The Phase Rule and Its Applications Part 24

ISOTHERMAL CURVES AND THE s.p.a.cE MODEL

In the preceding chapter we considered the changes in the solubility of double salts and of mixtures of their const.i.tuent salts with the temperature; noting, more especially, the relationships between the two systems at the transition point. It is now proposed to conclude the study of the three-component systems by discussing very briefly the solubility relations at constant temperature, or the isothermal solubility curves. In this way fresh light will be thrown on the change in the solubility of one component by the addition of another component, and also on the conditions of formation and stable existence of double salts in solution. With the help of these isothermal curves, also, the phenomena of crystallization at constant temperature--phenomena which have not only a scientific interest but also an important bearing on the industrial preparation of double salts--will be more clearly understood.[354]

A brief description will also be given of the method of representing the variation of the concentration of the two salts in the solution with the temperature.

Non-formation of Double Salts.--In Fig. 102 are shown the solubility curves of two salts, A and B, which at the given temperature do not form a double salt.[355] The ordinates represent the amount of A, the abscissae the amount of B in a _constant amount_ of the third component, the solvent. The {273} point A, therefore, represents the solubility of the salt A at the given temperature; and similarly, point B represents the solubility of B. Since we are dealing with a three-component system, one solid phase in contact with solution will const.i.tute a bivariant system (in the absence of the vapour phase and under a constant pressure). At any given temperature, therefore, the concentration of the solution in equilibrium with the solid can undergo change. If, now, to a pure solution of A a small quant.i.ty of B is added, the solubility of A will in general be altered; as a rule it is diminished, but sometimes it is increased.[356] The curve AC represents the varying composition of the solution in equilibrium with the solid component A. Similarly, the curve BC represents the composition of the solutions in contact with pure B as solid phase. At the point, C, where these two curves intersect, there are two solid phases, viz. pure A and pure B, in equilibrium with solution, and the system becomes invariant. At this point the solution is saturated with respect to both A and B, and at a given temperature must have a perfectly definite composition. To take an example, if we suppose A to represent sodium sulphate decahydrate, and B, magnesium sulphate heptahydrate, and the temperature to be 18.5 (_i.e._ below the transition point), the point C would represent a solution containing 2.16 gm.-molecules Na_{2}SO_{4} and 4.57 gm.-molecules MgSO_{4} per 100 gm.-molecules of water (p. 268). The curve ACB is the boundary curve for saturated solutions; solutions lying outside this curve are supersaturated, those lying within the area ACBO, are unsaturated.

[Ill.u.s.tration: FIG. 102.]

[Ill.u.s.tration: FIG. 103.]

[Ill.u.s.tration: FIG. 104.]

Formation of Double Salt.--We have already learned in the preceding chapter that if the temperature is outside[357] the {274} transition interval, it is possible to prepare a pure saturated solution of the double salt. If, now, we suppose the double salt to contain the two const.i.tuent salts in equimolecular proportions, its saturated solution must be represented by a point lying on the line which bisects the angle AOB; _e.g._ point D, Fig.

103. But a double salt const.i.tutes only a single phase, and can exist, therefore, in contact with solutions of varying concentration, as represented by EDF.

Let us compare, now, the relations between the solubility curve for the double salt, and those for the two const.i.tuent salts. We shall suppose that the double salt is formed from the single salts when the temperature is raised above a certain point (as in the formation of astracanite). At a temperature below the transition point, as we have already seen, the solubility of the double salt is greater than that of a mixture of the single salts. The curve EDF, therefore, must lie above the point C, in the region representing solutions supersaturated with respect to the single salts (Fig. 104). Such a solution, however, would be metastable, and on being brought in contact with the single salts would deposit these and yield a solution represented by the point C. At this particular temperature, therefore, the isothermal solubility curve will consist of only two branches.

[Ill.u.s.tration: FIG. 105.]

Suppose, now, that the temperature is that of the transition point. At this point, the double salt can exist together with the single salts in contact with solution. The solubility curve {275} of the double salt must, therefore, pa.s.s through the point C, as shown in Fig. 105.

From this figure, now, it is seen that a solution saturated with respect to double salt alone (point D), is supersaturated with respect to the component A. If, then, at the temperature of the transition point, excess of the double salt is brought in contact with water,[358] and if supersaturation is excluded, _the double salt will undergo decomposition and the component A will be deposited_. The relative concentration of the component B in the solution will, therefore, increase, and the composition of the solution will be thereby altered in the direction DC. When the solution has the composition of C, the single salt ceases to be deposited, for at this point the solution is saturated for both double and single salt; and the system becomes invariant.

