The Reaction of N-Methylimidazole with a Macrocyclic Ligand Complex of Fe(ll) An Undergraduate Experiment in the Spectrometric Analysis of Two ~uccessive~quilibria Nicholas K. Kildahl Worcester Polytechnic Institute, Worcester, MA 01609 The determination of equilibrium constants for adductformation reactions usim spectrometric methods is a n important problem that des&es attention a t the undergraduate level. Although several articles in this Journal have dealt with 1:l interactions (141, the problem of two successive adduct-formation equilibria has received far less attention (5-7) although it has been dealt with thoroughly in the inorganic literature (8-21). Introducing Students to Two-step Problems Two successive equilibria occur in avariety of situations, most notably in protonation processes and transition metal-ligand interactions. In particular, axial ligation reactions of transition metal porphyrin complexes, which commonly occur in two stages, are important model reactions for biological systems. Computer Methods The dimculty in the spectrometric analysis of two-step systems is that the equilibria almost invariably overlap; the second step begins before the first step is complete. This complicates the mathematics sufficiently to warrant the use of computer analysis to obtain the two equilibrium constants. Calculational methods for accomplishing this have been developed, and most involve sophisticated nonlinear least-squares (NLLSQ) routines (8).
of two molecules of N-methylimidazole (MeIm)to the axial sites of the complex, Fe(Me4[141tetraeneN4)2t(FeL), a s seen in Figure 1, in acetonitrile (An) solvent using spectrometric titration techniques. The ligation reactions are shown in reactions 1 and 2 below, which omit solvent molecules in the axial sites. FeL + MeIm
K'- FeL(Me1m)
-
FeL(Me1m)+ MeIm Kz FeL(MeIm)z
(1) (2)
This chemical system is ideal for a student experiment for several reasons. .All three complexes in reactions 1and 2 are intensely and beautifully colored, which often excites student interest. 'Isosbestic points are observed early and late in the titratian, which also interests students. The system is pushed ta completion at relatively low MeIm concentration (about 0.1 MI. The system is not sensitive to oxygen or to the presence of water in the solvent, which minimizes technical difficulties. Graphical Methods Also, the two-step equilibrium system is amenable to analysis by the elegant graphical methods of ref 9. These methods do not involve computer analysis and are quite simple for students to comprehend. Thus, they allow the student to concentrate on the chemistry of the system rather than on the black-box data analysis of the NLLSQ routines. I hope that this article will draw attention to the methods of ref 9, which are often neglected. Experimental Information for the Instructor Reagents
Figure 1. The complex ~e(~e~[14]tetraene~,)Zf, which is abbreviated as FeL. The complexity of such calculations has prevented this problem from being introduced a t the undergraduate level. Recently, however, a comprehensive treatment of the spectrometric analysis of multistep equilibria has appeared that presents greatly simplified methods for treatment of the two-step problem (9). An Ideal Chemical System In this article, I resent an experiment suitable for advanced undergradu3te laboratory courses in physical or inowanic chemistry The experiment involves the detennination of equilibrium constants for the successive binding
Reagent-grade An (Baker) and reagent-grade MeIm (Aldrich) may be used as received without significantly affecting results. However, if desired, An may be dried by refluxing over phosphorus pentoxide for 24 h under nitrogen, followed by distillation under nitrogen. MeIm may be purified and dried by refluxing over potassium hydroxide for 2 h under reduced pressure, followed by distillation under reduced pressure. All other materials usedin the synthesis of the iron complex were reagenbgrade and were used as received. Synthesis FeL(An)z(PFs)zwas synthesized by published procedures (12,13) and characterized by IR and electronic absorption spectroscopy (EM) to yield the following values: X,, ,(E), nm (M-'m-'): 551 (9.02 x lo3), 514 (sh). Because the synthesis requires a long reaction time, this should be carried out by the instructor. Volume 69 Number 7 July 1992
