Ternary and QuaternaryComposition Diagrams An Overview of the Subject Patrick MacCarthy Colorado School of Mines, Golden, CO 80401 The term composition refers to the relative or fractional amounts of the various components which comprise a system. The composition may be expressed in various ways, for example, on a mole fraction or on a weight fraction basis. For the purpose of this paper we will use mole fraction, X. A detailed discussion of concentration and composition units was presented in a previous paper (I). A unary (2) or singulary (3) system contains only one component and is thus of invariable composition ( X = 1). A binary system, A B, is univariant, that is, it is necessary to specify the mole fraction for only one of the components, say XA, since the second mole fraction term is automatically determined by the equation
+
Thus, the composition of a binary system can he depicted by means of a line extending from XA = 0 to XA = 1 (or from X B = 1 to XB = 0). All compositions of a system, prepared by mixing A and B, are uniquely represented by specific points on this line. This, of course, corresponds to the composition axis of binary phase diagrams, such as boiling point diagrams (4,5). In general, for an n-component system, " (2)
z x i = 1
i=l
Equation (2) shows that (n - 1) of the composition variables are independent, the nth mole fraction term being specified by "-1
x.=l-
x x, i=l
(3)
where x i is the mole fraction of the ith component. In order to define the comuosition of a ternary system it is nece.>arv 1,) spe,.ir, t h ~ni& , 1ra~1101is ~ 8 i t ~ w~mp~me lhv nl~, t h i r ~ lIwi11c dei)r1111+,111 , i i ~< I d ~ ~ i ~ iinmi ~ , l Jeqn. v 1:31; t h t is. a ternary system is bivariant. Conventionally: ternary compositions are represented by means of an equilateral triangle. The so-called trilinear or triangular coordinates are frequently encountered in the study of phase diagrams, and they find extensive application in chemistry (4-141, geology (15-20), metallurgy (2, 21), and chemical engineering (22-25). In general, ternary composition diagrams can be used for any three-component system which is bounded by a mass balance restriction as defined in e m . (2) with n = 3. A more general ~
~
~
~~~
triangularcoordinates (4-8,11-14,~8,19,>1-24j:~riangular comoosition diaerams are also used to classify coals (15,16, 26) and kerogen727) on the basis of their carbon, hydrogen, and oxveen . contents. Crude oils have been classified on the basis of the weight fraction of saturated hydrocarbons, aromatic hydrocarbons, and resins plus asphaltenes in the mixture (17). An equilateral triangle is used to classify common sediments on the basis of their Si02:(A1,Fe)20s:(Ca,Mg)COs contents (20). The proportions of Si02, CaO, and A120s in
sand, silt, and clay content< is a further application of the 922
Journal of Chemical Education
equilateral composition diagram (28). Triangles are also used to represent the composition of concrete mixes (29) and detergent blends (30). A recent paper uses a triangular composition diagram as the basis of a novel method for classifying experimental techniques in solution chemistry (31). Except for those who are thoroughly familiar with such diagrams, the use of triangular coordinates can often appear somewhat cumbersome, In particular, it may he difficult to estimate the composition represented by a specific point, simply by taking a quick glance at the diagram. Typically, one must engage in a more deliberate step-by-step mental exercise in order to establish the composition corresponding to a particular point on the equilateral triangle. To quote Smith, "it is somewhat tedious to locate compositionson such a diagram" (22). Similar, hut more pronounced difficulties are experienced in trying to visualize coordinates in a regular tetrahedron, which is commonly used to represent compositions of a quaternary system. These difficulties, as well as other topics, are treated in this paper, and alternative approaches, which minimize the conceptual difficulties involved, are discussed. In particular, the approach adopted in this paper should he of pedagogic value in the teaching of ternary and quaternary composition diagrams. Objectives of Paper This paper is basically a review of graphical methods for representing ternary and quaternary systems; and its ohjectives are: 1) to provide a generalized introduction to the use of triangular composition diagrams, and a guide to the literature on this subject; 2) to place the application of triangular coordinates, based on the equilateral triangle, into better perspective by examining some of the relevant geometry of triangles in general; 3) to show that right isosceles triangles, and right triangles in general, possess some very advantageous features for representing ternary systems; 4) to extend the discussions presented in this paper to a trirectangular tetrahedron for representing quaternary systems; and 5) to develop a pedagogically valuable tool