Page:Scientific Papers of Josiah Willard Gibbs.djvu/66

This page has been proofread, but needs to be validated.

30

GRAPHICAL METHODS IN THE

$A$ (fig. 13), let there be drawn the isometric $vv'$ and the isentropic $\eta \eta ^{\prime }$ , and let the positive sides of these lines be indicated as in the figure. The conditions $\left({\frac {dp}{d\eta }}\right)_{v}>0$ and $[dp:dv]_{\eta }<0$ require that the pressure at $v$ and $\eta$ shall be greater than at $A$ , and hence that the isopiestic shall fall as $pp'$ in the figure, and have its positive side turned as indicated. Again, the conditions $\left({\frac {dp}{d\eta }}\right)_{v}>0$ and $[dp:dv]_{\eta }<0$ require that the temperature at $\eta$ and at $p$ shall be greater than at $A$ , and hence, that the isothermal shall fall as $tt'$ and have its positive side turned as indicated. As it is not necessary that $\left({\frac {dp}{d\eta }}\right)_{v}>0$ , the lines $pp'$ and $tt'$ may be tangent to one another at $A$ , provided that they cross one another, so as to have the same order about the point $A$ as is represented in the figure; i.e., they may have a contact of the second (or any even) order.^[1] But the condition that $\left({\frac {dp}{d\eta }}\right)_{v}>0$ , and hence $\left({\frac {dt}{dv}}\right)_{v\eta }<0$ , does not allow $pp'$ to be tangent to $vv'$ , nor $tt'$ to $\eta \eta ^{\prime }$ .

If $\left({\frac {dp}{d\eta }}\right)_{v}$ be still positive, but the equilibrium be neutral, it will be possible for the body to change its state without change either of temperature or of pressure; i.e., the isothermal and isopiestic will be identical. The lines will fall as in figure 13, except that the isothermal and isopiestic will be superimposed. In like manner, if $\left({\frac {dp}{d\eta }}\right)_{v}<0$ , it may be proved that the lines will fall as in figure 14 for stable equilibrium, and in the same way for neutral equilibrium, except that $pp'$ and $tt'$ will be superposed.^[2]

↑ An example of this is doubtless to be found at the critical point of a fluid. See Dr. Andrews "On the continuity of the gaseous and liquid states of matter." Phil. Trans., vol 159, p. 575. If the isothermal and isopiestic have a simple tangency at $A$ , on one side of that point they will have such directions as will express an unstable equilibrium. A line drawn through all such points in the diagram will form a boundary to the possible part of the diagram. It may be that the part of the diagram of a fluid, which represents the superheated liquid state, is bounded on one side by such a line.
↑ When it is said that the arrangement of the lines in the diagram must be like that in figure 13 or in figure 14, it is not meant to exclude the case in which the figure (13 or 14) must be turned over, in order to correspond with the diagram. In the case, however, of diagrams formed by any of the methods mentioned in this article, if the

[1] An example of this is doubtless to be found at the critical point of a fluid. See Dr. Andrews "On the continuity of the gaseous and liquid states of matter." Phil. Trans., vol 159, p. 575. If the isothermal and isopiestic have a simple tangency at $A$ , on one side of that point they will have such directions as will express an unstable equilibrium. A line drawn through all such points in the diagram will form a boundary to the possible part of the diagram. It may be that the part of the diagram of a fluid, which represents the superheated liquid state, is bounded on one side by such a line.

[V1Page30-2] When it is said that the arrangement of the lines in the diagram must be like that in figure 13 or in figure 14, it is not meant to exclude the case in which the figure (13 or 14) must be turned over, in order to correspond with the diagram. In the case, however, of diagrams formed by any of the methods mentioned in this article, if the

[1]

[2]