On the Relation Between lambda Lines and Phase Separations

It has often been assumed that the slope of the isotherm involving a pair of secondary variable vanishes along a lambda line [for example, along a lambda 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 lambda line and so obeys the Buckingham-Fairbank relation. It is shown here by a heuristic theoretical argument that in (3)He-(4)He solutions ([unk]mu(4)/[unk]x(3))(T) and ([unk]mu(3)/[unk]x(3))(T) probably do not vanish and C(x3) does not have its maximum possible value, although it may become infinite when x(3) --> 0. (mu(4) and mu(3) are chemical potentials of (4)He and (3)He and x(3) is the molefraction of (3)He). Only at the tricritical point does ([unk]mu(4)/[unk]x(3))(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 NH(4)Cl 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 (3)He-(4)He solutions. There is evidence that any singular behavior at the tricritical point in (3)He-(4)He is already present along the lambda 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.