for the teaching of ternary and quaternary systems. The focus of this paper is on the representation of compositions, while temperature and pressure variables are not specifically treated. Furthermore, the detailed nature of phase transformations, or other properties which can he described by means of these diagrams (30) are not discussed in this paper; a discussion of projection methods is also omitted from this paper. Those aspects are more easily learned following a clear comprehension of the fundamental nature of the composition diagrams per se. General or Scalene Composition Triangle Throughout this paper, various geometrical relationships are stated without proof. The proofs are generally simple, and are left as an exercise to the interested reader; or they can he checked in textbooks on Euclidean or analytic geometry. In terms of increasing symmetry, triangles may be arranged
in the seouence scalene < isosceles < eauilateral. In order to shown that a scalene triangle can be used to represent threecomnonent svstems (14.19.32.33). Firure 1shows an arhit r a r i scalene-triangle, ABC. Within t h i triangle is P, an arbitrary point, and OPNS, TJPK, and MUPL are straight lines through the point P, and parallel to the sides AB, BC, and C A , respectively. The lines C I , A G , and BH are three arbitrary straight lines from the three apices to the opposite sides of the triangle, or to the continuation of those sides. PF, PD, and PE are three lines extending from the point P to the three sides of the triangle (or their extensions) and parallel to CI, AG, and B H , respectively. It can he readily shown that:
This geometrical property is the most general expression of the basic feature upon which all triangular composition diagrams are based. In order for the triangle to represent the ternary system, A B C, the following assignments are made:
19,32-35). One is sometimes left with the erroneous impression from textbooks that only an equilateral triangle can be used in this manner (36). Adopting this convention, it is clear that the apices must represent the respective pure components; for example, apex A represents pure A. I t is also evident that any point along a side of the triangle represents a binary mixture; for example, noint - - - - ~0 renresents a binarv mixture of B and C components. Finally, a little consideracon will show that any point, such as R , lying outside of the triangle, corresponds to a meaningless situation in the context of the specific system, A + B C, being considered. This, then, provides us with a means for evaluating the triangular coordinates, or composition, of a point, P; the composition is obtained by simply evaluating the three ratios specified in eqn. (4). This operation can be facilitated by setting the three lengths G A , H B , and IC arbitrarily equal to unity and by setting linear scales ranging from 0 to 1 along these three lines. Then,
. ~.
+
+ +
Where the symbol [T, G A ] represents a numerical value read directly at point T on a scale ranging from 0 at G to 1a t A . Similarly, XB and ,yc can he read directly from scales along HB and IC, respectively, leading to the set of equations: xa = LT,GAI
(9)
[U,HBl
(101
XB =
Equation (4) shows that the assignments specified in eqns. (5)-(7) satisfy the requisite mass balance condition stated in eqn. (2) or the more general expression given in Reference (I ). Thus, adopting this convention, all compositions of a ternary system are uniquely represented by individual points on a triangle; and conversely, each point on or within a triangle uniquely defines a specific ternary composition. I t should be emphasized that any triangle can he used for this purpose (14,
Thus, the composition corresponding to a point P can he read directly from the three numerical scales. In the general case, the three lines AG, BH, and CI are of unequal length, and thus the graduations on all three scales are not equivalent; however, this does not cause any fundamental problem (14, 24, 34). Compositions can be read directly in terms of percentages by setting the lengths A G , B H , and CI equal to 100. Methods for Reading Ternary Compositions An examination of the literature reveals various methods for reading triangular coordinates. While all of these methods are fundahentaily equivalent, the very existence of a variety of approaches can lead to some confusion if the reader does not understand the basis for them. The author identifies four main modes for reading triangular coordinates, and the geometric data underlying these modes are summarized in the table. Mode 1 This generalized mode for reading triangular coordinates is based on the set of eqns. (9)-(11) and has already been discussed above. Mode 2 This mode of reading ternary compositions is based on the following set of equations (see table):
F gJre 1 Scalene compos8I on sang e ABC Tne geometrical constrLctm mom yfng m e general lea metnod lor reaa ng lrlangu ar owyams mode 1 or
illustrated in this figure.
