254 Free Path Phenomena [CH. xn 



Variation of K with Temperature. 



* 308. Since c is proportional to the square root of the absolute tem- 

 perature, it is seen that if the molecules were true elastic spheres, the value 

 of K would be proportional to the square root of the temperature. 



As a matter of fact, it is found that K varies a good deal more rapidly 

 than this, as the temperature increases. The divergence between experiment 

 and the theoretical value obtained for elastic spheres is, however, one that 

 could have been predicted. The assumption that the molecules are elastic 

 spheres is, at best, only an approximation, and we must continually examine 

 what deviations are to be expected from the results obtained on this 

 assumption. 



The peculiarity of a system of elastic spheres is that the motion remains 

 geometrically the same if the velocity of each sphere is increased in the 

 same ratio. Thus in order to determine the motion of two spheres after 

 collision it is only necessary to know the directions of motion before collision, 

 and the ratio of their velocities ; we are not concerned with the actual values 

 of these velocities When, on the other hand, we suppose the molecules to 

 be surrounded by fields of force, this ceases to be true : the paths after 

 collision do not depend solely on the ratio of the velocities, but on the 

 absolute ma^njiiides of these velocities. For instance, in fig. 20 let OPQR, 

 O'P'Q'R be the paths described when two molecules surrounded by fields of 

 force meet one another, the figure being drawn for the sake of simplicity, for 

 the case in which the two velocities are equal and opposite. If now we 

 suppose the molecules moving along the same lines before encounter, but 

 each with double its former velocity, the paths will be different. For 

 obviously the higher velocity will carry each molecule further into the 

 other's field of force before the centres of the two molecules reach their 

 shortest distance apart, so that the path described, instead of being OPQR 

 will be, let us say, OPST. 



We can, however, by an obvious geometrical construction find the size 

 of two spheres which would describe paths having the same deflections as 

 OPQR or OPST. If we perform the construction for the two paths of 

 fig. 20 we find that the size of the sphere for the former path described 

 with small velocity is greater than for the latter path described with large 

 velocity. We can therefore see that, if we attempt to represent our mole- 

 cules by spheres, the size of these spheres must be supposed to decrease 

 as the mean molecular velocity increases, and therefore as the temperature 

 rises. 



