Thermodynamics and Chemistry

CHAPTER 6 THE THIRD LAW AND CRYOGENICS

6.2 MOLARENTROPIES 154

the molar entropy increases continuously with increasingTand has a discontinuity at each
phase transition.

As explained in Sec.6.1, by convention the zero of entropy of any substance refers to

the pure, perfectly-ordered crystal at zero kelvins. In practice, experimental entropy

values depart from this convention in two respects. First, an element is usually a

mixture of two or more isotopes, so that the substance is not isotopically pure. Second,

if any of the nuclei have spins, weak interactions between the nuclear spins in the

crystal would cause the spin orientations to become ordered at a very low temperature.

Above 1 K, however, the orientation of the nuclear spins become essentially random,

and this change of orientation is not included in the DebyeT^3 formula.

The neglect of these two effects results in apractical entropy scale, or conventional

entropy scale, on which the crystal that is assigned an entropy of zero has randomly-

mixed isotopes and randomly-oriented nuclear spins, but is pure and ordered in other

respects. This is the scale that is used for published values of absolute “third-law”

molar entropies. The shift of the zero away from a completely-pure and perfectly-

ordered crystal introduces no inaccuracies into the calculated value ofÅSfor any

process occurring above 1 K, because the shift is the same in the initial and final states.

That is, isotopes remain randomly mixed and nuclear spins remain randomly oriented.

6.2.2 Molar entropies from spectroscopic measurements

Statistical mechanical theory applied to spectroscopic measurements provides an accurate
means of evaluating the molar entropy of a pure ideal gas from experimental molecular
properties. This is often the preferred method of evaluatingSmfor a gas. The zero of
entropy is the same as the practical entropy scale—that is, isotope mixing and nuclear spin
interactions are ignored. Intermolecular interactions are also ignored, which is why the
results apply only to an ideal gas.

The statistical mechanics formula writes the molar entropy as the sum of a translational

contribution and an internal contribution:SmDSm;transCSm;int. The translational

contribution is given by the Sackur–Tetrode equation:

Sm;transDRln

.2M/3=2.RT /5=2

ph^3 NA^4

C.5=2/R (6.2.7)

Herehis the Planck constant andNAis the Avogadro constant. The internal contribu-

tion is given by

Sm;intDRlnqintCRT .d lnqint=dT / (6.2.8)

whereqintis the molecular partition function defined by

qintD

X

i

exp.i=kT / (6.2.9)

In Eq.6.2.9,iis the energy of a molecular quantum state relative to the lowest en-

ergy level,kis the Boltzmann constant, and the sum is over the quantum states of one

molecule with appropriate averaging for natural isotopic abundance. The experimental

data needed to evaluateqintconsist of the energies of low-lying electronic energy lev-

els, values of electronic degeneracies, fundamental vibrational frequencies, rotational

constants, and other spectroscopic parameters.