Page:Elementary Text-book of Physics (Anthony, 1897).djvu/249

§ 222] volume which contains $$n$$ molecules becomes $$v,$$ the pressure takes a new value, which we will still designate by $$p.$$ We have Since $$V$$ remains constant so long as the temperature is constant, and since $$m$$ and $$w$$ are fixed, we have $$pv$$ constant. Hence Boyle's Law follows from the kinetic theory of gases.

From Gay-Lussac's law (§ 211) it has been shown that if we reckon temperature from -273° C. as a zero, we have $$pv = RT$$ for all gases. Using the equation just proved, we have  Now $$\tfrac{1}{2}mV^2$$ is the mean kinetic energy of the molecule. The formula shows, therefore, that the temperature on the scale of the air-thermometer is proportional to the mean kinetic energy of the molecule. The zero of this scale will be that temperature at which $$V = 0,$$ or at which the molecules are at rest. There can be no temperature lower than this, and hence we obtain a warrant for calling this temperature a real or absolute zero. The final demonstration of the existence of such a zero will be given in § 231, where it is not based upon any particular theory of matter.

It was demonstrated by Maxwell that the mean kinetic energies of the molecules of difEerent gases at the same temperature are the same, or that $$\tfrac{1}{2}m_{1}V_{1}^2 = \tfrac{1}{2}m_{2}V_{2}^2.$$ If we consider equal volumes of two gases at the same pressure and temperature, for which, therefore, $$\tfrac{1}{3}m_{1}n_{1}V_{1}^2 = \tfrac{1}{3}m_{2}n_{2}V_{2}^2,$$ we obtain $$n_{1} = n_{2},$$ or the numbers of molecules of the two gases in the same volume are the same. This is Avogadro's law.

Up to this point we have considered the molecules as particles, and have supposed that all their energy exists as the kinetic energy of molecular motion. It is easy to show, however, that this supposition is in error, and that the molecules possess more energy than that given by $$\tfrac{1}{2}mnV^2.$$ Let us consider a unit mass of gas, the temperature of which is raised under constant pressure by a small amount $$\Delta T.$$ Then the work done by its expansion (§ 215) is