Page:Elementary Principles in Statistical Mechanics (1902).djvu/209

where indeed the individual values of which the average is taken would appear to human observation as identical. This gives

whence a value recognized by physicists as a constant independent of the kind of monatomic gas considered.

We may also express the value of ${\displaystyle K}$ in a somewhat different form, which corresponds to the indirect method by which physicists are accustomed to determine the quantity ${\displaystyle c_{v}}$. The kinetic energy due to the motions of the centers of mass of the molecules of a mass of gas sufficiently expanded is easily shown to be equal to

where ${\displaystyle p}$ and ${\displaystyle v}$ denote the pressure and volume. The average value of the same energy in a canonical ensemble of such a mass of gas is where ${\displaystyle \nu }$ denotes the number of molecules in the gas. Equating these values, we have whence Now the laws of Boyle, Charles, and Avogadro may be expressed by the equation where ${\displaystyle A}$ is a constant depending only on the units in which energy and temperature are measured. ${\displaystyle 1/K}$, therefore, might be called the constant of the law of Boyle, Charles, and Avogadro as expressed with reference to the true number of molecules in a gaseous body. Since such numbers are unknown to us, it is more convenient to express the law with reference to relative values. If we denote by ${\displaystyle M}$ the so-called molecular weight of a gas, that