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On the Quantum Theory of the Specific Heat of Hydrogen Part I. Relation to the New Mechanics, Band Spectra, and Chemical Constants

1926; American Institute of Physics; Volume: 28; Issue: 5 Linguagem: Inglês

10.1103/physrev.28.980

1536-6065

J. H. Van Vleck,

Experimental and Theoretical Physics Studies

1. If in the new quantum mechanics the hydrogen molecule is treated as a simple rotating dipole like HCl, the a priori probability is $2m$, and the rotational quantum number assumes the values $m=\frac{1}{2}, \frac{3}{2}, \ensuremath{\cdots}$. This, however, gives the ol\ifmmode \dot{}\else \.{}\fi{}d Planck specific heat curve, which rises to a maximum above the equipartition value, contrary to experiment. The failure of the simple theory is doubtless due to the non-polar character of the hydrogen molecule and is probably intimately connected with the alternating intensities found in the band spectra of certain non-polar molecules.2. The following ways of removing the specific heat dilemma are considered: (a) use of whole quanta, (b exclusion of the state $m=\frac{1}{2}$, (c) exclusion of every other state in accord with the type of quantization for non-polar molecules proposed by Ehrenfest and Tolman, (d) "weak" quantization of every other state, (e) a gyroscopic molecule. Satisfactory curves are obtained with (a) and (b), but the theoretical basis on the new quantum mechanics is obscure. Hypothesis (c) probably gives too steep a curve and an excessively large moment of inertia, (d) is questionable, and (e) is incompatible with the diamagnetism of hydrogen. The Einstein-Bose statistics do not affect the rotational specific heat of hydrogen appreciably. Recent experimental work of Bartels and Eucken and of Schreiner shows the moments of inertia of ${\mathrm{N}}_{2}$, ${\mathrm{O}}_{2}$, and CO cannot be deduced from existing specific heat data and so obviates the necessity of the absurdly small moments of inertia previously required by Scheel and Heuse's measurements for these elements.3. Recent theories of the hydrogen secondary spectrum give moments of inertia which are larger than the old values and which are hence more easily reconciled with specific heats. Ehrenfest and Tolman have suggested that the angular momentum of non-polar molecules equals only even multiples of $\frac{h}{2\ensuremath{\pi}}$, and according to Slater their proposal may be intimately related to the alternating intensities sometimes found in band spectra. It is shown that according to the correspondence principle the angular momentum can change by zero or $\frac{h}{2\ensuremath{\pi}}$ even in a non-polar molecule, provided there are simultaneous electron jumps. This fact partially destroys the quadrupolar symmetry and causes serious difficulties for their theory. The alternating intensities, however, seem difficult to explain otherwise, and may be related to the Heisenberg extension of the Pauli exclusion principle.Croze and Dufour find that, unlike the non-magnetic Fulcher bands, certain other lines of the hydrogen secondary spectrum are resolved by a magnetic field into doublets whose displacement is comparable with the normal Lorentz value and so surprisingly large for a molecule. It is suggested that these peculiar Zeeman doublets, whose polarization is sometimes anomalous, are due to loose coupling of the spin axis of the valence electron in either the initial or final state, but not in both.4. The Stern-Tetrode formula for chemical constants must be modified by including terms involving the a priori probability of the lowest state and the symmetry number. If the entropy of the solid phase vanishes at the absolute zero, observed vapor pressures apparently require a whole quantum specific heat curve for hydrogen and do not allow the specific heat curve (b); but if we abandon the Nernst heat theorem a more general interpretation of Simon's results is simply that in hydrogen the minimum a priori probability is the same in the gaseous and solid phases. The moment of inertia ${10}^{\ensuremath{-}41}$ gm. ${\mathrm{cm}}^{2}$ often deduced for the hydrogen molecule from chemical constants is erroneous. Present data on chemical constants are chaotic and the theory is uncertain owing to inadequate knowledge of the a priori probability in the solid phase and of the r\^ole of the symmetry number. Possibly this number is required in the solid as well as in the gas or else in neither.