Law of equipartition

Question

Law of equipartition predicts the heat capacity of gases correctly. It assumes that inter-molecular attraction in gases is negligible (which is true). But for solids, inter-molecular attraction is not negligible, the, how come it still predicts the correct value for molar heat capacity?

How can we ignore the potential energy due to inter-molecular attraction in solids?

Each oscillator has two degree of freedom (kinetic and potential). Solid has oscillators in all the 3 dimension. So, total degree for freedom will be $6$. Using equipartition, the internal energy of one mole a solid will be $ U = 6 \times \frac{1}{2}RT = 3RT$.

Heat capacity, $C = \dfrac{dQ}{dT} = \dfrac{dU + PdV}{dT} = \dfrac{dU}{dT}$ $(\because V \text{ is constant})$

$\implies C = \dfrac{d}{dT}[3RT] = 3R \approx 25 J mol^{-1} K^{-1}$

Which is a very close approximation of the actual values.

My question is why is this prediction in agreement with experimentally determined values, when we have ignored the potential energy due to inter-molecular attraction. Which is not negligible for solids?

Answer 1

Law of equipartition predicts the heat capacity of gases correctly. It assumes that inter-molecular attraction in gases is negligible (which is true). But for solids, inter-molecular attraction is not negligible, the, how come it still predicts the correct value for molar heat capacity?

You have to know what "negligible" means in the context of the equipartition theorem. For both gases and solids, the Boltzmann statistics and equipartition theorem seem valid only if the inter-particle interaction of two subsystems is negligible with respect to total energy of the system. That is true for gases and even for solids, but not for gravitational systems such as globular clusters.

The heat capacity $3R$ is a bad result for low enough temperatures - much better is the Debye formula, which does not depend on equipartition.