XVIII. A note on Hildebrand's approximation for thermal pressures in solids

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Abstract
The validity of a widely used approximation for thermal pressures due originally to Hildebrand is discussed. The distinction between the “vibrational” pressure and the thermal pressure is stressed; Hildebrand's expression is shown to lie intermediate between the two for all temperatures. Since the percentage difference between the two pressures diminishes monotonically with increasing temperature, the accuracy of the approximate expression improves with increasing temperature. But at temperatures comparable with the Debye temperature, Hildebrand's expression is much closer to the vibrational (10 per cent error) than the thermal pressure (40 per cent error). It follows further from the discussion that the linearly extrapolated volume from any temperature to the absolute zero lies intermediate between the real volume at absolute zero of temperature and the volume corresponding to static equilibrium.

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