Revival of Kinetic Theory by Clausius
(1857 - 1858)

The "kinetic theory of heat" -- the old idea that heat is directly related to the kinetic energy of atomic motion -- had to be given serious consideration as soon as energy conservation and thermodynamics had been introduced in the middle of the 19th century. Additional evidence that gas pressure is not caused by repulsive intermolecular forces (previously associated with the caloric fluid) came from experiments on the free expansion of gases by J. P. Joule and William Thomson; they confirmed that (as Gay-Lussac had found earlier) there is essentially no temperature change; more accurate measurements suggested that long-range forces are attractive, not repulsive. Another reason for favoring a kinetic theory of heat was the general adoption of the wave theory of light which -- combined with the view that heat and light are qualitatively the same phenomenon -- suggested that heat, like light, is a form of motion rather than a substance.

It was still logically possible to reject the kinetic theory of heat, as did J. R. Mayer and later Ernst Mach, by denying the need to reduce heat to any other form of energy. This antireductionist or positivist stance was the basis for the "energetics" movement at the end of the 19th century, but it was uncongenial to most scientists.

Having accepted the kinetic theory of heat, one still had several possible hypotheses to choose from. The molecular motion might be translational, rotational, or vibrational, or a combination of all three; the molecules might be small relative to the space in which they move, or large and thus crowded together; the motion might be similar for each molecule in the system or differ according to a definite pattern. Physicists still believed that an ether is needed to transmit energy between bodies in the form of light or radiant heat; if the ether also fills the space between molecules inside a body, it should have some effect on their motion. The old idea that molecules "swim" in the ether, or are suspended by it at definite equilibrium points around which they may vibrate, was not yet dead.

Among these possibilities the kinetic theory of gases was perhaps the simplest but by no means the most plausible. In fact, it seemed too simple to be true. It required that one ignore the ether and assert that molecules move through space at constant velocity, encountering no resistance except when they collide with each other or a boundary surface. The first scientist who was able to overcome the general reluctance to give serious consideration to this idea was the German physicist Rudolf Clausius (1822-1888).

In his first paper on kinetic theory, published in 1857, Clausius stated that he had been thinking about molecular motion even before writing his first article on thermodynamics in 1850, but had abstained from publishing his ideas because he wanted to establish the empirical laws of heat without making them appear to depend on any molecular hypothesis. Now that Kroenig had taken the lead with his 1856 paper, there was no question of priority, but the time seemed auspicious to attempt a unified description of several phenomena from the kinetic viewpoint. Kroenig had assumed that the molecules have only translatory motion (and, as Clausius was perhaps too polite to point out, had not even given the correct numerical factor in the pressure equation for that simple case). Clausius concluded that one must also include other kinds of molecular motion, such as rotation, and showed how one could estimate the fraction of the total energy which is translational by using heat data.

By including rotational motion in his kinetic theory, Clausius was compromising not with alternative theories but with empirical knowledge of gas properties. But the result of this compromise was damaging to the kinetic theory all the same: the ratio of translational energy to total energy came to be 0.6315 for the common gases whose ratio of specific heats is 1.421. Now 0.6315, as Maxwell and others intuitively realized, is not a very nice number. It is unlikely (though not impossible) that a direct calculation based on a plausible molecular model would lead to such a number. Perhaps the best that can be said for 0.6315 is that, despite the accuracy implied by its four significant figures, it is not too far away from 3/5, and we will see later there is some hope of making sense out of 3/5.

Clausius did take one step in this direction by reviving Avogadro's proposal that gaseous molecules may contain two or more atoms of the same kind. Some chemists had already come to the same conclusion, but Clausius was probably the first to introduce the idea into mid-19th century physics.

Another assumption dictated by experimental data was the extremely small size of molecules: Clausius stipulated that "the space actually filled by the molecules of the gas must be infinitesimal in comparison to the whole space occupied by the gas itself." Moreover, "the influence of the molecular forces must be infinitesimal" [Clausius 1857/1965, p. 116]. This means not only that the forces between molecules at their average distances are negligible but also that the short-range repulsive forces that cause molecules to rebound at collisions must act over a very small portion of the path of the molecule. If these conditions were not satisfied the gas would not obey the ideal gas laws. By this time it was well known from Regnauit's experiments that real gases do not obey the ideal gas laws, but Clausius was unable in 1857 to carry out the complex calculations needed to compute the deviations using a molecular model, so he limited his theory to ideal gases.

