Page:Scientific Papers of Josiah Willard Gibbs.djvu/339

gas-masses are known in terras of the temperature and potentials, equation (650) will give the value of $E$ in terms of the same variables together with $\lambda$.

If we write $G_{1}$ and $G_{2}$ for the total quantities of $S_{1}$ and $S_{2}$ per unit of area of the film, we have

$G_{1}=\lambda \gamma _{1},$

(652)

$G_{2}=\lambda \gamma _{2}+2\Gamma _{2(1)}.$

(653

Therefore,

$G_{2}=G_{1}r+2\Gamma _{2(1)},$

$\left({\frac {dG_{2}}{d\mu _{2}}}\right)_{G_{1}}=\lambda \gamma _{1}{\frac {dr}{d\mu _{2}}}+2{\frac {d\Gamma _{2(1)}}{d\mu _{2}}},$

(654)

where the differential coefficients in the second member are to be determined as in (650), and that in the first member with the additional condition that $G_{1}$ is constant. Therefore,

${\frac {4\Gamma _{2(1)}^{2}}{E}}=\left({\frac {dG_{2}}{d\mu _{2}}}\right)_{G_{1}},$

and

$E=4\Gamma _{2(1)}^{2}\left({\frac {dG_{2}}{d\mu _{2}}}\right)_{G_{1}},$

(655)

the last differential coefficient being determined by the same conditions as that in the preceding equation. It will be observed that the value of $E$ will be positive in any ordinary case.

These equations give the elasticity of any element of the film when the temperature and the potentials for the substances which are found in the contiguous gas-masses are regarded as constant, and the potentials for the other components, $\mu _{1}$ and $\mu _{2}$, have had time to equalize themselves throughout the element considered. The increase of tension immediately after a rapid extension will be greater than that given by these equations.

The existence of this elasticity, which has thus been established from a priori considerations, is clearly indicated by the phenomena which liquid films present. Yet it is not to be demonstrated simply by comparing the tensions of films of different thickness, even when they are made from the same liquid, for difference of thickness does not necessarily involve any difference of tension. When the phases within the films as well as without are the same, and the surfaces of the films are also the same, there will be no difference of tension. Nor will the tension of the same film be altered, if a part of the interior drains away in the course of time, without affecting the surfaces. In case the thickness of the film is reduced by evaporation, the tension may be either increased or diminished. (The evaporation of the substance $S_{1}$ in the case we have just considered, would diminish the tension.) Yet it may easily be shown that