G. F. Atkinson Univers~tyof Waterloo Waterloo, Ontario, Canada N2L 3G1
Equilibrium Composition, Additive Properties and Stoichiometry
One exception of long standing and general acceptance to the customary choice of rectangular Cartesian coordinates for presenting chemical data graphically is the use of trilinear cwrdinates for three-component phase diagrams. Such a diagram is derived by considering a triangular prism, the base of which is a composition plane with a point representing every possible mixture of the pure components located a t the vertices. The top of the prism is in general a complicated surface. The altitude above the composition plane a t each point represents the temperature a t which the first solids will appear if a melt of that composition is cooled. In reducing this three-dimensional model to a two-dimensional plan, either contours a t suitable temperature intervals or the valley-lines of the first-solids surface are projected down to the composition triangle. This communication arose from the initial thought that the formation of a complex AB (called for convenience of notation, C ) &om reagents A and B to the extent indicated by a formation constant 811 and according to the equation
CAB1 = 811fA1.[BI
(1)
where square brackets indicate equilibrium concentrations, should he conveniently represented on a similar diagram using trilinear coordinates. The vertical dimension could then be assigned to some additive property of the species present which is observed in a given type of experiment. By proceeding in this way, the behavior of the chemicals, which is invariant regardless of the kind of experiment performed, is singled out and dealt with in a composition plane, upon which the measurement variables of a variety of experiments can be erected as alternative third dimensions. In turn the manipulation of the composition and measurement data according to the strategy of a given method of determining stoichiometry can constitute a third stage in development of the model. The Composition Plane
A general equation for complex formation may be written as
with a corresponding mass law expression
The simplest case, when m = n = 1, should be examined first. The vertices of the composition triangle now represent the pure substances A, B, and C (which, here, is AB). All possible mixtures of these substances are represented by points within the triangle, and all possible mixtures of any pair of these substances are represented by points on one of the boundaries of the triangle. For instance, all starting mixtures of A and B are represented by points in the line AB. If A and B do not react, that is, if 011 = 0, the starting mixture is the only one needing representation. At the other extreme, if 811 is infinite so that A and B react until one is completely used up, the final composition will he a point on the line joining C to 792
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F qdre I Compos llon lr ang e snow ng eq. orurn cornposllons lor sewera1 r a .es 01 >,. when A 8 ( o m s from a passoble m xl.rer a1 A an0 0 See text lor a sc.ss on 01 po,nl R , , Tne abel a q I on cJrver s 6 on 3,
whichever of A or B represents the component present in excess. If A and B are present in stoichiometric amounts and 811 is infinite, the point corresponding to the final composition must lie on both AC and BC, hence a t C. Between these extreme cases, for all finite positive values of 811, the equilibrated system must be represented by a point within the composition triangle. For any given value of 811, the equilibrium compositions arising from all possible starting mixtures of A and B must lie on a curve extending from A to B, lying within the composition triangle, and approaching C most closely when the starting composition is the same as the stoichiometric composition ofC. Using a simple Fortran program, system compositions were computed for ratios of initial components A and B ranging from 1:99-991, and for various values of Dl]. Some of the results are shown in Figure 1. Examination of Figure 1 shows that for all values of 011, the equilibrium composition points arising from a given starting mixture of A and B lie on a straight line with that
B
D
Figure 2. Equilibrium compositions for several values of Bnn. Label digit as before, la) m = n = 2 (b) m = 2, n = 1 ( c ) rn = 3, n = 1 (d) m = 2. n = 3. I n Figure 2(d), tangentsdrawn from A osculate at the points of intersection of the curves with the line TT' which passes through R a .
