to it. For, owing to that conflict of conceptions, a new and interesting side of our theory has been brought forward which is worth a searching investigation. As in Part I. of this book, the speed of the molecular motion and the supply of energy therein contained were considered and calculated, so there remain for this Part II. the investigation of the character of the motion, the determination of the length of the path of a molecule between collisions, and the development of the consequences which will result from our knowledge of these free paths in respect of the different properties and phenomena that have been observed in gases.

63. Probability of Molecular Collisions

The pioneer investigation in which Clausius1 opened out this new wide field cannot, for the attainment of its first aim, viz. the determination of the length of the molecular path, dispense with mathematical expedients. As in Part I., we need again the calculus of probabilities in order to investigate processes which are conditioned by no other law than that of chance. But the demand on mathematical means may be more restricted than it was in the original memoir. It is not necessary to use the higher analysis, and the aid of elementary mathematics is sufficient; the following calculation presupposes no further mathematical knowledge than is needed for the calculation of compound interest.

Before we solve the actual question and determine the probable length of the straight path traversed by a molecule between two collisions, we follow Clausius' procedure in investigating a preparatory and more simple problem. Instead of the actual behaviour of the gaseous molecules, which are all moving about hither and thither, let us imagine the simpler case in which one particle (or a certain number of particles, but all with the same speed

1 Pogg. Ann. 1858, cv. p. 239; Abhandl. u. Wärmetheorie, 2. Abth. 1867, p. 260; Mech. Wärmetheorie, iii. 1889–91, p. 55; transl. Phil. Mag. [4] xvii. 1859, p. 81.

and in the same direction) is thrown into a space which is filled irregularly with molecules at rest, their distribution, however, being such that the density is the same everywhere.

For the solution of the question bound up with this idea, viz. What path will a particle so thrown in probably traverse without a collision? it is advisable to determine the density of distribution of the particles at rest and express it in terms of their mutual distances apart. If there are N molecules in unit volume, then, considering the volume of this unit to be divided into N equal parts, in fact into N small cubes, we have in each of these small cubes a space which contains on the average only a single molecule. If we denote by the letter λ the edge of one of these elemental cubes, which Clausius calls the mean distance between neighbouring molecules, the volume of one of the cubes is X3, and the relation

holds good.

Nx3 = 1

Since the density p may be expressed in terms of the molecular weight m and the number N by the formula (§ 13)

p = Nm,

the former formula shows that the density is related to the distance between neighbouring molecules by the equation

ρλ = m.

From the mean distance λ between neighbouring molecules Clausius deduces the mean probable length of free path by comparison of that mean distance with the smallest possible distance of separation, i.e. with the distance apart of their mean points or centres of gravity at a collision, and of the volume 3 of the elemental cube with the space which the moving particle must at least have for its motion.

If, on the collision of two particles, it happens that they come into actual contact, the least possible distance apart of their centres would be the diameter of either, if we could look upon the molecules as being spheres of equal size; if the molecules have any other shape, the calculus

of probabilities allows us to take for this distance the mean diameter of a molecule. But it is very conceivable that two molecules cannot come so near each other as to actually touch, but that they are repelled from each other, without actual contact occurring, by forces that come into play at certain, though very small, distances. On account of this possibility it is better that, as already suggested in § 44, we should not put the smallest distance apart of two molecules during a collision as absolutely equal to their diameter. With Clausius, we suppose each molecule to be surrounded by a spherical envelope which is called the sphere of action, meaning thereby that the mean point or centre of gravity of another molecule cannot penetrate into it. The radius of this sphere is thus equal to the smallest distance apart of the centres of the particles at the moment of a collision.

By introducing this conception we allow the possibility of the molecules exerting forces upon each other of sufficient strength to prevent actual contact and to cause mutual rebound from each other; we do not, however, thereby, on the other hand, bring in this hypothesis as necessary, as it still remains open to us to assume actual contact on collision; in the latter case we should have to define the sphere of action as eight times the volume of a molecule, and we might call the actual space occupied by a molecule its molecular sphere.

Denoting the radius of the sphere of action by s, and, therefore, the area of its central section by s2, we find that if the moving particle considered advances by the mean distance between neighbouring molecules, its anterior convex surface traverses a cylindrical space bounded by hemispherical ends, the anterior convex and the posterior concave, of volume equal to Ts. Since there is on the average only a single molecule in a volume equal to λ3, the probability that there is a molecule in the cylinder s described is as much smaller than 1 as s2 is less than λ3. The probability, therefore, that the particle moved strikes another as it passes over a path of length λ is determined by the ratio

or by the ratio of the central section of the sphere of action to the face of the elemental cube.

On the other hand, the probability that the particle does not undergo collision in its path λ, but passes through a layer of thickness λ without colliding with the other particles within it, is

We may therefore, to use an ordinary expression, bet πs2/(\2 — πs2) to 1 that the particle will undergo collision before it passes over a distance λ, and (x2 - πs2)/πs2 to 1 that it will not collide in this distance.

In these formula there lies a simple meaning. If we suppose that the molecule struck were pushed from the interior of the elemental cube containing it into the same face through which the moving particle entered, it would cut out of this face, whose area is λ2, a portion equal to TTS2, through which the entering molecule would not be free to pass; only the remainder λ2 — πs2 would allow free entrance for the molecule. The two probabilities therefore have, as indeed they must have, the same ratio as the not-free part to the free part of the face of the elemental cube.

64. Probability of a Longer Path being Traversed

From this value of the probability for the traversing of a path of length λ, or for the passage without collision through a gaseous layer of a thickness equal to that separating two neighbouring molecules, may be easily calculated the probability for the passage through a thicker layer or for a path of finite magnitude.

For this we suppose M moving particles, instead of one, to be simultaneously projected into the medium, consisting of particles at rest, which we suppose to be divided into a number of layers of thickness λ. Of these M particles Mπs2/2 will probably undergo collision in the first layer, while only the remainder M(1 - Ts2/2) will pass through it unhindered. Of these the number that collide in the second layer of the same thickness λ is M(1-πs2X2)πS2X2,

while M(1πs2/2)2 particles pass through this layer. So in the third layer M(1πs2/X2) 2πs2/2 collide, and M(1πs2/X2)3 pass through. Proceeding in this manner we see that in the xth layer there are probably

that collide, and thus have traversed a path of length xλ, while the remainder M(1 - πs2/λ2) do not probably suffer collision, and therefore traverse still longer paths.1

From this we at once obtain the probability that a single molecule will pass over a given path and then collide, by dividing the probability in the case of M particles by the number M, since the probability for one particle must be M times less than for M particles. The probability, therefore, that a moving particle traverses a path xλ and collides on its completion is

65. Calculation of the Mean Free Path under Simplified Assumptions

From the foregoing formulæ we can calculate by elementary methods and without great difficulty the probable mean value of the lengths of the paths traversed by all the particles. To find this it is only requisite to calculate the sum of all the different paths traversed by the M molecules and to divide it by their number, that is, by M.

Of the M particles there remain Ms2/X2 in the first layer, and these therefore traverse only the path λ; thus the sum of the paths traversed by these molecules is Mπs2/λ. Similarly, there remain in the second layer, after completing the path 2x, the number M(1 - πs2/λ2)πs2/λ2, the sum of whose paths is therefore 2M(1 - πs2/λ2)πs2/λ. In this way we find in general that the sum of the paths of the particles which collide in the xth layer is xM(1-πs2/λ2)-12/λ, and the total sum of the paths traversed by the whole of the M particles is therefore

This formula is developed in § 26* in a mathematically simpler form.