Although this view put forward by Heyn has not been conclusively proved, it must be said that there is much evidence in its favour. Further investigation is, however, required before a final decision as to the interpretation of the curves can be reached.

Determination of the Composition of Compounds, without a.n.a.lysis.--Since the equilibrium between a solid and a liquid phase depends not only on the composition of the liquid (solution) but also on that of the solid, it is necessary {229} to determine the composition of the latter. In some cases this is easily effected by separating the solid from the liquid phase and a.n.a.lyzing it. In other cases, however, this method is inapplicable, or is accompanied by difficulties, due either to the fact that the solid phase undergoes decomposition (_e.g._ when it contains a volatile const.i.tuent), or to the difficulty of completely separating the mother liquor; as, for example, in the case of alloys. In all such cases, therefore, recourse must be had to other methods.

In the first place, synthetic methods may be employed.[313] In this case we start with a solution of the two components, to which a third substance is added, which, however, does not enter into the solid phase.[314] We will a.s.sume that the initial solution contains _x_ gm. of A and _y_ gm. of B to 1 gm. of C. After the solution has been cooled down to such a temperature that solid substance separates out, a portion of the liquid phase is removed with a pipette and a.n.a.lyzed. If, now, the composition of the solution is such that there are _x'_ gm. of A and _y'_ gm. of B to 1 gm. of C., then the composition of the solid phase is _x_ - _x'_ gm. of A and _y_ - _y'_ gm. of B. When _x_ = _x'_, the solid phase is pure B; when _y_ = _y'_, the solid phase is pure A.

We have a.s.sumed here that there is only one solid phase present, containing A and B. To make sure that the solid phase is not a solid solution in which A and B are present in the same ratio as in the liquid solution, a second determination of the composition must be made, with different initial and end concentrations. If the solid phase is a solid solution, the composition will now be found different from that found previously.

The composition of the solid phase can, however, be determined in another manner, viz. by studying the fusion curve and the curve of cooling. From the form of the fusion curve alone, it is possible to decide whether the two components {230} form a compound or not; and if the compounds which may be formed have a definite melting point, the position of the latter gives at once the composition of the compounds (cf. p. 231).

This method, however, cannot be applied when the compounds undergo decomposition before the melting point is reached. In such cases, however, the form of the cooling curve enables one to decide the composition of the solid phase.[315] If a solution is allowed to cool slowly, and the temperature noted at definite times, the graphic representation of the rate of cooling will give a continuous curve; _e.g._ _ab_ in Fig. 76. So soon, however, as a solid phase begins to be formed, the rate of cooling alters abruptly, and the cooling curve then exhibits a break, or change in direction (point _b_). When the eutectic point is reached, the temperature remains constant, until all the liquid has solidified. This is represented by the line _cd_. When complete solidification has occurred, the fall of temperature again becomes uniform (_de_).

[Ill.u.s.tration: FIG. 76.]

[Ill.u.s.tration: FIG. 77.]

[Ill.u.s.tration: FIG. 78.]

The length of time during which the temperature remains constant at the point _c_, depends, of course, on the eutectic solution. If, therefore, we take equal amounts of solution having a different initial composition, the period of constant temperature in the cooling curve will evidently be greatest in the case of the solution having the composition of the eutectic point; and the period will become less and less as we increase the amount of one of the components. The relationship between initial composition of solution and the duration of constant temperature at the eutectic point is represented by the curve _a'c'b'_ (Fig. 77). When a compound possessing a definite melting point is formed, it behaves as a pure substance. If, therefore, the initial composition of the {231} solution is the same as that of the compound, no eutectic solution will be obtained; and therefore no line of constant temperature, such as _cd_ (Fig. 76). In such a case, if we represent graphically the relation between the initial composition of the solution and the duration of constant temperature, a diagram is obtained such as shown in Fig. 78. The two maxima on the time-composition curve represent eutectic points, and the minima, _a'_, _b'_, _e'_, pure substances. The position of _e'_ gives the composition of the compound.

When a series of compounds is formed, then for each compound a minimum is found on the time-composition curve.

[Ill.u.s.tration: FIG. 79.]

If the compound formed has no definite melting point, the diagram obtained is like that shown in Fig. 79. If we start with a solution, the composition of which is represented by a point between _d_ and _b_, then, on cooling, _b_ will separate out first, and the temperature will fall until the point _d_ is reached. The temperature then remains constant until the component _b_, which has separated out, is converted into the compound. After this the temperature again falls, until it again remains constant at the eutectic point c. In the case of the first halt, the period of constant temperature is greatest when the initial composition of the solution is the same as that of the compound; and it becomes shorter and shorter with {232} increase in the amount of either component. In this way we obtain the time-composition curve _b'e"d'_, of which the maximum point _e"_ gives the composition of the compound.

