Page:Scientific Papers of Josiah Willard Gibbs.djvu/272

The same physical relations may of course be deduced without giving up the use of the surface of tension as a dividing surface, but the formulæ which express them will be less simple. If we make $t,\mu _{3},\mu _{4}$, etc., constant, we have by (98) and (508)

$dp'=\gamma _{1}'d\mu _{1}+\gamma _{2}'d\mu _{2},$ $dp''=\gamma _{1}''d\mu _{1}+\gamma _{2}''d\mu _{2},$ $d\sigma =-\Gamma _{1}d\mu _{1}-\Gamma _{2}d\mu _{2},$

where we may suppose $\Gamma _{1}$ and $\Gamma _{2}$ to be determined with reference to the surface of tension. Then, if $dp'=dp''$,

$(\gamma _{1}'-\gamma _{1}'')d\mu _{1}+(\gamma _{2}'-\gamma _{2}'')d\mu _{2}=0,$

and

$d\sigma =\Gamma _{1}{\frac {\sigma _{2}'-\sigma _{2}''}{\sigma _{1}'-\sigma _{1}''}}d\mu _{2}-\Gamma _{2}d\mu _{2}.$

That is,

$\left({\frac {d\sigma }{d\mu _{2}}}\right)_{p'-p'',t,\mu _{3},\mu _{4},etc.}=-\Gamma _{2}+\Gamma _{1}{\frac {\gamma _{2}'-\gamma _{2}''}{\gamma _{1}'-\gamma _{1}''}}\cdot$

(515)

The reader will observe that ${\frac {\Gamma _{1}}{\gamma _{1}'-\gamma _{1}''}}$ represents the distance between the surface of tension and that dividing surface which would make $\Gamma _{1}=0$; the second number of the last equation is therefore equivalent to $-\Gamma _{2(1)}$.

If any component substance has the same density in the two homogeneous masses separated by a plane surface of discontinuity, the value of the superficial density for that component is independent of the position of the dividing surface. In this case alone we may derive the value of the superficial density of a component with reference to the surface of tension from the fundamental equation for plane surfaces alone. Thus in the last equation, when $\gamma _{2}'=\gamma _{2}''$, the second member will reduce to $-\Gamma _{2}$. It will be observed that to

in mass, will be equal to the sum of the superficial tensions of mercury in contact with water and of water in contact with its own vapor. This will be, according to the same authority, 42.58 + 8.25, or 50.83 grammes per meter, if we neglect the difference of the tensions of water with its vapor and water with air. As $p_{2}$, therefore, increases from zero to 236400 grammes per square meter (when water begins to be condensed in mass), $\sigma$ diminishes from about 55.03 to about 50.83 grammes per linear meter. If the general course of the values of $\sigma$ for intermediate values of $p_{2}$ were determined by experiment, we could easily form an approximate estimate of the values of the superficial density $\Gamma _{2(1)}$ for different pressures less than that of saturated vapor. It will be observed that the determination of the superficial density does not by any means depend upon inappreciable differences of superficial tension. The greatest difficulty in the determination would doubtless be that of distinguishing between the diminution of superficial tension due to the water and that due to other substances which might accidentally be present. Such determinations are of considerable practical importance on account of the use of mercury in measurements of the specific gravity of vapors.