CHAPTER XVI
ISOTHERMAL CURVES AND THE SPACE MODEL
In the preceding chapter we considered the changes in the solubility of double salts and of mixtures of their constituent 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 abscissæ the amount of B in a constant amount of the third component, the solvent. The
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 constitute 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 quantity 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 Na2SO4 and 4.57 gm.-molecules MgSO4 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.
Formation of Double Salt.—We have already learned in the preceding chapter that if the temperature is outside[[357]] the
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 constituent 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 constitutes 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 constituent 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.
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
of the double salt must, therefore, pass 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
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 passing the transition point to the region of stability for the double salt, the solution which is saturated for a mixture of the two constituent 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 pass 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).
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
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.
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 pass 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 constituent 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.
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.
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
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
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.
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
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 constitute 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]]
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 analogous 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.
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,
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.
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.
Representation in Space.
Space 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 space of the solubility relations of potassium and magnesium chlorides, and of the double salt which they form, carnallite.[[365]]
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
solution; along the Y-axis, the concentration of potassium 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 potassium chloride, and in the space between the two planes, there are represented the composition of solutions containing both magnesium and potassium 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).
We shall first consider the solubility relations of the single salts. The complete equilibrium curve for magnesium chloride and water is represented in Fig. 113 by the series of curves ABF1 G1 H1 J1 L1 N1. AB is the freezing-point curve of ice in contact with solutions containing magnesium chloride, and B is the cryohydric point at which the solid phases ice and MgCl2,12H2O can co-exist with solution. BFG is the solubility curve of magnesium chloride dodecahydrate. This curve shows a point of maximum temperature at F1, and a retroflex portion F1G1. The curve is therefore of the form exhibited by calcium chloride hexahydrate, or the hydrates of ferric chloride (Chapter VIII.). G1 is a transition point at which the solid phase changes from dodecahydrate to octahydrate, the solubility of which is represented by the curve G1H1. At H1 the octahydrate gives place to the hexahydrate, which is the solid phase in equilibrium with the solutions represented by the curve H1J1. J1 and L1 are also transition points at which the solid phase undergoes change, in the former case from hexahydrate to tetrahydrate; and in the latter case,
from tetrahydrate to dihydrate. The complete curve of equilibrium for magnesium chloride and water is, therefore, somewhat complicated, and is a good example of the solubility curves obtained with salts capable of forming several hydrates.
The solubility curve of potassium chloride is of the simplest form, consisting only of the two branches AC, the freezing-point curve of ice, and CO, the solubility curve of the salt. C is the cryohydric point. This point and the two curves lie in the YT-plane.
On passing to the ternary systems, the composition of the solutions must be represented by points or curves situated between the two planes. We shall now turn to the consideration of these. BD and CD are ternary eutectic curves (p. [284]). They give the composition of solutions in equilibrium with ice and magnesium chloride dodecahydrate (BD), and with ice and potassium chloride (CD). D is a ternary cryohydric point. If the temperature is raised and the ice allowed to disappear, we shall pass to the solubility curve for MgCl2,12H2O + KCl (curve DE). At E carnallite is formed and the potassium chloride disappears; EFG is then the solubility curve for MgCl2,12H2O + carnallite (KMgCl3,6H2O). This curve also shows a point of maximum temperature (F) and a retroflex portion. GH and HJ represent the solubility curves of carnallite + MgCl2,8H2O and carnallite + MgCl2,6H2O, G and H being transition points. JK is the solubility curve for carnallite + MgCl2,4H2O. At the point K we have the highest temperature at which carnallite can exist with magnesium chloride in contact with solution. Above this temperature decomposition takes place and potassium chloride separates out.
If at the point E, at which the two single salts and the double salt are present, excess of potassium chloride is added, the magnesium chloride will all disappear owing to the formation of carnallite, and there will be left carnallite and potassium chloride. The solubility curve for a mixture of these two salts is represented by EMK; a simple curve exhibiting, however, a temperature maximum at M. This maximum point corresponds with the fact that dry carnallite melts at this temperature with separation of potassium chloride. At all temperatures
above this point, the formation of double salt is impossible. The retroflex portion of the curve represents solutions in equilibrium with carnallite and potassium chloride, but in which the ratio MgCl2 : KCl is greater than in the double salt.
