Page:Elementary Text-book of Physics (Anthony, 1897).djvu/240

226 '''216. Ratio of the Elasticities and of the Specific Heats of a Gas.'''—The ratio of the two principal specific heats of a gas is the same as the ratio of its two principal elasticities. To show this, construct an adiabatic line $$\phi$$ and an isothermal line $$\theta$$ (Fig. 71) intersecting at the point $$O;$$ from that point draw a line parallel with the axis of volumes and take a point $$A$$ on that line very near the point $$O.$$ Through that point draw a line parallel with the axis of pressures, intersecting the isothermal and the adiabatic lines at $$B$$ and $$C$$ respectively. $$OA$$ is the diminution of volume, $$\Delta v,$$ caused by an increase of pressure $$AB = \delta p$$ if the compression is isothermal, or by the increase of pressure $$AC = \Delta p$$ if the compression is adiabatic. From the definition of elasticity (§ 102) we have the equations $$E_{t} = \frac{v. \delta p}{\Delta v}, E_{h} = \frac{v. \Delta p}{\Delta v},$$ and hence $$\frac{E_{h}}{E_{t}} = \frac{\Delta p}{\delta p}\cdot$$

We will now determine the value of the ratio $$\frac{\Delta p}{\delta p}$$ in terms of the principal specific heats. For convenience we assume that we are dealing with a unit mass of gas. The diminution of volume $$\Delta v$$ at constant pressure sets free the quantity of heat $$C_{p}. \Delta t,$$ where $$\Delta t$$ is the change of temperature that occasions the change of volume; the point $$A$$ then represents the condition of the gas. The gas may be brought into this same condition by an adiabatic compression from $$O$$ to $$C,$$ during which no heat either enters or leaves the gas, and by a diminution of pressure $$AC = \Delta p$$ while the volume is constant, caused by the abstraction of the heat produced by the compression. The heat which must be abstracted from the gas in order that it shall attain the condition denoted by $$A,$$ is to the heat that must be abstracted to cause the diminution of pressure $$BA = \delta p$$ in the ratio of $$\Delta p$$ to $$\delta p.$$ The heat which must be abstracted to cause the diminution of pressure $$BA = \delta p$$ at constant volume is $$C_{v}. \Delta t,$$ where $$\Delta t$$ has the same value as before, since the