On the Theory of the Temperature Variation of the Specific Heat of Hydrogen

Abstract

Quantum theory of rotational and vibrational specific heats of an elastic, non-gyroscopic model of a diatomic gas.---To account for the abnormally large specific heat of hydrogen at high temperatures, the molecule is assumed to have an internal vibrational degree of freedom. Assuming a dumb-bell model and the following law of force $F=\frac{a(r\ensuremath{-}{r}_{0})}{{r}^{3}}$, the energies of the stationary states are derived on the basis of the Bohr-Sommerfeld form of the quantum hypothesis, and an expression for the specific heat obtained. When suitable values of the two adjustable constants are chosen, satisfactory agreement is obtained with the experimental results for hydrogen throughout the entire range, to 1300\ifmmode^\circ\else\textdegree\fi{} K.Constants of the hydrogen molecule, computed from the empirical constants of the specific heat equation are: Nuclear spacing, 0.488 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}8}$ cm; moment of inertia, 1.975 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}41}$ gm ${\mathrm{cm}}^{2}$; wave-length corresponding to normal vibration, 2.05 \ensuremath{\mu}.Specific heats of hydrogen and water vapor; new empirical formulas for temperatures between 300\ifmmode^\circ\else\textdegree\fi{} and 2300\ifmmode^\circ\else\textdegree\fi{} K derived from Pier's data, are given: For hydrogen: ${c}_{v}=4.87+0.539\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}t+0.146\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}{t}^{2}$; for water vapor: ${c}_{v}=6.03+4.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}t\ensuremath{-}4.07\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}{t}^{2}+1.95\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}{t}^{3}$.