Throughout its whole course, the curve EMK represents solutions in which the ratio of MgCl2 : KCl is greater than in the double salt. As this is a point of some importance, it will be well, perhaps, to make it clearer by giving one of the isothermal curves, e.g. the curve for 10°, which is represented diagrammatically in Fig. 114. E and F here represent solutions saturated for carnallite plus magnesium chloride hydrate, and for carnallite plus potassium chloride. As is evident, the point F lies above the line representing equimolecular proportions of the salts (OD).
Summary and Numerical Data.—We may now sum up the different systems which can be formed, and give the numerical data from which the model is constructed.[[366]]
I. Bivariant Systems.
| Solid phase. | Area of existence. |
| Ice | ABDC |
| KCl | CDEMKLNO |
| Carnallite | EFGHJKM |
| MgCl2,12H2O | BF1G1GFED |
| MgCl2,8H2O | G1H1HG |
| MgCl2,6H2O | H1I1IH |
| MgCl2,4H2O | I1L1LKI |
| MgCl2,2H2O | L1N1NL |
II. Univariant Systems.—The different univariant systems have already been described. The course of the curves will be sufficiently indicated if the temperature and composition of the solutions for the different invariant systems are given.
III.—Invariant Systems—Binary and Ternary.
| Point. | Solid Phases. | Temperature. | Composition of solution. Gram- molecules of salt per 1000 gram- mol. water. | |||
| A | Ice | 0° | — | |||
| B | Ice; MgCl2,12H2O | -33.6° | 49.2 MgCl2 | |||
| C | Ice; KCl | -11.1° | 59.4 KCl | |||
| D | Ice; MgCl2,12H2O; KCl | -34.3° | 43 MgCl2; 3 KCl | |||
| E |
| MgCl2,12H2O; KCl; carnallite |
| -21° | 66.1 MgCl2; 4.9 KCl | |
| F1 | MgCl2,12H2O | -16.4° | 83.33 MgCl2 | |||
| F | MgCl2,12H2O; carnallite | -16.6° |
| Almost same as F1; contains small amount of KCl | ||
| G1 |
| MgCl2,12H2O; MgCl2,8H2O |
| -16.8° | 87.5 MgCl2 | |
| G |
| MgCl2,12H2O; MgCl2,8H2O; carnallite |
| -16.9° |
| Almost same as G1, but contains small quantity of KCl |
| H1 |
| MgCl2,8H2O; MgCl2,6H2O |
| -3.4° | 99 MgCl2 | |
| H |
| MgCl2,8H2O; MgCl2,6H2O; carnallite |
| ca. -3.4° |
| Almost same as H1, but contains small amount of KCl |
| J1 |
| MgCl2,6H2O; MgCl2,4H2O |
| 116.67° | 161.8 MgCl2 | |
| J |
| MgCl2,6H2O; MgCl2,4H2O; carnallite |
| 115.7° | 162 MgCl2; 4 KCl | |
| K |
| MgCl2,4H2O; KCl; carnallite |
| 152.5° | 200 MgCl2; 24 KCl | |
| L1 |
| MgCl2,4H2O; MgCl2,2H2O |
| 181° | 238.1 MgCl2 | |
| L |
| MgCl2,4H2O; MgCl2,2H2O; KCl |
| 176° | 240 MgCl2; 41 KCl | |
| M | Carnallite; KCl | 167.5° | 166.7 MgCl2; 41.7 KCl | |||
| [N1 | MgCl2,2H2O | 186° | ca. 241 MgCl2] | |||
| N | MgCl2,2H2O; KCl | 186° | 240 MgCl2; 63 KCl | |||
| [O | KCl | 186° | 195.6 KCl] | |||
With the help of the data in the preceding table and of the solid model it will be possible to state in any given case what will be the behaviour of a system composed of magnesium chloride, potassium chloride and water. One or two different cases will be very briefly described; and the reader should have no difficulty in working out the behaviour under other conditions with the help of the model and the numerical data just given.
