Quantitative Analysis of Continuous-Variation Plots with a Comparison of Several Methods Spectrophotometric Study of Organic and Inorganic 1:I Stoichiometry Complexes E. B N ~ M u ,D. Lavabre, G. Levy, and J. C. Micheau Laboratoire des IMRCP, Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France
Continuous-variation plots (1,2)are diagrams of a physical property that is related to the concentration of an equilibrium two-component complex against volume fraction X, or mole fraction X, of one of the two constituents. When the mother solutions are equimolar (x, = X, = X ) the diagrams are also called equimolar, and when X, # X, they are called nonequimolar. The diagrams have a pseudotriangular shape. The Uses of the Diagrams The stoichiometry of the complex is generally determined from equimolar diagrams. For 1:1 stoichiometry, the peak occurs a t x = 0.5, while for 1:2 stoichiometry, the peak will occur a t x = 0.33, etc. I Normally, further information is not extracted from equimolar diagrams. Thus, if the equilibrium constant K, is required, a nonequimolar diagram must be plotted in which the position of the peak ( x , # 0.5) provides information on K.,. However, for wmplexes of 1:l stoichiometry, equimolar diagrams can be readily analyzed numerically to give a good estimate of K., without recourse to nonequimolar diagrams (3). The principles of such calculations are based on the fact that a sharp peak indicates a stable complex, while a more rounded peak indicates a low value of the formation constant K, (4). Numerical analysis of the plot curvature near the peak gives the required value. I n this paper we do the following. We describe and compare three useful methods for the analysis of equimolar continuous-variation plots.
and The molar fraction of component Ais given by the following equation.
This gives m=
i s t h e molar fraction of t h e complex a t equilibrium throughout the continuous variation. Rewriting eq 2 as a quadratic equation in Y gives the following.
I n practice, a physical property that is characteristic of the mixture a t equilibrium is plotted against X . For example, using UV-vis spectroscopy, Beer-Lambert's law gives
.We then a ~ o l vthe methods ta the ouantitative determina-
plexes.
the organic complex of 3-aminopyridine with picxic acid in chloroform the inorganic complex of Fe(II1) with salicylic acid in aqueous solution Equation for the Method of Continuous Variation Consider the equilibrium for the formation of a complex of 1:1 stoichiometry.
Y
(x-Y)(I-x-~ (21 where m = K,,[C], is the equilibrium constant without units.
A,, = l(EAB[ABI+EA[AI+€dB])
(4)
where A, EAB, EA,and EB are the absorbance and the molar extinction coefficients of complex AB and constituents A and B a t the wavelength of observation h; and where 1 is the optical length of the cell (1cm). Although EA and ER are known, EB is generally unknown. When the following holds eq 4 can be rewritten a s below. AA =A,, -A,= YAel[CI,
(6)
where If [ABI is the concentration of complex AB, then the following equation is valid.
where [AIi and [BIi are initial concentrations of Aand B. I n the equimolar diagrams, solutions of A and B are prepared with equal concentration [CI,. Various aliquots of A and B are mixed a t constant volume V, by adding volume VAof solution A to volume V. - VAof solution B. Thus.
A, =
((EA - EB)X + EB1l[Clo
gives the absorbance of the mixed constituents, A and B, assuming there is no complexation. IA, is a function of x . ) Equation 6 is used to correct the baseline. I t transforms a n experimental continuous-variation plot (A,, vs. X ) to a corrected diagram (YAd [CI, vs. x ) . Over a scale factor, this latter expression is readily exploited using eq 3. Three different methods can be used to analyze the corrected diagrams. The first two are graphical. They do not require any computation because the requisite calibration Volume 69 Number 10 October 1992
833
Figure 1. Graphical representation of the initial tangents method on an equimolar Job diagram for a complex of 1:I stoichiometry;d = (Y, - YmaJY1.
Figure 2. Plot of the function (eq 9) as d = f (log m).This calibration curve enables determination of equilibrium constant m without units from an estimation of d = (Y, - Ym,,)/Y1.
curve and chart are supplied in the paper. The third method is based on a minimization algorithm using a nonlinear two-parameter best-fit method. This gives The Method of Initial Tangents The method of initial tangents (5) uses experimental data close to the starting points (X = 0 and = 1)to determine tangents. Points close to the peak are used to determine Y,,. (See Fig. 1.) Differentiation of eq 3 gives
x
which enables the calculation of the slopes of the initial tangents. Atx=O
This curve (see Fig. 21, which can be used to estimate m (where m = K,,[Clo) from ratio d, has the characteristic shape of the equimolar continuous-variation plot. Although this method is simple, quick, and sensitive for values of m from 1to TO3, it does not extract information about intermediate points (X = 0.25 or 0.75). This can be done using the method of chords. The Method of Chords Consider the graph shown in Figure 3. The following parameter 6 does not depend on a scale factor (7)withz = if < 0.5 or with z = 1- x if x > 0.5 (from symmetry of the diagram).
