Elinor M. Kartzmark University of Manitoba Winnipeg. Manitoba R3T 2N2 Canada
Models for the Representation of Four-Component Systems
The student learning phase rule experiences considerable difficulty in visualizing three-dimensional diagrams. The author is teaching a course in phase rule to Earth Science students whose work requires a knowledge of four-component and more complex systems. For these a three-dimensional representation cannot he avoided. In this paper, two inexpensive models, constructed of glass tuhing and colored plastic sheeting are described, which show how a point is plotted in a four-component system and how the composition of a point is deduced from its position in a model. A regular tetrahedron is the basis for both models. The four apexes of a regular tetrahedron represent the four pure components (A, B, C, D) (Fig. 1). The six binary systems (AR, AC, AD, BC, BD, CD) are represented on the edges and the four ternary systems (ABC, BCD, ACD, ABD) on the four equilateral faces. A point Q, within the tetrahedron, is a quaternary system. The plotting or fixing such a point is, of course. treated in phase rule texts ( 1 - 4 ) . but none points out t h a ~the ol'ten inystiiying directims given have as their hnsis a simple extension d r h e properties of theequilnternl triangle to those of the regular tetrahedron. Using the face ARC of Figure 1 as example, all systems having a fixed % B are to he found along a line ( u w ) parallel t.11AC and all systems having a fixed % A are to be found along a line paralled to BC b y ) .The intersection of these lines, 2, gives a unique system in the ternary A, B, C. Extended to the quaternary system, a point Q is found a t the intersection of three planes in the tetrahedron along each of which one comoonent has a fixed composition. In Figure 1, the plane, MNO, parallel to the base, represents all systems having a fixed percentage B. Figure 2 shows the intersection of three such &lanesatboint 01 It is obvious. from the comolexitv of the f&e, that'the stident would benefit greatly from a &lid model (see Fie. 3). A tetrahedron constructed of 5 mm glass tubing, 30 cm to an edee. is a convenient size for use with a small group. To fix of a point Q: (22.2%A; 44.4%B;2272%C; 11.2% the D), 2 mm glass rods were added to the tetrahedron, as follows (Fig. 2), to outline the planes (1) EFG (22.2%A), such that E B = FC = GD = 6.7 cm., (2) MNO (44.4%B), such that MA = NC = OD = 13.4 em., (3) HIJ (22.2%C ) ,such that H A = ZB
= J D = 6.7 cm. These lengths result by taking in the case of A, for instance, 22.2%of the 30 cm edge length. The points of intersection on faces ARD (R), and ARC (S) were joined by 2 mm tubing, as were those on faces ARC (T)and DBC ( U ) . The intersection Q is the point in question. This may be explained to the student as follows. The plane MNO (fixed in B) intersects the plane EFG (fixed in A), in the line RS; hence all points along RS are fixed in A and B. Similarly, all points
8
A
Figure 2. The quaternarysystem, 0.at the intersection of three planes in each of which one component has fixed composition.
Q
*
A Figure 1. A
quaternary system, 0, in the body of a regular tetrahedron.
Figure 3. Plastic model of system in Figure 2. Volume 57, Number 2,February 1980 1 125
Figure 5 . Plastic modal of system in Figure 4, photographed a t a different angle.
A Figure 4. The quaternaty system, 0,at the intersection of three planes of fixed ratio.
along TU are fixed in C and B. The intersection of R S and TU, the point Q, is therefore fixed in A, B and C and is a unique B C). The edges of the three system since D = 100 -(A planes may he painted in three different colors or, using the rods for support, transparent colored sheeting may he added to show the constant com~ositionplanes, relative to apexes of the same color. Alternativelv. .. to deduce the composition of a given point in a given model, it is only necessary to imagine or to insert three m lanes through the point parallel to three of the faces of thetetrahedronand toconvert the portion of edge length cut by the plane to the total edge length = 100%.For instance, if a plane through Q, parallel to the face ADC, cuts the edge AR so that MA = 13.4 cm, then the percent B = 13.4130 X 100. Of course, it is impossible to deduce the composition of Q from a. perspective drawing only. . A second model for a quaternary system also involves the intersection of three planes, in this case, along each of which
+ +
126 / Journal of Chemical Education
two components occur in a fixed ratio. This is the analogue of the line AK in the ternary system ABC of Figure 1where the ratio CIB = KBIKC for all systems on the line. In Figure 4 the three planes are ABX (ratio CID = 1/11, AIIM (ratio CIB = 112) and BDN (ratio CIA = 211). The planes ADM and RDN intersect in point F on face ARC and along FD within the tetrahedron where the ratio A:B:C is, therefore, fixed. Similarly, planes BDN and BAX intersect alone GB where the ratio A:C:D is fixed. The intersection Q, I-:>I unique point and can he deduced from the ratio* to he the watrni
