Page:Elementary Text-book of Physics (Anthony, 1897).djvu/263

§ 232] greater than unity. This temperature is therefore called the absolute zero.

The length of the degree on the absolute scale may be determined by designating the difference of temperature between two bodies by an arbitrarily chosen number and by measuring the efficiency of an engine working between the temperatures of those bodies. The most convenient assumption to make is that the absolute difference between the temperature of boiling water and the temperature of melting ice is 100 degrees. The temperature intervals or degrees on the scale thus formed are very nearly those of the Centigrade scale.

232. Relation of the Absolute Temperature to the Temperature of the Air Thermometer.—Let us assume that a substance exists which obeys perfectly the laws of Boyle and Gay-Lussac; such a substance is called a perfect gas. We wish to show that the temperatures indicated' by the expansion of a perfect gas, used as a thermometric substance, will be those of the absolute scale.

We must first prove that the work done by the expansion of a gas is equal to the area included between the lines representing its changes of pressure and volume, the two ordi nates representing its extreme pressures and the horizontal line of zero volume. The proof of this proposition does not depend on the properties of a perfect gas, and the proposition holds in all cases in which the body does work by expanding under a hydrostatic pressure which is the same at all points of its surface. Let us select a small area $$s$$ on the surface of the body. The pressure $$p$$ is applied to all points of the surface, and the force which acts on the area $$s$$ is therefore $$ps.$$ Let the body expand slightly, so that the area $$s$$ is displaced along its normal through the distance $$n.$$ The work done in displacing the area $$s$$ is $$psn,$$ and the work done in expanding the whole body is $$\textstyle \sum \displaystyle psn = p \textstyle \sum \displaystyle sn,$$ since $$p$$ is the same for all points on the surface. Now $$\textstyle \sum \displaystyle sn$$ is equal to the increase in the volume of the body, or to $$dv.$$ The work done during the small expansion is therefore $$pdv.$$ This expansion will, in general, involve an infinitesimal change in the pressure; but if the process here described be repeated for each