Page:Scientific Papers of Josiah Willard Gibbs.djvu/355

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EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.

319

the solid with isotropic stresses.^[1] It seems probable, however, that if the fluid in contact with the solid is not renewed, the system will generally find a state of equilibrium in which the outermost portion of the solid will be in a state of isotropic stress. If at first the solid should dissolve, this would supersaturate the fluid, perhaps until a state is reached satisfying the condition of equilibrium with the stressed solid, and then, if not before, a deposition of solid matter in a state of isotropic stress would be likely to commence and go on until the fluid is reduced to a state of equilibrium with this new solid matter.

The action of gravity will not affect the nature of the condition of equilibrium for any single point at which the fluid meets the solid, but it will cause the values of $p''$ and $\mu _{1}''$ in (661) to vary according to the laws expressed by (612) and (617). If we suppose that the outer part of the solid is in a state of isotropic stress, which is the most important case, since it is the only one in which the equilibrium is in every sense stable, we have seen that the condition (661) is at least sensibly equivalent to this:—that the potential for the substance of the solid which would belong to the solid mass at the temperature $t$ and the pressure $p''+(c_{1}+c_{2})\sigma$ must be equal to $\mu _{1}''$ . Or, if we denote by $(p')$ the pressure belonging to solid with the temperature $t$ and the potential equal to $\mu _{1}''$ , the condition may be expressed in the form

(p')=p''+(c_{1}+c_{2})\sigma .

(662)

Now if we write $\gamma ''$ for the total density of the fluid, we have by (612)

dp''=-g\gamma ''dz.

By (98)

d(p')=\gamma _{1}'d\mu _{1}'',

and by (617)

d\mu _{1}''=-gdz;

whence

d(p')=-g\gamma _{1}dz.

Accordingly we have

d(p')-dp''=g(\gamma ''-\gamma _{1}')dz,

and

(p')-p''=g(\gamma ''-\gamma _{1}')z,

$z$ being measured from the horizontal plane for which $(p')=p''$ . Substituting this value in (662), we obtain

c_{1}+c_{2}={\frac {g(\gamma ''-\gamma _{1}')}{\sigma }}z,

(663)

↑ The possibility that the new solid matter might differ in composition from the original solid is here left out of account. This point has been discussed on pages 79–82, but without reference to the state of strain of the solid or the influence of the curvature of the surface of discontinuity. The statement made above may be generalized so as to hold true of the formation of new solid matter of any kind on the surface as follows:—that new solid matter of any kind will not be formed upon the surface (with more than insensible thickness), if the second member of (661) calculated for such new matter is greater than the potential in the fluid for such matter.

[1] The possibility that the new solid matter might differ in composition from the original solid is here left out of account. This point has been discussed on pages 79–82, but without reference to the state of strain of the solid or the influence of the curvature of the surface of discontinuity. The statement made above may be generalized so as to hold true of the formation of new solid matter of any kind on the surface as follows:—that new solid matter of any kind will not be formed upon the surface (with more than insensible thickness), if the second member of (661) calculated for such new matter is greater than the potential in the fluid for such matter.

[1]