Graphical Representation on Logarithmic Triangular Coordinates Zvl Rlgbil University of Cincinnati. Cincinnati, OH 45221 I t is often instructive t o be able to represent the relationship between a property and its composition. Representation of three-component systems by means of triangular coordinates has become widespread in chemistry and other disciplines where it is used to plot phase diagrams and isoproperty lines and boundaries. Such diagrams are limited to components which add up to 100%. I t is not conceivable that in a mixture or compound a component may exist in negative ratio. Traditionally, therefore, a negative value could not he plotted on triangular coordinates. In what follows, however, negative values are explicitly allowed, and the results extend the use of triangular coordinates as will be described. Figure 1 shows an equilateral triangle ABC, of height h. The sum of the three perpendiculars a, b, and c drawn from any perpendicular to the three sides is equal to h. The proof for this can be found in textbooks on plane geometry. We now extend AB and AC as shown. Consider a point P I in the area between the extended arms, and draw the perpendiculars al, bl, and cl as shown. If we adopt the convention that a line drawn from an internal point directed to the side is positive, as shown by the arrow on a, then a l is negative being directed in the opposite direction. Values of bl and cl remain positive, their directions being identical to those of b and c. I t is again a trivial matter to show that a l bl + bl + cl = h. (For proof, draw aline throughPl parallelto CB, and bisect the angle a t A, cutting CB a t M and the parallel line at N. Now AM = h = AN - a1 r b~+ CI + (-a~), Q.E.D.). Hence a b c = h remains the general statement for the indenfinitely extended triangle ABC. If a is proportional to the logarithm of a physical or dimensionless parameter A, and similarly b, c, and h represent the logarithms of other parameters B, C, and H, we can write
point on the plane indicates the simultaneous values of p, V, and T. An isothermal ex~ansionis reoresented bv a straieht line parallel t o t h e ~ s i d e ' o the f triangie. An adiahaticchark willalw hea rtraieht line whirh will be inrlined to the three axes, as determine; by y,and the Carnot cycle will be simply a ~arallelomam.Of course. the ranee of values under studv foi the three variables may compietely remove the basic trianele from the actual limits of the firrure. he state of a real gas cannot be m&ed directly on these coordinates. However. use can made of the eeneralized compressibility factor, Z, and write @lZ)(V)(llr) = R
By applying values of Z taken from tables or graphs, the use of this coordinate system becomes obvious. In producing Figure 2, plots for different values of Z were constructed from published graphs.2
+
+ +
logA+logB+logC=logH Figure 1. Extended basic triangle showing directed lengths
where dimensional equality must be maintained. Of course, H can also receive the value zero, that is h = 1, and the triangle then degenerates into a single point from which three rays at 60" extend indefinitely. The one-to-one correspondence between a and A can be easily and conveniently established by dividing distances to a logarithmic scale. The immediately obvious application for this is the ideal gas law which can be written as
The basic triangle for this is shown in Figure 2, where it is included between the three sides marked 1.0 (log 1= O), and has a height proportional to log R, or 3.92 in the SI system. p and V increase in the value in the positive direction, while the values of T decrease, because log(l/T) = -log T. Any 'On leave from Technion-Israel Institute of Technology. Haifa
32000. Israel.
SU. G.J., Ind. Eng. Chem., 38, 803 (1946).
Figure 2. Basic triangle for ideal gas law, with values of the compresibility factor L
Volume 62 Number 10 October 1985
855
Figure 3. Two axes (a) over which the third axis (b) Is to be positioned to give me basic blangle of bight as required.
Another way of handling the behavior of real gases is to consider the coordinate system as representing the reduced values p,, V,, and T,.Here
Since 2, is a constant for any gas, this alternative simplifies the work considerably. The coordinates should, of course, he marked in the proper way for each case. Figure 2 has been marked for the first alternative. In order to facilitate the preparation of the diagram for
858
Journal of Chemical Education
any application, two transparent overlays have been pre~ a r e dIn . one. loaarithmic scales are plotted along two axes at 60". In another, asimple logarithmkscale is drawn. These are shown in Figures 3a and 3b on the printed page, but they are notvery useful because they are not transparent. For use, the two overlays are superimposed so that the lines of Figure 3b form 60" angles with those onFigure 3a, and one is moved relative to the other to give the desired height of the basic triangle. The overlays can be taped together and photocopied for further use. Another an~licationfor this method of presentation u , d d include ;he mapping of the lirnitsoiturb&nt flow Re = d u l u i v = kinematir visrositv = d o ) . 'l'he coordinate system lends itself to the pesent&ion i f other data which may occur in this form.