This mode can he considered a special case of mode 1where the three normally arbitrary lines A G , B H , and CI coincide with the three sides of the triangle AC, B A , and C B , respectively. In this case, scales ranging from 0 to 1 are arranged in a cvclic manner alone C A . A B . and BC. Each mole fraction term is read from a separate sidk of the triangle, as illustrated in Fieure 2(a). The fraction wn is determined hv drawine a line ; ingrcept throigh point P parallel toopposite side ( B C ~the of this line on the A-axis (i.e., C A ) gives XA. Likewise, X B and Volume 60 Number 1 1
November 1983
923
Figure 2. Various modes for reading triangular coordinates from the sides of a triangle. A scalene triangle is used far illustration: (a) mode 2 and mode 2': (b) mode 3; and (c' mode 4.
xc are determined in a similar manner along sides AB and BC, respectively. Alternatively, XA,XB. and x c could he read from sides BA, CB, and AC, respectively. A variation of mode 2 is based on the following set of equations (see tahle):
Figure 3. Geometrical construction illustrating methods for reading ternary composition^ from equilateral triangles: (a) mode 1, involving perpendiculars to opposite sides: (b) modes 2, 3, and 4 all involve the same geometrical construction. i.e., three straight lines through point P and parallel to the three sides of the triangle.
Summary of Selected Equalities in Scalene Triangle of Figure 1 Relevant to Reading Compositions Xa PDIGA TG'GA KCICA PMICA MNIBC
In this case, PM is evaluated in terms of the scale along C A , PO in terms of the scale,alongAB, and PJ in terms of the scale on side BC (Fig. 1). This will be referred to as mode 2'. Referring to Figure 2(a), XA can be established in mode 2' by drawing lines PM and P N through P parallel to AC and AB, respectively; these two lines represent XA,when read according to the appropriate scale (PM from CA scale and P N from a scale alone B A ) ; X B and xc can he determined in a similar manner. Mode 3
In this mode all three mole fraction quantities are read from a single, arbitrarily-chosen, side of the triangle. Consider the side BC; the appropriate set of equations (see table) is then:
Mode 3 is implemented by drawing lines through point P parallel to the other two sides and intersecting BC; XA,XB,and uc are read as indicated in Figure 2(b) In the case of a scalene triangle, mode 3 has an advantage over modes 1and 2, in that all three mole fraction quantities are read from a single scale, rather than from three nonequivalent scales. Mode 4 Of course, for all modes, it is necessary only to determine two mole fraction quantities since a ternary system is bivariant and the third fraction is obtainable through eqn. (3). The fourth method of reading triangular coordinates results from simply looking at the problem from a different, and in retrospect a very logical, perspective. The appropriate set of equations (see tahle) is: [LAB1
(21)
xc = [O,ACl
(22)
XB =
The fraction XB is read along the axis AB in the conventional manner of reading oblique coordinates; and x c is read from the axis AC (Fig. 2(c)). This removes the mystique and 924
Journal of Chemical Education
XB
[T,GA] [K,CA] [M.BC] - [N.BC]
PEIHB UHIHB LNAB POIAB CMBC
[U,HB] [L,AB] I
- [M,BC]
Xc PFIIC SIIIC [S,iC] NWBC [N.BC] PJIBC OAIAC [O,AC]
awkwardness often associated with reading ternary diagrams; in its simplest form, all one is doing is merely reading ohlique coordinates. Then, XA is ohtained by applying eqn. (3): XA=~-XB-xc
(23)
