# Einstein’s Theory of Specific Heats

## Einstein’s Theory of Specific Heats

for all solids is equal to 3R, where R is the gas constant. Obtained empirically in the

early nineteenth century, it was easily derived later from the equipartition theorem.

Each molecule was considered to be vibrating as a harmonic oscillator with three

degrees of freedom. Each degree of freedom had, on average, 1

2kT of kinetic energy

and 12kT of potential energy, so the average total energy per molecule is

Etotal = 6 *12kT = 3kT

The total energy per mole is then 3kTNA where NA is Avagadro’s number. The molar

heat capacity is then given by

CV = 0E

0T = 3kNA = 3R SH-1

Experimentally, as long as the temperature is above a critical value, different for each

material, Equation SH-1 works reasonably well for solids. However, when the tem-

perature falls below the critical value, the law of Dulong and Petit fails and CV S 0.

The equipartition theorem (kinetic theory) gives no hint as to why this occurs.

Einstein recognized that Planck’s quantization of the molecular oscillators in the

walls of the blackbody cavity was, in fact, a universal property of the molecular oscil-

lators in all solids. Accordingly, the average energy of the oscillators was not the 3kT

of kinetic theory, but rather that derived in Planck’s development of the emission

spectrum of a blackbody, given in Equation SH-2:

8E9 = hfehf>kT - 1SH-2

where f is the oscillation frequency of the molecules.

At high temperatures, hf>kT V 1, so

ehf>kT - 1 a1 +hfkT+ g b - 1 hfkT

SH-3

Substituting this result into Equation SH-2, we see that 〈E〉 S kT, as in kinetic theory.

However, at low temperatures the result is much different. The total energy for NA

( 1 mole) of oscillators is

E = 3NA 8E9 = 3NAhfehf>kT - 1

where 〈E〉 is given by Equation SH-2. The molar heat capacity is then

CV = 0E
0T = 3NAka hf kT b 2ehf>kT

ehf>kT - 1
SH-5

As T S 0 in Equation SH-5, CV S 0 also, and as T S , CV S 3NAk 3R.

Figure SH-1 illustrates the extent of the agreement between Einstein’s result, Equation
SH-5, and the low-temperature experimental data for diamond. Einstein’s approach to

the problem was clearly a significant improvement over the law of Dulong and Petit,

but note the deviations at very low temperatures. Peter Debye extended Einstein’s

work by replacing the solid whose molecules all oscillated with a single frequency

with a solid consisting of coupled oscillators with frequencies ranging from 0 to a

maximum value fD. Debye’s theory fits all solids very well,

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