Fig. 67Fig.67.
Phenol can also combine withp-toluidine in equimolecular proportions; and this compound is of interest, from the fact that it exists in two crystalline forms melting at 28.5° and 30°. Each of these forms now must have its own equilibrium curve, and it was found that the intermediate portion of the freezing point curve was duplicated, as shown in Fig. 68.[295]
Fig. 68Fig.68.
Fig. 69Fig.69.
Lastly, a curve is given, Fig. 69,[296]which corresponds with curve II., Fig. 64. Picric acid and benzene can form a compound, which, however, can exist only in contact with solutionscontaining excess of benzene. When the temperature is raised, a point (K) is reached at which the compound melts with separation of solid picric acid. The point, K, is, therefore, atransition point; analysis, however, showed that the composition of the solution at this point is very nearly that of the compound C6H2(NO2)3OH,C6H6, so that the melting point of the compound can almost be reached. The fusion of the compound of benzene and picric acid with separation of the latter is analogous to the (partial) fusion of Glauber's salt with separation of anhydrous sodium sulphate.
2.Optically Active Substances.
The question as to whether a resolvable inactive body is a mixture of the two oppositely active constituents (adl-mixture), or a racemic compound, is one which has given rise to considerable discussion during the past decade; and several investigators have endeavoured to establish general rules by which the question could be decided. In the case of inactive liquids it is a matter of great difficulty to arrive at a certain conclusion as to whether one is dealing with a mixture or a compound, for in this case the usual physical methods give but a dubious answer; and although the existence of a racemate in the liquid state (in the case of conine) has been asserted,[297]most chemists incline to the belief that such a thing is improbable.
Even in the case of crystalline substances, where the differences between the various forms is greater, it was not always easy to discriminate between thedl-mixture and the racemic compound. The occurrence of hemihedral faces was considered by Pasteur to be a sufficient criterion for an optically active substance. It has, however, been found that hemihedry in crystals, although a frequent accompaniment ofoptical activity, is by no means a necessary or constant expression of this property. Other rules, also, which were given, although in some cases reliable, were in other cases insufficient; and all were in so far unsatisfactory that they lacked a theoretical basis.
With the help of the Phase Rule, however, it is possible from a study of the solubility or fusion curves of the optically active and inactive substances, to decide the nature of the inactive substance, at least under certain conditions. On account of the interest and importance which these compounds possess, a brief description of the application of the Phase Rule to the study of such substances will be given here;[298]the two optical antipodes being regarded as the two components.
In the present chapter we shall consider only the fusion curves, the solubility curves being discussed in the next section on three-component systems. The rules which are hereby obtained, have reference only to the nature of the inactive substance in the neighbourhood of the melting points.
I.The inactive substance is adl-mixture.
In this case the fusion curves will have the simple form shown in type I, Fig. 63. A and B are the melting points of the two optical isomerides, and C the eutectic point at which the inactive mixture consisting of equal amounts of d- and l-form melts. Owing to the similar effect of the one form on the freezing point of the other, the figure is symmetrical. No example of this simple case has been investigated.
II.The two components form a racemic compound.
In this case there will be three melting point curves as in Fig. 64, type I. In this case also the figure must be symmetrical.
Examples.—As examples of this, may be taken dimethyl tartrate and mandelic acid, the freezing point curves of which are given in Figs. 70 and 71.[299]As can be seen, the curve for the racemic tartrate occupies a large part of the diagram,while that for racemic mandelic acid is much smaller. In the case of dimethyldiacetyl tartrate, this middle portion is still less.
Fig. 70Fig.70.
Fig. 71Fig.71.
Fig. 72Fig.72.
Active dimethyl tartrate melts at 43.3°; racemic dimethyl tartrate at 89.4°. Active mandelic acid melts at 132.8°; the racemic acid at 118.0°. In the one case, therefore, the racemic compound has a higher, in the other a lower melting point than the active forms.