While the strict mathematical deductions of his theory were thus limited to crases obeying the laws of Boyle and Gay-Lussac -- that is, temperatures and pressures not too far from those of the atmosphere -- Clausius did not hesitate to propose a qualitative description of molecular motion in other states of matter. In solids, the molecules vibrate about fixed equilibrium positions, while their constituent atoms vibrate and rotate within the molecule. In liquids the molecules no longer vibrate around fixed positions but may move around, yet without completing separating themselves from their neighbors. In the gaseous state the molecules move in straight lines, going beyond the reach of the attractive forces of other molecules but occasionally undergoing elastic collisions with them.

From this qualitative picture Clausius was able to develop a theory of changes of state. Thus, the evaporation of a liquid can be explained by assuming that even though the average motion of its molecules may not be sufficient to carry them beyond the range of the attractive forces of their neighbors, "we must assume that the velocities of the several molecules deviate within wide limits on both sides of this average value" (Clausius 1857/1965, p. 118) and therefore a few molecules will be moving fast enough to escape from a liquid surface even at temperatures below the boiling point.

The phenomena of latent heat could also be explained by the kinetic theory, if one adopted Clausius' description of the three states of aggregation:

"In the passage from the solid to the liquid state the molecules do not, indeed, recede beyond the spheres of their mutual action; but, according to the above hypothesis, they pass from a definite and, with respect to the molecular forces, suitable [ordered] position, to other irregular positions, in doing which the forces which tend to retain the molecules in the former position have to be overcome."
[Clausius 1857, p. 121]

Whenever a body is moved against the action of a force, mechanical work must be done, and therefore, according to the law of conservation of energy, heat must be supplied.

"In evaporation, the complete separation which takes place between the several molecules and the remaining mass evidently again necessitates the overcoming of opposing forces."
[Clausius 1857, p. 121]

and so heat must again be provided (latent heat of vaporization).

Near the end of his 1857 paper Clausius calculated the average speeds of molecules of oxygen, nitrogen, and hydrogen at the temperature of melting ice and found them to be 461 m/sec, 492 m/sec, and 1,844 m/sec respectively. A Dutch meteorologist, C.H.D. Buys-Ballot, looked at these numbers and realized a consequence that had escaped the notice of Herapath, Joule, Waterston, and Clausius: if the molecules of gases really move that fast, the mixing of gases by diffusion should be much more rapid than we observe it to be. For example, if you release an odorous gas like ammonia or hydrogen sulfide at one end of a room it may take a minute or so before it is noticed at the other end; yet according to the kinetic theory all the molecules should have traversed the length of the room several times by then.

Buys-Ballot apparently thought he had refuted the new theory by pointing out an obvious contradiction between its predictions and the real world. To meet this objection, Clausius had to make an important change in the theory. Abandoning his earlier postulate that the gas molecules have infinitesimal size, he now assumed that they have a large enough diameter or "sphere of action" so that a molecule cannot move very far without hitting another one.

Clausius now defined a new parameter: the mean free path (L) of a gas molecule, to be computed as the average distance a molecule may travel before interacting with another molecule. He argued that L may be large enough compared with molecular diameters so that the basic concepts of kinetic theory used in deriving the ideal gas law are unimpaired, yet small enough so that a molecule must change its direction many times every second, and may take a fairly long time to escape from a given macroscopic region of space. In this way the slowness of ordinary gas diffusion, compared with molecular speeds, could be explained.

The mean free path is inversely proportional to the probability that a molecule will collide with another molecule as it moves through the gas. For spheres of diameter d this probability is proportional to the collision cross section ( d^2) and to the number of molecules per unit volume (N/V). Thus the mean free path is determined by the formula

L = k V/Nd^2

where k is a numerical constant of order of magnitude 1 (its precise value was a matter of dispute for some time).

When Clausius introduced the mean free path in 1858 it may have looked like only an ad hoc hypothesis invented to save the theory, since he did not have any independent method for estimating the parameters N and d in the above formula. But before anyone had a chance to criticize it on those grounds, Maxwell incorporated the mean free path into his own kinetic theory and showed that it could be related to gas properties such as viscosity (see next section). As a result it soon became a valuable concept, not only for interpreting experimental data, but also for determining the size of molecules and thus justifying its own existence.

Updated 24 August 2001