On the other hand, the period of constant temperature for the eutectic point _c_ is greatest in the case of solutions having the same initial _composition_ as that corresponding with the eutectic point; and it decreases the more the initial composition approaches that of the pure component _a_ or the component e. In this way we obtain the time-composition curve _a'c'e'_. Here also the point _e'_ represents the composition of the compound. We see, therefore, that from the graphic representation of the freezing-point curve, and from the duration of the temperature-arrests on the cooling curve, for solutions of different initial composition, it is possible, without having recourse to a.n.a.lysis, to decide what solid phases are formed, and what is their composition.

Formation of Minerals.--Important and interesting as is the application of the Phase Rule to the study of alloys, its application to the study of the conditions regulating the formation of minerals is no less so; and although we do not propose to consider different cases in detail here, still attention must be drawn to certain points connected with this interesting subject.

In the first place, it will be evident from what has already been said, that that mineral which first crystallizes out from a molten magma is not necessarily the one with the highest melting point. The _composition_ of the fused ma.s.s must be taken into account. When the system consists of two components which do not form a compound, one or other of these will separate out in a pure state, according as the composition of the molten ma.s.s lies on one or other side of the eutectic composition; and the separation of the one component will continue until the composition of the eutectic point is reached. Further cooling will then lead to the simultaneous separation of the two components.

If, however, the two components form a stable compound (_e.g._ orthoclase, from a fused mixture of silica and pota.s.sium aluminate), then the freezing-point curve will resemble that {233} shown in Fig. 64; _i.e._ there will be a middle curve possessing a dystectic point, and ending on either side at a eutectic point. This curve would represent the conditions under which orthoclase is in equilibrium with the molten magma. If the initial composition of the magma is represented by a point between the two eutectic points, orthoclase will separate first. The composition of the magma will thereby change, and the ma.s.s will finally solidify to a mixture of orthoclase and silica, or orthoclase and pota.s.sium aluminate, according to the initial composition.

What has just been said holds, however, only for stable equilibria, and it must not be forgotten that complications can arise owing to suspended transformation (when, for example, the magma is rapidly cooled) and the production of metastable equilibria. These conditions occur very frequently in nature.

The study of the formation of minerals from the point of view of the Phase Rule is still in its initial stages, but the results which have already been obtained give promise of a rich harvest in the future.[316]

{234}

CHAPTER XIII

SYSTEMS OF THREE COMPONENTS

General.--It has already been made evident that an increase in the number of the components from one to two gives rise to a considerable increase in the possible number of systems, and introduces not a few complications into the equilibrium relations of these. No less is this the case when the number of components increases from two to three; and although examples of all the possible types of systems of three components have not been investigated, nor, indeed, any one type fully, nevertheless, among the systems which have been studied experimentally, cases occur which not only possess a high scientific interest, but are also of great industrial importance. On account not only of the number, but more especially of the complexity of the systems const.i.tuted of three components, no attempt will be made to give a full account, or, indeed, even a survey of all the cases which have been subjected to a more or less complete experimental investigation; on the contrary, only a few of the more important cla.s.ses will be selected, and the most important points in connection with the behaviour of these described.

On applying the Phase Rule

P + F = C + 2

to the systems of three components, we see that in order that the system shall be invariant, no fewer than five phases must be present together, and an invariant system will therefore exist at a _quintuple_ point. Since the number of liquid phases can never exceed the number of the components, and since there can be only one vapour phase, it is evident that in this case, {235} as in others, there must always be at least one solid phase present at the quintuple point. As the number of phases diminishes, the variability of the system can increase from one to four, so that in the last case the condition of the system will not be completely defined until not only the temperature and the total pressure of the system, but also the concentrations of two of the components have been fixed. Or, instead of the concentrations, the partial pressures of the components may also be taken as independent variables.