While this method appears to he the simplest approach to reading triangular coordinates, it is, in practice, seldom if ever, used or recommended. I t is different from mode 2 in that the axes are not labelled cyclically. In fact, the conventional manner of labeling the sides of the triangle in a cyclic manner serves to obscure this coordinate geometry viewoint, rather I h m render ir mow ,tpp;,rrnt. l';irioua method> iur n udinc ternary rumt~u.;itionshwe been illustrated in Figures 2(a), (b), add (c). conversely, in order to establish the point corresponding to a specified composition,the inverse procedures are adopted. For example, referring to Figure 2(h), the point corresponding to 0.20 mole fraction A, 0.50 mole fraction B, and 0.30 mole fraction C would be ohtained as follows: along BC mark off a point 0.50 units (=xB)from C, and through this point draw a straight line parallel to AC (i.e., side opposite point B); mark off a point 0.30 units (=xc) from B and through this point draw a s t aight line parallel to side AB. The intersection of these two lines represents the composition of interest. Special Case of Equilateral Triangle
The symmetry inherent in an equilateral triangle facilitates the use of modes 1 and 2. considerablv. Bv convention, in applying mode 1to the equilateral triangie, the three lines AG, BH, and CI (see Fig. 1) are set perpendicular to the three sides of the triangle. As a result, all three scales along AG, BH, and CI become equivalent. I t should now he clear that the common statement that: "the sum of the perpendiculars from any point within an eauilateral triande to the three sides of the triangle is equal to the altitude of t i e triangle" is simply a special (&d convenient) case of the more general condition stated by eqn. (4) which is applicable to all triangles. Mode 1, applied to an equilateral triangle, is the most common method advocated for reading ternary compositions (3,4,11-14,37,38i and is illustrated in Figure 3(a). This method was originallyproposed by Gibbs (321, who first introduced the concept of triangular
Figure 5. Two composition diagrams presenting equivalent information:(a) in terms of weight fraction:(b) in terms of mole fraction, for ternary system A, B, C, With molec~larweights of 100, 20. and 50. respectively.
Figure 4. Right isosceles triangular representation of ternary system in coniunction with mode 4. representation of ternary systems. The same ideas were later proposed, apparently independently, by Stokes (33,39) who compared the composition triangle to "that which Maxwell used for the composition of colonrs." Gibbs and Stokes actually developed their ideas in terms of a scalene triangle but .. .
.
triangle results from the fact that all three scales along the sides of the triangle become equivalent, again making a direct visual comparison easier. Mode 2 and mode 2' are also used extensively for reading ternary compositions from equilateral triangles (3,12-14,35,37,38,40-43) (Fig. 3(h)). Referring to Figure 3(b), XA, XB, and xc are listed, in that order, in the following sets: (CK, AL, BN) or (BJ, MC, OA) for mode 2; (PN, PK, PL) or (PM, PO, P J ) for mode 2'. These methods for reading triangular coordinates were proposed by Roozeboom (13,39,44), who introduced the use of equilateral triangular graph paper with a superimposed grid composed of three sets of equally spaced lines parallel to the three sides of the triangle. Modes3 and 4 are quite convenient to use even in the case of a scalene triangle. and no dramatic eain in convenience. other than that i f symmetry, is achi;ved by going to an equilateral triangle in these cases. Exam~lesof mode 3 amlied tdan esuilater&triangle are found in references (3,13, .%,36, 38) (Fig. 3(b)). In mode 3, XA, XB, and xc are given by any of the three sets: (JB, AL, LJ), (MN, MC, EN), or (CK, KO, OA). In mode 4 (Fig. 3(h)) AL and A 0 give XB and XC, respectively, BJ and B N give XA and x c , respectively, and CK and CM give XA and XB, respectively. The author is aware of only one example iron, the li;erature~'l~ where the equilateral triangle is read ,pwiiisally a(vurdiny tu mude 1 (Fig. 3\t1lj, is iIe.w~tcthe t h ~ thnt t "'l'hr euu~lnreral~rianrulnrdiaeram " but an adaptation of oblique coordinates for a particular ~ u r o o s e "(45). An eauilateral trianele with scales drawn according to mode 4 is illustrated in reference (24), but reading of the composition is discussed in terms of mode 1.
