Page:Scientific Papers of Josiah Willard Gibbs.djvu/54

Hence, in the diagrams of different gases, $CD\div BC$ will be proportional to the specific heat determined for equal volumes and for constant volume.

As the specific heat, thus determined, has probably the same value for most simple gases, the isentropics will have the same inclination in diagrams of this kind for most simple gases. This inclination may easily be found by a method which is independent of any units of measurement, for

$BD:CD::\left({\frac {d\log p}{d\log v}}\right)_{\eta }:\left({\frac {d\log p}{d\log v}}\right)_{t}:\left({\frac {dp}{dv}}\right)_{\eta }:\left({\frac {dp}{dv}}\right)_{t},$

i.e,. $BD\div CD$ is equal to the quotient of the co-efficient of elasticity under the condition of no transmission of heat, divided by the co-efficient of elasticity at constant temperature. This quotient for a simple gas is generally given as 1.408 or 1.421. As

$CA\div CD={\sqrt {2}}=1.414.$

$BD$ is very nearly equal to $CA$ (for simple gases), which relation it may convenient to use in the construction of the diagram.

In regard to compound gases the rule seems to be, that the specific heat (determined for equal volumes and for constant volume) is to the specific heat of a simple gas inversely as the volume of the compound is to the volume of its constituents (in the condition of a gas); that is, the value of $BC\div CD$ for a compound gas is to the value of $BC\div CD$ for a simple gas, as the volume of the compound is to the volume of its constituents. Therefore, if we compare the diagrams (formed by this method) for a simple and a compound gas, the distance $DA$ and therefore $CD$ being the same in each, $BC$ in the diagram of the compound gas will be to $BC$ in the diagram of the simple gas as the volume of the compound is to the volume of its constituents.

Although the inclination of the isentropics is independent of the quantity of gas under consideration, the rate of increase of $\eta$ will vary with this quantity. In regard to the rate of increase of $t$, it is evident that if the whole diagram be divided into squares by isopiestics and isometrics drawn at equal distances, and isothermals be drawn as diagonals to these squares, the volumes of the isometrics, the pressures of the isopiestics and the temperatures of the isothermals will each form a geometrical series, and in all these series the ratio of two contiguous terms will be the same.

The properties of the diagrams obtained by the other methods mentioned on page 17 do not differ essentially from those just described. For example, in any such diagram, if through any point we draw an isentropic, an isotherman and an isopiestic, which cut any isometric not passing through the same point, the ratio of the segments of the isometric will have the value which has been found for $BC:CD$.

In treating the case of vapors also, it may be convenient to use