Page:Scientific Papers of Josiah Willard Gibbs.djvu/57

$p=-{\frac {d\epsilon }{dv}},$

(11)

$t={\frac {d\epsilon }{d\eta }},$

(12)

$dW=pdv,$

$dH=td\eta .$

Eliminating $p$ and $t$ we have also

$dW=-{\frac {d\epsilon }{dv}}dv,$

(13)

$dH={\frac {d\epsilon }{d\eta }}d\eta \cdot$

(14)

The geometrical relations corresponding to these equations are in the volume-entropy diagram extremely simple. To fix our ideas, let the axe of volume and entropy be horizontal and vertical respectively, volume increasing toward the right and entropy upward. Then the pressure taken negatively will equal the ratio of the difference of volume of two adjacent points in the same horizontal line, and the temperature will equal the ratio of the difference of energy to the difference of entropy of two adjacent points in the same vertical line. Or, if a series of isodynamics be drawn for equal infinitesimal differences of energy, any series of horizontal lines will be divided into segments inversely proportional to the pressure, and any series of vertical lines into segments inversely proportional to the temperature. We see by equations (13) and (14), that for a motion parallel to the axis of volume, the heat received is 0 and the work done is equal to the decrease of energy, while for a motion parallel to the axis of entropy, the work done is 0, and the heat received is equal to the increase of the energy. These two propositions are true either for elementary paths or for those of finite length. In general, the work for any element of a path is equal to the product of the pressure in that part of the diagram into the horizontal projection of the element of the path, and the heat received is equal to the product of the temperature into the vertical projection of the element of the path. If we wish to estimate the value of the integrals $\int pdv$ and $\int td\eta$, which represents the work and heat of any path, by means of measurements upon the diagram, or if we wish to appreciate readily by the eye the approximate value of these expressions, or if we merely wish to illustrate their meaning by means of the diagram; for any of these purposes the diagram which we are now considering will have the advantage that it represents the differentials $dv$ and $d\eta$ more simply and clearly than any other.