In the first place it may be again noted that at a temperature above 167.5° (point M) carnallite cannot exist. If, therefore, a solution of magnesium and potassium chlorides is evaporated at a temperature above this point, the result will be a mixture of potassium chloride and either magnesium chloride tetrahydrate or magnesium chloride dihydrate, according as the temperature is below or above 176°. The isothermal curve here consists of only two branches.
Further, reference has already been made to the fact that all points of the carnallite area correspond to solutions in equilibrium with carnallite, but in which the ratio of MgCl2 to KCl is greater than in the double salt. A solution which is saturated with respect to double salt alone will be supersaturated with respect to potassium chloride. At all temperatures, therefore, carnallite is decomposed by water with separation of potassium chloride; hence all solutions obtained by adding excess of carnallite to water will lie on the curve EM. A pure saturated solution of carnallite cannot be obtained.
If an unsaturated solution of the two salts in equimolecular amounts is evaporated, potassium chloride will first be deposited, because the plane bisecting the right angle formed by the X and Y axes cuts the area for that salt. Deposition of potassium chloride will lead to a relative increase in the concentration of magnesium chloride in the solution; and on continued evaporation a point (on the curve EM) will be reached at which carnallite will separate out. So long as the two solid phases are present, the composition of the solution must remain unchanged. Since the separation of carnallite causes a decrease in the relative concentration of the potassium chloride in the solution, the portion of this salt which was deposited at the commencement must redissolve, and carnallite will be left on evaporating to dryness. (Incongruently saturated solution.)
Although carnallite is decomposed by pure water, it will be possible to crystallize it from a solution having a composition represented by any point in the carnallite area. Since during the separation of the double salt the relative amount of magnesium chloride increases, it is most advantageous to
commence with a solution the composition of which is represented by a point lying just above the curve EM (cf. p. [281]).
From the above description of the behaviour of carnallite in solution, the processes usually employed for obtaining potassium chloride will be readily intelligible.[[367]]
Ferric Chloride—Hydrogen Chloride—Water.—In the case of another system of three components which we shall now describe, the relationships are considerably more complicated than in those already discussed. They deserve discussion, however, on account of the fact that they exhibit a number of new phenomena.
In the system formed by the three components, ferric chloride, hydrogen chloride, and water, not only can various compounds of ferric chloride and water (p. [152]), and of hydrogen chloride and water be formed, each of which possesses a definite melting point, but various ternary compounds are also known. Thus we have the following solid phases:—
| 2FeCl3,12H2O | HCl,3H2O | 2FeCl3,2HCl,12H2O |
| 2FeCl3,7H2O | HCl,2H2O | 2FeCl3,2HCl,8H2O |
| 2FeCl3,5H2O | HCl,H2O | 2FeCl3,2HCl,4H2O |
| 2FeCl3,4H2O | ||
| FeCl3 |
From this it will be readily understood that the complete study of the conditions of temperature and concentration under which solutions can exist, either with one solid phase or with two or three solid phases, are exceedingly complicated; and, as a matter of fact, only a few of the possible equilibria have been investigated. We shall attempt here only a brief description of the most important of these.[[368]]
If we again employ rectangular co-ordinates for the graphic
representation of the results, we have the two planes XOT and YOT (Fig. 115): the concentration of ferric chloride being measured along the X-axis, the concentration of hydrogen chloride along the Y-axis, and the temperature along the T-axis. The curve ABCDEFGHJK is, therefore, the solubility curve of ferric chloride in water (p. [152]), and the curve A′B′C′D′E′F′ the solubility curve of hydrogen chloride and its hydrates. B′ and D′ are the melting points of the hydrates HCl,3H2O and HCl,2H2O. In the space between these two planes are represented those systems in which all three components are present. As already stated, only a few of the possible ternary systems have been investigated, and these are represented in Fig. 116. The figure shows the model resting on the XOT-plane, so that the lower edge represents the solubility curve of ferric chloride, the concentration increasing from right to left. The concentration of hydrogen chloride is measured upwards, and the temperature forwards. The further end of the model represents the isothermal surface for -30°. The surface of the model on the left does not correspond with the plane YOT in Fig. 115, but with a parallel plane which cuts the concentration axis for ferric chloride at a point representing 65 gm.-molecules FeCl3 in 100 gm.-molecules of water. The upper surface corresponds with a plane parallel to the axis XOT, at a distance corresponding with the concentration of 50 gm.-molecules HCl in 100 gm.-molecules of water.