-
Special Case of Right Isosceles Triangle: Cartesian Coordinates Mode 4 becomes uarticularlv advantageous when a rieht iio,rrlt:< tri;in& is I huien 1%) represent ternary ~r~mpusiriuni. Cunslder the right :ic.scelr triimele 1i.Y \' illustrated in Figure 4 and representing the ternary system 0 , X, Y;let OX =-OY = 1.Then,
xx = x
(24)
(25) Establishing the composition corresponding to a particular XY = Y
point, P , is equivalent to reading the conventional rectangular coordinates of the point; the third mole fraction quantity is given by: In other cases it may be advantageous to use a non-isosceles right triangle, for example when expressing concentrations in terms of weight molarity (1,31) or if one wishes to expand one axis relative to another (23,24,45). The right triangular representation was first introduced by Roozehoom (39,46). Comparison of Right Isosceles and Equilateral Triangular Representations of Ternary Systems Both methods of representation are fundamentally equivalent in that each one provides an unambiguous description of ternary mixtures. The major advantage of the right isasceles triangle is that it corresponds to the universally familiar rectangular coordinate system (when mode 4 is used) which is easy to visualize, and which allows the compasitions to be read off the triangle in a very simple and straightforward manner; in addition, conventional analytic geometry can he directly applied to calculations involving this representation.1 Of course, mode 4 can be similarly applied to other types of triangles, but i t then requires the use of oblique (47) and/or nonequivalent axes. A minor advantage of the right isosceles representation is that common rectangular graph paper is utilized, rather than the special triangular graph paper which is otherwise required for representing ternary systems. On the other hand, the right isosceles representation lacks the three-fold symmetry inherent in the conventional representation based on the equilateral triangle. Apparently, this lack of symmetry is the primary reason that the right isosceles triangle has not been more widely adopted in the past (3,4, 41). While this symmetry factor may be of more genuine concern in the case of some phase diagrams, it should not he a serious consideration in most cases. In particular, when one of the components is present in large excess, for example, an aqueous solution of two salts, the orthogonal coordinate system appears undoubtedly superior (34) Many of the phase diagrams found in the literature are expressed in terms of weiebt fraction. rather than mole frac-
termsof mole fraction, and vice versa. This is illukrated in Figure 5 where the components A, B, C have molecular weights of 100, 20, and 50, respectively. Figure 5(a) is expressed in terms of w e i ~ hfraction t and Fimre 5(b) Dresents the same information in terms of mole fraction. Themedian CD in diagram (a) is transformed into the line C'D' in diagram
Volume 60 Number 11 November 1983
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of different area, located near one side of the triangle A'B'C'. Accordinglv, the argument for retaining the eauilateral triangular r&resentat&n, on the basis of symmetiy, is not particularly convincing. Finally, it should he pointed out that even though some (relatively minor) distortion does occur on going from an equilateral to a right isosceles representation, the relative area occupied by a given region on the composition diagram does not change during this transformation; for example, in a triangle ABC, the area lying between 30%A and 100% A on the diagram represents 49% of the total area regardless of whether the triangle is equilateral, right isosceles, or scalene. The advantages inherent in the rectangular coordinate representation of composition diagrams will become even more pronounced when quaternary systems are considered in a later section. Surnmarv of Composition Triangle Properties I t is appropriate to summarize at this point some of the useful characteristics of the general composition triangle. All of these have been discussed, or can he readily deduced from the basic geometrical principles which were presented above. I t should be reemphasized that these characteristics are not limited to the equilateral triangle but are common to all triangular representations: 1) Each alwx repre,rnli ,I pure compunent. 2, Each nomt on a aide d t h e trianrlt: rrvrhpnrs a uniuue binary mixture. 3) Each woint within the triangle a unique ter- rewresents . nary mixture. 4) A straieht line parallel to one of the sides revresents variation in t i e relatiGe amounts of two components while the fractional amount of the third component remains con.stant. ....... 5 1 .\
.traiyht line ihnugh nti apex cdrrespundi I,, maintn:niny thc relntivr ;tntouttts , n i t a t , c l i the c
