J. W. Gibbs — Eqtcilibriuin of IJeterogeneoKs ySiibstances. 213 



regarded as defining an ideal gas. It will be observed that most of 

 these equations might be abbreviated by the use of different con- 

 stants. In (270), for example, a single constant might be used for 

 H—c—a 



— " C -\- €t 



a e '^ , and another for ■ ^ The equations have been given 



in the above form, in order that the relations between the constants 

 occurring in the different equations might be most clearly exhibited. 

 The sum c + a is the specific heat for constant pressure, as appears if we 

 diflerentiate (266) regarding jt> and in as constant.* 



* We may easily obtain the equation between the temperature and pressure of a 

 saturated vapor, if we know the fundamental equations of the substance both in the 

 gaseous, and in the liquid or solid state. If we suppose that the density and the specific 

 heat at constant pressure of the liquid may be regarded as constant quantities (for such 

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

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

 have for a unit of the liquid 



t dr/ = k dt, 

 whence 



7] = k log t + H\ 



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



dfi — - (k log t + R')dt+V dp, 

 whence 



11 = 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 sub- 

 stance in the gaseous state, the two equations will both hold true of coexistent liquid 

 and gas. Eliminating u we obtain 



p H—H' + k—c—a k—c—a, E—E' V p 



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 



C 

 log p=A — Blog t -, 



A, B, and C 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 wiU of course have the same form. 



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

 tent with two different kinds of solids, one of which can be formed by the combina- 

 tion 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 

 /x , -- k't-k't log t- H't + V'p + E', 



and for the other 



ji., = k"t-k"t log t-H"t+ V"p + E", 



and for the gas 



^;, = E-^t(c + a-H— (c + a) log f + a log — j. 



