Now, calculating the electronic partition function:qelect 1 exp
qelect2.03
1080
Let us check to see if we need to include any contribution of the first excited
state to the value ofqelect. Since the Defor H 2 is 7.61
10 ^19 J, the minimum
on the potential energy curve for the ground electronic state lies at 7.61
10 ^19 J with respect to the zero point, and there is an excited electronic state
1.822
10 ^19 J above that. Therefore, with respect to the zero point of en-
ergy, 2 has a value of (7.61
10 ^19 1.822
10 ^19 ) J, or 5.79
10 ^19
J. (It might be useful to draw some electronic potential energy curves to ver-
ify this.) Evaluation ofg 2 e^2 /kTgives usg 2 exp
kT
(^2)
^1 exp
1.26
1061
Granted, this is a large number, but it is still 19 orders of magnitude smaller
than the first term, and is negligible with respect to the first term in qelect.
Thus, it can be ignored, as can the contribution of any additional excited elec-
tronic states.
Since qelectis so large, what effect does it have on thermodynamic state func-
tions? Actually, very little at normal temperatures. The large values ofqelectare
just a consequence of where the zero point of energy was selected. For state
functions that are related to the derivative ofqwith respect to temperature and
pressure,qelectchanges only very, very slowly with changes in temperature and
pressure for most systems. Therefore, its effect on state functions is small. For
Aand G, which depend directly on q, the practical effect is small because we
are mostly interested in changesin Aand G, and the direct numerical conse-
quences ofqelectcancel. The translational partition function still has the ma-
jority of influence—but not all, as we will see—on the thermodynamic prop-
erties of diatomic molecules.
Finally, we should generalize qelectfor larger molecules. The issue is the
same: what is the zero point against which the electronic energy is measured?
For multiatomic molecules, the zero point of electronic energy is defined in the
same way as for diatomics: the energy where all atoms are separated (techni-
cally, to an infinite distance) from each other. Similar to the dissociation en-
ergy, the atomization energyis defined as the difference between this separa-
tion of atoms and the ground electronic state of the molecule. All other
treatment ofqelectfor molecules is as previously discussed for diatomics.
18.4 Molecules: Vibrations
In order to complete the definition of the partition function for molecules, we
must consider the two other ways a molecule can have energy. It can have ro-
tational energy, and it can have vibrational energy.
Molecules are composed of multiple atoms that are bonded by covalent
bonds. Quantum mechanics indicates that those atoms are constantly vibrat-
ing about some equilibrium position, even at absolute zero, having some
(5.79
10 ^19 J)
(1.381
10 ^23 J/K)(298 K)7.61^10 ^19 J
(1.381
10 ^23 J/K)(298 K)18.4 Molecules: Vibrations 623