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The Equipartition of Energy Theorem and Its Sad History

Perfectly spherical molecules have three degrees of freedom; one for each coordinate axis. In a gas made up of such molecules one third of the energy is embodied in fluctuations in each of the three directions. Such molecules could also spin but there is no way for translational movement to be converted into molecular spin. The term degrees of freedom is misleading. It is not potential degrees of freedom that is important; it is effective interacting energy modes.

For a gas made up of elongated but cylindrically symmetric molecules like hydrogen H2 there are an additional two degrees of freedom for rotation about two axes. There is no way translational or spinning motion about the other axes can be converted into spinning about the axis perpendicular to the circular cross sections.

A molecule may have internal structure and that internal structure may have vibrations. However if the energy required to initiate a vibration is vastly greated than the average energy involved in collisions of the molecules then the internal structure may not share in the equipartition of energy. For example, the average energy involved in molecular collisions at room temperature is about 1/40 of an electron voltl (ev); the energy required to move an electron from one state to another in an atom is several electron volts. Likewise the energy involved in vibrations in the separation distance or angle between atoms in a molecule may be too large to be effected by the collisions of molecules at some temperature range. Thus the effective degrees of freedom may depend upon the temperature under consideration.

With these considerations taken into account in assessing the degrees of freedom, in a gas of molecules having n effective degrees of freedom the total energy of the gas is divided equally among those n degrees of free of freedom. This is the Equipartition of Energy Theorem. It can be extended. In a gas made up of a mixture of molecules the energy of the gas would be divided among the different components of the gas such that the average energies would be the same.

The first glimmerings of the Equipartition of Energy Theorem came from a man named John James Waterston. In 1839 Waterston went to India to serve as a naval instructor for the East India Company in Bombay. In 1845 he sent an article to the Royal Society in England is which he argued that in a gas of mixed composition the mean squared velocities of the molecules of the different components are inversely proportional to the masses of the molecules. This is just another way of saying that their average kinetic energies are equal.

One of the two referees said of Waterston's paper that it was

[…] nothing but nonsense, unfit even for reading before the Society.

Waterston's rejected paper was filed in the Royal Society archives. Waterston was sure of the importance of his paper so sent an abstract of it to the people he thought m ight be interested and in 1850 a shortened version was read at a meeting of the British Association. This attracted no significant interest. It was not until 1895 that Waterston's article was published in the Philosophical Transactions due to the efforts of Lord Raleigh. Raleigh said in the introduction to that publication

The omission to publish it at that time was a misfortune which probably retarded the development of the subject by ten or fifteen years. It is singular that Waterston appears to have advanced no claim for subsequent publication, whether in the Transactions of the Society, or some other channel. At any time since 1860 reference would have been made to Maxwell, and it cannot be doubted that he would have at once recommended that everything possible should be done to atone for the original failure of appreciation.

In 1859 Jame Clerk Maxwell had presented a paper at a meeting of the British Association which gave results similar to those of Waterston and in 1860 published a paper that further extended the results. In 1868 Ludwig Boltzmann extended the Equipartition Theorem to include the case of molecules that are not rigid and thus have some internal degrees of freedom. In 1878 Maxwell published a paper in which he extended the theorem to include the case in which molecules interact at a distance, such as through their electric fields.

When the British scientist J.S. Haldane put together a collection of Waterston's paper he included the following remarks

It is probable that in the long and honorable history of the Royal Society no mistake more disasterous in its actual consequences for the progress of science and the reputation of British science than the rejection of Waterston's papers was ever made. … There is every reason for believing that had the papers been published physical chemistry and thermodynamics would have developed mainly in this country and along much simpler, more correct, and more intelligiblel lines than those of their actual development.

Reference:

Max Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill Book Co., New York, 1966.

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