On the Relation Between λ Lines and Phase Separations

Abstract

It has often been assumed that the slope of the isotherm involving a pair of secondary variable vanishes along a λ line [for example, along a λ line in the pressure-volume plane ([unk]P/[unk]V)_T vanishes], and therefore that the specific heat for constant extensive variable (e.g., C_V) has the greatest possible value on the λ line and so obeys the Buckingham-Fairbank relation. It is shown here by a heuristic theoretical argument that in ³He-⁴He solutions ([unk]μ₄/[unk]x₃)_T and ([unk]μ₃/[unk]x₃)_T probably do not vanish and C_x3 does not have its maximum possible value, although it may become infinite when x₃ → 0. (μ₄ and μ₃ are chemical potentials of ⁴He and ³He and x₃ is the molefraction of ³He). Only at the tricritical point does ([unk]μ₄/[unk]x₃)_T finally vanish and C_x3 have a value, which cannot be exceeded without the system's becoming unstable. In the case of the transition in solid NH₄Cl the experimental facts seem to indicate that at the higher temperatures, where the transition is of higher order, ([unk]P/[unk]V)_T does not become zero along the transition line. A statistical thermodynamic description of tricritical points is given, and shown to accord qualitatively with the experimental results for the ³He-⁴He solutions. There is evidence that any singular behavior at the tricritical point in ³He-⁴He is already present along the λ line. Finally, an analysis is made of the possible behavior of binary liquid solutions, and it is shown that a tendency of C_V to exceed its maximum value can result in a flat top on the coexistence curve.

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