163.

To ascertain the full physical significance of the quantity element of action, $h$, much further research work will be required. On the other hand, the value obtained for $k$ enables us readily to state numerically in the C. G. S. system the general connection between the entropy $S$ and the thermodynamic probability $W$ as expressed by the universal equation [eqn:(164)] (164). The general expression for the entropy of a physical system is $S=1.34·{10}^{-16}\log W\frac{\text{erg}}{\text{degree}}.\mathrm{\backslash Label}[eqn](294)\text{upshape (294)}$

This equation may be regarded as the most general definition of entropy. Herein the thermodynamic probability $W$ is an integral number, which is completely defined by the macroscopic state of the system. Applying the result expressed in [eqn:(293)] (293) to the kinetic

theory of gases, we obtain from equation [eqn:(194)] (194) for the ratio of the mass of a molecule to that of a mol, $\omega =\frac{k}{R}=\frac{1.34\times {10}^{-16}}{831\times {10}^5}=1.61\times {10}^{-24},\mathrm{\backslash Label}[eqn](295)\text{upshape (295)}$ that is to say, there are in one mol $\frac{1}{\omega }=6.20\times {10}^{23}$ molecules, where the mol of oxygen, $O_2$, is always assumed as $32\text{gr}$. Hence, for example, the absolute mass of a hydrogen atom ($\frac{1}{2}{H}_2=1.008$) equals $1.62\times {10}^{-24}\text{gr}$. With these numerical values the number of molecules contained in $1{\text{cm}}^3$ of an ideal gas at $0°\mathrm{C}.$ and $1$ atmosphere pressure becomes $N=\frac{76·13.6·981}{831·{10}^5·273\omega }=2.77·{10}^{19}.\mathrm{\backslash Label}[eqn](296)\text{upshape (296)}$ The mean kinetic energy of translatory motion of a molecule at the absolute temperature $T=1$ is, in the absolute C. G. S. system, according to [eqn:(200)] (200), $\frac{3}{2}k=2.01·{10}^{-16}.\mathrm{\backslash Label}[eqn](297)\text{upshape (297)}$ In general the mean kinetic energy of translatory motion of a molecule is expressed by the product of this number and the absolute temperature $T$.

The elementary quantity of electricity or the free charge of a monovalent ion or electron is, in electrostatic units, $e=\omega ·9654·3·{10}^{10}=4.67·{10}^{-10}.\mathrm{\backslash Label}[eqn](298)\text{upshape (298)}$ Since absolute accuracy is claimed for the formulæ here employed, the degree of approximation to which these numbers represent the corresponding physical constants depends only on the accuracy of the measurements of the two radiation constants $a$ and $c_2$.

[164.]164. Natural Units.–-All the systems of units which have hitherto been employed, including the so-called absolute C. G. S. system, owe their origin to the coincidence of accidental circumstances,

inasmuch as the choice of the units lying at the base of every system has been made, not according to general points of view which would necessarily retain their importance for all places and all times, but essentially with reference to the special needs of our terrestrial civilization.

Thus the units of length and time were derived from the present dimensions and motion of our planet, and the units of mass and temperature from the density and the most important temperature points of water, as being the liquid which plays the most important part on the surface of the earth, under a pressure which corresponds to the mean properties of the atmosphere surrounding us. It would be no less arbitrary if, let us say, the invariable wave length of Na-light were taken as unit of length. For, again, the particular choice of Na from among the many chemical elements could be justified only, perhaps, by its common occurrence on the earth, or by its double line, which is in the range of our vision, but is by no means the only one of its kind.