In the case of partially racemic compounds (i.e.the compound of a racemate with an optically active substance) the type of curve will be the same, but the figure will no longer be symmetrical. Such a curve has been found in the case of the l-menthyl esters of d- and l-mandelic acid (Fig. 72).[300]The freezing point of l-menthyl d-mandelate is 97.2°, of l-menthyl l-mandelate 77.6°, and of l-menthyl r-mandelate 83.7.° It will be observed that the summit of the curve for the partially racemic mandelate is very flat, indicating that the compound is largely dissociated into its components at the temperature of fusion.
III.The inactive substance is a pseudo-racemic mixed crystal.
In cases where the active components can form mixed crystals, the freezing-point curve will exhibit one of the forms given in Fig. 65. The inactive mixed crystal containing 50 per cent. of the dextro and laevo compound, is known as a pseudo-racemic mixed crystal.[301]So far, only curves of the types I. and II. have been obtained.
Examples.—The two active camphor oximes are of interest from the fact that they form a continuous series of mixed crystals,all of which have the same melting point. The curve which is obtained in this case is, therefore, a straight line joining the melting points of the pure active components; the melting point of the active isomerides and of the whole series of mixed crystals being 118.8°.
Fig. 73Fig.73.
In the case of the carvoximes mixed crystals are also formed, but the equilibrium curve in this case exhibits a maximum (Fig. 73). At this maximum point the composition of the solid and of the liquid solution is the same. Since the curve must be symmetrical, this maximum point must occur in the case of the solution containing 50 per cent.of each component, which will therefore be inactive. Further, this inactive mixed crystal will melt and solidify at the same temperature, and behave, therefore, like a chemical compound (p.187). The melting point of the active compounds is 72°; that of the inactive pseudo-racemic mixed crystal is 91.4°·
Transformations.—As has already been remarked, the conclusions which can be drawn from the fusion curves regarding the nature of the inactive substances formed hold only for temperatures in the neighbourhood of the melting points. At temperatures below the melting point transformation may occur;e.g.a racemate may break up into adl-mixture, or a pseudo-racemic mixed crystal may form a racemic compound. We shall at a later point meet with examples of a racemic compound changing into adl-mixture at a definite transition point; and the pseudo-racemic mixed crystal of camphoroxime is an example of the second transformation. Although at temperatures in the neighbourhood of the melting point the two active camphoroximes form only mixed crystals but no compound, a racemic compound is formed at temperatures below 103°. At this temperature the inactive pseudo-racemic mixed crystal changes into a racemic compound; and in the case of the other mixed crystals transformation to racemate and (excess of) active component also occurs, although at a lower temperature than in the case of the inactive mixed crystal. Although this behaviour is one of considerable importance, this brief reference to it must suffice here.[302]
3.Alloys.
One of the most important classes of substances in the study of which the Phase Rule has been of very considerable importance, is that formed by the mixtures or compounds of metals with one another known as alloys. Although in the investigation of the nature of these bodies various methods are employed, one of the most important is the determination of the character of the freezing-point curve; for from the form of this, valuable information can, as we have already learned, beobtained regarding the nature of the solid substances which separate out from the molten mixture.
Although it is impossible here to discuss fully the experimental results and the oftentimes very complicated relationships which the study of the alloys has brought to light, a brief reference to these bodies will be advisable on account both of the scientific interest and of the industrial importance attaching to them.[303]
We have already seen that there are three chief types of freezing-point curves in systems of two components, viz. those obtained when (1) the pure components crystallize out from the molten mass; (2) the components form one or more compounds; (3) the components form mixed crystals. In the case of the metals, representatives of these three classes are also found.
1.The components separate out in the pure state.
In this case the freezing-point curve is of the simple type, Fig. 63, I. Such curves have been obtained in the case of a number of pairs of metals,e.g.zinc—cadmium, zinc—aluminium, copper—silver (Heycock and Neville), tin—zinc, bismuth—lead (Gautier), and in other cases. From molten mixtures represented by one branch of the freezing-point curve one of the metals will be deposited; while from mixtures represented by the other branch, the other metal will separate out. At the eutectic point the molten mass will solidify to aheterogeneous mixtureof the two metals, forming what is known as theeutectic alloy. Such an alloy, therefore, will melt at a definite temperature lower than the melting point of either of the pure metals.