starting coIdposition. One such line out of the many which could be drawn, is shown in Figure 1. All such lines in this figure will meet, if extended beyond AB, in the point Rll which is the reflection of C i n AB. The equations of the curves shown are second degree, and must accordingly he conic sections ( I ) . Therefore, the set of curves for various values of 611 must be a family of conics having w i n t s A and B in common, and differing by the paramet&&l. For other eases, Fortran programs based on successive approximations by Newton's method (2) were prepared for: m = n = 2; m = 2, n = 1; m = 3, n = 1; and m = 2, n = 3. These cases cover most complex-forming reactions since m and n are interchangeable, that is A,B, and A,Bm yield mirror-image curves. Representative curves for some choices of m and n and for a variety of values of Om, are shown in Figure 2. In each case, it is assumed that only the complex named need be considered, and this pure species is plotted a t the third vertex of the equilateral composition triangle. In consequence of this convention, the point R,, which is analogously defined to the point Rm -- mentioned ~reviouslv,is different for each set of numerical values of m and n. The location of Rmn for any pair of values of m and n is shown in Figure 3. Essentially, if the perpendicular distance of C from AB is taken as unity, the perpendicular distance beyond AB a t which n - 1). The location of the appropriate R,, lies is l/(m R,, for any stoichiometry can also be found by this procedure
+
Locationsof R,n for variousvaluesof
and
Consider triangle ABRll. ark off along A R u line segments, each beginning at A, which are successively 'h, Yj, #, ete. of the distance A R n . Label the points terminating these segments with successive possible values of m, namely 2,3,4, etc., and join each point thus labelled to B. Next, repeat the process starting fmm B and labelling the points with successive passiVolume 51, Number 12, December 1974
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Figure 4. Surface showing value of a property of C (which is zero-valued for A and 8) for all compositions. Modified Job curve obtained by proposed method is also shown (see discussion of Fig. 7).
ble values of n and joining them to A. The intersection of the lines showing appropriate values of rn a n d n is the position of Rmn. One utility of locating R,, lies in the fact that, if a stoichiometry is postulated for a complex, the join of R,. to the starting composition point for an experiment can be extended to show the locus of equilibrium mixtures. In general, these loci will lie in different directions for different wstulated stoichiometries (and hence different points ~ , , j and if a measure of the concentration of a t lea& one constituent can be obtained in the equilibrium mixture, i t may be possible to choose between .&oichiometries. Such a choice may not require a knowledge of values for 0. The Additive Property Now, upon the composition triangle previously described, consider erecting everywhere normals representing values on a convenient scale of some property of the constituent species in the chemical system. The property should have a suitable experimental measure available. and should be additive over all species. though i t will 'often be of interest to seek a prope;ty which is zerovalued for one or two of the three species plotted. In the simplest case, consider a property zero-valued for A and for B, and hence for all unreacted combinations of A and B. A perpendicular can be erected a t C, and joined to A and B by curves representing the decrease in the measure of the property as the proportion of C present decreases. Thus, the value of the property for every point in the composition triangle is given by a ruled surface in which lie the line AB and the point C' terminating the perpendicular from C. If a linear relationship such as j3e;r's law prevails between measurement value and composition, the ruled surface becomes a plane and the figure erected on the composition triangle becomes a distorted tetrahedron. This is shown in Figure 4. If the property is non-zero-valued for A and B as well as C, and if a linear "law" is still relating measurement to composition, the distorted tetrahedron becomes a triangular prism truncated by a tilted plane. The contributions of A, B, and C to the measured quantity may be represented by a stack of three tetrahedra. Alternatively, the contributions of A and B may be combined to give an irregular four-sided pyramid, with its base in the plane erected upon AB and its vertex a t C, and the contribution of C then superimposed as a distorted tetrahedron with the perpendicular erected upon C as one edge. These relationships are shown in an exploded diagram of the three contributions in Figure 5 and analogously to the simpler case of Figure 4 in Figure 6. At any point on the composition curve for a given stoichiometry and a given 0, the normal 784
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Figure 5 . Exploded view of contributions of A and Band C to a measured properly.
erected will be intersected by the various surfaces separating the contributions of constituents to the measured quantity. The Determination of Stoichiometry The model thus far described consists of a solid standing on a triangular base, terminated by a surface showing all possible values of a measured property for a chemical system, and having traced in that surface the upward projection of the real compositions attainable in the given system and thus the real values attainable by the measured property. The purpose of creating this model is to use i t to interrelate various methods of determining values of m and n in the complex C. It is therefore appropriate to undertake to generate from the model some familiar graphical displays related to methods of determining stoichiometry.
