Page:Scientific Papers of Josiah Willard Gibbs.djvu/402

of discontinuity. Now if we compare the actual state of things with the supposed state, there will be in the former in the vicinity of the surface a certain (positive or negative) excess of energy, of entropy, and of each of the component substances. These quantities are denoted by $\epsilon ^{\text{S}},\eta ^{\text{S}},m_{1}^{\text{S}},m_{2}^{\text{S}}$, etc., and are treated as belonging to the surface. The ^{$^{\text{S}}$} is used simply as a distinguishing mark, and must not be taken for an algebraic exponent.

It is shown that the conditions of equilibrium already obtained relating to the temperature and the potentials of the homogeneous masses, are not affected by the surfaces of discontinuity, and that the complete value of $\delta \epsilon ^{\text{S}}$ is given by the equation

$\delta \epsilon ^{\text{S}}=t\delta \eta ^{\text{S}}+\sigma \delta s+\mu _{1}\delta m_{1}^{\text{S}}+\mu _{2}\delta m_{2}^{\text{S}}+{\text{etc.,}}$

(23)

in which $s$ denotes the area of the surface considered, $t$ the temperature, $\mu _{1},\mu _{2}$, etc., the potentials for the various components in the adjacent masses. It may be, however, that some of the components are found only at the surface of discontinuity, in which case the letter $\mu$ with the suffix relating to such a substance denotes, as the equation shows, the rate of increase of energy at the surface per unit of the substance added, when the entropy, the area of the surface, and the quantities of the other components are unchanged. The quantity $\sigma$ we may regard as defined by the equation itself, or by the following, which is obtained by integration:

$\epsilon ^{\text{S}}=t\eta ^{\text{S}}+\sigma s+\mu _{1}m_{1}^{\text{S}}+\mu _{2}m_{2}^{\text{S}}+{\text{etc.}}$

(24)

There are terms relating to variations of the curvatures of the surface which might be added, but it is shown that we can give the dividing surface such a position as to make these terms vanish, and it is found convenient to regard its position as thus determined. It is always sensibly coincident with the physical surface of discontinuity. (Yet in treating of plane surfaces, this supposition in regard to the position of the dividing surface is unnecessary, and it is sometimes convenient to suppose that its position is determined by other considerations.)

With the aid of (23), the remaining condition of equilibrium for contiguous homogeneous masses is found, viz.,

$\sigma (c_{1}+c_{2})=p'-p'',$

(25)

where $p',p''$ denote the pressures in the two masses, and $c_{1},c_{2}$ the principal curvatures of the surface. Since this equation has the same form as if a tension equal to $\sigma$ resided at the surface, the quantity $\sigma$ is called (as is usual) the superficial tension, and the dividing surface in the particular position above mentioned is called the surface of tension.

By differentiation of (24) and comparison with (23), we obtain

$d\sigma =-\eta _{\text{S}}dt-\Gamma _{1}d\mu _{1}-\Gamma _{2}d\mu _{2}-{\text{etc.,}}$

(26)