Variation of specific heat of solids with temperature
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According to Dulong and Petit's law, the molar specific heat of every solid must come out to be equal to 6 cal/goC or 3R (where R is universal gas constant for 1 mole). But the result is nearly 6 cal and not exactly 6 cal/goC. In addition, when the experiment is conducted at various temperatures, we note that the specific heat is not a constant quantity. Instead, it varies with temperature and only at a specific temperature depending upon the nature of the material, it approaches 6. If a graph is drawn between the temperature and the Cv for a solid, we actually get the result as depicted in the above figure.
According to the classical theory, the molar specific heat should always be a constant, regardless the temperature at which the experiment is performed. But, the actual result shows anomaly. This was not explained at that time. However, Planck propounded quantum theory in 1905. Einstein applied it to specific heat problem in 1907. Ultimately, Debye was successful in 1912, in fully explaining the thermal behavior of metals and non-metals.
If Cp is the specific heat at constant pressure,
where dW is the work done by the gas in expanding against external pressure, i.e., the work done by the gas in pushing the piston upwards through a distance 
x.
Work done, dW = force x distance = P x A x
D x
where A is the cross-sectional area of the piston.
-----(4)
Combining equations (1), (2), (3) and (4), we get
The equation of state of gas is PV =RT
where R is the gas constant for 1 mole of the gas.
Therefore from equation (5),