Ternary Systems.—We pass over the binary system FeCl3—H2O, which has already been discussed (p. [152]), and the similar system HCl—H2O (see Fig. 115), and turn to the discussion of some of the ternary systems represented by
points on the surface of the model between the planes XOT and YOT. As in the case of carnallite, a plane represents the conditions of concentration of solution and temperature under which a ternary solution can be in equilibrium with a single solid phase (bivariant systems), a line represents the conditions for the coexistence of a solution with two solid phases (univariant systems), and a point the conditions for equilibrium with three solid phases (invariant systems).
In the case of a binary system, in which 2FeCl3,12H2O is in equilibrium with a solution of the same composition, addition of hydrogen chloride must evidently lower the temperature at which equilibrium can exist; and the same holds, of course,
for all other binary solutions in equilibrium with this solid phase. In this way we obtain the surface I., which represents the temperatures and concentrations of solutions in which 2FeCl3,12H2O can be in equilibrium with a ternary solution containing ferric chloride, hydrogen chloride, and water. This surface is analogous to the curved surface K1K2k4k3 in Fig. 97 (p. [256]). Similarly, the surfaces II., III., IV., and V. represent the conditions for equilibrium between the solid phases 2FeCl3,7H2O; 2FeCl3,5H2O; 2FeCl3,4H2O; FeCl3 and ternary solutions respectively. The lines CL, EM, GN, and IO on the model represent univariant systems in which a ternary solution is in equilibrium with two solid phases, viz. with those represented by the adjoining fields. These lines correspond with the ternary eutectic curves k3K1 and k4K2 in Fig. 97. Besides the surfaces already mentioned, there are still three others, VI., VII., and VIII., which also represent the conditions for equilibrium between one solid phase and a ternary solution; but in these cases, the solid phase is not a binary compound or an anhydrous salt, but a ternary compound containing all three components. The solid phases which are in equilibrium with the ternary solutions represented by the surfaces VI., VII., and VIII., are 2FeCl3,2HCl,4H2O; 2FeCl3,2HCl,8H2O; and 2FeCl3,2HCl,12H2O respectively.
The model for FeCl3—HCl—H2O exhibits certain other peculiarities not found in the case of MgCl2—KCl—H2O. On examining the model more closely, it is found that the field of the ternary compound 2FeCl3,2HCl,8H2O (VII.) resembles the surface of a sugar cone, and has a projecting point, the end of which corresponds with a higher temperature than does any other point of the surface. At the point of maximum temperature the composition of the liquid phase is the same as that of the solid. This point, therefore, represents the melting point of the double salt of the above composition.
The curves representing univariant systems are of two kinds. In the one case, the two solid phases present are both binary compounds; or one is a binary compound and the other is one of the components. In the other case, either one or both solid phases are ternary compounds. Curves belonging
to the former class (so-called border curves) start from binary eutectic points, and their course is always towards lower temperatures, e.g. CL, EM, GN, IO. Curves belonging to the latter class (so-called medial curves) would, in a triangular diagram, lie entirely within the triangle. Such curves are YV, WV, VL, LM, MV, NS, ST, SO, OZ. These curves do not always run from higher to lower temperatures, but may even exhibit a point of maximum temperature. Such maxima are found, for example, at U (Fig. 116), and also on the curves ST and LV.
