Physical Chemistry , 1st ed.

Further, we approximated qqtrans, and were able to determine an expression

for qin terms of various quantities, including the masses of the atoms and sev-

eral universal constants. From that q, we were able to derive expressions for E,

S, and related state functions and show that the statistical thermodynamic val-

ues for these state functions were very close to experiment (for S) or agreed

with the values predicted by other theories (like E^32 kTfor a monatomic gas

as predicted by kinetic theory, which we will consider in a later chapter).

Does this imply that qtransis the overwhelming contribution to the overall

qand that all other partition functions make negligible contributions? No, not

for all gaseous species. This assumption worked well with monatomic gases

that have all-electron-paired singlet electronic states, and have no vibrations or

rotations (they are, after all, simply atoms). Vibrational and rotational parti-

tion functions don’t exist for atomic species, and since most of the gases of in-

terest have filled orbitals, no electronic states contribute substantially to the

partition function (an issue we will consider in more detail later), and our ap-

proximation ofq qtransis a very good one.

What about qnuc? What is the contribution of nuclear energy levels to the

overall partition function?

From the definition of partition function,qnucis defined as

qnuc 



ifirst

giei/kT (18.2)

nuclear level

where the summation explicitly states that it is taken over the possible nuclear

energy levels. Nuclear energy levels are dictated by the arrangement(s) of pro-

tons and neutrons in the nucleus. Nuclear energy levels are on the order of

millions of electron volts, or 10^11 joules, per mole! The very idea of nuclear

energy, and its immense magnitude with respect to chemical energies, sug-

gests that the spacing of nuclear energy levels is very large, so large that if the

first nuclear energy level were arbitrarily set to an energy of zero, equation 18.2

becomes

qnucg 1 



isecond

giei/kT (18.3)

nuclear level

Note that in equation 18.3, the exponential in the first term is equal to 1, and

the summation in equation 18.3 now starts at the secondnuclear energy level.

But if even the second nuclear energy level (that is, the first “nuclear excited

state”) is so high in energy, the negative exponential in even the first term of

the summation is very, very small; successive terms are even smaller. We sug-

gest that the i2 term, and all higher terms, are so negligible that they can

be ignored. The nuclear partition function is therefore

qnucg 1 (18.4)

That is, the (effective) nuclear partition function is the degeneracy of the

ground state of the nucleus.

For chemical purposes, the nuclear partition function is ignored. This is for

several reasons. First of all, a degeneracy is likely to be some small whole num-

ber. Although this has the overall effect of multiplying the overall qby that

whole number, it does not really affect the determination of thermodynamic

properties. For one thing, many thermodynamic properties are related to how

qchanges with temperature or pressure. Because changes in nuclear energy

618 CHAPTER 18 More Statistical Thermodynamics