Page:Scientific Papers of Josiah Willard Gibbs.djvu/302

evident that ${\frac {\Gamma }{\gamma '-\gamma ''}}$ represents the distance from the surface of tension to a dividing surface located so as to make the superficial density of the single component vanish (being positive, when the latter surface is on the side specified by the double accents), and that the coefficient of $dt$ (without the negative sign) represents the superficial density of entropy as determined by the latter dividing surface, i.e., the quantity denoted by $\eta _{S(1)}$ 8(l) on page 235.

When there are two components, neither of which is confined to the surface of discontinuity, we may regard the tension as a function of the temperature and the pressures in the two homogeneous masses. The values of the differential coefficients of the tension with respect to these variables may be represented in a simple form if we choose such substances for the components that in the particular state considered each mass shall consist of a single component. This will always be possible when the composition of the two masses is not identical, and will evidently not affect the values of the differential coefficients. We then have

$d\sigma =-\eta _{S}dt-\Gamma _{'}d\mu _{'}-\Gamma _{''}d\mu _{''}$ $dp'=\eta _{V}'dt+\gamma 'd\mu _{'},$ $dp''=\eta _{V}''dt+\gamma ''d\mu _{''},$

where the marks $_{'}$ and $_{''}$ are used instead of the usual $_{1}$ and $_{2}$ to indicate the identity of the component specified with the substance of the homogeneous masses specified by $'$ and $''$. Eliminating $d\mu _{'}$, and $d\mu _{''}$ a we obtain

$d\sigma =-\left(\eta _{S}-{\frac {\Gamma _{'}}{\gamma '}}\eta _{V}'-{\frac {\Gamma _{''}}{\gamma ''}}\eta _{V}''\right)dt-{\frac {\Gamma _{'}}{\gamma '}}dp'-{\frac {\Gamma _{''}}{\gamma ''}}dp''.$

(579)

We may generally neglect the difference of $p'$ and $p''$, and write

$d\sigma =-\left(\eta _{S}-{\frac {\Gamma _{'}}{\gamma '}}\eta _{V}'-{\frac {\Gamma _{''}}{\gamma ''}}\eta _{V}''\right)dt-\left({\frac {\Gamma _{'}}{\gamma '}}+{\frac {\Gamma _{''}}{\gamma ''}}\right)dp.$

(580)

The equation thus modified is strictly to be regarded as the equation for a plane surface. It is evident that ${\frac {\Gamma _{'}}{\gamma '}}$ and ${\frac {\Gamma _{'}}{\gamma ''}}$ represent the distances from the surface of tension of the two surfaces of which one would make $\Gamma _{'}$ vanish, and the other $\Gamma _{''}$, that ; ${\frac {\Gamma _{'}}{\gamma '}}+{\frac {\Gamma _{''}}{\gamma ''}}$ represents the distance between these two surfaces, or the diminution of volume due to a unit of the surface of discontinuity, and that the coefficient of $dt$ (without the negative sign) represents the excess of entropy in a system consisting of a unit of the surface of discontinuity with a part of each of the adjacent masses above that which the same matter would have if it existed in two homogeneous masses of the