Finally, whereas all the other ternary univariant curves run in valleys between the adjoining surfaces, we find at the point X a similar appearance to that found in the case of carnallite, as the univariant curve here rises above the surrounding surface. The point X, therefore, does not correspond with a eutectic point, but with a transition point. At this point the ternary compound 2FeCl3,2HCl,12H2O melts with separation of 2FeCl3,12H2O, just as carnallite melts at 168° with separation of potassium chloride.
The Isothermal Curves.—A deeper insight into the behaviour of the system FeCl3—HCl—H2O is obtained from a study of the isothermal curves, the complete series of which, so far as they have been studied, is given in Fig. 117.[[369]] In this figure the lightly drawn curves represent isothermal solubility curves, the particular temperature being printed beside the curve.[[370]] The dark lines give the composition of the univariant systems at different temperatures. The point of intersection of a dark with a light curve gives the composition of the univariant solution at the temperature represented by the light curve; and the point of intersection of two dark lines gives the composition of the invariant solution in equilibrium with three solid phases. The dotted lines represent metastable systems, and the points P, Q, and R represent solutions of
the composition of the ternary salts, 2FeCl3,2HCl,4H2O; 2FeCl3,2HCl,8H2O; and 2FeCl3,2HCl,12H2O.
The farther end of the model (Fig. 116) corresponds, as already mentioned, to the temperature -30°, so that the outline evidently represents the isothermal curve for that temperature. Fig. 117 does not show this. We can, however, follow the isothermal for -20°, which is the extreme curve on the right in Fig. 117. Point A represents the solubility of 2FeCl3,12H2O in water. If hydrogen chloride is added, the concentration of ferric chloride in the solution first decreases and then increases, until at point 34 the ternary double salt 2FeCl3,2HCl,12H2O is formed. If the addition of hydrogen chloride is continued, the ferric chloride disappears ultimately, and only the ternary double salt remains. This salt can coexist with solutions of the composition represented by the curve which passes through the points 173, 174, 175. At the last-mentioned point, the ternary salt with 8H2O is formed. The composition of the solutions with which this salt is in equilibrium at -20° is represented by the curve which passes through a point of maximal concentration with respect to HCl, and cuts the curve SN at the point 112, at which the solution is in equilibrium with the two solid phases 2FeCl3,4H2O and 2FeCl3,2HCl,8H2O. The succeeding portion of the isotherm represents the solubility curve at -20° of 2FeCl3,4H2O, which cuts the dark line OS at point 113, at which the solution is in equilibrium with the two solid phases 2FeCl3,4H2O and 2FeCl3,2HCl,4H2O. Thereafter comes the solubility curve of the latter compound.
The other isothermal curves can be followed in a similar manner. If the temperature is raised, the region of existence of the ternary double salts becomes smaller and smaller, and at temperatures above 30° the ternary salts with 12H2O and 8H2O are no longer capable of existing. If the temperature is raised above 46°, only the binary compounds of ferric chloride and water and the anhydrous salt can exist as solid phases. The isothermal curve for 0° represents the solubility curve for 2FeCl3,12H2O; 2FeCl3,7H2O; 2FeCl3,5H2O; and 2FeCl3,4H2O.
Finally, in the case of the system FeCl3—HCl—H2O, we find closed isothermal curves. Since, as already stated, the salt 2FeCl3,2HCl,8H2O has a definite melting point, the temperature of which is therefore higher than that at which this compound is in equilibrium with solutions of other composition, it follows that the line of intersection of an isothermal plane corresponding with a temperature immediately below the melting point of the salt with the cone-shaped surface of its region of existence, will form a closed curve. This is shown by the isotherm for -4.5°, which surrounds the point Q, the melting point of the ternary salt.
