Gibbs-Konovalow theorem

This is the first Gibbs-Konovalow theorem for equilibria at constant pressure. 

In the case of a set of isothermal equilibrium states we have already seen that if the pressure passes through an extreme value, then a sufficient condition for this is that the two phases shall have identical compositions. (Gibbs-Konovalow theorem). We now require to find the condition that this extreme value is a maximum or minimum, and to do this it is necessary to investigate the second differentials of the coexistence curves. 

We shall not repeat here any discussion of the properties already established for states of uniform composition. It will be recalled that a state of uniform composition corresponds to an extreme value (maximum, minimum, or inflexion with a horizontal tangent) of the equilibrium pressure at constant temperature, or of the equilibrium temperature at constant pressure (Gibbs-Konovalow theorems, chap. XVIII, 6 and 9). 

At the temperature corresponding to fig. 29.6, we have three phases a, e, j8 on a straight line. The system must therefore be in an indifferent state cf. fig. 29.1), and hence because of the Gibbs-Konovalow theorem the temperature of coexistence must pass through an extremum. If we suppose this to be a maximum, then the system will be represented by a diagram of the kind shown in fig. 29.7, which represents conditions of constant p, and in which temperature is plotted vertically. 

We see therefore that, in accordance with the Gibbs-Konovalow theorem, when In a (1 - a ) passes through the maximum value in the indifferent state, so also does p. 

The system is therefore bivariant, and of the first category. The curve passes through a maximum value of the temperature at A hence by the converse of the second Gibbs-Konovalow theorem the point A is an indifferent point. We may now set the determinant derived from (29.19) (of which there is only one in this case because the system is only bivariant) equal to zero ... 

Two theorems of Gibbs and of Konovalow —Under what circumstances shall we observe such a state of indifferent equilibrium Two important theorems, discovered by J. Willard Gibbs, found anew by D. Konovalow, give us this information. Here are these two theorems ... 

First Theorem of Gibbs and Konovalow.—Under a canr slant pressure cause the composition of the liquid mixture to vary in a wdl-defined way the boiling-point of this mixture changes] if, for a certain composition of the liquid mixture, the boUing-point passes through a maximum or minimum, this liquid mixture gives off a saturated vapor of the same composition, and reciprocally. 

Second Case Between the points and (Fig. 65) the curve D has a point I, OF ORDNATE P SMALLER THAN ALL THE others.—According to the second theorem of Gibbs and of Konovalow, this point is an indifferent point when the liquid and the saturated... 

G. Bruni has very well shown this opinion to be inadmissible. We may, in fact, apply to the systems we are studying the theorems of Gibbs and of Konovalow (Art. 194), and particularly the first. It suffices to substitute for the words mixed liquid, mixed vapor, the words mixed crystals, mixed liquid. 

The curve C, formed necessarily by two branches both S3mi-metrical with respect to the line X, has, for the abscissa a point of maximum ordinate. From the first theorem of Gibbs and Konovalow (Art. I94 which may be applied to the double mixture formed by the mixed crystals and the mixed liquid, this point belongs also o the line c, for which it is also a point of maximum or minimum ordinate. At this indifferent point I the mixed hquid, which is inactive by compensation, must give, on freezing, mixed holoedral crystals of composition x=i. 

Each of the points on this line may be regarded, if so wished, as a point of maximum ordinate the theorem of Gibbs and Konovalow may be applied to each of these points whatever the composition of the mixed liquid, it deposits mixed crystals of the same composition. 

The above equations enable us to derive two important theorems first enunciated by Gibbs, and later rediscovered by Konovalow and Duhem. 

These theorems, in which we consider the effect either of an isothermal or isobaric displacement were demonstrated by Gibbs and Konovalow for bivariant systems and generalized by Saurel. They can only be applied to systems having a variance of at least two systems of smaller variance are not susceptible to such displacements. 

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