Figure 6. Surface as in Figure 4 for case where the property is non-zero-valued for all ol A and B and C. Numbered curves: 1 = measured values. 2 = Contributions of equilibrium amounts of A and 0 . 3 = values of ( 1 2). 4 = incorrect practice of subtracting contributions of starting amounts of A and B (by using points on the line A'B' instead of pointson 2).
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Continuous Variations (Job's Methodot lsomolar Solutions) In this technique, samples are prepared with a wide range of ratios of A to B, hut always the same total amount (moles) of A and B. Absorbance of each sample is plotted as ordinate against mole ratio of starting materials A and B as abscissa. The resulting curve shows an extremum, usually a maximum, which in simple cases lies a t the abscissa corresponding to the stoichiometry of the c o m ~ l e xor is related to that abscissa hv a correction equation (3). References to some papers-outlining this method and its limitations and variations are found in the list of general references a t the end of this paper. Consider first the case of A and B not absorbing light. The prepared samples are represented by points on the AB side of the composition triangle. Reaction occurring is represented by the composition point moving into the triangle on the straight line away from R,, for the stoi-
chiometry involved to a point representing the equilibrifor the parum mixture corresponding to the value of "3(, ticular reaction. Measurement corresponds to raising the normals from the points thus obtained to give a new set of points lying in the absorbance plane, through which will be sketched a projection of the composition curve. This projection still lies in a tri-linear reference frame, and to obtain the familiar Job plot, i t must be re-projected into the plane of the prism face erected on AB. In order to preserve true absorbances, this projection must be made parallel to the composition plane, and in order to pair absorb a n c e ~correctly with starting compositions used, the pmjection (in plan view) must be toward R m n . This process is illustrated in Figures 4, 6, and 7 for the simple case m = n = 1. It will he noted that the upward-concave portions of Job curves thus generated "sharpen" the peak of the curve and compared to conventional Job curves make i t easier to assign a stoichiometry to the species unambiguously. Moreover, the correction of peak position for vari-
/ Figure 7. Proposed generation of a Job-type curve with sharpened peak. See text.
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Figure 8. Generation of traditionally shaped Job curve. See text.
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Figure 11. Error incurred by overcorrection using starting rather than equilibrium contributions from A and B.
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Figure 9. Less desirable scheme for generating Job curve. resulting in nonlinear abscissa Scale
Figure 10. Interrelationship of schemes for generating Job curves. Lettering agrees with that in Figure 7-9.
