158 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



and therefore, by (98), the quantity of any component gas G l in the 

 gas-mixture, and in the separate gas to which p lt q v etc. relate, is the 

 same and may be denoted by the same symbol m r Also 



whence also, by (93)-(96), 



All the same relations will also hold true whenever the value of \fs 

 for the gas-mixture is equal to the sum of the values of this function 

 for the several component gases existing each by itself in the same 

 quantity as in the gas-mixture and with the temperature and volume 

 of the gas-mixture. For if p lt r] l} e lt fa, Xi> fi> Pz> e ^ c - 5 e ^ c - are 

 defined as relating to the components existing thus by themselves, we 

 shall have 



whence 



Therefore, by (88), the potential // 1 has the same value in the gas- 

 mixture and in the gas G l existing separately as supposed. Moreover, 



\ u/i< / v, m 



*- 



whence ^ = 



Whenever different bodies are combined without communication of 

 work or heat between them and external bodies, the energy of the 

 body formed by the combination is necessarily equal to the sum of 

 the energies of the bodies combined. In the case of ideal gas-mixtures, 

 when the initial temperatures of the gas-masses which are combined 

 are the same (whether these gas-masses are entirely different gases, 

 or gas-mixtures differing only in the proportion of their components), 

 the condition just mentioned can only be satisfied when the tempera- 

 ture of the resultant gas-mixture is also the same. In such com- 

 binations, therefore, the final temperature will be the same as the 

 initial. 



If we consider a vertical column of an ideal gas-mixture which is 



*A subscript m after a differential coefficient relating to a body having several 

 independently variable components is used here and elsewhere in this paper to indicate 

 that each of the quantities m lt m 2 , etc., unless its differential occurs in the expression to 

 which the suffix is applied, is to be regarded as constant in the differentiation. 



