GAS, KINETIC THEORY OF. –
I. GENERALITIES
The word g. has been used since the seventeenth century to designate that particular state of aggregation of any matter in which its constituent particles (today called its molecules) not only experience no mutual cohesion, or semicoherence, such as that corresponding respectively to the solid and liquid states, but instead manifest a marked tendency toward the most unrestricted expansion. Just as mechanics studied the laws of motion of solids, so it also studied those of liquids and gases on the basis of their collective properties; physics likewise did so, especially with regard to their thermal behavior.But once the atomic and molecular hypothesis had become established in the first half of the last century, two major problems arose: the study of the dynamics of individual molecules and atoms, and the interpretation of the collective properties of bodies in their various states of aggregation, particularly gases, with reference to their molecular constitution. The first of these problems, now called microscopic, could not then be undertaken and only today constitutes one of the principal aims of modern physics; the second, however, was developed considerably and, as regards gases, led not only to the theoretical results anticipated but also to others that far transcend the narrow field of its original formulation. It is especially for this reason that it is now one of the most interesting and instructive chapters of physics.
II. CONSTITUTION OF G
As early as 1738, D. Bernoulli regarded g. as constituted by an aggregate of exceedingly numerous heavy material corpuscles, very far apart from one another in relation to their dimensions and each endowed with a remarkably high velocity; on this basis he succeeded in demonstrating that this aggregate must obey the well-known Boyle–Mariotte law. Bernoulli’s corpuscles are now thought of as replaced by molecules, in the immense number determined by Loschmidt on the basis of Avogadro’s law, and it is assumed that each molecule is subject to the laws of mechanics. Evidently, a first series of the properties of our g. must follow as a statistical consequence of this premise. For the sake of simplicity, monoatomic molecules are considered, all the more since it is known that, when these considerations are restricted to the statistical aspect alone, the transition to more complex cases presents no insurmountable difficulties. The molecules are generally quite far apart, but sometimes they also approach one another, collide, and rebound. Since, however, at extremely small distances they repel one another, it is probable that their collisions take place only between their so-called (repulsive) spheres of action and are therefore elastic. It was from this point of view that Joule (1848), Kroenig (1856), and Clausius (1857) began to give less vague form to Bernoulli’s ideas, followed by Maxwell, who in 1850 took the decisive step, which may be regarded as the foundation not only of the kinetic theory of g., but also of statistical mechanics in its generally accepted sense.III. THE MAXWELLIAN DISTRIBUTION OF VELOCITIES
Experience establishes that a gas enclosed in a vessel and shielded from external influences, whatever its initial distribution may have been, very quickly assumes a steady state. The pre-Maxwellian researchers had thought that, in order to interpret this steady state, it was sufficient to assume that the molecules possessed a certain mean velocity, constituted in any manner whatsoever (not even excluding, for example, the particular case in which all of them possessed that velocity as their own velocity, without more). Maxwell, on the other hand, intuitively understood that, in order to characterize this steady state, it was not enough to assign a mean velocity, but that it was also necessary for the number of molecules endowed with each of the innumerable velocities present to remain unchanged. This does not at all mean that every molecule must preserve its own velocity unchanged, but rather that the molecules which, in a short interval of time, lose that velocity as a result of collisions must be as numerous as those which, for the same reason, acquire II.Two important questions thus arose: 1) what was the particular distribution of velocities corresponding to the steady 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 that he had demonstrated that, in the steady state, the number of molecules having velocity components u, v, w (along Cartesian axes) is given by an expression of the form a e^{-b(u²+v²+w²)}, where ‘a’ and ‘b’ are positive constants dependent on the physical state of the gas.
But Maxwell himself recognized the defects in his demonstration and, in 1866, proposed a second one, based on the law governing the collisions of elastic bodies. He deduced the statistical condition that the number of molecules abandoning a certain velocity during each brief interval of time should equal the number of those acquiring it, and he rediscovered the preceding law, which from then on was called Maxwell’s law of distribution.
In 1870, however, L. Boltzmann examined Maxwell’s demonstration in great depth and first proved that the law is not only sufficient but necessary for the steady state; he then sought, and believed he had demonstrated, what experience confirms: that if the steady state of a gas is disturbed in any manner, the gas, left to itself, spontaneously returns to the steady state. Maxwell’s law would therefore represent the limiting distribution toward which every velocity distribution different from it tends.
IV. LOSCHMIDT’S OBJECTION AND BOLTZMANN’S PROBABILISTIC INTERPRETATION
Loschmidt observed (1876) that the irreversible transition from an arbitrary distribution to the Maxwellian one could not follow from the reversible laws of dynamics. Consequently, Boltzmann came to the conclusion that this transition was not rigorously necessary at every moment, but only extremely probable. Thus the stationarity of Maxwell’s law would not be a rigorous necessity, but only a requirement of extreme probability. Among the infinitely many possible distributions, those of the Maxwellian type represent the overwhelming majority; therefore, even if the g. happens to pass through all possible states, it will practically always appear in the Maxwellian distribution.Once the necessity of resorting to probabilistic considerations had been recognized, Maxwell’s law was derived more directly by means of them. The conclusion, however, always remains the same.
At this point it is very interesting to appreciate the extremely high values of the probabilities involved, which are practically equivalent to the greatest certainties. Attempts are made to illustrate them with intuitive examples, for instance by supposing that a monkey is placed at a typewriter and that all the letters and spaces are continuously recorded. What probability is there that the monkey will type a page of the Divina Commedia? The case cannot theoretically be ruled out, although it is practically impossible. Well, the probability that a g. contained in a one-litre vessel will spontaneously assume a non-Maxwellian distribution for one second is of the same order!