Equilibrium electron theory of thermionic emission.—The equation for the thermionic saturation current per unit area in the form $I=A{T}^{\alpha }{e}^{-\frac{b}{T}}$ is derived according to Richardson's thermodynamical method, assuming (1) the boundary condition that when $\psi$ the free electron evaporation constant is zero, the free electron concentration inside and outside the hot body is the same, and (2) the interior free electron concentration given by the author's equilibrium theory of conduction (Phys. Rev. 22, 259, 1923). $\alpha$ depends only upon the valence in the electron-forming reaction and always lies between 1/2 and 5/4. $\beta$ depends upon the valence and the two evaporation constants ${\phi }_0$ and ${\psi }_0$, referring respectively to a bound and a free electron. $A$ is found to depend upon the valence, the electronic chemical constant and the concentration of electron-forming particles in the hot body, varying from 4.16${\mathrm{}\left(10\right)\mathrm{}}^{14}$ for BaO to 8.13${\mathrm{}\left(10\right)\mathrm{}}^{14}$ for Ni. Values of $\beta$ calculated from the computed $A$ and experimental data show excellent agreement with $\beta$ observed for Pt, W, Mo, Ni, Ca, CaO, SrO, and BaO, the verification being more general than by use of Dushman's formula. The experimental data however are not sufficiently accurate to decide between the two formulas. The failure of certain hot bodies to agree with either formula is discussed and possible explanations offered. Assuming the theory correct, the free electron concentrations in several metals, Mo$S_2$ and CaO are calculated from experimental data. For metals at 0°C this concentration is of the order of magnitude of ${10}^{17}$ per cc, and decreases with rising temperature. A noteworthy consequence of this low concentration is that the contribution of the free electrons to the specific heat of the conductor remains negligible down to very low temperatures.

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