3 16 Lord Rayleigh on the Work that may be 



Since the process is reversible, (3) gives the work which may 

 be done during the mixing of a volume v of two gases under 

 the partial pressures C^ and Q 2 , with a large quantity of the 

 same gases under an equal total pressure, but with partial pres- 

 sures Pj and P 2 . 



The quantity denoted by W can never be negative. To verify 

 this from (3), write it in the form 



W 



g =yio g £ + yio g J,=.o g {g)^) y '}, w 



where 



*-p7+iY y -p7+P*' *~W+%' v '<£+%' 



so that x + y=x ! + y , — 1 . 



Now (Todhunter's c Algebra/ p. 392) if a, b, c, . . . be any 

 positive quantities, 



a+b+c+d+ ...'^/yj £ 

 > {abed ...)». 



Suppose that a, b, c. . . . consist of p equal quantities a and q 

 equal quantities /3; then 



^a+ ^/3>««/3«. 



If now we take 



n ' w y ' x" ^ y' 



we see that 



(5)'©''. 



and therefore, since # + y=l, that W is always positive, unless 

 u = /3, in which case the composition of the two mixtures is the 

 same, and W vanishes. 



We have now to show how the formula for the mixture of two 

 pure gases may be derived from (3). Let v x be the volume of 

 the first gas and v 2 of the second, at the constant pressure 

 P^Pg. The value of the interdiffusion of v } and v 2 must be 

 the same as that of their diffusion into a large quantity of a mix- 

 ture whose composition is identical with that of the mixture of 

 i\ and v T For, on this supposition, the separation of the two 

 mixtures spoken of would have no mechanical value. Now by 

 (3) the value of W for the diffusion of a quantity v of pure gas 

 into a large quantity of a mixture whose partial pressures are Pj 



