Page:Scientific Papers of Josiah Willard Gibbs.djvu/45

Rh the condition that $$\gamma = 1$$ throughout the whole diagram, may be seen by reference to page 5.

As the general equations (1), (2), (3) are not altered by interchanging $$v, -p$$ and $$-W$$ with $$\eta, t$$ and $$H$$ respectively, it is evident that, so far as these equations are concerned, there is nothing to choose between a volume-pressure and an entropy-temperature diagram. In the former, the work is represented by an area bounded by the path which represents the change of state of the body, two ordinates and the axis of abscissas. The same is true of the heat received in the latter diagram. Again, in the former diagram, the heat received is represented by an area bounded by the path and certain lines, the character of which depends upon the nature of the body under consideration. Except in the case of an ideal body, the properties of which are determined by assumption, these lines are more or less unknown in a part of their course, and in any case the area will generally extend to an infinite distance. Very much the same inconveniences attach themselves to the areas representing work in the entropy-temperature diagram.In neither diagram do these circumstances create any serious difficulty in the estimation of areas representing work or heat. It is always possible to divide these areas into two parts, of which one is of finite dimensions, and the other can be calculated in the simplest manner. Thus in the entropy-temperature diagram the work done in a path $$AB$$ (fig. 2) is represented by the area included by the path $$AB$$, the isometric $$BC$$, the line of no pressure and the isometric $$DA$$. The line of no pressure and the adjacent parts of the isometrics in the case of an actual gas or vapor are more or less undetermined in the present state of our knowledge, and are likely to remain so; for an ideal gas the line of no pressure coincides with the axis of abscissas, and is an asymptote to the isometrics. But, be this as it may, it is not necessary to examine the form of the remoter parts of the diagram. If we draw an isopiestic $$MN$$, cutting $$AD$$ and $$BC$$, the area $$MNCD$$, which represents the work done in $$MN$$, will be equal to $$p (v - v')$$, where $$p$$ denotes the pressure in $$MN$$ and $$v$$ and $$v'$$ denote the volumes at $$B$$ and $$A$$ respectively (eq. 5). Hence the work done in $$AB$$ will be represented by $$ABNM + p(v'' - v')$$. In the volume-pressure diagram, the areas representing heat may be divided by an isothermal, and treated in a manner entirely analogous.Or we may make use of the principle that, for a path which begins and ends on the same isodynamic, the work and heat are equal, as appears by integration of equation (1). Hence, in the entropy-temperature diagram, to find the work of any path, we may extend it by an isometric (which will not alter its work), so that it shall begin and end There is, however, a consideration of a