Application of Analytic Geometry to Ternary and Quaternary Diagrams Patrick MacCalthy Colorado School of Mines, Golden, CO 80401 Advantages to representing ternary and quaternary composition diagrams by means of rectangular coordinates were ~ o i n t e dout in a recent aaDer ( I ) . A further advantaee to that ipproach is that analytic geometry, based on rectangular coordinates, is directly applicable as demonstrated in this paper. Ternary Sydems
y = mx
+1
[--
x = my
+1
1-1 5 m 5 0; adding X to a mixture of 0 Y]
5 m 5 -1;
adding Y to a mixture of 0 + X]
+
(8)
(9)
All straight lines on the composition diagram can be similarly represented. Example No. 1: Center-of-Gravity Rule
Straight Lines on Composition Diagrams
With rectangular coordinates, straight lines are described without the need for transformation of axes. Nine specific cases are described by eqns. (1)-(9). Consider the right isosceles composition triangle OXY where OX and OY are of unit length (Fig. 1). The sides of the composition triangle, which represent binary mixtures, are described by the straight-line equations: y=O x =0 x
+y = 1
[binary mixture of 0 + XI [binary mixture of 0 + Y] [binary mixture of X + Y]
Referring to Figure 1, analytic geometry can be used to calculate the net composition of a system produced by mixing any number of individual components. Imagine a system produced by mixing ml moles of composition C1(xl,yl), m2 moles of compositions C&,yz), etc. The net composition is given by:
(1) (2)
(3)
In more general terms, a line parallel to any side of the triangle describes changes in a ternary system where the composition of one component is maintained constant as the relative amounts of the other two components are varied (1 2 k 2 0): y=k
x=k x
+y = k
[xuconstant; xx and xo vary1 [xx constant; xu and xo vary] [xo constant; xx and xy vary]
(4)
(5) (6)
A straight line through any apex corresponds to adding (or removing) the corresponding component with respect to a fixed quantity of the other two constituents: y=mr
[0 5 m 5 -;
adding 0 to a mixture of X + Y] (7)
Figure 1. Right isosceles triangular composition diagram OXY.
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Journal of Chemical Education
where m; is number of moles of com~osition(x;.v;): . . . s is number of contrihuring samples: y, represents mole fmrtion. When virwed in rhecontrxt o t a coordinate frame,eans. 110) and (11)are recognized as the coordinates of the center of gravity of a system having point masses of ml, mn,. . . m, located at points (xl,yl), (x2.y2), . . . (x,,~,).This is the socalled center-of-grau~tyrule, and i t presents itself in a very straightforward manner when analytic geometry is applied. The triangular representation of ternary systems, as originally stated by Gihbs (2), and later by Stokes (3), was couched in terms of what is essentially the center-of-gravity rule. There are a number of corollaries to the center-of-gravity rule: (1) it follows that for a ternary system the total composition must fall within the triangle defined by the three components; (2) if two samples are mixed, the resulting composition must fall on the straight line connecting the coordinates of the initial samples (the center of gravity of two point masses must lie on the straight line connecting them). This is a well-known and very useful property of composition diagrams, and is deduced here in a simple manner. All systems which can be prepared by mixing two ternary samples of compositions (xl,yl) and (x2,y2) are depicted by the straight-line equation:
I t is seen, by substitution, that eqns. (10) and (11) (with s =
2) do satisfy eqn. (12), confirming that the final system does
indeed lie on the straieht line connectine the two initial samples. Ah a specific example, imagine mixing two samples ot cumpositions C, t0.10.0.20.0.701 and C, t0.70.0.30.0.001 in terms of mole fraction. can systems of composition C; (0.60, 0.25, 0.15) and Cq (0.40, 0.25, 0.35) be prepared by mixing C1 and Cp? Applying eqn. (12) to C1and Cp, with x1 = 0.10, YI = 0.20; xz = 0.70, y2 = 0.30, gives: The coordinates of CRdo not satisfv eon. (13). whereas the coordinates of Cqdo; thus, the system ~3 c a m i t , but Cqcan, be produced by mixing of C1 and Cz.