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ous values of om, proposed by Likussar and Boltz in their Table 111 (3) becomes unnecessary. If a Job curve of the more familiar shape is desired, this can be constructed at the sacrifice of preservation of true ahsorbances by carrying out the projection from p directly toward R,, rather than horizontally toward the normal erected upon Rmn. This is shown in Figure 8 for the same situation used in Figure 7, while the result of projecting from p along a normal to the prism face raised on AB is shown in Figure 9. This latter pmcedure might appear to be worth consideration because it preserves true absorbance~,hut it does so by sacrificing linearity in the scale of starting compositions along AB. Figure 10 is lettered to agree with the three previous figures and shows the relationships of the three projection processes. When all of A and B and C contribute to the absorbance measurement, it is customary to correct the absorbance values read from the prepared samples before plotting and examining the Job curve. Such a correction process can be seen in Figure 6. The plane separating the
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four-sided pyramid of contributions from A and B from the tetrahedron of contributions from C (shown as a dotted plane in the figure) is defined as the new base plane from which vertical distances will be plotted, so the corrected plot shows contributions from C only. Inspection of Figure 6, or of the simplified oblique view of the same situation in Figure 11, shows that the correction thus made diminishes as the ~ r o.~ o r t i oofnC to A and B increases. A variation of this correction is sometimes recommended or reported, in which the entire absorbance of the unmixed starting reagents corresponding to each sample for measurement is subtracted rather than the remaining ahsorbance of the unreacted portions of starting reagents remaining in the equilibrium mixture. This correction is increasingly in excess, the more the composition point involved lies away from AB and toward C, and it is thus maximallv overcorrecting the art of the curve in which the maximum is sought; tending to flatten the peak. In spite of its being an overcorrection, this procedure is likely to continue popular, because i t is so readily made, either calculationally after measuring series of samples of the reactants separately (4), or instrumentally by using partitioned optical cells such as Hellma No. 238 in which the unmixed reactants serve as reference sample in a doublebeam spectrophotometer and thus are deducted automatically from the total absorbance of the mixed sample in the working position. In terms of Figure 11, this introduces an error equivalent to the fall of the plane A'B'C between points e and f . The overcorrection is easily seen to be made in Bbvillard's graphical form (5) of the method of Hagenmuller (6) as modified by Sakellaridis (7), and is also found implicitly in many student laboratory instructions. Mole Ratio In this technique, samples are prepared with a wide range of ratios of A to B as in Job's method, but always with the same constant amount of one component, say A. Absorhances are plotted against the amount of the varied component used in preparing the sample. The resulting curve, for large values of Om,, is composed of straight line segments intersecting a t an abscissa value corresponding to the mole ratio of A to B in the complex. If the formation constant Omnis small, that is the complex is weaker,
the measured property curve (at point p ) intersect$ the plane ACC' a t the corresponding point of the mole ratio curve (point m ) . Comparison with Figure 8 shows that this is the same projection process which gives a continuous variations curve of conventional shape at the point of intersection with the prism face erected on AB. It is beyond the scope of this paper to connect the new methods of Momoki (8) to the proposed model, hut such an exercise should lie within the capabilities of a mathematically alert senior undergraduate.
Figure 12. Proposed generation of a mole ratio-type curve
the point of intersection is replaced by a rounded corner and extrapolation of the linear parts becomes necessary to find the mole ratio required. Eventually if small enough values of 8," are considered, the method becomes of little use unless elaborate modifications such as those proposed by Momoki (8) are used. References to this method are also found in the list of general references. In Firmre 12. a mole ratio curve is derived for the same simple case used in considering continuous variations. Here. the orism face standina upon A ( ' is extended past C as required, and a projection frbm the point R,, through
Three Dimensional Representation The model used in this analysis was in fact developed through engineering drawing techniques on paper. Eventually it became helpful to construct a physical model, and this was done quite simply with a slab of 6-in. polystyrene insulation foam and a hot-wire jigsaw sold for use with this material (Pyro/kerf by Solofix Co., Loughhorough, England). A prism the exact size of the trilinear coordinate graph paper was first cut from a square block of foam having its side equal to the desired triangle side. One of the pieces thus removed was taped back into position as a temporary hase, the resulting quadrilateral turned to stand upon it, and a cut made diagonally on the top surface from one edge of the triangle base to the opposite edge a t the vertex to form the measured property plane. The two pieces of the prism were then taped together, the temporary base removed, and the prism placed once more upon one of its triangular faces. The desired composition curve, such as those shown in Figure 2, was traced on the top face of the prism and cut. The pieces thus obtained can be used in various ways. Figure 13 shows the construction of the foam model, and its use with a point source of light (at R) and two perpendicular plane surfaces as a shadow-casting model of the type proposed by Nyburg and Halliwell(9). Summary
A triangular prism model relating composition to measurements of additive properties can be used to analyze
Figure 13. Construction of polystyrene foam model, and its use as a shadow-casting device to illustrate generation of curves. In construction drawings, dotted iine is being cut, iine S-S shows orientation of hot-wire cutter. X represents tape temporarily bridging previous cut.
