If there is one single molecule at rest in a cubical space 1 cm. on a side, the chance that another molecule which is shot through the cube will impinge upon the one contained is clearly in which is the mean diameter of the two molecules. If there are contained molecules the chance is multiplied by , that is, it becomes . But on the average the chance of an impact in going a centimeter is the number of impacts actually made in traversing this distance. The mean free path is the distance traversed divided by the number of impacts made in going that distance. Hence This would be the correct expression for the mean free path of a molecule which is moving through a group of molecules at rest. If, however, the molecules are all in motion they will sometimes move into a collision which would otherwise be avoided, so that the collisions will be more numerous when the molecules are in motion than when at rest—how much more numerous will depend upon the law of distribution of the speeds of the molecules. It is through a consideration of the Maxwell distribution law that the factor is introduced [Pg 276] into the denominator (see Jeans, Dynamical Theory of Gases) so that equation (54) becomes [Pg 277] APPENDIX F NUMBER OF FREE POSITIVE ELECTRONS IN THE NUCLEUS OF AN ATOM BY RUTHERFORD’S METHOD If represents the number of free positive electrons in the nucleus, the electronic charge, the known charge on the -particle, namely , and the known kinetic energy of the -particle, then, since the inertias of the negative electrons are quite negligible in comparison with that of the -particle, if the latter suffers an appreciable change in direction in passing through an atom it will be due to the action of the nuclear charge. If represents the closest possible approach of the -particle to the center of the nucleus, namely, that occurring when the collision is “head on,” and the -particle is thrown straight back upon its course, then the original kinetic energy must equal the work done against the electric field in approaching to the distance , i.e., Suppose, however, that the collision is not “head on,” but that the original direction of the -particle is such that, if its direction were maintained, its nearest distance of approach to the nucleus would be (Fig. 41). The deflection of the a particle will now be, not 180°, as before, but some other angle . If follows simply from the geometrical properties of the hyperbola and [Pg 278] the elementary principles of mechanics that Fig. 41 For let represent the path of the particle and let . Also let = velocity of the particle on entering the atom and its velocity at . Then from the conservation of angular momentum [Pg 279] and from conservation of energy Since the eccentricity , and for any conic the focal distance is the eccentricity times one-half the major axis, i.e., , it follows that But from equations (58) and (59)