( 313 ) 

 directly from this definition. H= { [flog f do dm indicates, namely, 



that H of the whole = 2H of the parts; further we can always 

 think the molecules divided into two parts with densities ƒ, ami/,, 

 so that ƒ==ƒ, +ƒ„ 



(/i 4 /,) /( '." (/i f- /,) </, %/, + /, ^/, . 

 so that also // <^ //, + ^ Prom ihis latter way of treatment appears 

 at the same time that the three properties arc also valid for the 

 entropy of Bojltzmann if the gas is not in the stale of equilibrium. 

 To answer the same questions with regard to the entropy of Gi BBS 

 we consider the formula: 



3 3 



— q — - -A' -}- A log (2a m T) | A' log v. 



As to the tirst question for N x molecules in the volume v. 



3 3 



— ?j, = - AT, -f — A , log (2.tw7) + A , % r , 

 — z 



so 



— 2ÏJ, = 3 A', + 3A T t hg (2rr wT) -f- 2AT, log v 



and for 2 A", mol. in volume 2v : 



— ^— 3A', -f 3A T , Zo? (2-t m7') f 2A r , lor, 2v . 



so that the entropy of the whole is not equal to the sum of the 

 entropies of the parts. The increase in entropy or free energy) of 

 the whole, however, is always equal to the increase in entropy (or 

 free energy) of the parts '). 



As far as the second and third questions are concerned, we may 

 directly take the general case of a mixture of two different gases 

 and tind then : 



_♦ » 3* n- V ' .V 



7' 2 2 2 



e = (2.1 T) tn 1 ///, v 



from which : 



if? 3A^ 3A f x 3A T , , 



— ■== -„- log (2xT) 4- - log w, 4- - log m, f A log u 



J. Li — 1-1 



') Accordingly the equation derived by Dr. L. S. Ornstein in his Thesis for the 

 Doctorate: "Application of Gibbs's Statistical Mechanics to molecular- theoretical 

 problems", (Leiden 1908) p. 54: 

 k 



2 (J»/ -J-o/) = 4< — J*,, states only that the increase in J< of the joint elements 

 l 



forms the total increase in J/ of the whole. For the "zero-state"', for which vj-,,/ 

 and J,„ hold, viz. the state in which the potential energy o, z J,,,/ — j„ is by 

 no means valid. It is exactly this "zero-state", that has been considered above. 



