Page:Scientific Papers of Josiah Willard Gibbs.djvu/60

24 and in part vapor. The properties of such a mixture are very simple and clearly exhibited in the volume-entropy diagram.

Let the temperature and the pressure of the mixture, which are independent of the proportions of vapor, solid and liquid, be denoted by $$t'$$ and $$p'$$. Also let $$V, L$$ and $$S$$ (fig. 9) be points of the diagram which indicate the volume and entropy of the body in three perfectly defined states, viz: that of a vapor of temperature $$t'$$ and pressure $$p'$$, that of a liquid of the same temperature and pressure, and that of a solid of the same temperature and pressure. And let $$v_{V}, \eta_{V}, v_{L}, \eta_{L}, v_{S}, \eta_{S}$$ denote the volume and entropy of these states. The position of the point which represents the body, when part is vapor, part liquid, and part solid, these parts being as $$\mu, \nu$$, and $$1 - \mu - \nu$$, is determined by the equations  where $$v$$ and $$\eta$$ are the volume and entropy of the mixture. The truth of the first equation is evident. The second may be written or multiplying by $$t'$$  The first member of this equation denotes the heat necessary to bring the body from the state $$S$$ to the state of the mixture in question under the constant temperature $$t'$$, while the terms of the second member denote separately the heat necessary to vaporize the part $$\mu$$, and to liquefy the part $$\nu$$ of the body.

The values of $$v$$ and $$\eta$$ are such as would give the center of gravity of masses $$\mu, \nu$$ and $$1 - \mu - \nu$$ placed at the points $$V, L$$ and $$S$$. Hence the part of the diagram which represents a mixture of vapor, liquid and solid, is the triangle $$VLS$$. The pressure and temperature are constant for this triangle, i.e., an isopiestic and also an isothermal here expand to cover a space. The isodynamics are straight and equidistant for equal differences of energy. For $$\frac{d\epsilon}{dv} = -p'$$ and $$\frac{d\epsilon}{d\eta} = t'$$, both of which are constant throughout the triangle.