14-17] The Statistical Method 15 



V 



The changes produced by Collisions when the Molecules are 

 Elastic Spheres. 



16. The state of a gas is fully known from the statistical point of 

 view, when the density and the law of distribution of velocities at every 

 point of the gas are known. The main problem of this chapter, which we 

 now proceed to attack, is to search for a steady state : i.e., a state in which 

 the density and law of distribution of velocities remain the same at everv 

 point of the gas throughout all time. 



We begin, by discussing the simplest case. Not only are the molecules 

 supposed to be hard rigid spheres, but we suppose that the external physical 

 conditions are the same at every point of space, and that the gas fills infinite 

 space. The latter assumption is a temporary one, which enables us to 

 consider separately the elements of the problem which are introduced by 

 the presence of a containing vessel. 



Under the conditions now postulated, we may clearly begin by assuming 

 the gas to have the same molecular density v and the same law of distri- 

 bution of velocities / at every point of space. Since there is nothing to 

 distinguish the different regions in space, this uniformity in space will 

 obviously be maintained throughout all time, but the actual form of the 

 function / will change with the time. 



17. The first problem is to find an expression for the change in the 

 number of molecules belonging to class A (defined on p. 14) which occurs 

 during an interval of time dt. Since the motion of the molecules is one of 

 uniform velocity except when collisions take place, it appears that molecules 

 can only enter or leave class A through the occurrence of collisions. We 

 begin by considering molecules which leave class A through collisions. 



Let us consider a special kind of collision which we shall call a collision 

 of class a. This is to be defined as a collision in which the three following 

 conditions are satisfied. 



(i) One of the two colliding molecules is to be a molecule of class A. 

 (ii) The second colliding molecule is to be of class B (defined on p. 14). 



(hi) The direction of the line joining the centre of the former molecule 

 to that of the latter at the moment of impact is to be such that 

 a line drawn parallel to it from the centre of a fixed sphere of 

 unit radius to the surface of this sphere meets the surface inside 

 a small element of area da>, this element being such that the 

 direction-cosines of a line drawn to its centre from the centre of 

 the sphere are I, m, n. 



