as follows at once from the formulæ of § 29*, by employing his law of distribution of speeds. 34. Correction of the Formula by allowing for the Dimensions of the Molecules The formula obtained for the free path may be further improved by a correction which van der Waals1 first attempted to apply. The length as given by the formula is somewhat too great, since the calculation of the probability of collision in § 27* was conducted as if the space occupied by the sphere of action of a molecule between two collisions (or the fourfold volume of the path, as in § 69) were equal to a cylinder whose base is a central section of the sphere of action, and whose height is the free path -in a strict calculation we must remember that this space has hemispherical ends. Owing to this circumstance, the free path L which the centre of a molecule traverses between two collisions is diminished by a magnitude of the order of s, the radius of the sphere of action. In the case of central collisions the diameter of a molecule or the radius of its sphere of action would have to be subtracted, and a smaller amount in every other case. The mean value of the correction may be found by a simple integration over the hemisphere. The correction corresponding to an angle of impact s made by the direction of the relative motion with that of the line of centres is s cos s. The probability of this value of the angle of impact is obtained by projecting the ring-shaped element 2 sin s ds of the spherical surface on the plane of the section and dividing by ", the area of this section; its value is 2 cos s sin s ds, and consequently the mean value of the correction is Van der Waals, who instead of this had found the value s, has given a more exact calculation in a later memoir.2 He pointed out that the correction thus found does not refer to the absolute value of the molecular free path; it is rather the path of 1 Over de continuiteit van den gas- en vloeistoftoestand, Leiden 1873, p. 48. Abstract in Beiblätter, 1877, i. p. 10. An English translation of F. Roth's version has been published by the Physical Society of London. 2 Arch. Néerl. xii. p. 201, 1877. the colliding particle relatively to that encountered that has to be shortened by s, and, of this shortening, part falls on the striking particle and part on that struck. If the former moves with a velocity whose components are u, v, w, and the latter with a velocity whose components are U, V, W, the path of the striking particle is diminished by U2 + V2 + W2 (u — U)2 + (v — V)2 + (w — W)2 + (0 - + (WW)2 }· To find the average shortening of the paths of a particle with velocity components u, v, w for all its collisions, we have to multiply the former number by the collision-frequency (§ 29*) where πs2N(km/π)3d U dV dW re − km(U2 + y2 + W2), r = √ {(u — U)2 + (v − V)2 + (w — W)2}, and integrate the result is πs3N√(u2 + v2 + w2). If now we multiply this expression by the probability of occurrence of the velocity w = √ (u2 + v2 + w2), viz. 4π − 1 (km) 1 w2dw e- km2 (§ 18*), and integrate with respect to w between the limits 0 and ∞, we obtain the value s3NO, which we must divide by the mean collision-frequency r√27s No in order to find the mean value of the correction According to this calculation, on taking account of the magnitude of the molecules, we have to put for the value of the mean free path If instead of N, the number of molecules per unit volume, we introduce A, the edge of the elementary cube containing one molecule as given by the relation Advantageous use of this improved formula has been made in the investigation of the actual volume occupied by molecules (§ 117). G. Jäger has extended these considerations also to the theory of viscosity. 35*. Influence of Cohesion on the Free Path W. Sutherland2 has obtained a second correction of the formula which gives the molecular free path by calculating in what ratio the probability of collision between two particles is increased by their mutual attraction. We now proceed to give his calculation in order to put on a better footing what has been said in §§ 71 and 85 respecting this action of the forces of cohesion. Since we need not calculate the absolute motion of both particles, but only their relative motion with respect to each other, we may take one to be fixed, while we ascribe to the other a velocity which is equal to the relative velocity with which they move relatively to each other. The path of the moving particle which is attracted by the fixed one lies in a plane which contains also the position of the fixed particle, and we may therefore denote the position of the moving particle at time t with respect to the fixed particle at the origin by the coordinates p and in that plane. The attraction, which depends on the radius p only and is independent of the angle 7, being denoted by F(p), the motion is given by the differential equations T This constant h represents twice the area of the surface described Wien. Sitzungsanz. 1899, p. 89. 2 Phil. Mag. 1893 [5] xxxvi. p. 507. in unit time by the radius vector p, and we determine its value by consideration of a point on the orbit so distant from the origin that the attraction F(p) may be taken as vanishingly small; the velocity is then constant, being that of the relative velocity r with which the particles began to approach each other in straight paths, and the surface h is then equal to the product rb, where b is the length of the perpendicular from the fixed particle on the rectilineal part of the path of the moving particle. The angular velocity is therefore += brp-2, and the first differential equation, on introduction of this value, becomes -3 p = b2x2p¬3 — F(p), which on integration gives {p2 = C — } b2r2p¬2 — [dpF(»). The constant C may be determined by application of the equation at an infinitely great distance p where the total velocity r is given by the formula while from a former equation. ρ p+=h/p=0 We thus finally obtain p2 = r2 — b2r2p¬2 + 2[o dpF(o). The shortest distance to which the particles approach each other is determined by the vanishing of p, and thus by the equation 0 = r2 — b2r2p¬2 + 2√°° dpF(p). A collision ensues if this distance is less than the radius s of the sphere of action, and this occurs if the perpendicular distance b which satisfies the equation is less than a limiting value, which we may put as b2 <s2{1 + 2r-2fTM° dpF(p)} ; since for every value of p which falls within the sphere of action, and is therefore less than s, we may assume that the function F(p) is equal to 0, as this small distance is never reached. If these gases had no cohesion, then F) would be zero for every value of P, and the condition for the occurrence of a collision would be b2 <s2, as is obvious from the meaning of b, viz. the distance of the particle encountered from the path of the striking particle. The influence of the forces of cohesion on the frequency of the collisions and on the length of the free path therefore consists in the replacement in all calculations of the actual section s2 of the sphere of action by a larger area The magnitude by which the section has to be augmented depends on the temperature, and is, indeed, inversely proportional to the absolute temperature, as is shown by the occurrence in the denominator of the square of the molecular velocity r. This ratio was given in § 71, and its value was estimated in § 85 for the explanation of the observations on internal friction.1 In the next following investigations we shall for simplicity leave out of account both this correction and that given in § 34*. 36*. Free Path in Mixed Gases In the case of a mixture of gases composed of molecules of two different kinds, we find the free path L1 of a particle of the first kind and the free path L2 of a particle of the second kind from the formulæ of § 30*, viz. taking account of the meaning of the magnitudes Q we may write these {√2πs12N1 + ño2N2√(1 + k ̧m1/km2)} L1 2 =1 and, in case the temperature of both components of the mixture is {√2πs22 N2+ πo2N1√(1 + m2/m1)} L2 = 1. These equations were first established by Maxwell.2 Compare the account of the observations on the friction of vapours in § 87. |