Page:Scientific Papers of Josiah Willard Gibbs.djvu/189

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

$td\eta =kdt,$

whence, $\eta =k\log t+H',$

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

$d\mu =-(k\log t+H')dt+Vdp,$

whence $\mu =-kt-kt\log t-H't+Vp+E',$

(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 $\mu$ we obtain

$\log {\frac {p}{a}}={\frac {H-H'+k-c-a}{a}}-{\frac {k-c-a}{a}}\log t-{\frac {E-E'}{at}}+{\frac {V}{a}}{\frac {p}{t}}\cdot$

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

$\log p=A-B\log t-{\frac {C}{t}},$

$A,B$, 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 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

$\mu _{1}=k't-k't\log t-H't+V'p+E',$

and for the other $\mu _{2}=k''t-k''t\log t-H''t+V''p+E'',$

and for the gas $\mu _{3}=E+t\left(c+a-H-(c+a)\log t+a\log {\frac {p}{a}}\right).$

Now if a unit of the gas unites with the quantity $\lambda$ of the first solid to form the quantity $1+\lambda$ of the second it will be necessary for equilibrium (see pages 67, 68) that

$\mu _{3}+\lambda \mu _{1}=(1+\lambda )\mu _{2}.$

Substituting the values of $\mu _{1},\mu _{2},\mu _{3}$ given above, we obtain after arranging the terms and dividing by $at$

$\log {\frac {p}{a}}=A-B\log t-{\frac {C}{t}}+D{\frac {p}{t}},$

when $A={\frac {H+\lambda H'-(1+\lambda )H''-c-a\lambda k'+(1+\lambda )k''}{a}},$

$B={\frac {(1+\lambda )k''-\lambda k'-c-a}{a}},$

$C={\frac {E+\lambda E''-(1+\lambda )E''}{a}},\,\,\,\,D={\frac {(1+\lambda )V''-\lambda V''}{a}}\cdot$

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