153 



The preceding fundamental equations all apply to gases of constant 

 composition, for which the matter is entirely determined by a single 

 variable (m). We may obtain corresponding fundamental equations 

 for a mixture of gases, in which the proportion of the components 

 shall be variable, from the following considerations. 



moderate pressures as the liquid experiences while in contact with the vapor), and 

 denote this specific heat by k, and the volume of a unit of the liquid by V, we shall 



have for a unit of the liquid 



t drj = k dt, 



whence t\ k log t + H', 



where //' denotes a constant. Also, from this equation and (97), 



d/j. = - (k log t + H') dt + Vdp, 

 whence M = kt - kt log t - H't +Vp + E', (A) 



where E' denotes another constant. This is a fundamental equation for the substance 

 in the liquid state. If (268) represents the fundamental equation for the same substance 

 in the gaseous state, the two equations will both hold true of coexistent liquid and gas. 

 Eliminating fj. we obtain 



p H-H' + k-c-a k-c-a, E-E' Vp 



log- = logt + * 



6 a a a at a t 



If we neglect the last term, which is evidently equal to the density of the vapor divided 

 by the density of the liquid, we may write 



logp=A -Blogt--, 

 t 



A, B, and G denoting constants. If we make similar suppositions in regard to the 

 substance in the solid state, the equation between the pressure and temperature of 

 coexistent solid and gaseous phases will of course have the same form. 



A similar equation will also apply to the phases of an ideal gas which are coexistent 

 with two different kinds of solids, one of which can be formed by the combination of the 

 gas with the other, each being of invariable composition and of constant specific heat 

 and density. In this case we may write for one solid 



and for the other fj^=k"t- k"t log t - H"t + V"p + E\ 



and for the gas fj^ E+tl c + a-H-(c + a)logt + alog - ). 



\ a / 



Now if a unit of the gas unites with the quantity X of the first solid to form the 

 quantity 1+X of the second it will be necessary for equilibrium (see pages 67, 68) that 



Substituting the values of fjt^, fj^ t ^ given above, we obtain after arranging the terms 

 and dividing by at 



when A= H+\H'-(l + \)H"-c-a-\k' 



a 



D (l+\)k"-\k'-c-a 



^~ -' 



n E+\E'-(\+\)E" 



' - 



a 



We may conclude from this that an equation of the same form may be applied to 

 an ideal gas in equilibrium with a liquid of which it forms an independently variable 

 component, when the specific heat and density of the liquid are entirely determined 



