## Einstein’s Theory of Specific Heats

The law of Dulong and Petit states that the molar heat capacity at constant volume CV
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,