which are essentially different values [being in the ratio (Q1/N2)2 or m2/m1, where m denotes the molecular weight]. But in the conditions that have chiefly been employed in the experiments we can take for the but slightly variable coefficient the value

D=

2 Ω,

+

́8N\o2 √ (N ̧2+N22) + si2N1 √ 2* o2 √(N ̧2+N22)+$22Q2√2} which we find on putting N1 = N2.

101. Dependence on Pressure and Temperature

To test the formula we can also raise the question whether the formula correctly gives the law of dependence of the coefficient of diffusion on both pressure and temperature.

Since both of the mixing gases have the same temperature,

where mi and m2 denote the molecular weights of the two gases. If then we denote by m and 2 the values of the molecular weight and molecular speed for any normal gas which has the same temperature, we have

By means of this relation the formula for the total pressure p = }π(N ̧μ‚Ñ ̧2 + N ̧m¿Q2)

takes the simple form

2

p = }πNmQ2,
ἐπΝΩ,

wherein N = N, + N, signifies, as before, the total number of all the gaseous molecules of both kinds in unit volume. By this formula we may replace N in the formula for D by the pressure p, and we then obtain for the coefficient of diffusion D = 'π(m2)3p ̄1(Y1 ̄1 + Y1⁄2 ̃ ̄1),

where

4

1

-1

y1 = m11{√2s,2N1/N2 + o2√(1 + m1/m2)}
Y1⁄2=m,1{√/2s,2N2/N1 + o2√(1 + m2/m1)}

If we put s, 0 = s,, these general formulæ would turn

into that which would hold if the mutual collisions of particles of the same kind might be left out of account. We need not therefore pursue this improbable assumption, according to which the coefficient of diffusion would have the constant value

any further here, if we obtain an answer to our question from the more general formula.

Since the magnitudes denoted by y do not depend on the pressure p, the equations show directly that according to the theory the coefficient of diffusion is inversely proportional to the total pressure of the two diffusing gases. But this is the very law which Loschmidt deduced in 1870 from his observations, and which has been confirmed by all later experimenters.

Loschmidt has further concluded from his observations that the coefficient of diffusion increases as the temperature rises, and that it increases indeed proportionally to the square of the absolute temperature. The theoretical formula likewise requires an increase of the coefficient with rise of temperature. In the formula p is not variable when we are dealing with experiments that are made at constant temperature and pressure. If the magnitudes y were also independent of the temperature, D would increase proportionally to the power of the absolute temperature (since this on the kinetic theory is proportional to N2), and therefore less than is really the case according to Loschmidt's experiments.

But according to the observations on viscosity the radii and S2 of the spheres of action are dependent on the temperature, either actually or apparently; this is also possible, and, indeed, highly probable, for the newly-introduced magnitude σ, which has a similar meaning. If we may assume that and the magnitudes s are variable, at least approximately, in the same way with the temperature, the coefficient of diffusion D must increase with the temperature, according to the same law as the product of the coefficient

of viscosity, which, according to § 78, is given by the formula

n = 0·30967 mN/πs2√/2,

into the absolute temperature or the magnitude 1+ að, where is the centigrade temperature and a is the coefficient of expansion.

Now, the value of the coefficient of viscosity, according to the observations described in § 85, increases with the temperature in such wise that it may be taken, at least approximately, to be proportional to the function

(1+ad)",

where the exponent n has values which for different gases lie between 0.75 and 1; its value being greater for the easily condensible gases than for those which were formerly considered permanent. We should have then to expect that the coefficient of diffusion varies as

(1+ad)1+",

where n has the value appropriate to the gas.

The observations from which Loschmidt1 has concluded the law put forward by him were made with three pairs of gases diffusing into each other, viz. carbonic acid and air, carbonic acid and hydrogen, hydrogen and oxygen. The value of n for carbonic acid, which occurs in the first two pairs, has been found, from experiments on viscosity, to be nearly equal to 1, so that Loschmidt's observations on the diffusion of this gas are in excellent agreement with the investigations of the viscosity of the same gas. For hydrogen and oxygen the value of n given by the experiments on viscosity is certainly less; but the agreement in Loschmidt's experiments on diffusion is not so great that we are forced to assume the exponent n to be always exactly equal to 1.

In fact von Obermayer2 found later, by experiments

Wien. Sitzungsber. 1870, lxi. Abth. 2, p. 367; lxii. Abth. 2, p. 468. 2 Ibid. 1880, lxxxi. Abth. 2, p. 1102.

on diffusion, that the exponent has different values for different gases, and his observations give for

and these values so far agree with those which the same observer obtained for the dependence of the coefficient of viscosity on the temperature, viz.

that the exponent for the diffusion in the case of any pair lies, with one exception, between the exponents for the viscosity of the two gases concerned.

From these experiments we cannot doubt that the variability of the two phenomena, viscosity and diffusion, with the temperature is to be referred to exactly the same causes. Both coefficients, that of viscosity and that of diffusion, depend on the temperature, not only because their formulæ contain the molecular speed , but also in so far as the radiis and σ of the spheres of action are variable with the temperature. The only difference in their variability with the temperature consists in that for diffusion being greater than that for viscosity in the ratio of 1a9 to 1.

We may further conclude that the change of both magnitudes and with the temperature is effected by forces that are equal or, at least, of like kind. Both magnitudes are radii of spheres of action; s refers to the mutual

1 Wien. Sitzungsber. 1876, lxxiii. Abth. 2, p. 433; Carl's Repert. 1877, xiii. p. 130. Compare § 85.

action of similar molecules, while denotes the distance apart of dissimilar molecules when they collide and, therefore, the radius of this sphere for the action of dissimilar particles on each other. If, now, as we saw in our discussion of viscosity, the diminution of the radius s of the sphere of action with rise of temperature is only apparent, and is to be referred to forces of cohesion, as with Sutherland's explanation, we shall have to assume, in the same way, that the magnitude σ also appears to diminish with rise of temperature only because the forces of cohesion, or, more accurately, the attractive forces exerted by dissimilar molecules on each other, heighten the probability of an encounter, and, indeed, increase it the more strongly the less the speed of the molecular motion or the lower the temperature. Accordingly the law of dependence of the coefficient of diffusion on the temperature must be represented by a formula which must be formed quite similarly to that found for viscosity (§ 85), viz.

n = n(1 + a9)1 (1 + aC) / (1 + C/O);

we shall therefore have to put

D = Do(1 + a9)3 (1 + aC') / (1 + C' /O),

where a represents the coefficient of expansion, and the absolute temperature

℗ = I + a1 = 9+ 272·5,

while D stands for the value of D at 0° C., and C' is a constant which represents the measure of the cohesion between dissimilar particles, as does C for similar particles.

We may omit a detailed comparison of this theoretical formula with the observations; for, since Sutherland's formula has been proved for viscosity, the agreement of the values of the exponent n now shows that the formula for the diffusion can represent the results of experiment in the same excellent way. It is therefore sufficient to give the values.