140.

The connection between energy and entropy just obtained allows furthermore a certain conclusion as regards the temperature. For from the equation of the second principle of thermodynamics, $dS={\displaystyle \frac{dE}{T}}$ and from differentiation of [eqn:(222)] (222) with respect to $E$ it follows that $E = N \frac{h\nu}{2}\, \frac{1 + e^{-\tfrac{h\nu}{kT}}}{1 - e^{-\tfrac{h\nu}{kT}}} = Nh\nu \left(\frac{1}{2} + \frac{1}{e^{\tfrac{h\nu}{kT}} - 1}\right).

\Label[eqn]{(223)}\tag*{\upshape (223)}$ Hence, for the zero point of the absolute temperature $E$ becomes, not $0$, but $N{\displaystyle \frac{h\nu }{2}}$. This is the extreme case discussed in the preceding paragraph, which just allows thermodynamic equilibrium to exist. That the oscillators are said to perform vibrations even at the temperature zero, the mean energy of which is as large as ${\displaystyle \frac{h\nu }{2}}$ and hence may become quite large for rapid vibrations, may at first sight seem strange. It seems to me, however, that certain facts point to the existence, inside the atoms, of vibrations independent of the temperature and supplied with appreciable energy, which need only a small suitable excitation to become evident externally. For example, the velocity, sometimes very large, of secondary cathode rays produced by Roentgen rays, and that of electrons liberated by photoelectric effect are independent of the temperature of the metal and of the intensity of the exciting radiation. Moreover the radioactive energies are also independent of the temperature. It is also well known that the close connection between the inertia of matter and its energy as postulated by the relativity principle leads to the assumption of very appreciable quantities of intra-atomic energy even at the zero of absolute temperature.

For the extreme case, $T=\infty$, we find from [eqn:(223)] (223) that $E=NkT,\mathrm{\backslash Label}[eqn](224)\text{upshape (224)}$ i.e., the energy is proportional to the temperature and independent of the size of the quantum of action, $h$, and of the nature of the oscillators. It is of interest to compare this value of the energy of vibration $E$ of the system of oscillators, which holds at high temperatures, with the kinetic energy $L$ of the molecular

motion of an ideal monatomic gas at the same temperature as calculated in [eqn:(200)] (200). From the comparison it follows that $E=\frac{2}{3}L.\mathrm{\backslash Label}[eqn](225)\text{upshape (225)}$ This simple relation is caused by the fact that for high temperatures the contents of the hypothesis of quanta coincide with those of the classical statistical mechanics. Then the absolute magnitude of the region element, $G$ or $h$ respectively, becomes physically unimportant (compare [sect:125.] Sec. 125) and we have the simple law of equipartition of the energy among all variables in question (see below [sect:169.] Sec. 169). The factor $\frac{2}{3}$ in equation [eqn:(225)] (225) is due to the fact that the kinetic energy of a moving molecule depends on three variables ($\xi$, $\eta$, $\zeta$,) and the energy of a vibrating oscillator on only two ($f$, $\psi$).

The heat capacity of the system of oscillators in question is, from [eqn:(223)] (223), d E d T = N k ( h ν k T ) 2 e h ν k T ( e h ν k T − 1 ) 2 . \Label [ e q n ] ( 226 ) \upshape (226) It vanishes for T = 0 and becomes