31*. Number of Molecular Collisions in a

Current of Gas

To be able to apply these formula to the theory of internal friction we have yet to determine the influence which a forward motion of a gas exerts on the collision-frequency of its molecules. If this motion at all points of the gas is characterised by the same speed and the same direction, the frequency of collisions can neither increase nor diminish. But a perceptible influence may result if layers move near each other with different velocities, as is shown by the experiments made to determine the viscosity. A state of things then arises by the mixing of layers, which we can represent with tolerable accuracy by supposing two masses of gas of the same kind and at the same temperature to be flowing in the same enclosure with unequal speeds.

Consider, therefore, two groups of gaseous molecules in the same vessel, which they fill with unequal densities; they further differ in the unequal speed of their flow, but are otherwise completely alike: Maxwell's law of distribution of speeds therefore holds in both groups in exactly equal fashion, provided that we apply it only to that part of the molecular motion which shows itself as heat, and therefore provided that from the motions of the individual molecules we subtract the progressive motion of the group as a whole. In the formulæ referring to the separate groups we have consequently to introduce the same value, not only for the molecular weight m, but also for the constant k, and this holds, too, for the radius of the sphere of action s. Suppose, further, that the flow has the same direction for both groups, and take this direction to be that of one of the axes of coordinates. Then the number of collisions per unit time of a particle of the first kind, of which there are N, per unit volume, is

T1 = πs2(√2 N1Q + N27),

and that of a particle of the second kind, of which there are N2 per unit volume, is

r = √ {(u, · U2)2 + (v1

2

-

v2)2 + (w,

= km {(u, — a1)2 + v12 + w12 + (u2 — aş)2 + v22 + w22},

where a, and a, denote the speeds of flow of the two groups in the direction of u.

By substitution as before in the expression for y of

4 = 2km {(U — A)2 + V2 + W2} + {}km {(u − a)2 + v2 + w2}.

If we now introduce polar coordinates, the integrations with respect to U, V, W are easily performed, and those with respect to u, v, w partially so, the final shape of the integral being

which by development in powers of a gives

Y = 2√(2/km) e-ma*(1+kma2 + ...);

und this for a = 0 reduces to the known result

r = √20.

If we also develop the exponential function in powers of a we

obtain

y= 2√(2/km) (1 + ¿kma2 + ...),

which shows that the collision-frequency is increased by the flow of the gas by an amount which is of the order of the square of the difference a = a1 .. This difference, or the relative velocity of two neighbouring layers, is in the theory of internal friction always looked upon as very small, and its square as therefore negligible. Here, too, it is a very small magnitude of the order of the molecular free path; and in the formula, which by introduction of the mean speed becomes

we may neglect the correctional term as vanishingly small, and therefore apply to a flowing gas the same formulæ for the

collision-frequency of its molecules as to a gas which possesses no other than its molecular heat-motion.

32*. Collision-frequency of a Particular Molecule

The collision-frequency of a molecule which moves with a given speed may be calculated, but not quite so easily as the mean collision-frequency of all the molecules. To calculate this number B we can, in the case of a single gas composed of exactly similar molecules, make use of the formula obtained in § 29*, viz. :

V W

-00

81

B = =s2N(km/=)'{°_dUf°_dv[_dWre ̄v +r+m,

81

where

r = √ {(U — u)2 + (V − v)2 + (W — w)2} .

Since the velocity of the colliding molecule and its axial components are in general of finite magnitude we may put new variables U+u, V + v, W + w for U, V, W without altering the limits of the integrations; consequently

B = rs2N(km/x)1[°_dUf°_av[°_aw√(U2+V2+W2)e ̄*,

81

where for shortness is put

x = km{(U + u)2 + (V + v)2 + (W + w)2} .

Since there is no distinction as regards direction, we may choose our coordinate system as we like, and, therefore, take the direction, in which the particle under consideration moves with the speed

w = ✓ (u2 + v2 + w2),

as that of one of the axes.

If we further substitute polar coordinates, s, o in place of the Cartesian U, V, W we have

q = km(42 + w2 + 2w¥ cos s).

On integration with respect to s and this becomes

I arrived at this last expression in 1866 in a Latin dissertation, in which, starting with Clausius' formula, I deduced Maxwell's. From this value of B I calculated, by integration, the value of the mean collision-frequency

and, by developing this in a series, obtained the same value

r = 2Ns2√(2π/km) = √2 s2NQ,

which we have already found in a simple way.

The magnitude B denotes the number of collisions which a particle moving with speed w experiences in unit time from an assemblage of N other particles whose mean speed is . Closely allied to this expression is that of another magnitude

which we deduced in § 30*; this represents the number of collisions that occur in unit time between a particle of a group whose mean speed is 2, and the N2 particles of another group with mean speed 2. The chief difference between the two expressions is that a denotes a speed of arbitrary amount, while represents a mean value; but otherwise they are so similar that we might expect the formula

which we have formed from that last given, to represent the number B with at least approximate accuracy.

This expectation is fairly well justified by the comparison of a few values of the exact ratio

Dissertatio de Gasorum Theoria, Vratislaviæ 1866. Also in the first edition of this book, 1877, p. 294.

The following series of figures show that the two expressions agree remarkably well both for small and large values of w, while for middle values of a regular deviation occurs.

The values obtained from the approximate formula are rather too large, but the errors are in all cases less than 2 per cent. The simple approximate formula can therefore very well be used instead of the more complicated exact formula in all calculations when absolute accuracy is not desired.

33*. Molecular Free Path

With each collision a molecule starts on a fresh rectilineal piece of its zigzag path. The number of collisions is therefore the same as that of the straight bits of the path traversed. Consequently we find the mean length of one of these bits by dividing the length of path traversed per unit time (which is measured by the velocity) by the number of collisions experienced per unit time.

Since mean values are taken in this calculation, our first thought is to divide the mean speed by the mean collisionfrequency, and call the quotient the mean free path. In this way we have already obtained in § 28* the value

for the mean free path of the particles of a simple gas, from the assumption, which is approximately true, that the speeds of all the particles are the same. Instead of this value, which Clausius1 gave, Maxwell2 obtained the somewhat smaller value

Phil. Mag. 4, xix. p. 434, 1860; Abhandl. i. d. mech. Wärmetheorie, Abth. 2, note to p. 265; Mech. Wärmetheorie, iii. p. 61, 1889-91.

2 Phil. Mag. 4, xix. p. 28, 1860; Scientific Papers, Cambridge 1890, i. p. 387.