/kBT+e−^2 hν/kBT+e−^3 hν/kBT+... (8.15)

This equation can be rewritten using the series approximation:

(8.16)

Substituting the exponential term of eqn 8.15 for xin eqn 8.16 yields:

(8.17)

This partition function was used by Max Planck in developing the quantum
theory, as discussed in Chapter 9. The partition term is dependent upon
energy dependence with motions, such as rotation, having a different
associated partition function.

Statistical thermodynamics

Partition functions serve as the platform for the calculation of the ther-
modynamic properties of molecules. For example, the total energy of a
system, E, is given by the sum of the products of the number of molecules,
Ni, at each energy, Ei:

(8.18)

This expression can be rewritten since the number of molecules at an
energy Eiis given by the partition function using eqn 8.14:

(8.19)

E

Ne

q

E

N

q

Ee

EkT

i

ii

i

iB EkTiB

/

==/

−

−

∑∑

ENEii
i

=∑

q
ehkT/B

=

− −

1

1 ν

1

1

1 23 4...

+

=+ + + + +

x

xx x x

P

e

e

e

i q

EkT

EkT

i

iB EkT

iB

/ iB

/

/

==

−

−

−

∑

CHAPTER 8 STATISTICAL THERMODYNAMICS 167