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Physical Methods
IR spectra for characterization of FeL(An)z(PF& may be obtained from KBr pellets or Nujol mulls. We used a Perkin-Elmer 683 infrared spectrometer and PE 3600 data station. EAS may be performed with any high-quality spectrophotometer covering the visible wavelength range. The results presented here were obtained using a Shimadzu UV-2100U spectrometer system. Standard l-cm spectrometer cells equipped with tightfitting teflon plugs are preferable to cells with loose-fitting square plastic or teflon caps, which leak when shaken. Preparation of Solutions
Stock Solution of FeL(An)zfPFdz I recommend that students perform the experiment in pairs. Allow for 2 mL of stock solution for every pair of students. To prepare 100 mL of a 1.25 x lo3 M stock solution, weigh 0.0845 g FeL(An)z(PF& (MW = 676.3 glmol) into a 100-mLvolumetric flask, and add An to the mark. This solution is stable to air and moisture and may be prepared several days in advance ifdesired. 0.1M MeZm in An Using a 50- or 100-pL syringe, transfer 39.8 p L of MeIm to a 5-mL volumetric flask. Add An to the mark. Then stopper and shake to insure uniformity. 1.0M MeZm in An Using a 500-pL syringe, transfer 398 pL of MeIm to a 5-mL volumetric flask. Add An to the mark. Then stopper and shake. Requirements
Each pair of students will require ahout 30 FL of 0.1 M MeIm and 15 pL of 1.0 M MeIm for each titration. Thus, the 5-mL volumes recommended above will be sufficient for over 150 pairs of students. If desired, students may prepare their own MeIm solutions for titration. However, this will greatly increase the amount of solution for disposal after the experiment. Student Procedures Preparation of Solutions for Titration C a u t i o n : Preparation and transfer of carried out in a fume hood.
solutions should he
Working solution of FeL(AnhPF6h Use a l-mL pipet or a syringe to transfer 0.5 mL of iron stock solution to a 5-mL volumetric flask. Add An to the mark. Then stopper the flask and shake to mix. Record the concentrations of the iron stock solution and your working solution. Record the color of the solution. Solutions of MeIm in An (0.1M and 1.OM) will be supplied by the instructor. Spectrometric Titration
Using a graduated pipet or a syringe. transfer 3.00 mL of your working solution of FeL(An)z(PF&to a 1-cm spenrophotometer cell. Fill the reference cell with An. Record the ambient temperature. Set up t h e electronic absorption spectrophotometer to overlay successive spectra, then record the EAS of vour working solution ov& the interval 750-300 nm. ~ e i s u r e and record the absorbances at 551,604, and 663 nm. These are the-, ,? values for FeL, FeL(MeIm), and FeL(Me1m)~, respectively. 592
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Using a 10-pLsyringe, add a l y L aliquot of 0.1 M MeIm in An. Then stopper and shake the cell. Allow the solution to stand for 1min before recording the spectrum. The delay of 1min is necessary to ensure equilibration of reaction 1, particularly in the early stages of the titration (14). After addition of all subsequent aliquots, record any color changes. Record the spectrum so that it overlays the first spectrum. Then record the absorbances at h = 551, 604, and 663 nm. In similar fashion, add two more l-pL aliquots, two 2-pL aliquots, two 4 p L aliquots, and two 10-L aliquots of 0.1 M MeIm. Wait 1 min, and scan the spectrum. Record the absorbance at 551,604, and 663 nm aRer addition of each aliquot. Now change over to 1.0 M MeIm in An. Add a total of about 15 pL of this solution in aliquot sizes ranging from 1 to 4 pL. Record the spectrum and specified absorbances aRer each addition. Finally, change over to neat MeIm. Add a total of about 30 PL to the cell in aliquot sizes ranging from 1to 10 uL. ~ecordthe spectrum a i d absorbances aiier the addition of each aliquot. Continue addition of Melm until no further spectralchanges occur. The fmal concentration of MeIm should be about 0.125 M. Again record the ambient temperature, and average your two temperature measurements. Results and Method of Data Analysis
