Page:Scientific Papers of Josiah Willard Gibbs.djvu/218

Since a similar relation will hold true for $\eta '$, we obtain

${\frac {d}{dv}}{\frac {d\eta }{dt}}={\frac {d}{dv}}{\frac {d\eta '}{dt}},$

(339)

which must hold true within the given limits of temperature and density. Now it is granted that

${\frac {d\eta }{dt}}={\frac {d\eta '}{dt}}$

(340)

for very great values of $v$ at any temperature within 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 $t$), 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 $\eta '$, we shall have

${\frac {d\eta }{dv}}={\frac {d\eta '}{dv}}\cdot$

(341)

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

$d\epsilon =td\eta -pdv,$

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-mixture with convertible components, which shall relate only to the phases of equilibrium. For this purpose, we may use the equations 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 potentials for those components which it is convenient to regard as the ultimate components of the gas-mixture.

In the case of a binary gas-mixture with convertible components, the components will have the same potential, which may be denoted by $\mu$, and the fundamental equation will be

$p=a_{1}L_{1}t^{\frac {c_{1}+a_{1}}{a_{1}}}e^{\frac {\mu -E_{1}}{a_{1}t}}+a_{2}+L_{2}t^{\frac {c_{2}+a_{2}}{a_{2}}}e^{\frac {\mu -E_{2}}{a_{2}t}},$

(342)

where⁠$L_{1}=e^{\frac {H_{1}-c_{1}-a_{1}}{a_{1}}},$⁠$L_{2}=e^{\frac {H_{2}-c_{2}-a_{2}}{a_{2}}}.$

(343)