these lack the convenience of the right triangle as discussed here. Quaternary Systems
The visualization of quaternary systems (O,X,Y,Z) is also simplified considerably when depicted by means of rectangular coordinates, i.e., by using a trirectangular tetrahedron (Fie. 2). The aoolication of coordinate eeometrv in threedimensional s&ce facilitates computaticks relacng to quaternary systems (6). For example, the binary mixtures 0 X, 0 Y and X Yare again represented by eqns. (1)-(3), and the face representing a ternary mixture of X, Y, and Z is expressed as:
+
+
+
Example No. 2: The Lever Rule
The lever rule is also a consequence of the center of gravity rule applied to the mixing of two samples, or the breakingup of a system into two nonidentical samples. Since the net composition of the system must correspond to the center of gravity of the two samples, the product of the distance on the diagram from one sample composition to the total composition, hv the number of moles of that samole. . . must eoual the corresponding product for the second sample. That is a verbal expression of the lever rule, which mav he derived more explicitly as follows: If two systems CI(X&) and Cz(xz,yz) are mixed, the resulting net composition C&3,y3) is speclfied by eqns. (10) and (11) with s = 2. The distance between C1 and C3 is:
C,C,= [(x,
- x , ) ~+ b3- Y , ) ~ ] ' ~
(14)
and the distance between Cz and C3is:
All compositions which can be prepared by mixing two samples of specified composition must lie on the straieht line, in three-d~'mensiona1space, connecting the coordinkes of the original samples. Thus, if the coordinates of the two initial quaternary samples are (XI, yl, zl) and (xz, y2, zp), all "intermediate" compositions are represented by the set of equations depicting the straight line connecting these two points in three-dimensional space: '-XI
x,-x,
- Y-Y, - 1-21 Y2-Y,
(20)
22-21
The three individual equations embodied in eqn. (20), can be rearranged into sets of two independent equations, one set of which is comprised of eqn. (12) in conjunction with the following equation:
C,C,= [(x2- x3)' + CYz - y3)1'11 (15)
Substituting for x3 and y3 from eqns. (10) and (I]), respectively, and combining eqns. (14) and (15) gives:
which is the mathematical ex~ressionof the lever rule:. ml. and rn, are the numher of molrs of CI and C, respectively. The lever rulr may be simplified bv. ex~ressine . - it in terms of its components, or projections, along the x and y axes:
- X3 - m, r 3 - 11 m2 x2
(17)
and
The lever rule is frequently used to calculate the relative amounts of two phases of specified compositions in mutual equilibrium, correspond in^ to a stated total com~osition. ~ h a,mvositions r of the two phases must be known br determined experimentdlv. The followinr example illustrates the application of analsic geometry to this type of problem. Consider a ternary system of composition 21.16 mole % A, 63.26 mole % B and 15.58 mole % C which disproportionates into two phases of compositions 15.00 mole %A, 34.00 mole % B, 51.00 mole % C (phase No. 1) and 23.00 mole % A, 72.00 mole % B and 5.00 mole % C (phase No. 2). What are the relative amounts of the two phases? Application of eqn. (17) with XI = 15.00, xz = 23.00, and x3 = 21.16, gives mllmz = 0.299, where ml and m2 are the total numbers of moles of phase 1 and phase 2 produced, respectively. Equation (18) could he used instead of eqn. (17). The center-of-gravity rule and the lever rule are valid even when nonequivalent axes are used for the ternarv diaeram (4). Analy& geometry based on oblique coordinates can be employed with other types of composition triangle (5), but
Thus, any composition which can be prepared by mixing the two quaternary samples (XI, yl, z,) and (xz, y2, zz) must satisfy both eqn. (12) and eqn. (21). For example, one can prepare a quaternary system of composition C~(0.30,0.30, 0.10, 0.30) by mixing two quaternary systems of compositions Cs(0.40.0.20. 0.05. 0.35) and Cn(O.10. 0.50. 0.20. 0.20). whereas a system of composition (0.20,0.30,0.10,0.40) can: not be prepared in this manner. (See Example No. 1 for methodbf Ealcu~ation). The center-of-eravitv rule can be readilv a ~ o l i e dto auaternary systems ;sing rectangular coordinates simpli by considering eons. (10) and (11) in coniuction with a corresponding equition in 2. The lever rule can also be readily applied in three dimen-
-
&.
V
Figure 2. Trirectangular tebahedral representation of quaternary system 0.X.Y.Z; ox = OY = 0 2 = 1.
Volume 63
Number 1 January 1986
41
sions (quaternary systems) since it is a special case of the center-of-gravityrule. For example, if a quaternary system is subdivided, hv human intervention or bv a natural ~ h a s e separation, into two separate subsystems of known compositions, the relative amounts of the two final svstems is eiven by eqn. (17) or (18) or by a corresponding eq"ation in tkrms of 2 . Thus, it can be readily calculated that if a quaternary system of 14 mole %A,27 mole % B, 31 mole % C and 28 mole % D disproportionates into two phases of compositions 16 mole % A, 24 mole % B, 33 mole % C and 27 mole % D (phase No. 1) and 8 mole % A, 36 mole % B, 25 mole % C and 31 mole % D (phase No. 2), mllmz is equal to 311.
42
Journal of Chemical Education
In summary, the Cartesian representation of ternary and quaternary composition diagrams facilitates calculations relatine to these diamams.
Literature Cited (1) MscCsnhy,P., J. CHEW E ~ ~ c . , 6 0 . 9 2 (1983). 2
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~ ~R.~E.. .b -s ~~ ~. ~operations:. ~ - ~ 3 ~r d. d ~ MG.~W-H~II ~ ~ r B~ W~company, ~ NovYork, 1980, Chapter 10. o, ,are y,G, ,,J,Phys, Chem,,34, 1745(1930,, (6) wiegsnd. J. H..I"~.EW C ~ ~ ~ . , A ~ 15.380 I . E ~(1943). . . (4) ~
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