1. The heat capacity per one mole  3R
HEAT CAPACITY OF SOLIDS: BRIEF HISTORY
Almost 200 years ago, in 1819, Messrs. Dulong and
Petite (or, rather Messieurs Dulong and Petite, because
they were Frenchmen), studied the heat capacity of a
large number of elementary solids, and they found
an amazing regularity: namely, that in all cases
The heat capacity per one mole  3R
Where R = J/K-mole is the “gas constant”.
R had been known to the early XIX-century physicists
from the equation of state of ideal gas: pV=NRT
This empirical rule became widely known as the
Dulong – Petite Law
(althogh from the formal viepoint it’s not a “law”)

2. Some 40 or 50 years later, when statistical mechanics
Right from the beginning scientists suspected that such a
relation with the gas constant was not coincidental…
Some 40 or 50 years later, when statistical mechanics
emerged (i.e., the “old” theory, fully based on Newtonian
mechanics), a theoretical explanation of the Dulong-Petite
Law was obtained.
Namely, as was shown by classical stat-mech calculation,
If the classical Hamiltonian of a micro-object (i.e., a particle)
consists of several “quadratic” terms:
where pi and qi are the generalized position and momentum
coordinates, and Ai and Bi are constants, then each quadratic
term contributes ½ kBT to the object energy, regardless of
A and B constant values! This is the FAMOUS “EQUIPAR-
TITION THEOREM” of classical statistical mechanics.

3. So, if we take one mole of such particles, then each
“quadratic term” contributes
to the total system energy, and its contribution to
The heat capacity is:

4. In the classical statistical mechanical model each atom in
the solid is thought of as an “elementary harmonic oscillator”
with a Hamiltonian:
Where  is the “spring constant”, and m is the atom’s mass.
Note that there are SIX “QUADRATIC TERMS”, which
indeed yields the total of Cv = 3R !!! ☻☻☻
NOTE, in addition: this result predicts that that the heat capacity is
CONSTANT over the entire T range form 0 to infinity!

5. After such a spectacular
success of statistical
mechanics, the moods
of physicists were jubilant…

6. However, not long afterwards
something happened
that had
the effect
of a……

7. After the first successful liquefaction of air and nitrogen in
the second half of the XIX century, low-temperature studies
became possible. Scientist started measuring heat capacity
In the low-T range, and… what they found was obviously
a major embarrassment:
Dulong-Petite value and Equipartion Theorem prediction