Page:Scientific Papers of Josiah Willard Gibbs.djvu/338

surfaces, by ${\displaystyle \sigma }$ and ${\displaystyle s}$ the tension and area of one of these surfaces, and by ${\displaystyle E}$ the elasticity of the film when extended under the supposition that the total quantities of ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ in the part of the film extended are invariable, as also the temperature and the potentials of the other components. From the definition of ${\displaystyle E}$ we have

${\displaystyle 2d\sigma =E{\frac {ds}{s}},}$

(643)

and from the conditions of the extension of the film

${\displaystyle {\frac {ds}{s}}=-{\frac {d(\lambda \gamma _{1})}{\lambda \gamma _{1}}}=-{\frac {d(\lambda \gamma _{2}+2\Gamma _{2(1)})}{\lambda \gamma _{2}+2\Gamma _{2(1)}}}\cdot }$

(644)

Hence we obtain

${\displaystyle \lambda \gamma _{1}{\frac {ds}{s}}=-\gamma _{1}d\lambda -\lambda d\gamma _{1},}$ ${\displaystyle (\lambda \gamma _{2}+2\Gamma _{2(1)}){\frac {ds}{s}}=-\gamma _{2}d\lambda -\lambda d\gamma _{2}-2d\Gamma _{2(1)},}$

and eliminating ${\displaystyle d\lambda }$,

${\displaystyle 2\gamma _{1}\Gamma _{2(1)}{\frac {ds}{s}}=-\lambda \gamma _{1}d\gamma _{2}+\lambda \gamma _{2}d\gamma _{1}-2\gamma _{1}d\Gamma _{2(1)}.}$

(645)

If we set

${\displaystyle r={\frac {\gamma _{2}}{\gamma _{1}}},}$

(646)

we have

${\displaystyle dr={\frac {\gamma _{1}d\gamma _{2}-\gamma _{2}d\gamma _{1}}{\gamma _{1}}},}$

(647)

and

${\displaystyle 2\Gamma _{2(1)}{\frac {ds}{s}}=-\lambda \gamma _{1}dr-2d\Gamma _{2(1)}.}$

(648)

With this equation we may eliminate ${\displaystyle d\sigma }$ from (643). We may also eliminate do- by the necessary relation (see (514))

${\displaystyle d\sigma =-\Gamma _{2(1)}d\mu _{2}.}$

This will give

${\displaystyle 4\Gamma _{2(1)}^{2}d\mu _{2}=E(\lambda \gamma _{1}dr+2d\Gamma _{2(1)}),}$

(649)

or

${\displaystyle {\frac {4\Gamma _{2(1)}^{2}}{E}}=\lambda \gamma _{1}{\frac {dr}{d\mu _{2}}}+2{\frac {d\Gamma _{2(1)}}{d\mu _{2}}},}$

(650)

where the differential coefficients are to be determined on the conditions that the temperature and all the potentials except ${\displaystyle \mu _{1}}$ and ${\displaystyle \mu _{2}}$ are constant, and that the pressure in the interior of the film shall remain equal to that in the contiguous gas-masses. The latter condition may be expressed by the equation

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

(651)

in which ${\displaystyle \gamma _{1}'}$ and ${\displaystyle \gamma _{2}'}$ denote the densities of ${\displaystyle S_{2}}$ and ${\displaystyle S_{2}}$ in the contiguous gas-masses. (See (98).) When the tension of the surfaces of the film and the pressures in its interior and in the contiguous