( 314 ) 



and 



3N 3 A 7 ::.V, :;.V 3 



— t} = --+ - log (2x7) • - log m l + — log m, -\- N log v. 



For the first substance alone we find : 

 3 A', 3A T , :'..\ 



*J> = 



1 h,,(2.tT) | ! A..////, I- A'. Aw r, 



2 2' 2 



for the second alone : 



3/V, 3 A; 3 A', 



— n, = ~ö~ + ö lo 9( 23tT ) ^ ., / "' / '"* + A > / "' / ''• 



From this appears — i t = — »?, -j- - »/, or the entropy of the 

 mixture is equal to the sum of the entropies of the gases forming 

 the mixture, which now holds too if the component parts consist of 

 the same gas. So there is no perfect harmony with thermodynamics : 

 lor this entropy GlBBs's paradox no longer holds. 



That the - »] of the whole volume is not equal to the i 1 - i t of 

 the parts is a consequence of the fact that the extension in phase 

 v \ Nl X v t Ns * s 11(,t equal to the extension (v } -\- v,) N * + - N - . In the 

 total volume there are more possibilities of combination of place 

 than when the volume has been divided into two separate parts. We 

 may also say that Boltzmann's entropy just as in thermodynamics, 

 may be divided with regard to the volume, Gibbs's entropy with 

 regard to the molecules. It' we compare the formulae : 



ƒƒ — N I log - f C ) and ij = A 7 (- lo : , v + < '), 



it appears that ( i inits's entropy can be brought into harmony with 

 thermodynamics by augmenting r t by N log N or N (log N -\- C). 

 This may be done by multiplying the density e r ' by A' A . c - x or A'/ 

 by approximation. So we should have to take for j; the mean log 

 of the density, not with respect to the specific, but with respect to 

 the generic phases. l ) 



l ) When I had written the above, I observed that the last remarks are not 

 new In the last sentence of bis book Gibbs himself has already made the obser- 

 vation that we shall have to take — 7, and not — r. as equivalent for the 

 entropy, "except in the thermodynamics of bodies in which the number of molecules 

 of the various kinds is constant." So it will always have to be done where the 

 entropy of the whole is compared with that of the parts. 



Nevertheless considering thai Gibbs devotes so few words lo the matter, 1 feel 

 justified in not suppressing my remarks. 



