If we denote by a the probability that this molecule will traverse a path equal to 1 unhindered, a is a proper fraction which, from the assumption made, is so far of constant magnitude that for every position of the starting-point it has one and the same value. If the gas as a whole has no motion of translation, the value of a is also the same for every direction in which the molecule considered can move.

It therefore follows that the probability of traversing a path equal to 2, that is, the path 1 twice over, is a.a or a2. So, too, the probability of its traversing without collision a path three times as long is a3; and we thus see that in general the probability of an unhindered passage through a length x is given by the function

so that, as a is a proper fraction and thus log a negative, l is positive.

This formula agrees in form and meaning with the expression established in the elementary theory (§ 66), viz.

in which q denotes the ratio of the path traversed to the mean free path. We can also now easily see that the constant means nothing else than the mean probable value of the molecular free path which the molecule considered can attain.

For out of n molecules which move in the same way as the given molecule, that is, with the same velocity and in the same direction, the number

traverse the length x without collision, but only

ne

(x + dx)/l

pass over the length x + dx; hence in the length dr

molecules undergo collision from among those that have traversed the path x. The sum of all the paths traversed by these molecules

and hence as each particle must certainly collide after traversing some distance between the limits x = 0 and x = ∞, the sum of the paths traversed by all the n molecules before collision is

Thus the mean value of these n free paths is l.

This mean probable value of the free path is to be understood as corresponding only to particles that move with a certain definite velocity, since we assumed the same motion for all the n particles; it is therefore denoted by l, so as to be different from the symbol L used in § 65. In addition to altering with the speed of the particle, may in general depend also on position, time, and direction, if the molecular motion of the gas alters with these magnitudes.

27*. Probability of an Encounter

Before we determine the value of the free path 7 for a particle of a real gas, let us solve, by Clausius' method, a preliminary problem.

Into a space filled with molecules at rest, of which n are contained in each unit of volume, let a molecule enter with the velocity w. What is the probability that this molecule may in a given interval t, say the unit of time, collide with one of those at rest, the radius of the sphere of action being s?

In the time t the molecule traverses the length wt; its sphere of action therefore moves through the volume swt. Since in this space there are answt molecules at rest, the probable number of encounters which the molecule meets with in the interval t is also

πns2wt,

and the probable number of encounters in unit time is therefore given by the product

πηςω,

the value of which may also be interpreted as the probability of an encounter in unit time.

To this simple problem another, which better corresponds to reality, may be reduced.

Suppose a multitude of particles in motion, and all with the same velocity in the same direction, so that all the particles have the same velocity-components U, V, W; assume further that the particles fill the space with equal density on the average, and that there are n of these particles per unit volume. Into this swarm let another, or even a number of other particles, enter, which move with a different velocity in a different direction; let the velocity of this second group when resolved in the same three directions have the components u, v, w. We have to find the probability of an encounter, and the probable time that elapses before an encounter occurs.

The probability of an encounter in this case is the same as if, instead of allowing both systems to move in two different directions, we had, more simply, assumed that the one swarm was at rest and the other moved relatively to it with the relative velocity

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

The probability, therefore, that a given particle of the one system should collide with any particle of the other in the unit of time is to be represented by the same formula as before when for the absolute velocity the relative velocity r is substituted. Thus the probability sought is

πns2r.

28*. Number of Encounters

From this simple formula we obtain that which holds for the case of a real gas by simply finding the mean value of the relative velocity of two of its molecules. In this calculation we first of all assume that all the particles are moving with the same speed. This assumption is certainly not quite true, as we know from our former investigations; since, however, it has shown itself very serviceable in the calculation of the pressure and in other problems, we may here, too, expect by its help to obtain formulæ that are approximately correct.

If, as before, we denote by G the velocity which all the

molecules possess, the components into which the velocity of any particle can be resolved are expressed by the formulæ

in which s and denote two angles which determine the direction of the motion. By formulæ of the same kind we may express the velocity-components u, v, w of the particle whose collisions with others we wish to count; but these formulæ are substantially simplified if we so choose the system of coordinates that one of the three axes coincides with the direction of motion of this molecule. We may therefore put

u = G, v = 0, w = 0,

and the relative velocity of this particle with respect to the other taken is

r = G√(2 − 2 cos s)

= 2G sin s.

On substitution, then, we get

πns2r = 2′′ns2G sin s,

and this magnitude denotes the number of particles with which in unit time any particle so collides that the directions of motion of the colliding particles make the angle s with each other.

In order to calculate the total number of collisions which a particle suffers in the unit of time we have to take the sum of the values of the above expression for all values of the angles s. It is therefore necessary to know how great is the number n of the particles for which the angle of encounter with the particle considered has the value s, or, better expressed, a value differing infinitely little from s, so as to lie between s and s + ds as its limits. We find this number by making use of the property of heat-motion, that it goes on in the same way in all directions without distinction, so that equal numbers of particles move in every direction.

Consider all the particles with their directions of motion to be so displaced the latter parallel to themselves-that all move towards the colliding molecule, which for the instant is considered at rest; then the paths of all the particles which make an angle between s and s+ds with the colliding particle fall in the space included between two infinitely close cones whose vertices lie on the colliding particle and whose axes coincide with the direction

of its motion. The number of particles with a given direction, therefore, when all directions occur equally, bears the same ratio to the whole number of particles as the surface of the zone intercepted between these two cones on a sphere constructed with the colliding molecule as centre bears to the whole surface of the sphere, viz.

2 sin s ds: 4π.

The number, therefore, n of the particles in the unit of volume which move in the direction defined by the angle s is

n = N sin s ds,

where N is the whole number contained in unit volume.

It is now easy to find, in the way required above, the total number of the collisions. Since the angle s can increase from 0° to 180°, the value of this sum is

A =

π

= #s2NG(" sin }s sin s ds = 2-s2NG (′′ sin2 }s cos §s ds ;

and the evaluation of this integral gives the value

A=TS2NG

for the number of collisions which a particle undergoes in unit time in a large group of other similar particles, when all the particles have the same velocity G, and there are on the average N particles in unit volume.

Compare this number with that first found

which holds for the case of a particle when it moves with the speed among a multitude of particles at rest, of which there are n in the unit of volume. If we assume the speed and the number of particles to be the same in both cases, or ∞ = G and n = N, we see that the number of collisions denoted by A is greater than the other in the ratio 4: 3. A gaseous particle, therefore, as Clausius1 first perceived, meets with others more frequently when they are all in motion than when one only is in motion and the others are at rest.

Inversely, the mean length of the straight path which a particle traverses between two successive collisions is smaller in Phil. Mag. [4] xix. 1860, p. 434; Abhandl. über die mech. Wärmetheorie, 2. Abth. 1867, Note on p. 265.