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itself during the time in which the molecule wanders past it, and therefore comes more easily into the condition of colliding with it. The probability is increased in the same measure as the relative velocity of the two moving particles with respect to each other is greater than that of the particle moving alone. In the case supposed the magnitude of the relative velocity is easily obtained; the absolute velocities of the two particles (since they are perpendicular to each other) form the shortest pair of sides of a rightangled triangle, the hypotenuse of which represents the relative velocity of each with respect to the other. If these absolute velocities are represented by N, and N, as hitherto, the magnitude of the relative velocity is

√(Q,2 + N22).

2

This consideration may be directly extended to the more general case such as really occurs. If the molecules do not move only at right angles to the particles coming in among them, but hither and thither in all possible directions, the probability of a collision is increased by this motion in exactly the same measure as by that to-and-fro motion which we have hitherto assumed. We have, therefore, in this case too, nothing further to alter than to substitute the above expression for the relative velocity instead of the absolute velocity of the particles which throng into the medium at rest.

This consideration, which Maxwell1 seems to have employed several times, puts us now in a position to find an expression for the free path not only for a homogeneous gas, but also for the case before us of a molecule of one gas moving in a different gas.

The probable number of collisions experienced by a particle in unit time is equal to the mean number of the other particles which come within the range of its sphere of action during this time as it moves along. The path of the particle in unit time is measured by its velocity, for which, in the case in which all the particles are in motion, we must substitute the expression for the relative velocity

Phil. Mag. [4] xix. p. 28; Scientific Papers, i. p. 387.

just given. While the particle traverses this path its sphere of action moves over a cylindrical path of equal length, the section of which is equal to that of the sphere of action, viz. To2, In this volume, whose magnitude is

there are

πσον (Ω, + Ω),

Νηπσον (Ω, + Ω)

2

molecules of the second kind, if N, denotes the number of these molecules in unit volume; and the number of the collisions that ensue is just the same. If we divide the whole path, passed over in unit time by the molecule by this number of collisions, we obtain the mean free path of a particle of the first kind amid a crowd of particles of the second kind for which we are looking, viz.

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In the same way the free path of a molecule of the second kind in a medium consisting of molecules of the first kind is

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In the particular case of the molecules being all of one kind, these expressions turn into the value of the free path already given in § 68 on Maxwell's theory, viz.

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This comparison of the general formula with the special one already known shows the mechanical meaning of the numerical factor √2, which was not explained before.

97. Molecular Free Path in a Gaseous Mixture

By the help of these formulæ it is now easy to write down the value of the mean free path for the case first mentioned in § 96, where a molecule moves in a mixture of molecules, some of which are of the same and some of a different kind.

If there are N, particles of the first kind and N, particles of the second in unit volume, we see at once from the formulæ we have established that the whole number of collisions which a particle of the first gas undergoes in unit time is given by the sum

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represents the whole number of collisions which a molecule of the second gas experiences in unit time in the mixture. Hence for the mean free path of a particle of the first kind we obtain the value

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and similarly for a particle of the second gas,

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Both values are dependent on the numbers N, and N,, and are therefore variable with the ratio of the amounts of the two gaseous components in the mixture.

98. Coefficient of Diffusion

If we insert these values of the two free paths 2, and L2 in the formula of § 95, viz.

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we obtain the value of the coefficient of diffusion D, of which we said in the introductory explanations of § 93 that it possesses the same meaning for the process of diffusion as the conductivity does for the propagation of heat. If this analogy were allowed to be perfect we should expect that, just as the conductivity is a constant magnitude, so too is the coefficient of diffusion, which will always keep the same value in all experiments made with the same pair of gases.

But this expectation is not justified by our formula.1

See also Tait, Trans. Roy. Soc. Edin. 1887, xxxiii. p. 266; Phil. Mag. [5] xxiii. p. 141.

For by substitution we obtain for the value of the coefficient of diffusion

D= {π(N2Q ̧2/г1 + N1Q22/г)/N,

wherein the magnitudes I have the meanings

г1 = πs12Ñ1Q1√2 + πo2Ñ¿√(Q,2 + N,2)

1

г2 = πs22Ñ2Q√2 + πo2Ñ‚√(Q,2 + N‚2)

given before. This value is by no means constant; for it depends not only on the temperature by reason of N, and N, and on the pressure of the mixture by reason of N, but it varies also with N, and N2, and thus with the ratio in which the two gases have already mixed with each other.

The dependence of the coefficient of diffusion on the temperature and pressure of the gaseous mixture can raise no doubt. The coefficient, however, is to be looked on as a constant in this respect, that it has the same value everywhere throughout the space filled by the gas in an experiment on diffusion, provided that the temperature is kept constant; for the pressure equalises itself everywhere if the diffusion proceeds without a separating partition.

It is also in agreement with experiment if the coefficient of diffusion varies in value with the temperature and total pressure of the gaseous mixture. Loschmidt observed that the value of the coefficient was found to be the greater the higher the temperature in the experiment; and this is shown by the formula too. From the formula also it follows that the value of the coefficient of diffusion alters with the total pressure of the mixture in such wise that D is inversely proportional to this pressure, because the number N of the molecules contained in unit volume varies directly as the pressure. This too is in perfect agreement with Loschmidt's observations.

But our theoretical formula exhibits still another variation of the coefficient of diffusion, which we should not a priori expect. The value of D is dependent also on the values of N, and N2, and these numbers, which express how many molecules of the one or other kind there are in unit volume of the mixture, alter on their side with x, as in § 95, and therefore with the position in the tube. The coefficient

D therefore assumes a different value at every different place in the mixture that is being formed by the diffusion; its values consequently alter with the rate of fall of pressure of each of the two gases, so that for the same reason the coefficient is variable with the time during the whole period of the experiment. So the observed values of the coefficient D will not be able to exhibit any perfect agreement together if the observations are made for unequal values of the ratio of mixture N, N2

The theoretical calculation of an experiment on diffusion therefore becomes very complicated, and so excessively laborious that we easily comprehend why the foregoing formulæ, which were deduced in the first edition of this book, have scarcely at all been used up to now. People have for the most part preferred to employ less exact but more convenient formula, which give a constant value for the coefficient of diffusion.

99. Another Theory of Diffusion

The theory leads to a constant value of the coefficient of diffusion if a somewhat different fundamental assumption is made as to the cause of the slowness with which the mixture of the two gases proceeds. There can be no doubt but that this cause must be sought in the mutual encounters of the particles, which prevent a forward path in a straight line. But the question may be raised whether the two kinds of molecules take part in these processes in the same way and in equal measure.

It does not seem improbable to assert that the encounters between molecules of the same kind have little influence on the velocity of the current with which each of the two gases flows against the other; that just as often as a molecule loses velocity by an encounter with another of the same kind, it happens that it gains velocity in the direction of the flow. Quite otherwise, on the contrary, is the case with encounters between particles of different kinds. Since both gases are streaming in different directions, the final result of the encounters between particles of different kinds must

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