Page:Scientific Papers of Josiah Willard Gibbs.djvu/48

method is applied. On this, however, depend the forms of the isometrics, isopiestics and isodynamics in the entropy-temperature diagram, and of the isentropics, isothermals and isodynamics in the volume-pressure diagram. As the convenience of a method depends largely upon the ease with which these lines can be drawn, and upon the peculiarities of the fluid which has its properties represented in the diagram, it is desirable to compare the methods under consideration in some of their most important applications. We will commence with the case of a perfect gas.

Case of a perfect gas.

A perfect ideal gas may be defined as such a gas, that for any constant quantity of it the product of the volume and the temperature varies as the temperature, and the energy varies as the temperature, i.e.,

$pv=at,$

(a)[1]

$\epsilon =ct.$

(b)

The significance of the constant $a$ is sufficiently indicated by equation (a). The significance of $c$ may be rendered more evident by differentiating equation (b) and comparing the result

$d\epsilon =cdt$

with the general equations (1) and (2), viz:

$d\epsilon =dH-dW,\,\,dW=pdv.$

If $dv=0$, $dW=0$, and $dH=cdt$, i.e.,

$\left({\frac {dH}{dt}}\right)_{v}=c,$[2]

(c)

i.e., $c$ is the quantity of heat necessary to raise the temperature of the body one degree under the condition of constant volume. It will be observed, that when different quantities of the same gas are considered, $a$ and $c$ both vary as the quantity, and $c\div a$ is constant; also, that the value of $c\div a$ for different gases varies as their specific heat determined for equal volumes and for constant volume.

With the aid of (a) and (b) we may eliminate $p$ and $t$ from the general equation (4), viz:

$d\epsilon =td\eta -pdv,$

1. In this article, all equations which are designated by arabic numerals subsist for any body whatever (subject to the condition of uniform pressure and temperature), and those which are designated by small capitals subsist for any quantity of a perfect gas as defined above (subject of course to the same conditions).
2. A subscript letter after a differential co-efficient is used in this article to indicate the quantity which is made constant in the differentiation.