Analysis of titration data can he accomplished using absorbance diagrams, or A diagrams, and absorbance-difference-quotient (ADQ)diagrams (9). A Diagrams
An A diamam is a d o t of the absorbance at one wavelength against the absorbance a t a second wavelength using one point for each of the s~ectralscans taken durine the &ation. The diagram thus 'directly shows the relativz absorbance changes at two wavelengths as a function of titrant concentration. For a one-step system, it can be shown (9) that the absorbance at any wavelength must be proportional to the absorbance at any other wavelength, so that an A diagram for such a system will be linear. Thus, a single-step system is quickly diagnosed from such a plot. However, if a system is governed by two or more equilibria, the Adiagrams will "change direction" each time a new equilibrium becomes dominant in the system. This will occur over particular ranges of titrant concentration. If successive K values differ by a factor that is greater than or equal to lo3 or the A diagram will consist of linear segments. In other words, if the successive equilibria overlap very little or not at all, the A diagram will consist of linear segments, one corresponding to each of the equilibrium steps. These segments are connected by curved regions in which we fmd the dominance shift from one equilibrium to the next. The linear segments in the A diagram give way gradually to a smooth curve as ApK decreases, that is, as the amount of overlap of the successive equilibria increases. ADQ Diagrams
An ADQ diagram is constructed using absorbances at three wavelengths: h,, b,and &. First, a value is calculated for each of the titration spectra using the following expression.
Finally, one quotient is plotted against the other over the series of titration spectra. As shown in ref 9, ADQ diagrams are linear for systems involvingtwo successive equilibria. They are curved if there are more than two steps. Determining the Number of Equilibria To determine whether one or more successive equilibria govern a system, it is first necessary to construct several A diagrams, using data fmm several wavelengths. Linearity indicates a single equilibrium, and construction of ADQ diagrams is unnecessary. Linear A diagrams are easily analyzed to obtain the single K value, as discussed in ref 9. On the other hand, curvature of A diagrams indicates two or more successive equilibria, and it is then necessary to construct a number of ADQ diagrams. Linearity indicates two successive equilibria. In this case, the Adiagrams can be analyzed with relatively little effort to obtain the following. Extinction m&cients of intermediate species at the plotted wavelengths, which are usually not directly measurable. Equilibrium mnstants for the two equilibria involved, as discussed below. Ifthe ADQ diagrams are not linear, three ormore successive eauilibria are indicated. This situation will not be discussed further because our focus is on the two-step situation. The reader is referred to ref 9 for derivations of relevant equations and for detailed explanations of the uses of A diagrams for multistep processes. The FeUMelm System Titration Spectra
Figure 2. Titration spectra obtained during the spectrometric titration of FeL with Melm. Inset in the direc- - a~shows - scans -~~ - 1 4 . oroceedina ~~~~toon of the arrow. lnset b shows scans 8-22. See ~abqi1 for corresponding absomance and concentrat on data. ~~
~
-.7
~
~
~~
where A, is the absorbance a t hl after addition of the nth aliquot oftitrant, and A. is the initial absorbance a t hl (before addition of titrant). This series of absorbance differences a t 4 is called AA&. Corresponding series of absorbance differences are cald a t e d a t b and &, again using the initial absorbance as a reference point. These series of differences are indicated by the following terms.