Graphic Representation.--Hitherto the concentrations of the components have been represented by means of rectangular co-ordinates, although the numerical relationships have been expressed in two different ways. In the one case, the concentration of the one component was expressed in terms of a fixed amount of the other component. Thus, the solubility of a salt was expressed by the number of grams of salt dissolved by 100 grams of water or other solvent; and the numbers so obtained were measured along one of the co-ordinates. The second co-ordinate was then employed to indicate the change of another independent variable, _e.g._ temperature. In the other case, the combined weights of the two components A and B were put equal to unity, and the concentration of the one expressed as a fraction of the whole amount. This method allows of the representation of the complete series of concentrations, from pure A to pure B, and was employed, for example, in the graphic representation of the freezing point curves.

Even in the case of three components rectangular co-ordinates can also be employed, and, indeed, are the most convenient in those cases where the behaviour of two of the components to one another is very different from their behaviour to the third component; as, for example, in the case of two salts and water. In these cases, the composition of the system can be represented by measuring the amounts of each of the two components in a given weight of the third, along two co-ordinates at right angles to one another; and the change of the system with the temperature can then be represented by a third axis at right angles to the first two. In those cases, {236} however, where the three components behave in much the same manner towards one another, the rectangular co-ordinates are not at all suitable, and instead of these a _triangular diagram_ is employed. Various methods have been proposed for the graphic representation of systems of three components by means of a triangle, but only two of these have been employed to any considerable extent; and a short description of these two methods will therefore suffice.[317]

[Ill.u.s.tration: FIG. 80.]

In the method proposed by Gibbs an equilateral triangle of unit height is used (Fig 80).[318] The quant.i.ties of the different components are expressed as fractional parts of the whole, and the sum of their concentrations is therefore equal to unity, and can be represented by the height of the triangle. The corners {237} of the triangle represent the pure substances A, B, and C respectively. A point on one of the sides of the triangle will give the composition of a mixture in which only two components are present, while a point within the triangle will represent the composition of a ternary mixture. Since every point within the triangle has the property that the sum of the perpendiculars from that point on the sides of the triangle is equal to unity (the height of the triangle), it is evident that the composition of a ternary mixture can be represented by fixing a point within the triangle such that the lengths of the _perpendiculars_ from the point to the sides of the triangle are equal respectively to the fractional amounts of the three components present; the fractional amount of A, B, or C being represented by the perpendicular distance from the side of the triangle _opposite_ the corners A, B, and C respectively.

The location of this point is simplified by dividing the normals from each of the corners on the opposite side into ten or one hundred parts, and drawing through these divisions lines at right angles to the normal and parallel to the side of the triangle. A network of rhombohedra is thus obtained, and the position of any point can be read off in practically the same manner as in the case of rectangular co-ordinates. Thus the point P in Fig. 80 represents a ternary mixture of the composition A = 0.5, B = 0.3, C = 0.2; the perpendiculars P_a_, P_b_, and P_c_ being equal respectively to 0.5, 0.2, and 0.3 of the height of the triangle.

Another method of representation, due to Roozeboom, consists in employing an equilateral triangle, the length of whose _side_ is made equal to unity, or one hundred; the sum of the fractional or percentage amounts of the three components being represented therefore by a side of the triangle. In this case the composition of a ternary mixture is obtained by determining, not the _perpendicular_ distance of a point P from the three sides of the triangle, but the distance in a direction _parallel_ to the sides of the triangle (Fig. 81). Conversely, in order to represent a mixture consisting of _a_, _b_, and _c_ parts of the components A, B, and C respectively, one side of the triangle, say AB, is first of all divided into ten or one {238} hundred parts; a portion, B_x_ = _a_, is then measured off, and represents the amount of A present. Similarly, a portion, A_x'_ = _b_, is measured off and represents the fractional amount of B, while the remainder, _xx'_ = _c_, represents the amount of C. From _x_ and _x'_ lines are drawn parallel to the sides of the triangle, and the point of intersection, P, represents the composition of the ternary mixture of given composition; for, as is evident from the figure, the distance of the point P from the three sides of the triangle, when measured in directions _parallel_ to the sides, is equal to _a_, _b_, and _c_ respectively. From the division marks on the side AB, it is seen that the point P in this figure also represents a mixture of 0.5 parts of A, 0.2 parts of B, and 0.3 parts of C. This gives exactly the same result as the previous method. The employment of a right-angled isosceles triangle has also been suggested,[319] but is not in general use.

[Ill.u.s.tration: FIG. 81.]