This diagram explains very clearly the phenomenon of the decomposition of a double salt at the transition point. As is evident, this decomposition will occur when the solution which is saturated at the temperature of the transition point, with respect to the two single salts (point C), does not contain these salts in the same ratio in which they are present in the double salt. If point C lay on the dotted line bisecting the right angle, then the pure saturated solution of the double salt would not be supersaturated with respect to either of the single salts, and the double salt would, therefore, not be decomposed by water. As has already been mentioned, this behaviour is found in the case of optically active isomerides, the solubilities of which are identical.

At the transition point, therefore, the isothermal curve also consists of two branches; but the point of intersection of the two branches now represents a solution which is saturated not {276} only with respect to the single salts, but also for the double salt in presence of the single salts.

We have just seen that by a change of temperature the two solubility curves, that for the two single salts and that for the double salt, were made to approach one another (_cf._ Figs. 104 and 105). In the previous chapter, however, we found that on pa.s.sing the transition point to the region of stability for the double salt, the solution which is saturated for a mixture of the two const.i.tuent salts, is supersaturated for the double salt. In this case, therefore, point C must lie above the solubility curve of the pure double salt (Fig. 106), and a solution of the composition C, if brought in contact with double salt, will deposit the latter. If the single salts were also present, then as the double salt separated out, the single salts would pa.s.s into solution, because so long as the two single salts are present, the composition of the solution must remain unaltered.

If one of the single salts disappear before the other, there will be left double salt plus A or double salt plus B, according to which was in excess; and the composition of the solution will be either that represented by D (saturated for double salt plus A), or that of the point F (saturated for double salt plus B).

[Ill.u.s.tration: FIG. 106.]

In connection with the isothermal represented in Fig. 106, it should be noted that at this particular temperature a solution saturated with respect to the pure double salt is no longer supersaturated for one of the single salts (point D); so that at the temperature of this isothermal the double salt is not decomposed by water. At this temperature, further, the boundary curve consists of three branches AD, DF, and FB, which give the composition of the solutions in equilibrium with pure A, double salt, and pure B respectively; while the points D and F represent solutions saturated for double salt plus A and double salt plus B.

On continuing to alter the temperature in the same direction {277} as before, the relative shifting of the solubility curves becomes more marked, as shown in Fig. 107. At the temperature of this isothermal, the solution saturated for the double salt now lies in a region of distinct unsaturation with respect to the single salts; and the double salt can now exist as solid phase in contact with solutions containing both relatively more of A (curve ED), and relatively more of B (curve DF), than is contained in the double salt itself.

[Ill.u.s.tration: FIG. 107.]

Transition Interval.--From what has been said, and from an examination of the isothermal diagrams, Figs. 104-107, it will be seen that by a variation of the temperature we can pa.s.s from a condition where the double salt is quite incapable of existing in contact with solution (supersaturation being excluded), to a condition where the existence of the double salt in presence of solution becomes possible; only in the presence, however, of one of the single salts (_transition point_, Fig. 105). A further change of temperature leads to a condition where the stable existence of the pure double salt in contact with solution just becomes possible (Fig. 106); and from this point onwards, pure saturated solutions of the double salt can be obtained (Fig. 107). _At any temperature, therefore, between that represented by Fig. 105, and that represented by Fig. 106, the double salt undergoes partial decomposition, with deposition of one of the const.i.tuent salts._ The temperature range between the transition point and the temperature at which a stable saturated solution of the pure double salt just begins to be possible, is known as the _transition interval_ (p. 270).

As the figures show, the transition interval is limited on the one side by the transition temperature, and on the other by the temperature at which the solution saturated for double salt and the less soluble of the single salts, contains the component salts in the same ratio as they are present in the double salt. The greater the difference in the solubility of the single salts, the larger will be the transition interval. {278}

Isothermal Evaporation.--The isothermal solubility curves are of great importance for obtaining an insight into the behaviour of a solution when subjected to isothermal evaporation. To simplify the discussion of the relationships found here, we shall still suppose that the double salt contains the single salts in equimolecular proportions; and we shall, in the first instance, suppose that the unsaturated solution with which we commence, also contains the single salts in the same ratio. The composition of the solution must, therefore, be represented by some point lying on the line OD, the bisectrix of the right angle.

From what has been said, it is evident that when the formation of a double salt can occur, three temperature intervals can be distinguished, viz. the single-salt interval, the transition interval, and the double-salt interval.[359] When the temperature lies in the first interval, evaporation leads first of all to the crystallization of one of the single salts, and then to the separation of both the single salts together. In the second temperature interval, evaporation again leads, in the first place, to the deposition of one of the single salts, and afterwards to the crystallization of the double salt. In the third temperature interval, only the double salt crystallizes out. This will become clearer from what follows.

[Ill.u.s.tration: FIG. 108.]

[Ill.u.s.tration: FIG. 109.]

