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

Rh 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