During the early part of the nineteenth century, studies on the heat capacity of materials tended to indicate that it was a rather uninteresting property, somewhat independent of temperature. These results, arising mainly because of a rather limited experimental temperature range and by working around room temperature and above, gave rise to the Dulong-Petit law, which referred to the constant pressure heat-capacity as being approximately constant at about 26.8 J/K mol. The constant volume heat capacity is even more nearly the same for all elements, since the difference done in expansion pV work in the constant pressure case adds additional minor variations. Lewis and Gibson (1917) measured the constant volume heat capacities and noted that they were all within 1.5% of 24.69 J/K mol for elements heavier than potassium and for which data were available.
This can be readily seen from classical physics where the equipartition theorem would assign a vibrational energy of 3kT to each atom in a solid. For N atoms, this would give 3NkT for the total vibrational energy. The molar internal energy arising from vibrational motion would be
From here the molar constant volume heat capacity is easily predicted to be
This is precisely the Dulong-Petit Law and readily shows its independence from temperature.
But when experimental apparati improved and measurements were made for lower temperatures it became apparent that the heat capacity actually drops with decreasing temperature and in fact approaches zero at absolute zero temperature. Einstein first tackled this problem by invoking Planck's quantum ideas. He suggested that each atom was an oscillator of frequency n and then asserted that any oscillation would have to be an integer multiple nhn. From this he calculated the molar vibrational energy and, by differentiating with respect to T, arrived at what is known as the Einstein Formula.
At high temperatures, you can expand the exponentials and see how the expression reduces to the Dulong-Petit Law. But at low temperatures it indeed approaches zero as observed. Numerical agreement is not excellent, however, and Peter Debye produced a further correction by not assuming that all atoms had oscillated at the same frequency, but rather averaged over all frequencies present. This Debye Equation nicely predicts the observed experimental results, lending further credence to Planck's Quantum Hypothesis.