possibility of separating Isotopes. 185 



§4. Whether or not this method of separation by diffusion 

 is of convenient practical application depends largely on the 

 time taken for the attainment of the steady state. An 

 estimate of this can be made, somewhat roughly. 



Let v denote the number of molecules of both kinds 

 (^i + v 2 ) per unit volume at the point a* measured along a 

 uniform cylinder, the two ends of which are maintained at 

 different fixed temperatures. The pressure being the same 

 throughout, v T is constant along the cylinder. If diffusion 

 is taking place, the number of molecules m 1 which per unit 

 time cross unit area of a normal section of the cylinder in 

 the positive direction is v uq, equal to the number of 

 molecules m 2 crossing in the opposite direction, where 



Wo 



■--■*■© +J-& 



Here D 12 is the ordinary coefficient of diffusion of the 

 .mixture. In the present case, where the molecules of the 

 two sorts are so nearly alike, we can replace D 12 by D n , 

 'the coefficient of self -diffusion of the gas ; if the molecules 

 behave like rigid elastic spheres D n = 1-200 fi/p, where fi is 

 the coefficient of viscosity and p the density of the gas. 



The equation giving the rate of variation of the relative 

 concentrations at any point is 



This cannot be exactly integrated, because (on account of 

 the temperature gradient) v and D 12 vary along the tube. 



We know, however, that initially, when the mixture of 

 gas is uniform, 'ft\J'd% = 0, and ?/ '= — ^tD 12 ^ logT/da? ; 

 finally, u ' tends to zero, while X x tends to its equilibrium 

 value C — k T log T. Clearly, the greater the value of Di 2 , the 

 more rapid is the progress towards the steady state. It seems 

 likely that the order of magnitude of the time taken may be 

 estimated by making the calculation from the last equation 

 as if v and D ]2 were constant along the cylinder, giving 

 D 12 its minimum value. 



In this case, since Z't log T is independent of the time, we 

 may write 



| t Oi + fa log T) = D„ g, O, + i, log T). 



The solution of this is 



A.i + k T log T = A + ZApe-rt+^f^ 1 \ 

 where the constants p arc determined by the initial and end 