In the following table are given the temperature and the composition of the liquid at the eutectic point, for three pairs of metals:—
The melting points of the pure metals are, zinc, 419°; cadmium, 322°; silver, 960°; copper, 1081°; aluminium, 650°.
2.The two metals can form one or more compounds.
In this case there will be obtained not only the freezing-point curves of the pure metals, but each compound formed will have its own freezing-point curve, exhibiting a point of maximum temperature, and ending on either side in an eutectic point. The simplest curve of this type will be obtained when only one compound is formed, as is the case with mercury and thallium.[304]This curve is represented in Fig. 74, where the summit of the intermediate curve corresponds with a composition TlHg2. Similar curves are also given by nickel and tin, by aluminium and silver, and by other metals, the formation of definite compounds between these pairs of metals being thereby indicated.[305]
Fig. 74Fig.74.
A curve belonging to the same type, but more complicated, is obtained with gold and aluminium;[306]in this case, several compounds are formed, some of which have a definite melting point, while others exhibit only a transition point. The chief compound is AuAl2, which has practically the same melting point as pure gold.
3.The two metals form mixed crystals (solid solutions).
The simplest case in which the metals crystallize out together is found in silver and gold.[307]The freezing-point curve in this case is an almost straight line joining the freezing points of the pure metals (cf.curve I., Fig. 65, p.210). These two metals, therefore, can form an unbroken series of mixed crystals.
In some cases, however, the two metals do not form an unbroken series of mixed crystals. In the case of zinc and silver,[308]for example, the addition of silverraisesthe freezing point of the mixture, until a transition point is reached. This corresponds with curve IV., Fig. 65. Silver and copper, and gold and copper, on the other hand, do not form unbroken series of mixed crystals, but the freezing-point curve exhibits an eutectic point, as in curve V., Fig. 65.
Not only may there be these three different types of curves, but there may also be combinations of these. Thus the two metals may not only form compounds, but one of the metals may not separate out in the pure state at all, but form mixed crystals. In this case the freezing point may rise (as in the case of silver and zinc), and one of the eutectic points will be absent.
Iron-Carbon Alloys.—Of all the different binary alloys, probably the most important are those formed by iron and carbon: alloys consisting not of two metals, but of a metal and a non-metal. On account of the importance of these alloys, an attempt will be made to describe in brief some of the most important relationships met with.
Before proceeding to discuss the applications of the Phase Rule to the study of the iron-carbon alloys, however, the mainfacts with which we have to deal may be stated very briefly. With regard to the metal itself, it is known to exist in three different allotropic modifications, calledα-,β-, andγ-ferrite respectively. Like the two modifications of sulphur and of tin, these different forms exhibit transition points at which the relative stability of the forms changes. Thus the transition point forα- andβ-ferrite is about 780°; and below this temperature theα- form, above it theβ- form is stable. Forβ- andγ-ferrite, the transition point is about 870°, theγ- form being the stable modification above this temperature.
The different modifications of iron also possess different properties. Thus,α-ferrite is magnetic, but does not possess the power of dissolving carbon;β-ferrite is non-magnetic, and likewise does not dissolve carbon;γ-ferrite is also non-magnetic, but possesses the power of dissolving carbon, and of thus giving rise to solid solutions of carbon in iron.
Various alloys of iron and carbon, also, have to be distinguished. First of all, there ishard steel, which contains varying amounts of carbon up to 2 per cent. Microscopic examination shows that these mixtures are all homogeneous; and they are therefore to be regarded as solid solutions of carbon in iron (γ-ferrite). To these solutions the namemartensitehas been given.Pearlitecontains about 0.8 per cent. of carbon, and, on microscopic examination, is found to be a heterogeneous mixture. If heated above 670°, pearlite becomes homogeneous, and forms martensite. Lastly, there is a definite compound of iron and carbon, iron carbide orcementite, having the formula Fe3C.
