Page:Scientific Papers of Josiah Willard Gibbs.djvu/197

Now if the gas is an ideal gas-mixture,

$d\mu _{1}={\frac {a_{1}t}{p_{1}}}dp_{1}={\frac {dp_{1}}{\gamma _{1}}},$ and $d\mu _{2}={\frac {a_{2}t}{p_{2}}}dp_{2}={\frac {dp_{2}}{\gamma _{2}}}$

therefore$\left({\frac {\gamma _{1}^{(L)}}{\gamma _{1}}}-1\right)dp_{1}=\left(1-{\frac {\gamma _{2}^{(L)}}{\gamma _{1}}}\right)dp_{2}.$

(288)

We may now suppose that $S_{1}$ is the principal component of the liquid, and $S_{2}$ is a gas which is absorbed in the liquid to a slight extent. In such cases it is well known that the ratio of the densities of the substance $S_{2}$ in the liquid and in the gas is for a given temperature approximately constant. If we denote this constant by $A$, we shall have

$\left({\frac {\gamma _{1}^{(L)}}{\gamma _{1}}}\right)dp_{1}=(1-A)dp_{2}.$

(289)

It would be easy to integrate this equation regarding $\gamma _{1}$ as variable, but as the variation in the value of $p_{1}$ is necessarily very small we shall obtain sufficient accuracy if we regard $\gamma _{1}$ as well as $\gamma _{1}^{(L)}$ as constant. We shall thus obtain

$\left({\frac {\gamma _{1}^{(L)}}{\gamma _{1}}}-1\right)(p_{1}-p'_{1})=(1-A)p_{2},$

(290)

where $p'_{1}$ denotes the pressure of the saturated vapor of the pure liquid consisting of $S_{1}$. It will be observed that when $A=1$, the presence of the gas $S_{2}$ will not affect the pressure or density of the gas $S_{1}$. When $A<1$, the pressure and density of the gas $S_{1}$ are greater than if $S_{2}$ were absent, and when $A>1$, the reverse is true.

The properties of an ideal gas-mixture (according to the definition which we have assumed) when in equilibrium with liquids or solids have been developed at length, because it is only in respect to these properties that there is any variation from the properties usually attributed to perfect gases. As the pressure of a gas saturated with vapor is usually given as a little less than the sum of the pressure of the gas calculated from its density and that of saturated vapor in a space otherwise empty, while our formulæ would make it a little more, when the gas is insoluble, it would appear that in this respect our formulæ are less accurate than the rule which would make the pressure of the gas saturated with vapor equal to the sum of the two pressures mentioned. Yet the reader will observe that the magnitude of the quantities concerned is not such that any stress can be laid upon this circumstance.

It will also be observed that the statement of Dalton's law which we have adopted, while it serves to complete the theory of gas-mixtures (with respect to a certain class of properties), asserts nothing