210 J. W. Gibhs — Equilibrium of Heterogeneous Substances. 



potentials have the same values as at that point. We may also 

 express this condition by saying that the pressure nnist be as great 

 as is consistent with equations (246), (247). This condition with the 

 equations mentioned will always be sufficient foi- equilibrium ; when 

 the condition is not satisfied, if equilibrium subsists, it will be at 

 least practically unstable- 



Hence, the phase at any point of a fluid mass, which is in stable 

 equilibrium under the influence of gravity (whether this force is due 

 to external bodies or to the mass itself), and which has throughout 

 the same independently variable components, is completely deter- 

 mined by the phase at any other point and the difierence of the 

 values of the gravitation potential for the two points. 



FUNDAMENTAL EQUATIONS OF IDEAL GASES AND GAS-MIXTUKES. 



For a constant quantity of a perfect or ideal gas, the product of 

 the volume and pressure is proportional to the temperature, and the 

 variations of energy are proportional to the variations of tempera- 

 ture. For a unit of such a gas we may write 



p v:= a t^ 

 de z=. c dt, 



a and c denoting constants. By integration, we obtain the equation 



e= ct+E, 

 in which S also denotes a constant. If by these equations we elimin- 

 ate t and p from (11), we obtain 



s-E , a £-E , 



de =z d?} dv, 



C V c 



or 



d€ , dv 



c vt = dv - (/ — . 



The integral of this equation may be written in the form 



c log =: // — a log V — JI, 



where ^denotes a fourth constant. We may regard ^as denoting the 

 energy of a unit of the gas for ^=0 ; ^its entropy for ^=1 and v=zl ; 

 a its pressure in the latter state, or its volume for t=l and p=zl ; 

 c its specific heat at constant volume. We may extend the application 

 of the equation to any quantity of the gas, without altering the 



values of the constants, if we substitute — , -, — for e, ri, v. respec- 



m m m i i-> i y 



tively. This will give 