Spectra obtained during a typical titration of FeL with MeIm in An a t 25 'C are shown in Figures 2a and 2b. Figure 2a shows the evolution of spectra during the fwst half of the titration, in which reaction 1dominates. Similarly, Figure 2b shows spectral changes due primarily to reaction 2. Isosbestic points are observed a t 567 and 624 nm. The ~ o i nat t 567 nm is observed earlv in the titration. It corresponds to crossover of the spectra of FeL and FeL(Me1mJ. It ~ersistsuntil the concentration of FeL(Me1mh becomes si&ficant. Then it blurs and disappears. It is replaced by the point a t 624 nm, which is the spectral crossover for FeL(Me1m) and FeL(Me1m)~. As the titration proceeds, the pink solution, which contains FeL, turns purple, then blue, and finally cyan due to
Two ratios (quotients) of these absorbance differences are then calculated for each titration spectrum by dividing one of the differences into each of the other two. For example, the quotients
might be dculated.
A at 551 nm Figure3. Absorbance at 604 nm versus absorbance at 551 nmforthe FeUMelm System.
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Table 1. Concentration and Absorbance Data for the Titratlon of FeL with Melm in An
volved. For such systems, A diagrams often quickly disclose this.) The linear ADQ diaeram shown in Figure 4 confirms two successive equilibria. The Adiagrams can now be ana551 604 663 Scan pL of WL of pL of [Melmltp log [Melml lyzed to obtain the extinction coeffl0.1 M 1.OM neat (x10) cients for FeL(Me1m) at 551, 604, and Melm Melm Melm 663 nm, and to obtain the values of Kl and Kz. This is accomplished using the so-called absorbance triangle. Fimre 5 reproduces the Am ~~-- ASST ~ -A diagram of figure 3. The point corre0.814 0.533 0.040 0.0999 4.000 sponding to the beginning of the titra0.072 0.166 -3.779 0.724 0.703 tion is labelled B, a i d that correspond0,107 0.798 -3.633 0.666 0.233 ing to the end of the titration is E . The 0.365 -3.437 0.590 0.886 0,179 absorbance triangle is obtained by wn0.498 -3.303 0.543 0.914 0.246 structing tangents to the A diagram at 0.761 -3.H9 0.478 0.913 OS7 points B and E . and then extendine 10 10 1.088 -2.963 0.426 0.887 0.468 them until they 'intersect at the 11 1 1.42 -2.849 0.390 0.860 0.554 labelled I . for intermediate. 0.831 0.627 (The construction of tangents may be 12 1 1.75 -2.758 0.360 carried out by a number of methods. 0.787 0,732 13 2 2.40 -2.619 0.321 Those in Figure 5 were done using the 0.755 0,809 14 2 3.06 -2.514 0.296 mirror technique.' More rigorous meth0.709 0.903 15 4 4.37 -2.359 0.260 ods based on differentiation of the 0.679 0.963 equation describing the best polyno16 4 5.68 -2.246 0.239 0.632 1.061 17 1 9.80 -2.009 0.204 mial fit of the A diagram can be used if 0.606 1.114 desired.) 18 1 13.9 -1.856 0.186 0.585 1.I 58 The triangle is completed by conned19 2 22.2 -1.655 0.171 ing points B and E. This procedure ef0.565 94 20 5 42.7 -1.370 0.157 fectively resolves the observed A dia1'21 0.554 21 10 83.5 -1.078 0.151 gram into three linear A diagrams, one 0.547 22 10 124 -0.906 0.146 corresponding to each of the equilibria The spwrophotometer cell initially containsd 3.00 mL of 1.26 x lo4M FeL(An)z(PF& T = 25 'C. in reactions 1and 2, and the third corresponding to direct conversion of FeL to FeL(MeI& in one step, without involvement of the intermediate complex. FeL(Me1mh. Concentration and absorbance data are colThus, the line connecting B and I, that is, line BI, is the lected in Table 1. A diagram that would be observed if reaction 1were the only process occurring in the system. A similar corresponAnalyzhg the Diagram deuce exists between reaction 2 and line IE, while line BE Figure 3 shows one of the three possible A diagrams for is the A diagram that would be observed if the sum of reacthe data in Table 1.The plot clearly wnsists of two segtions 1and 2 were to occur in a single step. ments separated by a region in which the plot changes diThus. the ooint I wrresoonds to the intermediate comreetion, suggesting at a glance that the FeLMeIm system plex, F ~ L ( M ; I ~which ,, isihe terminal species for reaction is governed by two successive overlapping equilibria. 1 and the initial soecies for reaction 2. Point I reveals the (For this system the number of equilibria is readily perabsorbances that &odd be observed at 551 and 604 nm if ceived from the progression of titration spectra in Figure 2. the intermediate complex were the only Fe-containingspeHowever, there are many systems for which the spectra cies present in solution. alone do not clearly reveal the number of equilibria inFrom these absorbance values and from the known concentration of Fe, extinction coefficients for the intermediate at 551 and 604 n m are found: 4.4 x lo3 and 9.0 x lo3, respectively. These values and those obtained from similar analysis of the remaining Adiagrams (Ase3vs. Aeo4and Am vs. A551),which are not shown, are presented in Table 2. By this method, extinction coefficients for FeL may be obtained even though they cannot be directly measured experimentally.