x
The equation of the tangent a t x = 0 is
x
I t is closely related to the shape of the diagram. When the peak is sharp, which occurs with a stable complex, 6 is
These two straight lines cross a t = 0.5 and
At x = 0.5, Y,,on the continuous-variation plot is given by the following relationship.
c a ~ o be t obtained directly The values of Yl and Y, from the experimental continuous-variation plot because they are both proportional to a scale factor that can be eliminated by considering the following ratio (6). 834
Journal of Chemical Education
Figure 3. Graphical representation of the chords method; 6 Y/(Y,,, - 24;z = x if x < 0.5 and z = 1 - x if x > 0.5, from the symmetry of the Job diagram.
Figure 4. Chart required for the method of chords. All points of the initial Job diagram near x = 0.25 or 0.75 (z = 0.25) must be included on this diagram. Each experimental pint provides an independent estimate of m. small. When the peak is rounded, which occurs with a less stable complex, 6 is large. Thus, from eqs 3 and 8, we get the following.
The plot of 6 as a function ofz for differentvalues of m can be used to draw a chart that is useful in estimating the value of m. (See Fig. 4.) Each experimental point from an equimolecular 1:l Job diagram can be plotted on this chart, which can be used to estimate the value of m, and thus K,,. The range of sensitivity is 1< m < lo3
Then the molar extinction coefficientof the complex EAB is caleulated from the extremum of the Job diagram (X = 0.51, by combining eqs 3,5, and 6.
Figure 5: Equimolar Job diagram of a mmplex of t :t stoichiometry between 3-aminopyridine and picric acid in solution in chloroform ([C], = 1od M), Showing the change in absorbance at 400 nm of the mixture during the continuous variation. The squares numbered from 1 to 9 are the experimental pints. At 400 nm. the molar extinction coefficients of picric acid and 3-aminopyridineare negligible. Thus, their intrinsic absorbances were not substracted from the measured values. The experimental Job diagrams can be analyzed directly without correction. The continuous line represents the best fit using the numerical method (m = 30: EAB= t t ,000). the two parameters to be fitted. This method requires the estimated values of these two parameters. They can be obtained from either of the graphical methods mentioned above. Applications The methods of analysis described here are quite general. They are also independent of the molecular structures of the species or of the chemical nature of the complexation when a single complex is formed i n 1:l stoichiometry. They have been applied to both organic and inorganic wmplexes. Organic Complex between Picric Acid and 3-Aminopyridine
A bright-yellow color is observed upon mixing the two almost colorless chloroformic solutions. Thus, UV-vis spectroscopy is suitable for studying this equilibrium. Figure 5 shows an experimental equimolar Job diagram taken at 400 nm. Plotting the nine experimental points from Figure 5 on a chart similar to that in Figure 4 allows the estimation of parameter m by the method of chords. n'l: m = 30 n'2: m = 25
To directly obtain K, and method.
EX,,
one can use a numerical
n'4: m = 15 n'6: m = 9
The Numerical Method
The starting point is eq 6. A4 =A,,-A,=
n'3: m = 30
n'7: m = 21
YAEI[CI.
(6)
A minimization procedure using a nonlinear leastsquares method is applied to the following function F.
where n is the number of experimental points. Y is calculated from eq 3 by replacing m with K,[C], and by replacing AE by its equivalent from eq 5. K,, and EAB are
n ~ 8m : = 19.5
n'9: m = 30
This gives an averaged value of m = 22 i 8, which is comparable to that obtained by the initial tangents method: m = 31 f 8. These results were also compared to those obtained using Hildebrand-Benesi's plot (8).(See Table 1.) The results are within experimental error of each other for the equilibrium constants, but not for the extinction coefficients. The HB method differs.Apossible explanation is Volume 69 Number 10 October 1992
835
Table 1. The Comoarison of Various Methods for the Determination of ~hermod~namic and Spectroscopic Parameters of Com~lexFormation between Picric Acid (1 x 10" M) in CHC13 at 25 'C. and 3-~mino~yridine
HB
(2.0
* 0.4) x lo5
8800 i 800
IT
(3.1 f 0.8) x lo5
11800 f 800
CM
(2.2 f 0.8) x lo5
12000 i 1000
LSF
(3.0 f 0.2) x lo5
I1000 i 300
-
--
nB: melnod of Hloabrana an0 Beneo: I T m e m o of nt1a;tJngenls methao of cnords -SF: oasl-sq~arcsI1 rnetnao r r a a1 400 nm.