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and to teach the interrelationships of techniques for establishing stoichiometry. This method is shown for continuous variations and mole ratio experiments, and can he extended to deal with other schemes of experimental work. Acknowledgment
This work was made possible by an unknown curriculum-maker who in pre-relevance days decreed that the author should learn projective geometry in various coordinate systems. Computation was supported by a National Research Council of Canada grant, and performed by Dr. Keith Curtis, whose help is gratefully acknowledged.
General References with Notes
Continuous Variations Job, P., Ann. Chim., 9. 113 (19281 (the clsssie early papan lmore referred ta than read)). Vo~burgh,W. C..andCooper. G. R..J Ampr Chom. Sor., 63.437(19411. Gould, R. K.. and Vosburgh, W. C., J. A r n e Chom. Soc, 64, 1630 (19421 (opened s new wave af interest in the method and made it availsble in an Endish language de3eription1. K. K., J Woldbye, F.,Aclo Cham. Scond., 9. 299 (1955): Jones, M. M., and In-, Php. Chrm.. 62, IM)5(1958l (ouflinepr~autioniendrestrictionsonthemethodl. Asmus. E.. 2. Awl. Chsm.. 183. 321 (Part I) and 101 (Pert Dl, (19611. la comprehen. which often "sure eonfurion when omitted fmm rive treatment not ouer1oo!ing simpli6ed prerentaiionsl. Klausen. K. S., and Lanpmyhr, F. J., Awl. Chim Acro. 28. 335 (19631 !scheme for diptinguishing AB hom A,B.): Awl. Chim. Arfo. 40. 167 (19681 (prrsents computed cvlves for various 8toichiometri.s and comment. on the behavior of termins1 tan-
Literature Cited (11 LO"-Y. ~ . , - ~ ~ ~ d i ~2 ~~ ~t ~~ ~~ i ~1 1~ 1946,chap.m. ~~ ~ t ~ ~, l~ o , ~ " d ~~ ~ Mole .~ t Ratio (21 S h o d . G. E. F., and Taylor. A. E., "Caleulur." Rev. Ed.. Prentice-Hall. New York. 1947. p. 180. Ym,J. H.. and Jones, A. L., Ind. En8 Chem., Awl. Ed., 16, 111 (1944): Halvey, E.. (31 Likusssr. W..andBoltz. 0. F..Anal. Cham.. 43.1265l19711. and Manning. D.L.. J. Amer Chsm. Sac.. 72. 4488 I19501 (show basic development ( 4 Vsrga. L. P., sndVeateh.F. C.,Awi. Chem., 39.1101 119671. of tho methodl (51 Beviilard. P..Bull. S o c Chim. Fr., LS09 (19551. b m u s . E..2. Amlyt. Chem.. 178,104 l19Wl la comprehensivetreatment1 (61 Hagenmuller, P.. Comple8rmdus. 230. 2190!1950l. Klausen. K. S.. and Lanmvhr. F. J.. AM!. Chim. Aclo. 28 501 I19631 (extends trestment by Aamusta polynuelear complexel. (71 Sskellsridis,P.. Compte$rendur, 216.25W (19531. I81 Momoki, K.. Sekino. J., Safo, H., and Yamsguehi, N., Am!. Chm., 41, 1286 Momoki, K.. et sl., see Ref. (8) (pmpose the notion of normalhsd absorbance, and (19691. introduces new plotting methods: complicsted notation m a k e this valuabi~peper 191 Nyburg.S. C.,andHalliuell. H.F., J. CHEM.EDUC.,38. 123 (1961l. dimcult taread: thereare numerouaerrata for thiapapexl
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