where the sign of summation denotes a summation for all integral values of x from 1 to ∞. The mean probable length of the free path L of a molecule is therefore given by the formula :-1 Σ.x11 = 1 + 26 + 342 + . . . = (1 − 6) ̃2, or, in our case, 1 Σ.x(1 — πs2/X31-1 = (λ2 / πs2)2. Hence it results from this calculation of probability that the length of the path which a moving particle would traverse without collision amid a multitude of particles at rest is on the average This formula, which Clausius deduced in the memoir referred to in a similar way, but with the use of the integral calculus, assumes an intelligible form if we write it Ι:λ= λ : πς; it then expresses that the mean free path bears the same ratio to the mean distance separating two neighbouring particles as the area of a face of the elemental cube has to the central section of the sphere of action. From this proposition Clausius draws a very important conclusion. The above proportion shows indeed that the free path L is greater than the distance of molecular separation A, and that it must be very much greater than the latter in a rarefied gas. For by definition λ is the edge of the elemental cube in which a single molecule is contained, and s, the radius of the sphere of action, is a distance within which the force exerted by the molecule is sensible. It would be in contradiction of our theory, no less than of experiment, which has shown an almost perfect absence of cohesion in gases, if we were not to assume the latter length s to be considerably smaller than the former λ; consequently also the proportion shows that L is considerably larger than λ. A molecule therefore passes by many molecules like itself before it collides with another. 66. Probability of Particular Free Paths Now that we have determined the mean value of the molecular free path, the probability-formulæ obtained in § 64 present a simple meaning which makes it possible for us to numerically answer the question, How much more probable is a shorter path than a longer? The expression M(1 - πs2/x2), which we have found for the probable number of those among the M projected particles that traverse a path greater than xλ, becomes when the mean free path L is introduced into it, so that we can see how the number of the particles which collide in each layer and that of those which pass through it unobstructed depend only on the ratio of the average distance of molecular separation to the mean free path. If we wish also to refer to the mean free path the actual path traversed, which hitherto has been given by the number x, we can put for the path y = xλ = qL, where the number q gives the number of times by which the path already traversed by the particle under consideration exceeds the mean free path. If we also put we shall have L = Qλ, x = Qq, and the probable number of particles which do not undergo collision in a path of length qL is given by M(1 — 1/Q). We do not indeed know the number Q, i.e. the ratio of the mean free path to the distance of molecular separation, which occurs in this expression, but we do know that its value must be very great, so great indeed that we may look upon it as almost infinitely great. If Q were actually infinitely great, we should have (1 Q-10 = e-1 = 1 1/2.718... 0.36788, where e is the base of natural logarithms. = Therefore the number of particles which traverse a path at least q times greater than that passed over in the mean is, with this assumption, a According to this formula we have calculated the following table. Out of every 100 particles The table shows that the mean free path is considerably exceeded extremely seldom. The only play of chance, therefore, is to set up in this case too, as well as in that of the distribution of speeds, a uniformity that is maintained with great exactness. 67. Free Path if all the Molecules have Equal Speed The value we have found for the mean free path needs still a correction, which the considerations of the last paragraph do not, however, touch. The value of the mean free path found in § 65, viz. L = X3/πs2, holds only for a simple hypothetical case described fully in § 63; it was supposed that only the one particle whose free path we were calculating was in motion, all the others being at rest. All the particles, however, are in motion in the actual case. It is easy to see that this general motion must increase the probability of a collision of one particle with the others; for the particle can also be struck by another that moves from the side and with which it would not come into contact as a result of its own motion. By the general motion, therefore, of all the particles the probability of a collision is increased, and thus the mean value of the molecular free path is diminished. Clausius has calculated this shortening of the path for the case in which all the particles move with equal speeds but in all possible directions in space. With this supposition we find the number of collisions increased in the ratio of 4: 3, and, therefore, the free path shortened in the ratio 34. We obtain then, as is proved in § 28*, the value for the mean free path of a particle in the uniformly moving medium, and this differs only by the factor from the former value. From this equation also we can deduce a proportion like the former and of similarly simple meaning, viz. L:s=λ3: πs3. 68. Molecular Free Path with an Unequal But these calculations do not correspond exactly to the real state of things, since the underlying assumption as to the way in which the speeds are distributed among the molecules cannot possibly be right. The supposition that all the particles of a gaseous medium are to have equal speeds gives no real picture of the motion which exists in a gas that is in equilibrium under a pressure which is everywhere the same and at a temperature which is everywhere the same. The true law according to which the molecules arrange their speeds is, as we know (§ 24), that discovered by Maxwell. M Maxwell has also calculated the mean value of the free path on the assumption of this law. The calculation cannot be here given; a deduction of the formula will be found in § 29*. 2 The result of the calculation agrees almost exactly with that just mentioned which Clausius obtained on the assumption of equal speeds in the molecules. In this case, too, the formula demonstrated in § 65 undergoes no further alteration than the addition of a numerical factor, and there results for the mean free path The factor, the value of which will be more closely indicated in § 96, is nearly the same as that in Clausius' formula; for the latter is or 0.75, and the former 1/2 or 0·707, so that they are approximately in the ratio of 17 to 16 [or, still more nearly, of 35 to 33]. The value of the free path that follows from Maxwell's law is somewhat the smaller; there also results from this law a smaller value of the mean speed than that given by Clausius' theory; both results are explained on the simple ground that a shorter path and a slower speed occur more frequently than a longer path and a higher speed. 69. Molecular Path-volume The name of molecular path-volume has been given by Loschmidt to the content of the cylindrical space which a molecule describes when it traverses its mean free path. The magnitude of this volume is s2L, since the radius of the sphere of action is equal to the distance apart of the middle points of two molecules during collision, and is, therefore, equal to the diameter of a molecule in the case of actual contact during collision; hence by the foregoing formula it is equal to X3/4/2. If we replace in this expression the size of the elemental cube, or of the space that contains a single molecule only, by the number N of molecules 1 Phil. Mag. 1860 [4] xix. p. 28; Scientific Papers, 1895, i. p. 387. 8 Wiener Sitzungsber. 1865, lii. Abth. 2, p. 397. |