132.

This determination of the entropy of an ideal monatomic gas is based solely on the general connection between entropy and probability as expressed in equation [eqn:(164)] (164); in particular, we have at no stage of our calculation made use of any special law of the theory of gases. It is, therefore, of importance to see how the entire thermodynamic behavior of a monatomic gas, especially the equation of state and the values of the specific heats, may be deduced from the expression found for the entropy directly by means of the principles of thermodynamics. From the general thermodynamic equation defining the entropy, namely, $dS=\frac{dE+pdV}{T},\mathrm{\backslash Label}[eqn](187)\text{upshape (187)}$ the partial differential coefficients of $S$ with respect to $E$ and $V$ are found to be ${\left(\frac{\partial S}{\partial E}\right)}_V=\frac{1}{T},{\left(\frac{\partial S}{\partial V}\right)}_E=\frac{p}{T}.$

Hence, by using [eqn:(186)] (186), we get for our gas ${\left(\frac{\partial S}{\partial E}\right)}_V=\frac{3}{2}\frac{kN}{E-E_0}=\frac{1}{T}\mathrm{\backslash Label}[eqn](188)\text{upshape (188)}$ and ${\left(\frac{\partial S}{\partial V}\right)}_E=\frac{kN}{V}=\frac{p}{T}.\mathrm{\backslash Label}[eqn](189)\text{upshape (189)}$ The second of these equations $p=\frac{kNT}{V}\mathrm{\backslash Label}[eqn](190)\text{upshape (190)}$ contains the laws of Boyle, Gay Lussac, and Avogadro, the last named because the pressure depends only on the number $N$, not on the nature of the molecules. If we write it in the customary form: $p=\frac{RnT}{V},\mathrm{\backslash Label}[eqn](191)\text{upshape (191)}$ where $n$ denotes the number of gram molecules or mols of the gas, referred to $O_2=32\text{gr}.$, and $R$ represents the absolute gas constant $R=831\times {10}^5\frac{\text{erg}}{\text{degree}},\mathrm{\backslash Label}[eqn](192)\text{upshape (192)}$ we obtain by comparison $k=\frac{Rn}{N}.\mathrm{\backslash Label}[eqn](193)\text{upshape (193)}$ If we now call the ratio of the number of mols to the number of molecules $\omega$, or, what is the same thing, the ratio of the mass of a molecule to that of a mol, $\omega ={\displaystyle \frac{n}{N}}$, we shall have $k=\omega R.\mathrm{\backslash Label}[eqn](194)\text{upshape (194)}$ From this the universal constant $k$ may be calculated, when $\omega$ is given, and vice versa. According to [eqn:(190)] (190) this constant $k$ is nothing but the absolute gas constant, if it is referred to molecules instead of mols.

From equation [eqn:(188)] (188) $E-E_0=\frac{3}{2}kNT.\mathrm{\backslash Label}[eqn](195)\text{upshape (195)}$

Now, since the energy of an ideal gas is also given by $E=An{c}_{v}T+E_0\mathrm{\backslash Label}[eqn](196)\text{upshape (196)}$ where $c_v$ is the heat capacity of a mol at constant volume in calories and $A$ is the mechanical equivalent of heat: $A=419\times {10}^5\frac{\text{erg}}{\text{cal}}\mathrm{\backslash Label}[eqn](197)\text{upshape (197)}$ it follows that $c_v=\frac{3}{2}\frac{kN}{An}$ and further, by taking account of [eqn:(193)] (193) $c_v=\frac{3}{2}\frac{R}{A}=\frac{3}{2}·\frac{831\times {10}^5}{419\times {10}^5}=3.0\mathrm{\backslash Label}[eqn](198)\text{upshape (198)}$ as an expression for the heat capacity per mol of any monatomic gas at constant volume in calories.Compare F. Richarz, Wiedemann's Annal., 67, p. 705, 1899.

For the heat capacity per mol at constant pressure, $c_p$, we have as a consequence of the first principle of thermodynamics: $c_p-c_v=\frac{R}{A}$ and hence by [eqn:(198)] (198) $c_p=\frac{5}{2}\frac{R}{A},\frac{c_p}{c_v}=\frac{5}{3},\mathrm{\backslash Label}[eqn](199)\text{upshape (199)}$ as is known to be the case for monatomic gases. It follows from [eqn:(195)] (195) that the kinetic energy $L$ of the gas molecules is equal to $L=E-E_0=\frac{3}{2}NkT.\mathrm{\backslash Label}[eqn](200)\text{upshape (200)}$