If an unsaturated solution of the two single salts in equimolecular proportion (_e.g._ point _x_, Fig. 108) is evaporated at a temperature at which the formation of double salt is impossible, the component A, the solubility curve of which is {279} cut by the line OD, will first separate out; the solution will thereby become richer in B. On continued evaporation, more A will be deposited, and the composition of the solution will change until it attains the composition represented by the point C, when both A and B will be deposited, and the composition of the solution will remain unchanged. The result of evaporation will therefore be a mixture of the two components.

If the formation of double salt is possible, but if the temperature lies within the transition interval, the relations will be represented by a diagram like Fig. 109. Isothermal evaporation of the solution X will lead to the deposition of the component A, and the composition of the solution will alter in the direction DE; at the latter point the double salt will be formed, and the composition of the solution will remain unchanged so long as the two solid phases are present. As can be seen from the diagram, however, the solution in E contains less of component A than is contained in the double salt. Deposition of the double salt at E, therefore, would lead to a relative decrease in the concentration of A in the solution, and to counterbalance this, _the salt which separated out at the commencement must redissolve_.

Since the salts were originally present in equimolecular proportions, the final result of evaporation will be the pure double salt. If when the solution has reached the point E the salt A which had separated out is removed, double salt only will be left as solid phase. At a given temperature, however, a single solid phase can exist in equilibrium with solutions of different composition. If, therefore, isothermal evaporation is continued after the removal of the salt A, double salt will be deposited, and the composition of the solution will change in the direction EF. At the point F the salt B will separate out, and on evaporation both double salt and the salt B will be deposited. In the former case (when the salt A disappears on evaporation) we are dealing with an _incongruently saturated solution_; but in the latter case, where both solid phases continue to be deposited, the solution is said to be _congruently saturated_.[360]

A "congruently saturated solution" is one from which the {280} solid phases are continuously deposited during isothermal evaporation to dryness, whereas in the case of "incongruently saturated solutions," at least one of the solid phases disappears during the process of evaporation.

[Ill.u.s.tration: FIG. 110.]

Lastly, if the temperature lies outside the transition interval, isothermal evaporation of an unsaturated solution of the composition X (Fig. 110) will lead to the deposition of pure double salt from beginning to end. If a solution of the composition Y is evaporated, the component A will first be deposited and the composition of the solution will alter in the direction of E, at which point double salt will separate out. Since the solution at this point contains relatively more of A than is present in the double salt, both the double salt and the single salt A will be deposited on continued evaporation, in order that the composition of the solution shall remain unchanged. In the case of solution Z, first component B and afterwards the double salt will be deposited. The result will, therefore, be a mixture of double salt and the salt B (congruently saturated solutions),

It may be stated here that the same relationships as have been explained above for double salts are also found in the resolution of racemic compounds by means of optically active substances (third method of Pasteur). In this case the single salts are doubly active substances (_e.g._ strychnine-_d_-tartrate and strychnine-_l_-tartrate), and the double salt is a partially racemic compound.[361]

Crystallization of Double Salt from Solutions containing Excess of One Component.--One more case of isothermal crystallization may be discussed.

It is well known that a double salt which is decomposed by pure water can nevertheless be obtained pure by crystallization from a solution containing excess of one of the single salts (_e.g._ in the case of carnallite). Since the double salt is partially decomposed by water, the temperature of the experiment must be within the transition {281} interval, and the relations will, therefore, be represented by a diagram like Fig. 109. If, now, instead of starting with an unsaturated solution containing the single salts in equimolecular proportions, we commence with one in which excess of one of the salts is present, as represented by the point Y, isothermal evaporation will cause the composition to alter in the direction YD', the relative amounts of the single salts remaining the same throughout. When the composition of the solution reaches the point D', pure double salt will be deposited. The separation of double salt will, however, cause a relative decrease in the concentration of the salt A, and the composition of the solution will, therefore, alter in the direction D'F. If the evaporation is discontinued before the solution has attained the composition F, only double salt will have separated out. Even within the transition interval, therefore, pure double salt can be obtained by crystallization, provided the original solution has a composition represented by a point lying between the two lines OE and OF. Since, as already shown, the composition of the solution alters on evaporation in the direction EF, it will be best to employ a solution having a composition near to the line OE.