- -
Evaluating Equilibrium Constants The absorbance triangle may also be used to evaluate the equilibrium constants for reactions 1and 2, using the bisection of sides method (9).A line, or ray, is drawn con-
"
'A mirror IS held perpendicular to the curve at the polnt of merest. It is al gned ~ n t i tne l curve and ts reflect on in tne mirror appear to
0.5
1.O
1.5
Delta A at 604lDelta A at 551 Figure 4. ADO diagram for the FeUMelm System.
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Journal of Chemical Education
2.0
2.5
join smoothly. Astraight line perpendicular to thecurve is then drawn using the edge of the mirror. Finally, the tangent is constructed perpendicular to the first straight line and touching the curve at the point
Table 2. Results of Analysis of Absorbance Triangles Constructed from A Diagrams A Diagram
Used
604 vs 551 (Figure 3) 604 vs 663 663 vs 551
Figure 5. Absorbance triangle forthe Adiagram of Figure 3. necting each vertex of the triangle with the midpoint of the opposite side. These rays are shown in Figure 5. The ray B M connects the point representing the beginning of the titration with the midpoint of the linear absorbance diagram for the equilibrium of reaction 2. The intersection of B M with the experimental A diagram gives the half-equivalence absorbances at both plotted wavelengths for the second equilibrium. The half-equivalence absorbance is the absorbance at the half-equivalence point of the equilibrium step concerned. At the half-equivalence point of the second equilibrium
Thus, pKz is found by determining the mncentration of MeIm corresponding to the half-equivalence absorbance. plot of the This is done using an absorban-ncentration type shown in Figure 6. In Figure 5, BM intersects the experimental A diagram at Ass] = 0.36. From Figure 6, this absorbance corresponds to
h nrn 551 604 604 663 551
0.55 1.14 1.13 0.044 0.57
. . .
- ---
r (~'cm-')for FeL(Melm)x 103
4.4 9.0 9.0 0.35 4.5
Mean r Values for FeL(Melm): 4.4 x 10' at 551 nm 9.0 x lo3 at 604 nm 0.33 x lo3 at 663 nm A Diagram PK~ (pKl + pK2)/2 PKI Used 604 vs 551 2.76 3.34 3.92 (Figures 3,5) 604 vs 663 2.77 3.40 4.03 2.76 3.42 4.08 663 vs 551 4.01 f 0.08 Mean: 2.76f 0.01
equilibrium. pKl may be obtained by the same procedure used to obtain pKz. Finally, intersection of the ray ZP with the diagram gives absorbance values that correspond to the following equation, which is explained below.