A short description may now be given of the application of the Phase Rule to the two-component system iron—carbon; and of the diagram showing how the different systems are related, and with the help of which the behaviour of the different mixtures under given conditions can be predicted. Although, with regard to the main features of this diagram, the different areas to be mapped and the position of the frontier lines, there is general agreement; a final decision has not yet been reached with regard to the interpretation to be put on all the curves.
Fig. 75Fig.75.
The chief relationships met with in the case of theiron-carbon alloys are represented graphically in Fig. 75.[309]The curve AC is the freezing-point curve for iron,[310]BC the unknown freezing-point curve for graphite. C is aneutecticpoint. Suppose, now, that we start with a mixture of iron and carbon, represented by the pointx. On lowering the temperature, a point,y, will be reached at which solid begins to separate out. This solid phase, however, is not pure iron, but a solid solution of carbon in iron, having the composition represented byy′(cf. p.185). As the temperature continues to fall, thecomposition of the liquid phase changes in the direction ofyC, while the composition of the solid which separates out changes in the directiony′D; and, finally, when the composition of the molten mass is that of the point C (4.3 per cent. of carbon), the whole mass solidifies to a heterogeneous mixture of two solid solutions, one of which is represented by D (containing 2 per cent. of carbon), while the other will consist practically of pure graphite, and is not shown in the figure. The temperature of the eutectic point is 1130°.
Even below the solidification point, however, changes can take place. As has been said, the solid phase which finally separates out from the molten mass is a solid solution represented by the point D; and the curve DE represents the change in the composition of this solid solution with the temperature. As indicated in the figure, DE forms a part of a curve representing the mutual solubility of graphite in iron and iron in graphite; the latter solutions, however, not being shown, as they would lie far outside the diagram. As the temperature falls below 1130°, more and more graphite separates out, until at E, when the temperature is 1000°, the solid solution contains only 1.8 per cent. of carbon. At this temperature cementite also begins to be formed, so that as the temperature continues to fall, separation of cementite (represented by the line E′F′) occurs, and the composition of the solid solution undergoes alteration, as represented by the curve EF. Below the temperature of the point F (670°) the martensite becomes heterogeneous, and forms pearlite.
From the above description, therefore, it follows that if we start with a molten mixture of iron and carbon, the composition of which is represented by any point between D and C (from 2 to 4.3 per cent. of carbon), we shall obtain, on cooling the mass, first of all solid solutions, the composition of which will be represented by points on the line AD; that then, after the mass has completely solidified at 1130°, further cooling will lead to a separation of graphite and a change in the composition of the martensite (from 2 to 1.8 per cent. of carbon). On cooling below 1000°, however, the martensite and graphite will give rise to cementite and solid solutionscontaining less carbon than before, until, at temperatures below 670°, we are left with a mixture of pearlite and cementite.
We have already said that iron consists in three allotropic modifications, the regions of stability of which are separated by definite transition points. The transition point forα- andβ-ferrite (780°) is represented in Fig. 75 by the point H; and the transition point forβ- andγ-ferrite (870°) by the point I. Since neither theα- nor theβ-ferrite dissolves carbon, the transition point will be unaffected by addition of carbon, and we therefore obtain the horizontal transition curve HG. In the case of theβ- andγ-ferrite, however, the latter dissolves carbon, and the transition point is consequently affected by the amount of carbon present. This is shown by the line IG.
If a martensite containing less carbon than that represented by the point G is cooled down from a temperature of, say, 900°, then when the temperature has fallen to that, represented by a point on the curve IG,β-ferrite will separate out, and, as the temperature falls, the composition of the solid solution will alter as represented by IG. On passing below the temperature of HG, theβ-ferrite will be converted intoα-ferrite, and, as the temperature falls, the latter will separate out more and more, while the composition of the solid solution alters in the direction GF. On passing to still lower temperatures, the solid solution at F (0.8 per cent. of carbon) breaks up into pearlite. If the percentage of carbon in the original solid solution was between that represented by the points G and F, then, on cooling down, noβ-ferrite, but onlyα-ferrite would separate out.