The following table gives some of the numerical data from which the curves and the model have been constructed:—
| Point. | Solid Phases. | Temperature. | Composition of the solution in gm.-mols. salt to 100 gm.-mols. water. | |||
| HCl | FeCl3 | |||||
| A | 2FeCl3,12H2O | -20° | — | 6.56 | ||
| C | 2FeCl3,12H2O; 2FeCl3,7H2O | 27.4° | — | 24.30 | ||
| E | 2FeCl3,7H2O; 2FeCl3,5H2O | 30° | — | 30.24 | ||
| G | 2FeCl3,5H2O; 2FeCl3,4H2O | 55° | — | 40.64 | ||
| J | 2FeCl3,4H2O; FeCl3 | 66° | — | 58.40 | ||
| L |
| 2FeCl3,12H2O; 2FeCl3,7H2O; 2FeCl3,2HCl,8H2O |
| -7.5° | 19.22 | 23.72 |
| M |
| 2FeCl3,7H2O; 2FeCl3,5H2O; 2FeCl3,2HCl,8H2O |
| -7.3° | 23.08 | 28.55 |
| N |
| 2FeCl3,5H2O; 2FeCl3,4H2O; 2FeCl3,2HCl,8H2O |
| -16° | 28.40 | 31.89 |
| S |
| 2FeCl3,4H2O; 2FeCl3,2HCl,8H2O; 2FeCl3,2HCl,4H2O |
| -27.5° | 32.33 | 34.21 |
| O |
| 2FeCl3,4H2O; FeCl3; 2FeCl3,2HCl,4H2O |
| 29° | 33.71 | 49.84 |
| U |
| 2FeCl3,7H2O; 2FeCl3,2HCl,8H2O |
| -4.5° | 20.66 | 25.74 |
| V |
| 2FeCl3,12H2O; 2FeCl3,2HCl,12H2O; 2FeCl3,2HCl,8H2O |
| -13° | 22.40 | 18.00 |
| X |
| 2FeCl3,12H2O; 2FeCl3,2HCl,12H2O |
| -12.5° | 22.14 | 16.69 |
| Q | 2FeCl3,2HCl,8H2O | -3° (melting point) | ||||
Basic Salts.—Another class of systems in the study of
which the Phase Rule has performed exceptional service, is that of the basic salts. In many cases it is impossible, by the ordinary methods of analysis, to decide whether one is dealing with a definite chemical individual or with a mixture. The question whether a solid phase is a chemical individual can, however, be answered, in most cases, with the help of the principles which we have already learnt. Let us consider, for example, the formation of basic salts from bismuth nitrate, and water. In this case we can choose as components Bi2O3, N2O5, and H2O; since all the systems consist of these in varying amounts. If we are dealing with a condition of equilibrium at constant temperature between liquid and solid phases, three cases can be distinguished,[[371]] viz.—
1. The solutions in different experiments have the same composition, but the composition of the precipitate alters. In this case there must be two solid phases.
2. The solutions in different experiments can have varying composition, while the composition of the precipitate remains unchanged. In this case only one solid phase exists, a definite compound.
3. The composition both of the solution and of the precipitate varies. In this case the solid phase is a solid solution or a mixed crystal.
In order, therefore, to decide what is the nature of a precipitate produced by the hydrolysis of a normal salt, it is only necessary to ascertain whether and how the composition of the precipitate alters with alteration in the composition of the solution. If the composition of the solution is represented by abscissæ, and the composition of the precipitate by ordinates, the form of the curves obtained would enable us to answer our question; for vertical lines would indicate the presence of two solid phases (1st case), horizontal lines the presence of only one solid phase (2nd case), and slanting lines the presence of mixed crystals (3rd case). This method of representation cannot, however, be carried out in most cases. It is, however,
generally possible to find one pair or several pairs of components, the relative amounts of which in the solution or in the precipitate undergo change when, and only when, the composition of the solution or of the precipitate changes. Thus, in the case of bismuth, nitrate, and water, we can represent the ratio of Bi2O3 : N2O5 in the precipitate as ordinates, and N2O5 : H2O in the solution as abscissæ. A horizontal line then indicates a single solid phase, and a vertical line two solid phases. An example of this is given in Fig. 118.[[372]]
Bi2O3—N2O5—H2O.—Although various systems have been studied in which there is formation of basic salts,[[373]] we shall content ourselves here with the description of some of the conditions for the formation of basic salts of bismuth nitrate, and for their equilibrium in contact with solutions.[[374]]
Three normal salts of bismuth oxide and nitric acid are known, viz. Bi2O3,3N2O5,10H2O(S10); Bi2O3,3N2O5,4H2O(S4); and Bi2O3,3N2O5,3H2O(S3). Besides these normal salts, there are the following basic salts:—
| Bi2O3,N2O5,2H2O | (represented by B1-1-2) |
| Bi2O3,N2O5,H2O | ( ,, ,, B1-1-1) |
| 6Bi2O3,5N2O5,9H2O | ( ,, ,, B6-5-9) |
| 2Bi2O3,N2O5,H2O | ( ,, ,, B2-1-1) |
Probably some others also exist. The problem now is to find the conditions under which these different normal and basic salts can be in equilibrium with solutions of varying concentration of the three components. Having determined the equilibrium conditions for the different salts, it is then possible to construct a model similar to that for MgCl2—KCl—H2O or for FeCl3—HCl—H2O, from which it will be possible to determine the limits of stability of the different salts, and to predict what will occur when we bring the salts in contact with solutions of nitric acid of different concentrations and at different temperatures.
