460
DIFFUSION
the smaller planets to retain only the denser gases whose molecular velocity at a given temperature is least. Yet the present writer’s calculations indicate that the molecular velocities assigned by the kinetic theory are insufficient in themselves to remove helium from our atmosphere, a point discussed by Stoney in 1897 and 1900. Very similar in principle is the explanation of dissociation. If we assume the molecules of a compound gas to be made up of atoms which are bound together by their mutual attraction, and adopt Boltzmann’s hypothesis that the heat-energy of a molecule is due partly to translation and partly to rotation, it is evident that with increase of temperature there will be an increasing number of molecules, in which the “centrifugal force” due to rotation exceeds the force of attraction between the atoms, causing these atoms to break asunder. The kinetic theory of dissociation is treated at some length in Boltzmann’s book. The distribution of velocity among the molecules of a gas, worked out by Watson and Burbury in their article Boltzmann- Molechue (Ency. Brit., 9th ed., xvi. 612), is Maxwell called the Boltzmann-Maxwell distribution, and distribu- when it exists the gas is said by Tait to be ti°n. in the “special state.” There is abundant evidence that this distribution holds good in any gas in which (a) the actions between the molecules resemble the collisions of elastic bodies; (6) the sum of the volumes of the individual molecules is very small compared with the volume of the gas. Much doubt still exists as to how far the distribution is applicable to gases whose molecules are of appreciable volume, or repel one another with finite forces which are functions of the distances between them. Burbury is of opinion that in such cases the motions of the molecules tend to become correlated, by which he means that two neighbouring molecules are more likely to move in the same than in opposite directions. For monatomic molecules (material particles or smooth spheres), he finds that the probability of the velocity components {u^v^w-d, lying within the limits of the multiple differential dvridv-[dw1. du^dv^vj^... is proportional to where Q= + Vj2 + Wi) + S2612(w1m2 + ^2 + wiw'z) 6j2 being a negative function of the distance between the two molecules designated by the suffixes 1, 2, which is inappreciable except when this distance is small. The motion changes its character when Q ceases to be essentially positive, and this change may possibly be the condition for liquefaction. If the b12 coefficients vanish we have the Boltzmann-Maxwell distribution. The special case of the Boltzmann-Maxwell distribution for molecules regarded as non-spherical, elastic, rigid bodies, and capable of having angular velocities oj2, w3 about their principal axis of inertia, besides translatory velocities v, v, w of their centre of gravity, is interesting. The kinetic energy is given by T= + v2 + vF) + 5 (Aon2 + Bo)22 + Cw32), and the distribution being given by the expression du dv dw t/ojpiw./foj.j, it follows that the mean values of ^mv1, lAco,2, 1Bo)22, 1Cco32 are each equal to 1/A. This is a particular case of a statement known as Maxwell’s Law of Partition of Energy, according to which in certain cases, “ if the kinetic energy of a system be expressed as a sum of squares, the mean values of these squares are equal.” The applicability of this law to dynamical systems in general, and indeed to any systems other than groups of molecules arranged according to the Boltzmann-Maxwell distribution, has for many years been a source of controversy, and was discussed in 1900 by Lord Rayleigh. When C = 0, or the molecule consists of a distribution of masses along a straight line, the energy is equally divided between the five remaining components ; this gives a specific heat ratio of L4 approximating to that of most gases, while the specific heat ratio, on the hypothesis that the energy
OF
GASES
is wholly translational, is If, or nearly that of argon. The tendency of the molecules of a gas to assume the Boltzmann-Maxwell distribution, if their velocities are initially distributed in any other manner, has been investigated by Boltzmann and Watson in a theorem known as Boltzmann’s Minimum Theorem. Let fdv denote the probability that the co-ordinates and momenta of a molecule shall lie within the limits of the multiple differential dv{ = dp1,...dqn of Watson and Bur bury’s article Molecule). Let H denote the integral ff log fdv, or for a mixture of gases let H = 2// log fdv, the summation extending over the several components. Then it is proved in the Minimum Theorem that, as the result of intermolecular collisions or encounters, H tends to decrease until the “ special state” is reached, when it becomes a minimum. This proposition is further shown by Boltzmann to admit of both mathematical and physical interpretations. (1) According to the theory of probability, Boltzmann finds that if W is proportional to the probability that the distribution of co-ordinates and momenta may be defined by the function f, then log W is proportional to -H. Hence, as H decreases, W increases ; in other words [a) the molecular motions tend to pass from less probable to more probable distributions, and (6) the Boltzmann-Maxwell distribution is the most probable of all distributions. (2) The entropy of the gas is proportional to - H + a constant, and the tendency of H to decrease to a minimum thus represents the physical property that the entropy of a system tends to increase to a maximum. When the molecules of a gas are thoroughly mixed (as assumed in the Minimum Theorem), it appears, from the calculations of Tait and Watson, that a very small fraction of a second is sufficient, in ordinary cases, to restore the molecules to the Boltzmann-Maxwell distribution. We now have to consider diffusion and allied phenomena in which the distribution, instead of being uni- Diffusion‘ form, varies at different points of the gas. In such cases the process of equalization, which is comparatively slow, depends on the free paths of the molecules between collisions. We proceed to investigate the general problem by a method based on the work of Boltzmann and used subsequently by Burbury. If a molecule A moving with velocity w impinge normally on a stratum of gas of thickness x containing n molecules per unit volume, whose velocities are distributed about a mean velocity q, it is easy to see that, if nx is small, the probability of its encountering another molecule in traversing the layer is proportional to mrs2x, where s is the sum of the radii of the impinging and interfering molecules when these are spherical, and s is determined by the linear dimensions of the molecules in other cases. Since, moreover, the probability is evidently unaltered by increasing the velocities of the impinging2 and interfering molecules in the same ratio it may be written mrs xf/(wlq), where ^(w/q) is a function, calculated by Meyer, Tait, and others. If nx is not small we divide the nx molecules into a large number r of equal groups, and the chance of A escaping collision is the product of its chances of escaping collision with the groups, and therefore Lf, nxws2 , (wV -xnnsmw/q) -x/l Say r=o=!1 —^{qjj-6 ~6 ' Here l is the constant defined in the article Molecule {Ency. Brit. vol. xvi. p. 616). The mean free path may be found as in that article. Tait, on the other hand, defines the mean free path as the mean value of l, when the molecule A is replaced by a number of molecules in the “special state,” and the value he finds fora simple gas is OTIT/mvs2. Now let G be any physical quantity {e.g., mass, charge of electricity, momentum, energy, &c.) wdiich may be carried by a molecule. Let the quantity of G per unit volume vary uniformly in the direction of the axis of z, and be denoted by G(z); also ci f{w)dw denote the proportion of molecules whose translational speeds are between w and w + dw. It is required to find the rate at which G is being carried across the plane 2 = z0. Now the total number of molecules crossing the plane z0 in unit of time, and having their velocities between w and w-tdw,^ in directions at inclinations to the axis of z between 0 and d + dd is = wf{w) sin# cos# dd dw. The number of those that have travelled a distance between X and + d since their last encounter is found by multiplying this number by e~Xlldjl. These molecules must have collided between the planes z = 20 + X cos # and z=20-l-( + d) cos #, and they are assumed to carry their average share of the quantity G between