Formation of Mixed Crystals.--If the two single salts A and B do not crystallize out pure from solution, but form an unbroken series of mixed crystals, it is evident that an invariant system cannot be produced. The solubility curve will therefore be continuous from A to B; the liquid solutions of varying composition being in equilibrium with solid solutions also of varying composition. If, however, the series of mixed crystals is not continuous, there will be a break in the solubility curve at which two solid solutions of different composition will be in equilibrium with liquid solution. This, of course, will const.i.tute an invariant system, and the point will correspond to the point C in Fig. 108. A full discussion of these systems would, however, lead us too far, and the above indication of the behaviour must suffice.[362]

{282}

Application to the Characterization of Racemates.--The form of the isothermal solubility curves is also of great value for determining whether an inactive substance is a racemic compound or a conglomerate of equal proportions of the optical antipodes.[363]

As has already been pointed out, the formation of racemic compounds from the two enantiomorphous isomerides, is a.n.a.logous to the formation of double salts. The isothermal solubility curves, also, have a similar form. In the case of the latter, indeed, the relationships are simplified by the fact that the two enantiomorphous forms have identical solubility, and the solubility curves are therefore symmetrical to the line bisecting the angle of the co-ordinates. Further, with the exception of the partially racemic compounds to be mentioned later, there is no transition interval.

In Fig. 111, are given diagrammatically two isothermal solubility curves for optically active substances. From what has been said in the immediately preceding pages, the figure ought really to explain itself. The upper isothermal _acb_ represents the solubility relations when the formation of a racemic compound is excluded, as, _e.g._ in the case of rubidium _d_- and _l_-tartrates above the transition point (p. 265). The solution at the point _c_ is, of course, inactive, and _is unaffected by addition of either the _d_- or _l_- form_. The lower isothermal, on the other hand, would be obtained at a temperature at which the racemic compound could be formed.

The curve _a'e_ is the solubility curve for the _l_- form; _b'f_, that for the _d_- form; and _edf_, that for the racemic compound in presence of solutions of varying concentration. The point _d_ corresponds to saturation for the pure racemic compound.

[Ill.u.s.tration: FIG. 111.]

From these curves now, it will be evident that it will be possible, in any given case, to decide whether or not an inactive body is a mixture or a racemic compound. For this purpose, {283} two solubility determinations are made, first with the inactive material alone (in excess), and then with the inactive material plus excess of one of the optically active forms. If we are dealing with a mixture, the two solutions thus obtained will be identical; both will have the composition corresponding to the point _c_, and will be inactive. If, however, the inactive material is a racemic compound, then two different solutions will be obtained; namely, an inactive solution corresponding to the point _d_ (Fig. 111), and an _active_ solution corresponding either to _e_ or to _f_, according to which enantiomorphous form was added.

_Partially racemic compounds._[364] In this case we are no longer dealing with enantiomorphous forms, and the solubility of the two oppositely active isomerides is no longer the same. The symmetry of the solubility curves therefore disappears, and a figure is obtained which is identical in its general form with that found in the case of ordinary double salts (Fig.

112). In this case there is a transition interval.

[Ill.u.s.tration: FIG. 112.]

The curves _acb_ belong to a temperature at which the partially racemic compound cannot be formed; _a'dfb'_, to the temperature at which the compound just begins to be stable in contact with water, and _a"ed'f'b"_ belongs to a temperature at which the partially racemic compound is quite stable in contact with water. Suppose now solubility determinations, made in the first case with the original material alone, and then with the original body plus each of the two compounds, formed from the enantiomorphous substances separately, then if the original body was a mixture, identical solutions will be obtained in all three cases (point _c_); if it was a partially racemic compound, three different solutions (_e_, _d'_, and _f'_) will be obtained if the temperature was outside the transition interval, and two solutions, _d_ and _f_, if the temperature belonged to the transition interval.

{284}

_Representation in s.p.a.ce._

s.p.a.ce Model for Carnallite.--Interesting and important as the isothermal solubility curves are, they are insufficient for the purpose of obtaining a clear insight into the complete behaviour of the systems of two salts and water. A short description will, therefore, be given here of the representation in s.p.a.ce of the solubility relations of pota.s.sium and magnesium chlorides, and of the double salt which they form, carnallite.[365]

[Ill.u.s.tration: FIG. 113.]

Fig. 113 is a diagrammatic sketch of the model for carnallite looked at sideways from above. Along the X-axis is measured the concentration of magnesium chloride in the {285} solution; along the Y-axis, the concentration of pota.s.sium chloride; while along the T-axis is measured the temperature. The three axes are at right angles to one another. The XT-plane, therefore, contains the solubility curve of magnesium chloride; the YT-plane, the solubility curve of pota.s.sium chloride, and in the s.p.a.ce between the two planes, there are represented the composition of solutions containing both magnesium and pota.s.sium chlorides. Any _surface_ between the two planes will represent the various solutions in equilibrium with only one solid phase, and will therefore indicate the area or field of existence of bivariant ternary systems. A _line_ or _curve_ formed by the intersection of two surfaces will represent solutions in equilibrium with two solid phases (viz. those belonging to the intersecting surfaces), and will show the conditions for the existence of univariant systems. Lastly, _points_ formed by the intersection of three surfaces will represent invariant systems, in which a solution can exist in equilibrium with three solid phases (viz. those belonging to the three surfaces).