For our present purpose it is sufficient to pick out only some of the equilibria which have been studied, and which are represented in the model (Fig. 119). In this case use has been made of the triangular method of representation, so that the surface of the model lies within the prism.
This model shows the three surfaces, A, B, and C, which represent the conditions for the stable existence of the salts B1-1-1, S10, and S3 in contact with solution at different
temperatures. The front surface of the model represents the temperature 9°, and the farther end the temperature 75.5°. The dotted curve represents the isotherm for 20°. The prominences between the surfaces represent, of course, solutions which are saturated in respect of two solid phases. Thus, for example, pabc represents solutions in equilibrium with B1-1-1 and S10; and the ridge qdc, solutions in equilibrium with S10 and S3. The point b, which lies at 75.5°, is the point of maximum temperature for S10. If the temperature is raised above this point, S10 decomposes into the basic salt B1-1-1 and solution. This point is therefore analogous to the point M in the carnallite model, at which this salt decomposes into potassium chloride and solution (p. [284]); or to the point at which the salt 2FeCl3,2HCl,12H2O decomposes into 2FeCl3,12H2O and solution (p. [294]). The curve pab has been followed to the temperature of 72° (point c). The end of the model is incomplete, but it is probable that in the neighbourhood of the point c there exists a quintuple point at which the basic salt B1-2-2 appears. In the neighbourhood of e also there probably exists another quintuple point at which S4 is formed. These systems have, however, not been studied.
The following tables give some of the numerical data:—
Isotherm for 20°.
| Solid Phase. | Composition of the solution. Gram-mols. in 1000 gm.-mols. of water. | |
| Bi2O3 | N2O5 | |
| B1-1-1 | 10.50 | 38.65 |
| — | 27.20 | 83.84 |
| B1-1-1; S10 | 30.15 | 97.97 |
| S10 | 29.70 | 96.57 |
| — | 19.65 | 98.76 |
| — | 10.51 | 162.58 |
| — | 33.51 | 355.87 |
| S10; S3 | 51.00 | 403.0 |
| S3 | 14.35 | 492.0 |
| — | 7.45 | 592.9 |
Systems in Equilibrium with B1-1-1 and S10 (Curve pabc).
| Solid Phase. | Composition of the solution. Gram-mols. in 1000 gm.-mols. of water. | |
| Bi2O3 | N2O5 | |
| 9° | 26.7 | 88.2 |
| 20° (point a) | 30.15 | 97.97 |
| 30° | 33.6 | 112.3 |
| 50° | 41.8 | 148.4 |
| 65° | 57.21 | 190.8 |
| 75.5° (point b) | 87.9 | 288.4 |
| 72° (point c) | 96.0 | 327.0 |
Systems in Equilibrium with S10 and S3 (Curve qde).
