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 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 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,
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 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]]
In the method proposed by Gibbs an equilateral triangle of unit height is used (Fig 80).[[318]] The quantities 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
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 Pa, Pb, and Pc 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
hundred parts; a portion, Bx = a, is then measured off, and represents the amount of A present. Similarly, a portion, Ax′ = 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.
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 ACx and BCx are similar to the two triangles HCx′ and KCx′; and,
therefore, Ax : Bx = Hx′ : Kx′. But Ax = Dx and Bx = Ex; further Hx′ = Fx′ and Kx′ = Gx′. Therefore, Dx : Ex = Fx′ : Gx′ = b : a. At all points on the line Cx, therefore, the ratio of A to B is the same.
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.