Gas, Kinetic Theory of

GAS, Kinetic Theory of. - I. GENERALITIES. - Since the 17th century the word g. has designated that particular state of aggregation of any matter in which its constituent particles (today one says its molecules) not only feel no mutual cohesion, or semicohesion, similar to those corresponding respectively to the solid state and the liquid state, but instead display a marked tendency toward the most unconditional expansion. Mechanics, just as it studied the laws of motion of solids, also studied those of liquids and of gases on the basis of their collective properties; so too did physics, especially as regards their thermal behavior.

But once the atomic and molecular hypothesis became established in the first half of the last century, two great problems arose: that of the dynamic study of individual molecules and individual atoms, and that of the interpretation of the collective properties of bodies in their various states of aggregation, and in particular of gases, with reference to their molecular constitution. The first of the aforementioned problems, which today is called microscopic, could not be tackled at that time and constitutes only at present one of the major aims of modern physics; the second, on the other hand, was considerably developed, and, as regards gases, led not only to the theoretical results that had been foreseen, but also to others that go far beyond the narrow field of its original formulation. Especially for this reason it currently constitutes one of the most interesting and instructive chapters of physics.

II. CONSTITUTION OF GASES

As early as 1738 D. Bernoulli considered gases as made up of a collection of very numerous heavy material corpuscles, very far apart from one another in relation to their dimensions and each endowed with a notably high velocity; on this basis he succeeded in demonstrating that this collection had to obey the well-known Boyle-Mariotte law. The corpuscles of Bernoulli are now thought to be replaced by molecules in the enormous number determined by Loschmidt on the basis of Avogadro's law, and it is supposed that each molecule is subject to the laws of mechanics. Evidently a first series of the properties of our gas will have to follow as a statistical consequence of this position. For simplicity, monatomic molecules are considered, all the more so since it is known that, limiting these considerations to the statistical side alone, the transition to more complex cases presents no insurmountable difficulties. The molecules are in general notably far apart, but at times they also approach one another, collide, bounce back. Since, however, at very small distances they repel one another, it is probable that their collisions occur only between their so-called (repulsive) spheres of action and are therefore elastic. It is according to this point of view that Joule (1848), Kroenig (1856) and Clausius (1857) began to give a less vague form to Bernoulli's ideas, followed by Maxwell, who in 1850 took the decisive step, which can be considered the foundation not only of the kinetic theory of gases, but also of statistical mechanics in the generally understood sense.

III. THE MAXWELLIAN DISTRIBUTION OF VELOCITIES

Experience assures us that a gas enclosed in a vessel and removed from external actions, whatever its initial distribution may have been, very quickly assumes a stationary state. Pre-Maxwellian researchers had thought that to interpret that stationary state it was enough to admit that the molecules possessed a certain mean velocity, constituted in any manner whatsoever (not excluding, for example, even the particular case in which all possessed that velocity outright as their own velocity). Maxwell, on the other hand, intuited that to characterize that stationary state it was not enough to assign a mean velocity, but that it was also necessary that the number of molecules endowed with each of the countless velocities present should remain unchanged. This by no means signifies that every molecule must keep its own velocity unaltered, but rather that the molecules which in a short time lose that velocity as a result of collisions are as many as those which, for the same reason, acquire it.

Thus two important questions arose: 1) what was the particular distribution of velocities that corresponded to the stationary state of the gas? 2) what was the chief reason why this distribution of velocities was always reached rapidly?

Maxwell, aided by intuitive analogies with the law of errors of V. Herschell, believed he had demonstrated that in the stationary state the number of molecules having velocity components u, v, w (along Cartesian axes) is given by an expression of the type a e^b (u²+v²+w²), where 'a' and 'b' are positive constants depending on the physical state of the gas.

But Maxwell himself recognized the defects of his demonstration and in 1866 proposed a second one, based on the law of collisions of elastic bodies. He deduced the statistical condition whereby the number of molecules that abandon a certain velocity is, during every instant, equal to the number of those that acquire it, and he rediscovered the previous law, which from then on was called Maxwell's law of distribution.

In 1870, however, L. Boltzmann examined Maxwell's demonstration very thoroughly and first proved that the law is not only sufficient but necessary for the stationary state; then he sought and believed he had demonstrated — what experience confirms — that if the stationary state of a gas is disturbed in any manner whatsoever, the gas, left to itself, spontaneously returns to the stationary state. Maxwell's law would therefore represent the limiting distribution toward which every distribution of velocities different from it tends.

IV. LOSCHMIDT'S OBJECTION AND BOLTZMANN'S PROBABILISTIC INTERPRETATION

Loschmidt observed (1876) that the irreversible passage from any distribution to the Maxwellian one could not follow from the reversible laws of dynamics. Consequently Boltzmann reached the conclusion that this passage was not at every time rigorously necessary, but only extremely probable. And so the stationary character of Maxwell's law would not be a rigorous necessity, but only a requirement of extreme probability. Among the infinite possible distributions, those of the Maxwellian type represent the vast majority; therefore, even if the gas passes at random through all possible states, it will practically always appear in the Maxwellian distribution.

Now that the necessity of resorting to probabilistic considerations has been established, Maxwell's law is deduced by means of them more directly. The conclusion, however, is always the same.

At this point it is very interesting to realize the extremely high values of the probabilities that come into play, practically equivalent to the greatest certainties. One tries to illustrate them with intuitive examples, e.g., by supposing that a monkey is placed at a typewriter and that all the letters and spaces remain continuously recorded. What probability is there that the monkey will type a page of the Divine Comedy? Theoretically that case, which is practically impossible, cannot be excluded. Well then, the probability that a gas contained in a one-liter vessel spontaneously assumes a non-Maxwellian distribution for one second is of the same order!

V. PHYSICAL REPERCUSSIONS

The fact that a physical law that had always been considered certain and that had always been verified is instead an extremely probable law, yes, but not certain, led to revisions of various other physical notions and laws. And by now even classical physics admits many fundamental laws only as probable. See the entry TERMODINAMICA. Modern physics, moreover, is predominantly dominated by laws that are only probabilistic.

Paolo Strancio