318 On the Work that may be gained during the Mixing of Gases. 



when there is free diffusion than when the gases are uniformly 

 mixed ; for if the gases be allowed to rise from the reservoir 

 tolerably quickly (or if a series of movable pistons be interpo- 

 lated), the composition in the tube will be the same as in the 

 reservoir. If free diffusion be now allowed, there must be dissi- 

 pation. The original state of things will be restored if the 

 mixture be slowly forced back into the reservoir ; and accordingly 

 the work consumed in condensation must be greater than that 

 gained in the expansion. In fact it may be proved algebraically 

 by a process somewhat similar to that applied to equation (3), 

 that the pressure of the gases under free diffusion p } where 



p=P 1 e-'*i*H- P 2 €-^% (7) 



is greater than the pressure of a uniform mixture p\ where 



y^ + PJe-'-lVhPr (8) 



It is, however, possible to imagine other distributions which 

 shall give a pressure greater than (7). The mechanical equili- 

 brium gives one equation involving the two quantities p x andj» 2 , 

 viz. 



£ + &W,+*ft=<>; .... (9) 



and the subsidiary conditions are that jOj=P„ jo 2 = P 2 , when 

 2" = 0. Hence we may take as the most general solution, 



jo^Pje-^ + e-^J €^ x Xdx, 

 Jo 



p 2 = ¥ 2 €-W -€-*>*{ V^Xfifo, 



where X is an arbitrary function of w. Thus the total pressure 



jP l +ft==P 1 «-«' + *?*€-»* 



+ 1 *{e-^*-*> — e-^<*-*>}X& . (10) 

 Jo 



For free diffusion X=0 ; but it could always be taken so as to 

 make the integral either positive or negative, as might be de- 

 sired. 



The work required to decompose a mixture of gases is in ge- 

 neral small, and could scarcely be of much importance from an 

 industrial point of view. When, however, the proportion of one 

 ingredient is very insignificant, more work is required. Thus 

 the separation of the carbonic anhydride from the atmosphere 

 would require, relatively to the quantity obtained, a much larger 

 expenditure of work than the separation of the oxygen. This 



