Page:Scientific Papers of Josiah Willard Gibbs.djvu/56

The properties of the part of the entropy-temperature diagram representing a mixture of vapor and liquid, which are given on page 14, will evidently not be altered if the ordinates are made proportional to the logarithms of the temperatures instead of the temperatures simply.

The representation of specific heat in the diagram under discussion is peculiarly simple. The specific heat of any substance at constant volume or under constant pressure may be defined as the value of

$\left({\frac {dH}{dt}}\right)_{v}\,\,or\,\,\left({\frac {dH}{dt}}\right)_{p},\,\,i.e.,\,\,\left({\frac {d\eta }{d\log t}}\right)_{v}\,\,or\,\,\left({\frac {d\eta }{d\log t}}\right)_{p},$

for a certain quantity of the substance. Therefore, if we draw a diagram, in which $x=\eta$ and $y=\log t$, for that quantity of the substance which is used for the determination of the specific heat, the tangents of the angles made by the isometrics and the isopiestics with the ordinates in the diagram will be equal to the specific heat of the substance determined for constant volume and for constant pressure respectively. Sometimes, instead of the condition of constant volume or constant pressure, some other condition is used in the determination of specific heat. In all cases, the condition will be represented by a line in the diagram, and the tangent of the angle made by this line with an ordinate will be equal to the specific heat as thus defined. If the diagram be drawn for any other quantity of the substance, the specific heat for constant volume or constant pressure, or for any other condition, will be equal to the tangent of the proper angle in the diagram, multiplied by the ratio of the quantity of the substance for which the specific heat is determined to the quantity for which the diagram is drawn.^[1]

The Volume-entropy Diagram.

The method of representation, in which the co-ordinates of the point in the diagram are made equal to the volume and entropy of the body, presents certain characteristics which entitle it to a somewhat detailed consideration, and for some purposes give it substantial advantages over any other method. We might anticipate some of these advantages from the simple and symmetrical form of the general equations of thermodynamics, when volume and entropy are chosen as independent variables, viz:—^[2]

1. From this general property of the diagram, its character in the case of a prefect gas might be immediately deduced.
2. See page 2, equations (2), (3) and (4). In general, in this article, where differential coefficients are used, the quantity which is constant in the differentiation is indicated by a subscript letter. In this discussion of the volume-entropy diagram, however, $v$ and $\eta$ are uniformly regarded as the independent variables, and the subscript letter is omitted.