Page:Scientific Papers of Josiah Willard Gibbs.djvu/64

28 Of the two superposed diagrams, that which represents simple liquid is a continuation of the diagram on the left of $$MM$$. The isopiestics, isothermals and isodynamics pass from one to the other without abrupt change of direction or curvature. But that which represents a mixture of vapor and liquid will be different in its character, and its isopiestics and isothermals will make angles in general with the corresponding lines in the diagram of simple liquid. The isodynamics of the diagram of the mixture, and those of the diagram of simple liquid, will differ in general in curvature at the line $$MM$$, but not in direction, for $$\frac{d\epsilon}{dv} = -p$$ and $$\frac{d\epsilon}{d\eta} = t\cdot$$

The case is essentially the same with some substances, as water, for example, about the line which separates the simple liquid from a mixture of a liquid and solid.

In these cases the inconvenience of having one diagram superposed upon another cannot be obviated by any change of the principle on which the diagram is based. For no distortion can bring the three sheets, which are united along the line $$MM$$ (one on the left and two on the right), into a single plane surface without superstition. Such cases, therefore, are radically distinguished from those in which the superposition is caused by an unsuitable method of representation.

To find the character of a volume-entropy diagram of a perfect gas, we may make $$\epsilon$$ constant in equation on page 13, which will give for the equation of an isodynamic and isothermal  and we may make $$p$$ constant in equation, which will give for the equation of an isopiestic  It will be observed that all the isodynamics and isothermals can be drawn by a single pattern and so also with the isopiestics.

The case will be nearly the same with vapors in a part of the diagram. In that part of the diagram which represents a mixture of liquid and vapor, the isothermals, which of course are identical with the isopiestics, are straight lines. For when a body is vaporized under constant pressure and temperature, the quantities of heat received are proportional to the increments of volume; therefore, the increments of entropy are proportional to the increments of volume. As $$\frac{d\epsilon}{dv} = -p$$ and $$\frac{d\epsilon}{d\eta} = t$$, any isothermal is cut at the same angle by all the isodynamics, and is divided into equal segments by equidifferent isodynamics. The latter property is useful in drawing systems of equidifferent isodynamics.