Negative Degrees of Freedom and the Phase Rule

Abstract

THE first part of Brynestad's comments on our communication¹ is concerned with the 1-component system: two solids, liquid and vapour. We mentioned this in our original paper² only in passing, since we could not cite any example for discussion. Brynestad's comment raises an issue which we did not consider, namely, if such a system existed, whether the relative amounts of the two solid phases present would be in a fixed ratio. Brynestad, following Ricci³, cites the case of d- and l-sodium chlorate. Before we turn to this interesting case, let us first examine Brynestad's analysis of a system which we did discuss in our original article, namely, a 2-component, five-phase system. We cited a number of ‘near misses’ such as the system mercury (m.p. − 38.9°) and diethylaniline (− 38.8°) and said that deuteration of the organic moiety in this and in other examples of ‘near misses’ could conceivably bring the triple points of the two components into exact coincidence. We interpreted Brynestad's original remarks⁴ to mean that the chance of such a system existing was comparable to the chance of failure of the second law of thermodynamics. We see now that Brynestad did not mean this and that he does not rule out the possibility of such systems existing. He goes on to say that ‘most probably the two solids would occur in fixed proportions’. We do not believe this to be correct for reasons given as follows. Moreover, Brynestad claims that the system would be ‘a normal 4-phase system’. To say that solid mercury and solid diethylaniline are thermo-dynamically the same phase is to so alter the meaning of ‘phase’ as to derive it of any further value. One can easily trace the reasoning leading up to this conclusion: the phase rule is true; the mercury system + diethylaniline (for example) would, if considered to have five phases, imply a negative degree of freedom; negative degrees of freedom are not possible; therefore, at least two of the phases must be thermodynamically the same. To us it makes more sense to concede that negative degrees of freedom are possible; their existence is easily explained and the phase rule is not violated. We turn now to the system: d-NaClO₃(c) + l-NaClO₃(c) + melt + vapour.