CM:
that the experimental conditions are not exactly the same because the HB plot needs a large excess of one of the two components: A molar fraction X, > 0.95 for 3-aminopyridine has been used. Such a large value could induce some solvent effects that are neglected in classical variation plots. Inorganic Complex between SalicylicAcid andfe(ll1)
An intense violet color is observed upon mixing the two almost colorless aqueous solutions (9).UV-vis spectroscopy is used to study this equilibrium. The results obtained by the three methods are shown in Table 2. Tables 1and 2 show that they give similar values with comparable accuracy. Range of Applicability
These methods are specific to symmetrical continuousvariation plots from equilibria with a unique complex of 1 : l stoichiometry. Using W-vis absorption spectroscopy and assuming that &4 0.5 is sufficient for very accurate measurements, we can derive the usable limits ofthe characteristic ratio.
-
I n our case, the values of this ratio are 5 x lo-' for the complex of picric acid and 3-aminopyridine, and 0.17 for the complex of Fe(II1) and salicylic acid. Many inorganic (metal ligand) or organic (charge transfer, hydrogen bond, etc.) equilibria of pedagogic interest (10) are likely to fall within this range. Conclusion We presented three methods that can he used to analyze continuous-variations plots such as Job diagrams. These methods are specific to equimolar diagrams involving a unique complex of 1 : l stoichiometry. Their main advantaee is that thev can he used directlv on ~reliminarvdata obiained during the determination ofthe'stoichiom& Each of these methods can be used to estimate the eauilibrinm constant K, and the characteristic scale facto; of the measurement technique kmfor UV-vis spectroscopy). Although somewhat more lengthy, the numerical method appeared to be superior to the initial tangent or chord method because i t takes account of the complete set of experimental points. The range of application of these numerical methods is not limited to W-vis spectroscopy. Any technique that can detect complex formation could, in principle, be used. Experimental Compounds
The picric acid, the 3-aminopyridine, and the salicylic acid used were commercial products (Aldrich, most-high purity). The chloroform used was W-spectroscopic-grade, and the water was double-distilled. Ammonium iron alum [Fe(NH4)(S04)21.12Hz0 was the source of Fe(II1). Preparation of Solutions -
The solutions used (11)were prepared directly in the quartz sample cuvettes of l-cm optical path length (V, = 3 mL) using automatic pipets. The same procedure was used for the Fe(II1)Isalicylic acid system. Solution Number
Using eqs 6 and 8, we get m = K&I,
Using the workable sensitivity of m, in which I < rn < lo3
~
I
2.7 mL picric acid and'0.3 mL of 3-aminopyridine
9
0.3 mL picric acid and 2.7 rnL of 3-aminopyridine 3 mL 3-aminopyridine
~ e d w
The UV-vis spectra were recorded a t 25 'C using a HP8451A diode array spectrophotometer. The cuvettes contained a magnetic bar stirrer.
~
MathematicalTreatment
EAB(M'C~")
We used the minimization algorithm described by Powell
(5.7 f I ) x lo3
890 i 150
(12).To estimate uncertainty after optimization ofK, and
CM
(6.3 i I )
lo3
850 f 100
LSF
(5.7 i0.3) x lo3
900 f 20
IT
x
IT: method of initial tangents: CM: method of chords: LSF: least squares fit method, CAB at 526 nm.
836
3 mL picric acid
UV-vis Spectroscopy
Table 2. Comparison of Three Methods Described for Determination of Thermodynamic and Spectroscopic Parameters of Comolex Formation between Salicviic ~, Acid and Fe(lll) ([C], = lo3 M).
method
0
10
we get the following relation
Component
Journal of Chemical Education
E-, we randomly simulated five typical Job diagrams. We found that a relative Gaussian error of Y2.570 distributed a t random over ten experimental points led to a n uncertainty of f2.5% in the determination of EAB, and a n umcertainty of %% in K.,
- - - - - - -I itcratura .. . . . Citcrl ..
I. Ostromiaalensky, I. Rusr. PAW. Chem. Soc. 1910,42,1312, 1500 B o lkilt ~ Cham. &s ~
~~
1911 6 2 6 8 .
2. Job, P A n n Chim. 1928.9. 113-202.
7. ~ikusurr, w.;~ o l t zD. . ~ . ~ n acham. l . I~TI,A?(IO), 1265-1272. 8. Benesi, H.A : Hildehand, J.H. J. A m r Cham. Soc. 1949, 71,2703-2707. 9. Pass. G.: Sutcliffe. H. Pmcfim! Inoreanic Chernisim. 2nd ed.: Chamnan snd Hall: Inndon. 1974: 1 8 8 ~ 412 0 12690 7rp 188.
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