bei48482_FM

convenience. These vibrations may be resolved into components along three perpen-

dicular axes, so that we may represent each atom by three harmonic oscillators. As we

know, according to classical physics a harmonic oscillator in a system of them in ther-

mal equilibrium at the temperature Thas an average energy of kT. On this basis each

atom in a solid should have 3kTof energy. A kilomole of a solid contains Avogadro’s

number N 0 of atoms, and its total internal energy Eat the temperature Taccordingly

ought to be

E 3 N 0 kT 3 RT (9.50)

where R N 0 k8.31  103 J/KmolK 1.99 kcal/kmol K

is the universal gas constant. (We recall that in an ideal-gas sample of nkilomoles,

pV nRT.)

The specific heat at constant volume is given in terms of Eby

cV

V

and so here

cV 3 R5.97 kcal/kmolK (9.51)

Over a century ago Dulong and Petit found that, indeed, cV 3Rfor most solids at

room temperature and above, and Eq. (9.51) is known as the Dulong-Petit lawin

their honor.

However, the Dulong-Petit law fails for such light elements as boron, beryllium, and

carbon (as diamond), for which cV3.34, 3.85, and 1.46 kcal/kmol K respectively

at 20°C. Even worse, the specific heats of allsolids drop sharply at low temperatures

and approach 0 as Tapproaches 0 K. Figure 9.10 shows how cVvaries with Tfor sev-

eral elements. Clearly something is wrong with the analysis leading up to Eq. (9.51),

and it must be something fundamental because the curves of Fig. 9.10 share the same

general character.

Dulong-Petit law

E



T

Specific heat at

constant volume

Classical internal

energy of solid

Statistical Mechanics 321

7

Lead

6 5 4 3 2 1

0 200 400 600 800 1000 1200

Absolute temperature, K

cV

, kcal/kmol

• K

Aluminum

Silicon

Carbon (diamond)

Figure 9.10The variation with temperature of the molar specific heat at constant volume CVfor several

elements.

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