Page:Scientific Papers of Josiah Willard Gibbs.djvu/190

It is a rule which admits of a very general and in many cases very exact experimental verification, that if several liquid or solid substances which yield different gases or vapors are simultaneously in equilibrium with a mixture of these gases (cases of chemical action between the gases being excluded), the pressure in the gas-mixture is equal to the sum of the pressures of the gases yielded at the same temperature by the various liquid or solid substances taken separately. Now the potential in any of the liquids or solids for the substance which it yields in the form of gas has very nearly the same value when the liquid or solid is in equilibrium with the gas-mixture as when it is in equilibrium with its own gas alone. The difference of the pressure in the two cases will cause a certain difference in the values of the potential, but that this difference will be small, we may infer from the equation

$\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}=\left({\frac {dv}{dm_{1}}}\right)_{t,p,m},$

(272)

which may be derived from equation (92). In most cases, there will be a certain absorption by each liquid of the gases yielded by the

by its composition, except that the letters $A,B,C,$ and $D$ must in this case be understood to denote quantities which vary with the composition of the liquid. But to consider the case more in detail, we have for the liquid by (a)

${\frac {\zeta }{m}}=\mu =kt-kt\log t-Ht+Vp+E',$

where $k,H',V,E'$, denote quantities which depend only upon the composition of the liquid. Hence, we may write

$\zeta ={\text{k}}t-{\text{k}}t\log t-{\text{H}}t+{\text{V}}p+{\text{E}},$

where $k,H,V$, and $E$ denote functions of $m_{1},m_{2}$, etc. (the quantities of the several components of the liquid). Hence, by (92),

$\mu _{1}={\frac {d{\text{k}}}{dm_{1}}}t-{\frac {d{\text{k}}}{dm_{1}}}t\log t-{\frac {d{\text{H}}}{dm_{1}}}t+{\frac {d{\text{V}}}{dm_{1}}}p+{\frac {d{\text{E}}}{dm_{1}}}\cdot$

If the component to which this potential relates is that which also forms the gas, we shall have by (269)

$\log {\frac {p}{a}}={\frac {H-c-a}{a}}+{\frac {c+a}{a}}\log t+{\frac {\mu _{1}-E}{at}}\cdot$

Eliminating $A,B,C$, $D$ we obtain the equation

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

in which $A,B,C$, $D$ denote quantities which depend only upon the composition of the liquid, viz.:

$A={\frac {1}{a}}\left(H-{\frac {d{\text{H}}}{dm_{1}}}-c-a+{\frac {d{\text{k}}}{dm_{1}}}\right),$

$B={\frac {1}{a}}\left({\frac {d{\text{k}}}{dm_{1}}}-c-a\right),$

$C={\frac {1}{a}}\left(E-{\frac {d{\text{E}}}{dm_{1}}}\right),$$D={\frac {1}{a}}{\frac {d{\text{V}}}{dm_{1}}}\cdot$

With respect to some of the equations which have here been deduced, the reader may compare Professor Kirchhoff "Ueber die Spannung des Dampfes von Mischungen aus Wasser und Schwefelsäure," Pogg. Ann., vol. civ. (1858), p. 612; and Dr. Rankine "On Saturated Vapors," Phil. Mag., vol. xxxi. (1866), p. 199.