Joseph 6. Dance
Florida State University Tallahassee, Florida 32306
Heat Capacity and the Equipartition Theorem
O n e of the most important theorems to be discussed in a general chemistry class is the equipartition theorem (1). Historically speaking, the failure of many molecules to obey even approximately the equipartition theorem as originally stated was one more example of a phenomenon which clearly exposed the limitations of classical mechanics in understanding molecular properties. Pedagogically speaking, a modern discussion of the theorem, taking into account modifications introduced by quantum mechanics, permits an instructor to bring in and tie together important concepts from the fields of thermodynamics, quantum mechanics, spectroscopy, and statistics. Discussions of the theorem can he found in texts on statistical mechanics (8); the purpose of this paper will be to simplify and, hopefully, clarify the ideas found therein. The Classical Equipartition Theorem
The equipartition theorem deals with the heat capacity a t constant volume of substances. I n general, we know that unless a phase change is occurring, the absorption of heat by a substance results in a rise in temperature. However, in the case of heat capacity we are concerned not only with the temperature change, but also with the energy that is absorbed. The classical principle of the equipartition of energy may be stated as follows The average kinetic energy lo be associated with each degree of fmedmnfor a system in thermal epuilibrium is 1/&T per molecule.
The principle was expressed in this form in 1857 by Clausius, and it was used to explain the specific heats of monatomic gases, some diatomic gases and some solids. It also represented a modification of the kinetic molecular theory, which regarded molecules as hard elastic spheres with no internal stwcture. A word about terminology is necessary a t this point. Most texts define a degree of freedom of a molecule as an independent coordinate necessary to completely fix its position; on this basis a diatomic molecule has 6 degrees of freedom, and hence, from the statement above, it might be thought to possess a molar heat capacity of 3R (recall R = Nok). Nitrogen, a typical diatomic, is found to have a value of about 3.3R at 2000°K, indicating roughly one additional degree of freedom. This comes about because it is possible to store energy as potential energy in the N-to-N triple bond. Some authors (3) have found it convenient to think of a degree of freedom as an independent pathway in which energy could be absorbed. Thus, nitrogen could hypothetically absorb energy to translate in either of three orthogonal directions, to rotate about either of two orthogonal axes, and to vibrate 798 / lourno1 of Chemiccd Education
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0 Figure 1.
kT
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2kT kinetic energy
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3kT
Maxwellion distribution of kinetic energies.
(in which the energy absorbed is partitioned between a kinetic form and a potential form). This definition of degrees of freedom gives us seven instead of six. If one were to adopt this way of thinking, then the equipartition theorem could be economically stated as follows The heat capacitv, C,, of a system a1 thermal ~quilibriumand -posed o j parlicles possessing s degrees of freedom is ' I & per molecule.
The following simple arguments may be used to help understand the origin of the theorem. I n 1859 James Clerk Maxwell deduced the distribution law for velocities and translational kinetic energies in a macroscopic (molar) sample of a gas (4) (see Fig. 1). - ~NO
