Page:Scientific Papers of Josiah Willard Gibbs.djvu/157

measured toward the top of the page from $P_{1}P{2},m_{1}$ toward the left from $P_{2}Q_{2},$ and $m_{2}$ toward the right from $P_{1}Q_{1}.$ It is supposed that $P_{1}P_{2}=1.$ Portions of the curves to which these points belong are seen in the figure, and will be denoted by the symbols $(A),(B),(C).$ We may, for convenience, speak of these as separate curves, without implying anything in regard to their possible continuity in parts of the diagram remote from their common tangent $AC$. The line of dissipated energy includes the straight line $AC$ and portions of the primitive curves $(A)$ and $(C)$. Let us first consider how the diagram will be altered, if the temperature is varied while the pressure remains constant. If the temperature receives the increment $dt$, an ordinate of which the position is fixed will receive the increment $\left({\frac {d\zeta }{dt}}\right)_{p,m}$ or $-\eta dt.$ (The reader will easily convince himself that this is true of the ordinates for the secondary line $AC$, as well as of the ordinates for the primitive curves.) Now if we denote by $\eta '$ the entropy of the phase represented by the point $B$ considered as belonging to the curve $(B)$, and by $\eta ''$ the entropy of the composite state of the same matter represented by the point $B$ considered as belonging to the tangent to the curves $(A)$ and $(C)$, $t(\eta '-\eta '')$ will denote the heat yielded by a unit of matter in passing from the first to the second of these states. If this quantity is positive, an elevation of temperature will evidently cause a part of the curve $(B)$ to protrude below the tangent to $(A)$ and $(C)$, which will no longer form a part of the line of dissipated energy. This line will then include portions of the three curves $(A),(B),$ and $(C),$ and of the tangents to $(A)$ and $(B)$ and to $(B)$ and $(C)$. On the other hand, a lowering of the temperature will cause the curve $(B)$ to lie entirely above the tangent to $(A)$ and $(C)$, so that all the phases of the sort represented by $(B)$ will be unstable. If $t(\eta '-\eta '')$ is negative, these effects will be produced by the opposite changes of temperature.

The effect of a change of pressure while the temperature remains constant may be found in a manner entirely analogous. The variation of any ordinate will be $\left({\frac {d\zeta }{dp}}\right)_{t,m}$ or $vdp$. Therefore, if the volume of the homogeneous phase represented by the point $B$ is greater than the volume of the same matter divided between the phases represented by $A$ and $C$, an increase of pressure will give a diagram indicating that all phases of the sort represented by curve