Page:Scientific Papers of Josiah Willard Gibbs.djvu/65

Arrangement of the Isometric, Isopiestic, Isothermal and Isentropic about a Point.

The arrangement of the isometric, the isopiestic, the isothermal and the isentropic drawn through any same point, in respect to the order in which they succeed one another around that point, and in respect to the sides of these lines toward which the volume, pressure, temperature and entropy increase, is not altered by any deformation of the surface on which the diagram is drawn, and is therefore independent of the method by which the diagram is formed.^[1] This arrangement is determined by certain of the most characteristic thermodynamic properties of the body in the state in question, and serves in turn to indicate these properties. It is determined, namely, by the value $\left({\frac {dp}{d\eta }}\right)_{v}$ as positive, negative, or zero, i.e., by the effect of heat as increasing or diminishing the pressure when the volume is maintained constant, and by the nature of the internal thermodynamic equilibrium of the body as stable or neutral,—an unstable equilibrium, except as a matter of speculation, is of course out of the question.

Let us first examine the case in which $\left({\frac {dp}{d\eta }}\right)_{v}$ is positive and the equilibrium is stable. As $\left({\frac {dp}{d\eta }}\right)_{v}$ does not vanish at the point in question, there is a definite isopiestic passing through that point, on one side of which the pressures are greater, and on the other less, than on the line itself. As $\left({\frac {dt}{dv}}\right)_{v}=-\left({\frac {dp}{d\eta }}\right)_{v}$, the case is the same with the isothermal. It will be convenient to distinguish the sides of the isometric, isopiestic, etc., on which the volume, pressure, etc., increase, as the positive sides of these lines. The condition of stability requires that, when the pressure is constant, the temperature shall increase with the heat received,—therefore with the entropy. This may be written $[dt:d\eta ]_{p}>0.$^[2] It also requires that, when there is no transmission of heat, the pressure should increase as the volume diminishes, i.e., that $[dp:dv]_{\eta }<0$. Through the point in question,

1. It is here assumed that, in the vicinity of the point in question, each point in the diagram represents only one state of the body. The propositions developed in the following pages cannot be applied to points of the line where two superposed diagrams are united (see pages 25–28) without certain modifications.
2. As the notation ${\frac {dt}{d\eta }}$ is used to denote the limit of the ratio of $dt$ to $d\eta$, it would not be quite accurate to say that the condition of stability requires that the $\left({\frac {dt}{d\eta }}\right)_{p}>0$. This condition requires that the ratio of the differences of temperature and entropy between the point in question and any other infinitely near to it and upon the same isopiestic should be positive. It is not necessary that the limit of this ratio should be positive.