| Solid Phase. | Composition of the solution. Gram-mols. in 1000 gm.-mols. of water. | |
| Bi2O3 | N2O5 | |
| 11.5° | 44.5 | 396 |
| 20° | 51.0 | 405.4 |
| 50° | 66.5 | 444.2 |
| 65° | 80.0 | 454.4 |
Basic Mercury Salts.—The Phase Rule has also been applied by A. J. Cox[[375]] in an investigation of the basic salts of mercury, the result of which has been to show that, of the salts mentioned in text-books, quite a number are incorrectly stated to be chemical compounds or chemical individuals (p. [92]). The investigation, which was carried out essentially in the manner described above, included the salts mentioned in the following table; and of the basic salts said to be derived from them, only those mentioned really exist. In the following table, the numbers in the second column give the minimum values of the concentration of the acid, expressed in equivalent normality, necessary for the existence of the
corresponding salts in contact with solution at the temperature given in the third column:—
| Salt. | Normality of acid. | Temperature. |
| HgCrO4 | 1.41 | 50° |
| 3HgO.CrO3 | 2.6 × 10-4 | 50° |
| Hg(NO3)2.H2O | 18.72 | 25° |
| 3HgO.N2O5 | 0.159 | 25° |
| HgSO4 | 6.87 | 25° |
| 3HgO.SO3 | 1.3 × 10-3 | 25° |
| HgF2 | 1.14 | 25° |
| HgNO3.H2O | 2.95 | 25° |
| 5Hg2O.3N2O5.2H2O | ca. 0.293 | 25° |
| 2Hg2O.N2O5(?) | 0.110 | 25° |
| 3Hg2O.N2O5.2H2O(?) | 1.7 × 10-3 | 25° |
| Hg2SO4 | 4.2 × 10-3 | 25° |
| 2Hg2O.SO3.H2O | 5.6 × 10-4 | 25° |
Mercuric fluoride does not form any basic salt.
Since two succeeding members of a series can coexist only in contact with a solution of definite concentration, we can prepare acid solutions of definite concentration by bringing an excess of two such salts in contact with water.
Indirect Determination of the Composition of the Solid Phase.—It has already been shown (p. [228]) how the composition of the solid phase in a system of two components can be determined without analysis, and we shall now describe how this can be done in a system of three components.[[376]]
We shall assume that we are dealing with the aqueous solution of two salts which can give rise to a double salt, in which case we can represent the solubility relations in a system of rectangular co-ordinates. In this case we should obtain, as before (Fig. 120), the isotherm adcb, if we express the
composition of the solution in gram-molecules of A or of B to 100 gram-molecules of water.
Let us suppose, now, that the double salt is in equilibrium with the solution at a definite temperature, and that the composition of the solution is represented by the point e. The greater part of the solution is now separated from the solid phase, and the latter, together with the adhering mother liquor, is analyzed. The composition (expressed, as before, in gram-molecules of A and B to 100 gram molecules of water) will be represented by a point (e.g. f) on the line eS, where S represents the composition of the double salt. That this is so will be evident when one considers that the composition of the whole mass must lie between the composition of the solution and that of the double salt, no matter what the relative amounts of the solid phase and the mother liquor.
If, in a similar manner, we analyze a solution of a different composition in equilibrium with the same double salt (not necessarily at the same temperature as before), and also the mixture of solid phase and solution, we shall obtain two other points, as, for example, g and h, and the line joining these must likewise pass through S. The method of finding the
composition of an unknown double salt consists, therefore, in finding, in the manner just described, the position of two lines such as ef and gh. The point of intersection of these lines then gives the composition of the double salt.
If the double salt is anhydrous, the point S lies at infinity, and the lines ef and gh are parallel to each other.
The same result is arrived at by means of the triangular method of representation.[[377]] If we start with the three components in known amounts, and represent the initial composition of the whole by a point in the triangle, and then ascertain the final composition of the solution in equilibrium with the solid phase at a definite temperature, the line joining the points representing the initial and end concentration passes through the point representing the composition of the solid phase. If two determinations are made with solutions having different initial and final concentrations in equilibrium with the same solid phase, then the point of intersection of the two lines so obtained gives the composition of the solid phase.