Physical Chemistry , 1st ed.

so it will have to be included explicitly in the determination ofqelect. Therefore,

we have the electronic partition function as the sum of two negative expo-

nentials:

qelect 3 exp 

3 exp 

Note that we are defining the energy of the ground state as 0 J. We get

qelect 3 1.000    3 0.997

qelect5.991

If you were to compare qnucand qelectwith qtransas determined in the pre-

vious chapter, you would see that qtransis usually huge compared to qnucand

qelect, often by a dozen orders of magnitude or more. This suggests that qnuc

and qelecthave a very small impact on the overall thermodynamic properties of

gaseous species. This is in fact the case. We have already mentioned how we

can virtually neglect qnuc; as you might expect, in many cases we can virtually

neglect the contribution ofqelectas well. The exception to this is for molecules

that have a large number of low-lying excited states that might have a large de-

generacy (for example, they might be very symmetric molecules). And at very

high temperatures, low-lying electronic states can contribute significantly to

qelect. Generally speaking, it is a good idea to evaluate each system and its rel-

ative characteristics before determining qelect. We will see shortly how much

qelectaffects the calculated values of thermodynamic state functions.

18.3 Molecules: Electronic Partition Functions

We explicitly excluded molecules in our earlier treatment of the electronic par-

tition function. Let us consider qelectfor molecules now, starting with a di-

atomic molecule and generalizing the result to other molecules.

The key to getting an electronic partition function for molecules depends

on how we define the “zero” position for energy. Virtually all numerical scales

have benchmarks that are used to define certain numerical values. For exam-

ple, for atoms we defined the zero point as the ground electronic state.

Vibrations and rotations have well-defined minimum-energy points that serve

as starting points. But what about electronic energy?

Keep in mind, also, that the electronic potential energy curve isn’t just a

given, specific value. Because all molecules are constantly vibrating, even in

their lowest-energy state, the electronic energy curve is shaped like a (har-

monic-oscillator) potential energy curve. Figure 18.1 shows a representative

curve for a ground electronic and first excited electronic state of a typical di-

atomic molecule. In the ground electronic state, the energy is at a minimum at

some equilibrium distance labeled re. At very small internuclear distances, the

potential energy increases as internuclear repulsion becomes strong. At long

internuclear distances, the bond between the two atoms will become nonexis-

tent and the molecule will actually exist as two separated atoms. This point is

known as the dissociation limitfor the molecule.

The depth of the electronic potential energy well is called the dissociation

energy.However, there are two ways to define a dissociation energy. The energy

3.97

10 ^23 J



(1.381 

10 ^23 J/K)(1000 K)

0 J



(1.381 

10 ^23 J/K)(1000 K)

18.3 Molecules: Electronic Partition Functions 621

E  0

Internuclear separation

Ground

state

Excited

state

Dissociation

limit

Energy

D 0 De

Figure 18.1 The electronic potential energy

diagram for a hypothetical diatomic molecule. In

the ground state, some of the lower vibrational

energy levels are indicated. How is the “zero” point

of energy defined for a molecule that has elec-

tronic energy with this behavior?