MOLECULE 617 Hence the chance for such a molecule of free path between x and x + dx is B -? e u with the above definition of B. The chance of a molecule whose velocity is u having free path x is of course the same as the chance of its free path having the duration . If t = , the chance of duration between t and t + dt <j) U is thus ft e- Bt udt; or Be~ Bt dt. u Meyer determines the value of B, if the molecules be spheres, in the form B = 11 yi-lj 1_ 1 1 71 2/i - 1 2n + 1 J^ _1_ L.2 3.5 35 where ft = ~T== , and s is the sum of the radii of two molecules. It will be observed that the series converges very rapidly if u 2 h is less than unity, the successive coefficients being + 3 30 210 _ __ 1512 11880 Having found B for the number of encounters experienced per unit of time by a molecule having velocity w, we have for the average number of encounters experienced by any molecule per unit of time, which we denote by C, From which Meyer deduces Hence the mean value of the free path for all molecules, irrespec tive of velocity, is L = = C Thus the kinetic theory of gases presents to us the conception of apparently perfect rest, as the result of motion irregular in detail but permanent and stable on the average. Whatever difficulty may be felt at first sight in the acceptance of this theory in the case of a medium at rest is greatly enhanced when we pass to the contemplation of a disturbed medium like a mass of gas through which a wave of sound is passing. In our ordinary investigations of such a disturbance the gas is treated as a continuous body, sub jected to small relative motions of its parts, accompanied by corre sponding variations of internal pressure. When a disturbance or a local condensation or rarefaction is set up in any portion of this gas we calculate the resulting effects by the well-known equations of sound motion. But on this kinetic theory the medium is supposed to consist of a number of discrete masses elastic spheres or the like which preserve the physical properties of the medium merely by the recurrence of their mutual collisions, such collisions obeying no law in individual cases, but preserving a certain average uni formity in the motion of the whole aggregate ; and we need some further investigation to assure ourselves of the applicability of the ordinary treatment of wave motion to such a medium. Now we observe that the physical properties of our medium, so far as the relation between pressure, density, and temperature is concerned, merely require that the temperature be measured by the mean total kinetic energy of translation, and that the mean kinetic energy of translation parallel to any fixed line be equal to one-third of the mean total energy of translation. If the molecules constitut ing any portion of this medium were animated by a common velocity or acceleration, the physical properties of this portion would be similarly determined by the velocities and kinetic energies relative to the common motion. When the distribution of such relative velo cities is stable or permanent, the average relative kinetic energy in any fixed direction is one-third of the average relative total kinetic energy, such property constituting normal distribution. Suppose that in any portion of a medium, consisting of equal elastic spheres, this distribution has been disturbed that is, Swm 2 , Zmv 2 , and 2,mw 2 are unequal. If F"were the relative velo city of any pair of spheres after such disturbance and before they collide, and 6 the angle between V and the common normal at the point of impact, then the normal and tangential relative velocities before impact are Fcos and Fsin 0, and after impact they become - Fcos and Fsin respectively. The relative velocity after im pact, resolved in the direction of relative velocity before impact is therefore - F cos 2 0+ Fsin 2 0, or- Fcos 20; and the chance of being between and + d0 is sin 20 dO. Therefore the average square relative velocity resolved in the original direction becomes after impact r- f* / cos- J i 2 20sin20d0, or . The relative velocity after impact in the plane of F, and the normal perpendicular to the direction of F before impact is F sin cos + F sin cos 0, or F sin 20. And, if a fixed line be taken in the plane perpendicular to F, the average value of the square of the relative velocity after impact, resolved parallel to this line, is IT j/2 r2T r? y Z ^ I I sin 3 20 cos 2 c/> dd d<f>, or as before. "" ./o ./o Hence we conclude that, in whatever manner the distribution is disturbed in any portion of the medium at any instant, it will, for all those pairs of spheres which within any given interval encounter each other, have assumed the normal distribution after that interval. If T denote the average time between two collisions for any given sphere, the chance that this sphere shall continue for any time t t_ free from collisions is, as we have seen, e r . If, therefore, D be the number of spheres within any region whose total relative velocity is between w and w + dw, but so distributed that the mean square of their relative velocities along any fixed line is not , then after a time t considerably greater than T, say ten times r, the number of the D spheres which have escaped col lision will be utterly inconsiderable, and the distribution will have become normal throughout the region. Suppose, for instance, that a sound wave is passing along a tube filled with air, G R P C R ) the air in the tube is, at any instant, in a state of alternate com pression and rarefaction, as at C, R, C, R above. If the note sounded be (say) 500 vibrations per second, the length of the wave OR is about &* feet, and the time taken by the wave in traversing that distance is about r^th of a second. The air in any section of the tube near P has alternately a small positive momentum and an equal small negative momentum, the reversal taking place in every ^th of a second ; also the same cause which produces the average momentum in either case disturbs the distribution of energy among the x, y, and z directions, i.e., it is always producing an excess or defect in mu 2 above or below that of mv 2 and mw 2 . By what has been proved above, this abnormal distribution of energy becomes inappreciable, owing to molecular collisions in a time considerably less than nnnsth of a second in fact, in about ^-o-jnnnniTjth of a second, when the value of T for atmospheric air is considered. It is therefore legitimate, in calcu lating the velocity of sound in air (at least on the elastic sphere hypothesis), to regard the distribution as always normal in any section of the tube, the air in that section or in any elementary portion of it possessing, as a whole, any given velocity or accelera tion, estimated as if we were dealing with a continuous mass. DIFFUSION OF GASES. If any further light is to be thrown on the physical nature of a molecule from investigations, experimental or analytical, concerning gases, it will most probably be by means of experiments on the diffusion of gases, or else on the internal friction or viscosity of gases, and the com parison of these results with those obtained analytically the methods of the kinetic theory. Such investiga tions have been undertaken experimentally by Graham, Loschmidt, Maxwell, O. E. Meyer, and others. An ac- ount of them will be found in O. E. Meyer s work above referred to. The same problems have also been discussed analytically by Maxwell, 1 and by Stefan, O. E. Meyer, and Boltzmann in the treatises referred to below. We pro- eed to give a short account of Meyer s results. Phil. Mag., July 1860, and Feb. and March 1868.
virr -,Q