John A. Bornmann Lindenwood College st. Charles, Missouri
Triangular Coordinates Versus Rectangular Coordinates
Students of physical chemistry often have difficulty when they encounter triangular coordinates in the phase diagrams of three components. This is probably their first, and only, use of triangular coordinates and therefore they seldom use the system with the same ease with which they use rectangular coordinates. Their problems in triangular coordinates are generally two-fold. First of all, they usually have difficulty with the scales of the triangular plot in which the scale begins a t the base of the triangle and increases toward the apex. Secondly, they fail to appreciate that the condition z y z = 1 (or 100) is not only a physical condition imposed by nature but is also a mathematical condition of triangular coordinates. Triangular coordinates in various forms were suggested in the late 19th century by several authors. In the midst of a 140-page discourse on heterogeneous equilibria, J. Willard Gibbsl proposed triangular coordinates in which the triangles could be any shape, although he recognized certain advantages in using equilateral triangles. G. G. Stokeq2 in a note ac-
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companying an article by Wright, Thon~pson and Leon, proposed the use of equilateral triangular coordinates for the graphical representation of alloys. Stokes' proposal was apparently made independently of Gibbs, but he does acknowledge a relationship to Maxwell's treatment of the three color systems. At about the same time, H. W. Bakhuis Roozeboom3had been using right triangles, but on the suggestion of Gibhs he switched to equilateral triangles. Roozeboom also proposed the use of an equilateral pyramid to plot four component systems. Specially printed graph paper for equilateral triangular coordinates is readily available. Figure 1 shows an example of the method of plotting in which there are three variables: x, y, and z. The maximum value for each variable is 1, and that value is given a t each apex of the triangle. For example, z is equal to 1 (and x = y = 0) a t the apex marked z, and z = 0 a t the base of the triangle. Increasing values of 2 are represented by lines parallel to the base. I n Figure 1 there are four horizontal lines inside the triangle representing z = 0.2, 0.4, 0.6, and 0.8. Similar lines are
GIBBS,J . W., Tmm. Cmn. A d . , 3 , 108 (1876) see page 176.
l S ~ o ~G.s G., , Proe. Roy. Soe. (London), 49,174
(1891).
ROOZEBOOM, H. W . B., Z . Physik. Chem., 15, 143 (1894).
Volume 43, Number 7, July 1966
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given for the other variables. Thus the coordinates of any point can be obtained by the length of lines drawn perpendicular to each side of the triangle. I n Figure 1 the value of z of a point used as an example is equal to 0.2, and the other coordinates of the point are y = 0.2 and x = 0.6.
the triangle and progressing upward to the apex. Likewise, the intersection of planes of constant x and of constant y provide the scales for the other two coordinates. Thus, the scales of triangular coordinates are directly related to the familiar scales for x, y, and z in rectangular coordinates. It is possible to prove by geometry3 that the plane which includes the triangle in Figure 2 is the loci of points in which x y z = a. This equation, therefore, is a mathematical restriction on triangular coordinates. When the variables are mole fractions or weight percentages, there is also the physical restriction that the sum of the variables must be equal to one or one hundred. (The restriction explains why it is not possible to use other concentration expressions such as molarities with triangular coordinates.) The matching of the physical restriction with the mathematical r e striction of triangular coordinates makes this coordinate system ideal for phase diagrams with three components. Using the same technique, it is possible to derive other, specialized triangular coordinate systems in which the variables obey an equation of the type ax by cz = d. Consider as an example a case in which the maximum values are x = 1, y = 1,and z = 2 and each maximum arises when the other variables are zero. The maximum values are plotted on rectangular coordinates, and the points are joined to give a triangle,
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Figure 1. Triangular coordinates in which x y z = 1. The coordinotes of the ~ o l n are t x = 0.6, y = 0.2, and r = 0.2. See text for method of obtaining Re coordinates.
The students would benefit if they knew the relationship between rectangular coordinates and triangular coordinates. This relationship was briefly presented by D. R. Cruise,' but it is worth elaborating. As shown in Figure 2, triangular coordinates are a special case within rectangular coordinates. The triangular
Figure 2. The equilateral triangle of trlongulor coordinates 0s a part of three-dimensional rectongulor coordinates. The plmnes pardlei to Re xy plane ore planer of r = 014, o/2, ond 3-14, It can be shown that the equilateral triangle iv the loci of points in the Rrrt quodrmnt in which
x+y+z=o.
coordinate system is a part of a plane which intersects the axes of the rectangular system a t the points (a, 0,0), (O,a,O) and (O,O,a). I n common usage a = 1 or 100. I n triangular coordinates, the lines of constant z result from the intersection with the triangle of planes of constant z parallel to the xy plane, as shown in Figure 2. These give the familiar lines parallel to the base of CRUISE,D. R.,J. CHEM. EDUC., 43,30 (1966). 388
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Journal o f Chemical Edumfion
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Figure 3. (lefl) The derivation of o specidired triongular coordinate system In which the maximvm volues of x, y, and r are 1, 1, and 2, rerpectireiy. Figure 4. kight) The specialized triangular coordinde system derived In Figure 3 in which Zx 2y z = 2.
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as in Figure 3. It is then possible to see that the triangle will be isosceles and to calculate that the sides will have lengths 4% and 45. By means of geometrical argumentss it is possible to show that the plane of the triangle is the loci of points in which 2x 2y z = 2. The resultant triangular coordinate system is given in Figure 4. No particular use is foreseen for this special triangular coordinate system; it is presented to show that other triangular coordinate systems do exist and can be derived easily from rectangular coordinates.
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The geometrical proofs m e ttvdilable from the author
