EINSTEIN AND SPECIFIC HEATS 397to an average over the range £ ~ 0.02-0.07. This value is much too large to be
accommodated (simultaneously with Weber's points) by Einstein's Eq. 20.4: the
exponential drop of cv as T —*• 0, predicted by that equation, is far too steep.
Einstein did become aware of this discrepancy in 1911, when the much
improved measurements by Nernst showed that Eq. 20.4 fails at low T [N2].
Nernst correctly ascribed the disagreement to the incorrectness of the assumption
that the lattice vibrations are monochromatic. Einstein himself explored some
modifications of this assumption [ E4]. The correct temperature dependence at low
temperatures was first obtained by Peter Debye; for nonmetallic substances, cv
—* 0 as T"^3 [D3]. Einstein had ended his active research on the specific heats of
solids by the time the work of Debye and the more exact treatment of lattice
vibrations by Max Born and Theodore von Karman appeared [B7]. These further
developments need therefore not be discussed here.
However, in 1913 Einstein returned once again to specific heats, this time to
consider the case of gases. This came about as the result of important experimental
advances on this subject which had begun in 1912 with a key discovery by Arnold
Eucken. It had long been known by then that c, ~ 5 for molecular hydrogen at
room temperature. Eucken showed that this value decreased with decreasing T
and that cv « 3 at T «s 60 K [E5]. As is well known today, this effect is due to
the freezing of the two rotational degrees of freedom of this molecule at these low
temperatures. In 1913 Einstein correctly surmised that the effect was related to
the behavior of these rotations and attempted to give a quantitative theory. In a
paper on this subject, we find another instance of curve fitting by Einstein [E6].
However, this time he was wrong. His answer depended in an essential way on
the incorrect assumption that rotational degrees of freedom have a zero point
energy.*
In 1925 Einstein was to turn his attention one last time to gases at very low
temperatures, as we shall see in Section 23b.20c. Nernst: Solvay I**
'As the temperature tends to absolute zero, the entropy of a system tends to a
universal constant that is independent of chemical or physical composition or of
other parameters on which the entropy may depend. The constant can be taken
to be zero.' This modern general formulation of the third law of thermodynamics
implies (barring a few exceptional situations) that specific heats tend to zero as T
—* 0 (see [H2]). The earliest and most primitive version of the 'heat theorem' was
presented in 1905, before Einstein wrote his first paper on specific heats. The final
*In 1920 Einstein announced a forthcoming paper on the moment of inertia of molecular hydrogen
[E7]. That paper was never published, however,
**The preparation of this section was much facilitated by my access to an article by Klein [K4] and
a book by Hermann [HI].