Page:Scientific Papers of Josiah Willard Gibbs.djvu/79

curvatures, but somewhere falls below the tangent plane drawn through the point, the equilibrium although unstable in regard to discontinuous changes of state is stable in regard to continuous changes, as appears on restricting the test of stability to the vicinity of the point in question; that is, if we suppose a body to be in a state represented by such a point, although the equilibrium would show itself unstable if we should introduce into the body a small portion of the same substance in one of the states represented by points below the tangent plane, yet if the conditions necessary for such a discontinuous change are not present, the equilibrium would be stable. A familiar example of this is afforded by liquid water when heated at any pressure above the temperature of boiling water at that pressure.^[1]

Leading Features of the Thermodynamic Surface for Substances which take the forms of Solid, Liquid, and Vapor.

We are now prepared to form an idea of the general character of the primitive and derived surfaces and their natural relations for a substance which takes the forms of solid, liquid, and vapor. The primitive surface will have a triple tangent plane touching it at the three points which represent the three states which can exist in contact. Except at these three points, the primitive surface falls entirely above the tangent plane. That part of the plane which forms a triangle having its vertices at the three poits of contact, is the derived surface which represents a compound of the three states of the substance. We may suppose the plane to roll on the under side of the surface, continuing to touch it in two points without cutting it. This it may do in three ways, viz: it may commence by turning about any of the sides of the triangle aforesaid. Any pair of points which the plane touches at once represent states which can exist permanently on contact. In this way six lines are traced upon the surface. These lines have in general a common property, that a tangent plane at any point in them will also touch the surface in another point. We must say in general, for, as we shall see hereafter, this statement does not hold good for the critical point. A tangent plane at any point of the surface outside of these lines has the surface

1. If we wish to express in a single equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when surrounded by a medium of constant pressure $P$ and temperature $T$, this equation may be written

   $\delta (\epsilon -T\eta +Pv)=0,$

   when $\delta$ refers to the variation produced by any variations in the state of the parts of the body, and (when different parts of the body are in different states) in the proportion in which the body is divided between the different states. The condition of stable equilibrium is that the value of the expression in the parenthesis shall be a minimum.