J. W. Gibbs — Equilibrimn of Heterogeneous Substances. 245 

 dn dp 

 therefore 



(338) 



Since a similar relation will hold true for //, we obtain 



d dy] d dtf 



d^ ~dt~d^ ~dV (^'^^^ 



which must hold true within the given limits of temperature and 

 density. Now it is granted that 



dt - dt ^^^^^ 



for very great values of o at any temperature Avithin the given limits, 

 (for the two members of the equation represent the thermal capacities 

 at constant volume of the real and ideal gases divided by ^,) hence, 

 in virtue of (339), this equation must hold true in general within the 

 given limits of temperature and density. Again, as an equation like 

 (337) will hold true of ?/', we shall have 



drj dii 



dv dv' ^ ' 



From the two last equations it is evident that in all calorimetrical 

 relations the ideal and real gases are identical. Moreover the energy 

 and entropy of the ideal gas are evidently so far arbitrary that we 

 may suppose them to have the same values as in the real gas for any 

 given values of t and v. Hence the entropies of the two gases are 

 the same within the given limits ; and on account of the necessary 

 relation 



ds-=. t dt] — p dv, 



the energies of the two gases are in like manner identical. Hence 

 the fundamental equation between the energy, entropy, volume, and 

 quantity of matter must be the same for the ideal gas as for the 

 actual. 



We may easily form a fundamental equation for an ideal gas-mix- 

 ture with convertible components, which shall relate only to the 

 phases of equilibrium. For this purpose, we may use the e(piations 

 of the form (312) to eliminate from the equation of the form (273), 

 which expresses the relation ' between the pressure, the temperature, 

 and the potentials for the proximate components, as many of the 

 potentials as there are equations of the former kind, leaving the 



