133.

The preceding relations, obtained simply by identifying the mechanical expression of the entropy [eqn:(186)] (186) with its thermodynamic expression [eqn:(187)] (187), show the usefulness of the theory developed. In them an additive constant in the expression for the entropy is immaterial and hence the size $G$ of the region element of probability does not matter. The hypothesis of quanta, however, goes further, since it fixes the absolute value of the entropy and thus leads to the same conclusion as the heat theorem

of Nernst. According to this theorem the "characteristic function" of an ideal gasE.g., M. Planck, Vorlesungen über Thermodynamik, Leipzig, Veit und Comp., 1911, Sec. 287, equation 267. is in our notation $\Phi =S-\frac{E+pV}{T}=n(A{c}_p\log T-R\log p+a-\frac{b}{T}),$ where $a$ denotes Nernst's chemical constant, and $b$ the energy constant.

On the other hand, the preceding formulæ [eqn:(186)] (186), [eqn:(188)] (188), and [eqn:(189)] (189) give for the same function $\Phi$ the following expression: $\Phi =N(\frac{5}{2}k\log T-k\log p+a^{\prime })-\frac{E_0}{T}$ where for brevity $a^{\prime }$ is put for: $a^{\prime }=k\log \{\frac{kN}{eG}(2\pi mk{)}^{\frac{3}{2}}\}.$

From a comparison of the two expressions for $\Phi$ it is seen, by taking account of [eqn:(199)] (199) and [eqn:(193)] (193), that they agree completely, provided $a=\frac{N}{n}{a}^{\prime }=R\log \{\frac{N{k}^{\frac{5}{2}}}{eG}(2\pi m{)}^{\frac{3}{2}}\},b=\frac{E_0}{n}.\mathrm{\backslash Label}[eqn](201)\text{upshape (201)}$ This expresses the relation between the chemical constant $a$ of the gas and the region element $G$ of the probability.Compare also O. Sackur, Annal. d. Physik 36, p. 958, 1911, Nernst-Festschrift, p. 405, 1912, and H. Tetrode, Annal. d. Physik 38, p. 434, 1912.

It is seen that $G$ is proportional to the total number, $N$, of the molecules. Hence, if we put $G=Ng$, we see that $g$, the molecular region element, depends only on the chemical nature of the gas.

Obviously the quantity $g$ must be closely connected with the law, so far unknown, according to which the molecules act microscopically on one another. Whether the value of $g$ varies with the nature of the molecules or whether it is the same for all kinds of molecules, may be left undecided for the present.

If $g$ were known, Nernst's chemical constant, $a$, of the gas could be calculated from [eqn:(201)] (201) and the theory could thus be tested. For the present the reverse only is feasible, namely, to calculate $g$ from $a$. For it is known that $a$ may be measured directly by the tension of the saturated vapor, which at sufficiently low temperatures satisfies the simple equationM. Planck, l. c., Sec. 288, equation 271. $\log p=\frac{5}{2}\log T-\frac{A{r}_0}{RT}+\frac{a}{R}\mathrm{\backslash Label}[eqn](202)\text{upshape (202)}$ (where $r_0$ is the heat of vaporization of a mol at $0°$ in calories). When $a$ has been found by measurement, the size $g$ of the molecular region element is found from [eqn:(201)] (201) to be $g=(2\pi m{)}^{\frac{3}{2}}{k}^{\frac{5}{2}}{e}^{-\frac{a}{R}-1}.\mathrm{\backslash Label}[eqn](203)\text{upshape (203)}$ Let us consider the dimensions of $g$.