Page:Scientific Papers of Josiah Willard Gibbs.djvu/61

This case cane be but very imperfectly represented in the volume-presure, or in the entropy-energy diagram. For all points in the same vertical line in the triangle $VLS$ will, in the volume-pressure diagram, be represented by a single point, as having the same volume and pressure. And all the points in the same horizontal line will be represented in the entropy-temperature diagram by a single point, as having the same entropy and temperature. In either diagram, the whole triangle reduces to a straight line. It must reduce to a line in any diagram whatever of constant scale, as its area must become 0 in such a diagram. This must be regarded as a defect in these diagrams, as essentially differens states are represented by the same point. In consequence, any circuit within the triangle $VLS$ will be represented in any diagram of constant scale by two paths of opposite directions superposed, the appearance being as if a body should change its state and then return to its original state by inverse processes, so as to repass through the same series of states. It is true that the circuit in question is like this combination of processes in one important particular, viz: that $W=H=0$, i.e., there is no transformation of heat into work. But this very fact, that a circuit without transformation of heat into work is possible, is worthy of distinct representation.

A body may have such properties that in one part of the volume-entropy diagram ${\frac {1}{\gamma _{v,\eta }}}$, i.e., ${\frac {dp}{d\eta }}$ is positive and in another negative. These parts of the diagram may be separated by a line, in which ${\frac {dp}{d\eta }}=0$, or by one in which ${\frac {dp}{d\eta }}$ changes abruptly from a positive to a negative value.^[1] (In part, also, they may be separated by an area in which ${\frac {dp}{d\eta }}=0$.) In the representation of such cases in any diagram of constant scale, we meet with a difficulty of the following nature.

Let us suppose that on the right of the line $LL$ (fig. 10) in a volume-entropy diagram, ${\frac {dp}{d\eta }}$ is positive, and on the left negative. Then, if we draw any circuit $ABCD$ on the right side of $LL$, the direction

1. The line which represents the various states at its maximum density for various constant pressures is an example of the first case. A substance which as a liquid has no proper maximum density for constant pressure, but which expands in solidifying, affords an example of the second case.