2.3 Heat capacity and equipartition of energy  (Page 3/6)

Polyatomic molecules typically have one additional rotational degree of freedom at room temperature, since they have comparable moments of inertia around any axis. Thus, at room temperature, they have $d=6,$ and at high temperature, $d=8.$ We usually assume that gases have the theoretical room-temperature values of d .

As shown in [link] , the results agree well with experiments for many monatomic and diatomic gases, but the agreement for triatomic gases is only fair. The differences arise from interactions that we have ignored between and within molecules.

What about internal energy for diatomic and polyatomic gases? For such gases, $C_V$ is a function of temperature ( [link] ), so we do not have the kind of simple result we have for monatomic ideal gases.

Molar heat capacity of solid elements

The idea of equipartition leads to an estimate of the molar heat capacity of solid elements at ordinary temperatures. We can model the atoms of a solid as attached to neighboring atoms by springs ( [link] ).

Analogously to the discussion of vibration in the previous module, each atom has six degrees of freedom: one kinetic and one potential for each of the x -, y -, and z -directions. Accordingly, the molar specific heat of a metal should be 3 R . This result, known as the Law of Dulong and Petit , works fairly well experimentally at room temperature. (For every element, it fails at low temperatures for quantum-mechanical reasons. Since quantum effects are particularly important for low-mass particles, the Law of Dulong and Petit already fails at room temperature for some light elements, such as beryllium and carbon. It also fails for some heavier elements for various reasons beyond what we can cover.)