At the half-equivalence point for the conversion of FeL directly to FeL(Me1m)~,by adding reactions 1and 2, we get the following expression. [ ~ e ~ m=]KIKz '
In Figure 5, ZP intersects the curve at A m = 0.55, which translates to the following equation. -log [MeIml = fl'
The ray EN connects the end of the titration with the midpoint of the linear A diagram for the first equilibrium. Similarly, the intersection of EN with the experimental A diagram gives the half-equivalence absorbance for the first
A
+
s z = 3.34
Substitution of pK2 from above gives pK, = 3.92. Application of the bisection of sides method to all three Adiwams for the FeL/MeIm system gives the results collected in Table 2. Values of pl(l have been determined indirectly from pKz, rather than directly using ray EN. Direct determination would require knowledge of the wncentration of unbound MeIm a t the point of the titration represented by the intersection of EN with the Adiagram. At this early point in the titration, a significant and unknown fraction of the total concentration of MeIm is bound, so that the total and unbound concentrations differ.At later points in the titration, where the concentration of bound MeIm is insignificant with respect to the total, use of the total MeIm concentration to determine pK values introduces negligible error. The values of pKl and pKz obtained spectrometrically compare favorably with earlier estimates obtained using NMR spectroscopy.
pK1- pKz= 0.85 which comes from ref 12. log [Melm]
Figure 6. Absorbance at 551 nm versus the total wncentration of Melm.
pK, > 2.70 and pK2 > 1.85 which both come from ref 15. Volume 69 Number 7 July 1992
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Summary The graphical procedures of ref 9, though somewhat awkward to describe narratively, are simple to apply. After some initial assistance in constructing A diagrams, ADQ diagrams, and absorbance triangles, students will readily accomplish the analysis of the two-step FeL'MeIm system. Hopefully they will then realize that quite complex equilibrium systems can be analyzed by simple methods that do not require the use of a computer for data analysis. Concluding Remarks The experimental procedures, including the preparation of the iron workine solution and the titration. can readilv be accomplished Fn a 3-h laboratory period: The A diagrams, ADQ diagrams, and absorbance-concentration plots may be generated using any of several plotting mutines now available. Figures 3-6 in this article were generated using ProPLOT Scientific Graphics, Version 1.0, from Cogent Software.
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Journal of Chemical Education
Acknowledgments I thank my colleague, Ladislav H. Berka, for reading the manuscript and making many helpful suggestions. Literature Cited 1. Ramette.R. W.J C h . Edue. 1987.44.647. 2. L0ng.J. R: Drap, R.S. J. Chem Edue. 1982,69,1037. 3. Callins, M.J. J.C k m . Edm. l W ,63,457459. 4. Omten,A.;W0idk.J. F J. C k m . Educ. 1987,M, 814-816. L J ; Heyns, J. 6. B. chem. Ed*. ISST, 66,861-863. 5. Crum-, 6. Geiger, D.K: Paulek,E. J.;&as, L.T J. Chem E&. lml,68,357339. 7. Gi1.Y M.S.; Olilieira,N. C. J. C k m . Edm. 1990.67.473-478. 8. C o m p u l d i o ~M l e t h d ~for t k D~iamximtionofFormntion C o ~ l o n t sLeggee, : J. J..Ed.:Plenum:New York. 1985. 9. Polster,J.;Lachmann,H.Sprefmm~tric~rotionr:Ady~iaofCkmiwlEpuilib~ VCH: Weinhelm,Germany,1989:Cheptera 1-7. 10. E1dahl.N. K;Zimmer,M. J. Cmrd Chem 18W,19.71-42. 11. Maier.. T.0.:. Drwo. .. R S. Inwz. C k m . 1972.11.1861-1868. 12. Baldwin, D ; Weiffed, R.M.;Reiehplt, D. W: Roee, N J. J. A m Chem. Sac. 1075, 96.5152. 13. Raiehgotf D. W.;Roae, N. J. J.Anwr. Chem Soc. 1817.99,1813. 14. Hamilton,D. E.: Le-, T X;Kildshl N . K horg. C k m . 1978,18,3 3 M 3 6 9 15. Hollowa%C.E.;Styne.D.V.;Vuik,C.P J.J.C,S.;Daltoto, 1979,1%130.
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