EINSTEIN AND SPECIFIC HEATS 3936 cal/mol • deg. Thus, after half a century, the Dulong-Petit value had found a
theoretical justification! As Boltzmann himself put it, his result was in good agree-
ment with experiment 'for all simple solids with the exception of carbon, boron*
and silicon.' Boltzmann went on to speculate that these anomalies might be a con-
sequence of a loss of degrees of freedom due to a 'sticking together' at low tem-
peratures of atoms at neighboring lattice points. This suggestion was elaborated
by others [R3] and is mentioned by Wigand in his 1906 review as the best expla-
nation of this effect. I mention this incorrect speculation only in order to stress one
important point: before Einstein's paper of 1906, it was not realized that the dia-
mond anomaly was to be understood in terms of the failure (or, rather, the inap-
plicability) of the classical equipartition theorem. Einstein was the first one to state
this fact clearly.
By sharp contrast, it was well appreciated that the equipartition theorem was
in trouble when applied to the specific heat of gases. This was a matter of grave
concern to the nineteenth century masters. Even though this is a topic that does
not directly bear on Einstein's work in 1906, I believe it will be useful to complete
the nineteenth century picture with a brief explanation of why gases caused so
much more aggravation.
The reasons were clearly stated by Maxwell in a lecture given in 1875:
The spectroscope tells us that some molecules can execute a great many differ-
ent kinds of vibrations. They must therefore be systems of a very considerable
degree of complexity, having far more than six variables [the number charac-
teristic for a rigid body]... Every additional variable increases the specific heat.
... Every additional degree of complexity which we attribute to the molecule
can only increase the difficulty of reconciling the observed with the calculated
value of the specific heat. I have now put before you what I consider the greatest
difficulty yet encountered by the molecular theory. [M2]Maxwell's conundrum was the mystery of the missing vibrations. The follow-
ing oversimplified picture suffices to make clear what troubled him. Consider a
molecule made up of n structureless atoms. There are 3« degrees of freedom,
three for translations, at most three for rotations, and the rest for vibrations. The
kinetic energy associated with each degree of freedom contributes k.T/2 to cv. In
addition, there is a positive contribution from the potential energy. Maxwell was
saying that this would almost always lead to specific heats which are too large. As
a consequence of Maxwell's lecture, attention focused on monatomic gases, and,
in 1876, the equipartition theorem scored an important success: it found that cj
cv « 5/3 for mercury vapor, in accordance with cv = 3R/2 and the ideal gas rule
cp — cv = R [Kl]. It had been known since the days of Regnault** that several
"The good professor wrote bromine but meant boron.
**A detailed review of the specific heats of gases from the days of Lavoisier until 1896 is found in
Wullner's textbook [W4].