37*. Free Path of a Particle with a Given Speed

The values of the molecular free paths calculated in the preceding article are of the nature of average values, since they are deduced from the mean value of the speed and from that of the collision-frequency г. We may therefore, for instance, look on the value

found for a simple gas, as a mean value of the paths which are traversed by the whole lot of particles moving with different speeds. But it is in no way to be considered as the probable or mean length of free path which any one single particle, moving with a particular speed, passes over without a collision.

To find the probability that any particle moving with speed w traverses a path of length x (or rather of a length between x and x+dx) between successive collisions, we go back to the formulæ of § 26, which give

e-11 dx/l

for this probability, I being the mean length of the paths traversed by the molecules which move with the speed w. Since a particle

with speed collides on an average B times in unit time with other particles, where B has the value given in § 32*, the path travelled in unit time is

Bl=w.

Thus the mean free path of the particle with speed w is

l = w/B,

and the probability of a length x being traversed with speed w without disturbance, and for a collision to occur at its extremity, is

(B/w)e-B/dx;

also the probability of the particle's traversing a path which exceeds the limit x is

e- Bx/w

So as to show more clearly how the probability and mean free path depend on the speed w, in accordance with the above formulæ, I have calculated a few values of B/T and the corresponding values of l/L (I/B) (w/2) from the formula

and these I subjoin in the following table ' :

The first, third, and fourth columns contain the ratios of the actual speed to the mean speed, of the probable collisionfrequency B to its mean value г, and of the probable free path l to the mean free path L. The values of these ratios all increase together, as is shown by the curves on p. 430, which are plotted from the given numbers, the first representing the collisionfrequencies, the second the free paths, as functions of the speed. It is worth while to notice that the equalities w = =Q, B =г, and l = L do not occur simultaneously; but the mean collisionfrequency is that of a particle which moves with a speed somewhat greater than the mean speed, and the mean free path is attained by a particle whose speed is rather less than the mean speed.

38*. Different Mean Values of the Free Path

In addition to the mean value of the molecular free path already calculated in § 33*, we can, by a slightly different calculation founded on these last considerations, deduce another mean value which is important for the development of our theory.

1 [Here kmQ2 = 4/π = 1·27324.—TR.].

Of the N molecules contained in unit volume, Maxwell's law gives

4π- N(km) 13 e-kmw3 w2dw

as the number which move with a speed or with a speed differing infinitely little from w. According to the preceding article these B/r

particles attain on the average a free path w/B. If therefore 7: = N is a number sufficiently great, the sum of all the paths which

are traversed between successive collisions by the particles in question is

4π N(km)le-km2 w2dw = 4π N(km) e ́

– km2 (w3/B)dw.

The sum, therefore, of all the paths which all the N particles traverse in a straight line, i.e. between successive collisions, is given by the integral

4x − 3N(lm)} [® le ̄ kmw* w2dw = 4r ̄3N(km)3 [® dw(w3/B)e-km",

0

0

From this total length of the paths of all the particles we obtain their average length, which we shall express by M(1), by dividing by the number of particles N, viz.

The mean value given by this formula is thus expressible as the arithmetic mean of all the values of the free path l at any moment for the whole number N of the molecules contained in unit volume. We may thus take all the N particles as starting at a given moment, each with its speed w, and then determine the mean value of the lengths of the paths attained at this single

start.

We must distinguish this mean value from that which we obtain by considering the paths traversed by the particles in the course of a prolonged time. To find the mean in this other case we have to consider not only a single path traversed by any particle, but the whole of the B paths which it passes over backwards and forwards in the unit time. The sum of all the paths traversed in unit time is therefore given by the integral

- ' N (km)} [" dw w2Ble¬kmw2 = 4x ̄3N(km)} ["* dw w3c¬kmw”,

4π-'N(km) }

which is at once integrable, and leads to the value

But, according to § 32*, the number of these paths is.

Consequently the mean value which we obtain from this other consideration is the value already obtained,

L=2/г.

This mean value L is greater than the other M(1), since, in the summation of the B free paths of any particle, the larger values l of the faster particles come more into account than when only one path for each particle is considered, as B too increases with the speed w.

We must give up the idea of calculating with exactness the mean value M(1), by reason of the complicated form of the function B. But we can obtain an approximate value for M(1) by a tolerably simple calculation if we substitute for B its approximate value B = T√(} + km2)

as given in § 32*. If, then, we put

and from this we may calculate the mean value. By the help of tables of this integral we find

M(1) = 0·937 L.

This value is certainly less than L, but we must still remember that it is only approximate. For we have put too large a value for B, and have consequently got too small a value for M(1). For the values of w, which occur the most frequently, the error in the approximation to B is about 2 per cent., and thus the factor 0.937 is too small by this amount. The mean value M(1) is therefore about 4 per cent. smaller than L.

39*. Interval between Two Collisions

In calculating the average interval between two successive collisions of a particle with others, we can arrive at two different

1 Bessel, Fundamenta Astronomia, 1818, p. 36. Encke, Astronomisches Jahrbuch für 1834.