Page:Scientific Papers of Josiah Willard Gibbs.djvu/404

The term $\sigma s$ represents the work spent in forming the surface, and the term $(p''-p')v''$ the work gained in forming the interior mass. The second of these quantities is always equal to two-thirds of the first. The value of ${\text{W}}$ is therefore positive, and the phase is in strictness stable, the quantity ${\text{W}}$ affording a kind of measure of its stability. We may easily express the value of ${\text{W}}$ in a form which does not involve any geometrical magnitudes, viz.,

${\text{W}}={\frac {16\pi \sigma ^{2}}{3(p''-p')^{2}}},$

(29)

where $p'',p'$ and $\sigma$ may be regarded as functions of the temperature and potentials. It will be seen that the stability, thus measured, is infinite for an infinitesimal difference of pressures, but decreases very rapidly as the difference of pressures increases. These conclusions are all, however, practically limited to the case in which the value of $r$, as determined by equation (27), is of sensible magnitude.

With respect to the somewhat similar problem of the stability of the surface of contact of two phases with respect to the formation of a new phase, the following results are obtained. Let the phases (supposed to have the same temperature and potentials) be denoted by ${\text{A}},{\text{B}},$, and ${\text{C}}$; their pressures by $p_{\text{A}},p_{\text{B}}$ and $p_{\text{C}}$; and the tensions of the three possible surfaces by $\sigma _{\text{AB}},\sigma _{\text{BC}},\sigma _{\text{AC}}$. If $p_{\text{C}}$ is less than

${\frac {\sigma _{\text{BC}}p_{\text{A}}+\sigma _{\text{AC}}p_{\text{B}}}{\sigma _{\text{BC}}+\sigma _{\text{AC}}}},$

there will be no tendency toward the formation of the new phase at the surface between ${\text{A}}$ and ${\text{B}}$. If the temperature or potentials are now varied until $p_{\text{C}}$ is equal to the above expression, there are two cases to be distinguished. The tension $\sigma _{\text{AB}}$ will be either equal to $\sigma _{\text{AC}}+\sigma _{\text{BC}}$ or less. (A greater value could only relate to an unstable and therefore unusual surface.) If $\sigma _{\text{AB}}=\sigma _{\text{AC}}+\sigma _{\text{BC}},$ a farther variation of the temperature or potentials, making $p_{\text{C}}$ greater than the above expression, would cause the phase ${\text{C}}$ to be formed at the surface between ${\text{A}}$ and ${\text{B}}$. But if $\sigma _{\text{AB}}<\sigma _{\text{AC}}+\sigma _{\text{BC}},$ the surface between ${\text{A}}$ and ${\text{B}}$ would remain stable, but with rapidly diminishing stability, after $p_{\text{C}}$ has passed the limit mentioned.

The conditions of stability for a line where several surfaces of discontinuity meet, with respect to the possible formation of a new surface, are capable of a very simple expression. If the surfaces ${\text{A-B}},{\text{B-C}},{\text{C-D}},{\text{C-D}},{\text{D-A}}$, separating the masses ${\text{A, B, C, D}}$, meet along a line, it is necessary for equilibrium that their tensions and directions at any point of the line should be such that a quadrilateral $\alpha ,\beta ,\gamma ,\delta$ may be formed with sides representing in direction and length the normals and tensions of the successive surfaces. For the stability