Physical Chemistry Third Edition

144 3 The Second and Third Laws of Thermodynamics: Entropy

∆S◦(373.15 K)− 172 .88JK−^1 mol−^1 +(− 13 .402JK−^1 mol−^1 )ln

(

373 .15 K

298 .15 K

)

− 172 .88JK−^1 mol−^1 − 3 .007JK−^1 mol−^1  175 .89JK−^1 mol−^1

Exercise 3.20

a.Calculate the entropy change at 298.15 K for the reaction

2H 2 (g)+O 2 (g)→2H 2 O(l)

b.Calculate the entropy change at 400.0 K for the same reaction except that H 2 O(g) is the

product.

Chapter 2 presented the approximate calculation of energy changes of chemical

reactions, using average bond energies. There is an analogous estimation scheme for

the entropy changes of chemical reactions, in which contributions from bonds and

contributions from groups of atoms are included.^8 We do not discuss this scheme, but

the interested student can read the article by Benson and Buss.

Statistical Entropy and the Third Law of Thermodynamics

As mentioned earlier, there are some substances such as carbon monoxide that do not

obey the third law of thermodynamics in their ordinary forms. The absolute entropy

of these substances determined by an integration such as in Eq. (3.5-1) turned out to

be too small to agree with values inferred from entropy changes of chemical reactions

and absolute entropies of other substances. Carbon monoxide molecules have only a

small dipole moment (small partial charges at the ends of the molecule) and the two

ends of the molecule are nearly the same size, so a carbon monoxide molecule fits into

the crystal lattice almost as well with its ends reversed as in its equilibrium position.

Metastable crystals can easily form with part of the molecules in the reversed position.

If we assume that the occurrence of reversed molecules is independent of the rest of

the state of the crystal, we can write

ΩΩorientΩrest (3.5-10)

whereΩorientis the number of ways of orienting the molecules in ways compatible with

our knowledge of the state of the system, andΩrestis the number of possible states of

the crystal if the orientation of the molecules is ignored.

Statistical mechanics predicts that at absolute zero the various vibrations of a crystal

lattice all fall into a single lowest-energy state, as do the electronic motions. If there is

no entropy of isotopic mixing,

lim

T→ 0

Ωrest1 (equilibrium crystal) (3.5-11)

(^8) S. W. Benson and J. H. Buss,J. Chem. Phys., 29 , 546 (1958); S. W. Bensonet al., Chem. Rev., 69 , 279
(1969).