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

${\frac {d\Theta }{dT}}={\frac {2c_{v}}{3\nu }},$

(X)

whence

${\frac {1}{K}}={\frac {2c_{v}}{3\nu }}.$

(493)

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

We may also express the value of $K$ in a somewhat different form, which corresponds to the indirect method by which physicists are accustomed to determine the quantity $c_{v}$. 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

${\tfrac {3}{2}}pv,$

where $p$ and $v$ denote the pressure and volume. The average value of the same energy in a canonical ensemble of such a mass of gas is

${\tfrac {3}{2}}\Theta \nu ,$

where $\nu$ denotes the number of molecules in the gas. Equating these values, we have

$pv=\Theta \nu ,$

(494)

whence

${\frac {1}{K}}={\frac {\Theta }{T}}={\frac {pv}{\nu T}}.$

(495)

Now the laws of Boyle, Charles, and Avogadro may be expressed by the equation

$pv=A\nu T,$

(496)

where $A$ is a constant depending only on the units in which energy and temperature are measured. $1/K$, 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 $M$ the so-called molecular weight of a gas, that