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 = 8.314 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”)

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.

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:

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!

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

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

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