We see, therefore, that when martensite is allowed to coolslowly, it yields a heterogeneous mixture either of ferrite and pearlite (when the original mixture contained up to 0.8 per cent. of carbon), or pearlite and cementite (when the original mixture contained between 0.8 and 2 per cent. of carbon). These heterogeneous mixtures constitute soft steels, or, when the carbon content is low, wrought iron.
The case, however, is different if the solid solution of carbon in iron israpidlycooled (quenched) from a temperature above the curve IGFE to a temperature below thiscurve. In this case, the rapid cooling does not allow time for the various changes which have been described to take place; so that the homogeneous solid solution, on being rapidly cooled, remains homogeneous. In this way hard steel is obtained. By varying the rapidity of cooling, as is done in the tempering of steel, varying degrees of hardness can be obtained.
The interpretation of the curves given above is that due essentially to Roozeboom, who concluded from the experimental data that at temperatures below 1000° the stable systems are martensite and cementite, or ferrite and cementite, graphite being labile. It has, however, been pointed out, more especially by E. Heyn,[311]that this is not in harmony with the facts of metallurgy, which show that graphite is undoubtedly formed on slow cooling, and more especially when small quantities of silicon are present in the iron.[312]While, therefore, the relationships represented by Fig. 75 are obtained under certain conditions (especially when manganese is present), Heyn considers that all the curves in that figure, except ACB, representmetastablesystems—systems, therefore, akin to supercooled liquids. Rapid cooling will favour the production of the metastable systems containing cementite, and therefore give rise to relationships represented by Fig. 75; whereas slow cooling will lead to the stable system ferrite and graphite. Presence of silicon tends to prevent, presence of manganese tends to assist, the production of the metastable systems.
Althoughthis 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 Analysis.—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 necessaryto determine the composition of the latter. In some cases this is easily effected by separating the solid from the liquid phase and analyzing 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 constituent), 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 assume that the initial solution containsxgm. of A andygm. 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 analyzed. If, now, the composition of the solution is such that there arex′gm. of A andy′gm. of B to 1 gm. of C., then the composition of the solid phase isx-x′gm. of A andy-y′gm. of B. Whenx=x′, the solid phase is pure B; wheny=y′, the solid phase is pure A.
We have assumed 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 componentsform 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.abin 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 (pointb). When the eutectic point is reached, the temperature remains constant, until all the liquid has solidified. This is represented by the linecd. When complete solidification has occurred, the fall of temperature again becomes uniform (de).
Fig. 76Fig.76.
Fig. 78Fig.78.
Fig. 77Fig.77.
The length of time during which the temperature remains constant at the pointc, 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 curvea′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 thesolution is the same as that of the compound, no eutectic solution will be obtained; and therefore no line of constant temperature, such ascd(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 ofe′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.
Fig. 79Fig.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 betweendandb, then, on cooling,bwill separate out first, and the temperature will fall until the pointdis reached. The temperature then remains constant until the componentb, which has separated out, is converted into the compound. After this the temperature again falls, until it again remains constant at the eutectic pointc. 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 withincrease in the amount of either component. In this way we obtain the time-composition curveb′e″d′, of which the maximum pointe″gives the composition of the compound.
On the other hand, the period of constant temperature for the eutectic pointcis greatest in the case of solutions having the same initialcompositionas that corresponding with the eutectic point; and it decreases the more the initial composition approaches that of the pure componentaor the componente. In this way we obtain the time-composition curvea′c′e′. Here also the pointe′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 analysis, 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. Thecompositionof the fused mass 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 mass 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 potassium aluminate), then the freezing-point curve will resemble thatshown 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 mass will finally solidify to a mixture of orthoclase and silica, or orthoclase and potassium 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]
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 constituted 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 classes 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 aquintuplepoint. 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,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,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 atriangular diagramis 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]