Physical Chemistry , 1st ed.

constants of reactions knowing little other than the energy levels of the equi-

librium species. This is a powerful predictive tool, one that chemists use to es-

timate the equilibrium extent of reactions under conditions that might not be

directly measurable.

18.10 Crystals

The success of statistical thermodynamics as applied to gaseous systems in-

spired scientists to try to apply it to other systems. Some of the more instruc-

tive attempts occurred at the beginning of the twentieth century. During this

period, scientists made the first in-depth studies of matter at very low temper-

atures, approaching absolute zero. Hydrogen and helium gases were first lique-

fied in 1898 and 1908, respectively, and the techniques used to generate such

low temperatures were used to cool matter down and investigate its properties.

At such low temperatures, most matter is solid, and the best type of solid

sample to study is a crystal. Studies of crystals showed some intriguing thermo-

dynamic behavior. For instance, in the measurement of entropy it was found

that absolute entropy approached zero as the temperature approached absolute

zero. This is experimental verification of the third law of thermodynamics. But

a measurement of the heat capacity of the solid showed something interesting:

the heat capacity of the solid approached zero as the temperature approached

absolute zero, also. But for virtually all crystalline solids, the heat-capacity-

versus-temperature plot took on a similar shape at low temperatures, typified

by Figure 18.3: the curves have the distinct shape of a cubic function, that is,

yx^3. In this case, the variable is absolute temperature, so experimentally it

was found that the constant-volume heat capacity CVwas directly related to T^3 :

CVT^3 (18.64)

(where means “directly related” or “proportional to”). How can this behav-

ior be explained?

Scientists attempted to use statistical thermodynamics to understand the

heat capacities of crystals at low temperatures. Given the success of statistical

thermodynamics to gases, in which gas molecules are indistinguishable and

644 CHAPTER 18 More Statistical Thermodynamics

25

0

0

Temperature (K)

300

Cp

[J/(mol

• K

)]

20

15

10

5

200

Silicon

Germanium

100

Figure 18.3 Measurement of the heat capacity of crystals at very low temperatures shows a

curve that looks like a ykT^3 curve. Any theory of heat capacities of crystals should predict this

kind of behavior.