of Gases and Molecular Force. 527 



xxxv.) and introduce the necessary modifications for attracting 

 molecules. 



The starting-point is Clausius's equation of the Virial, 



where R is the force acting between two molecules at distance 

 r apart. Now the forces R consist of two sets, first the con- 

 tinuously acting attractions to be denoted by R a , and second, 

 the discontinuous repulsions that act during the collision of 

 two molecules to be denoted by R 6 , thus 



22Rr=22R^-22R & r. 



Of the actual values of R& we know nothing, but if v is the 

 number of encounters of a molecule in a second and /.', is the 

 average momentum communicated to a molecule in an en- 

 counter, then we can treat the virial of the unknown impulsive 

 forces as equivalent to that of an average continuous force of 

 repulsion pv acting at points 2a apart (the distance of two 

 centres at collision), so that X%R b r becomes %2ajbLv. The 

 value of n for forceless spheres is m (7rv 2 /Sy. 



When molecular attraction acts, the molecules at the instant 

 of collision have on the average a relative velocity greater 

 than that which rules amongst molecules that are remote from 

 one another's influence ; calling it Y e , then 



imV7 = i^V 2 + m 9 f(l/2a) ~ ™y(l/D) , 

 where D is the distance apart of molecules remote from one 

 another's influence, so that my(l/D) can be neglected, and 

 then 



Y e = {Y* + 2mf{l/2a) }*=V{1 + 2mf(l /2a)/V 2 }*. 



Thus, then, for attracting molecules the value of /ul is got by 

 supposing the velocities of forceless molecules increased in the 

 ratio \l + mf(l/2a)/v 2 \K The value of v for N forceless 

 molecules in volume B when the diameter of a molecule is so 

 small that it can be neglected in comparison to the mean free 



N — 



path is 2 =n 7r(2a) 2 (4,v 2 /37r)i. But when the diameter 2a can- 

 not be so neglected this must be increased in the ratio 



2N 

 1 : 1— o =d 7r(2a) 3 , so that 



O -D 



v=2^(2a) 2 (4^/37r)i/{l- § §w(2a)'l ; 

 hence for forceless molecules, 



t . ^2,2,-rV = Si oat L v3/o-n =m77K 



l-2N7r(2a) 3 /3B ~'"2B 1-6/B 7 