In employing the triangular diagram, it will be of use to note a property of the equilateral triangle. A line drawn from one corner of the triangle to the opposite side, represents the composition of all mixtures in which the _relative_ amounts of two of the components remain unchanged. Thus, as Fig. 82 shows, if the component C is added to a mixture x, in which A and B are present in the proportions of _a_ : _b_, a mixture _x'_, which is thereby obtained, also contains A and B in the ratio _a_ : b. For the two triangles AC_x_ and BC_x_ are similar to the two triangles HC_x'_ and KC_x'_; and, {239} therefore, A_x_ : B_x_ = H_x'_ : K_x'_. But A_x_ = D_x_ and B_x_ = E_x_; further H_x'_ = F_x'_ and K_x'_ = G_x'_. Therefore, D_x_ : E_x_ = F_x'_ : G_x'_ = _b_ : a. At all points on the line C_x_, therefore, the ratio of A to B is the same.

[Ill.u.s.tration: FIG. 82.]

[Ill.u.s.tration: FIG. 83.]

If it is desired to represent at the same time the change of another independent variable, _e.g._ temperature, this can be done by measuring the latter along axes drawn perpendicular to the corners of the triangle. In this way a right prism (Fig. 83) is obtained, and each section of this cut parallel to the base represents therefore an _isothermal surface_.

{240}

CHAPTER XIV

SOLUTIONS OF LIQUIDS IN LIQUIDS

We have already seen (p. 95) that when two liquids are brought together, they may mix in all proportions and form one h.o.m.ogeneous liquid phase; or, only partial miscibility may occur, and two phases be formed consisting of two mutually saturated solutions. In the latter case, the concentration of the components in either phase and also the vapour pressure of the system had, at a given temperature, perfectly definite values. In the case of three liquid components, a similar behaviour may be found, although complete miscibility of three components with the formation of only one liquid phase is of much rarer occurrence than in the case of two components. When only partial miscibility occurs, various cases are met with according as the three components form one, two, or three pairs of partially miscible liquids. Further, when two of the components are only partially miscible, the addition of the third may cause either an increase or a diminution in the mutual solubility of these. An increase in the mutual solubility is generally found when the third component dissolves readily in each of the other two; but when the third component dissolves only sparingly in the other two, its addition diminishes the mutual solubility of the latter.

We shall consider here only a few examples ill.u.s.trating the three chief cases which can occur, viz. (1) A and B, and also B and C are miscible in all proportions, while A and C are only partially miscible. (2) A and B are miscible in all proportions, but A and C and B and C are only partially miscible. (3) A and B, B and C, and A and C are only partially miscible. A, B, and C here represent the three components.

1.--_The three components form only one pair of partially miscible liquids._ {241}

An example of this is found in the three substances: chloroform, water, and acetic acid.[320] Chloroform and acetic acid, and water and acetic acid, are miscible with one another in all proportions, but chloroform and water are only partially miscible with one another. If, therefore, chloroform is shaken with a larger quant.i.ty of water than it can dissolve, two layers will be formed consisting one of a saturated solution of water in chloroform, the other of a saturated solution of chloroform in water. The composition of these two solutions at a temperature of about 18, will be represented by the points _a_ and _b_ in Fig. 84; _a_ representing a solution of the composition: chloroform, 99 per cent.; water, 1 per cent.; and _b_ a solution of the composition: chloroform, 0.8 per cent.; water, 99.2 per cent. When acetic acid is added, it distributes itself between the two liquid layers, and two conjugate _ternary_ solutions, consisting of chloroform, water, and acetic acid are thereby produced which are in equilibrium with one another, and the composition of which will be represented by two points inside the triangle. In this way a series of pairs of ternary solutions will be obtained by the addition of acetic acid to the mixture of chloroform and water. By this addition, also, not only do the two liquid phases become increasingly rich in acetic acid, but the mutual solubility of the chloroform and water increases; so that the layer _a_ becomes relatively richer in water, and layer _b_ relatively richer in chloroform. This is seen from the following table, which gives the percentage composition of different conjugate ternary solutions at 18.

------------------------------------------------------------------------- Heavier layer.

Lighter layer.

------------------------------------------------------------------------- Chloroform.

Water.

Acetic acid.

Chloroform.

Water.

Acetic acid.

------------------------------------------------------------------------- 99.01

0.99

0

0.84

99.16

0 91.85

1.38

6.77

1.21

73.69

25.10 80.00

2.28

17.72

7.30

48.58

44.12 70.13

4.12

25.75

15.11

34.71

50.18 67.15

5.20

27.65

18.33

31.11

50.56 59.99

7.93

32.08

25.20

25.39

49.41 55.81

9.58

34.61

28.85

23.28

47.87 -------------------------------------------------------------------------

{242}