Photochemistry and Photobiology, 1999, 69(4): 443-447
Determination of Equilibrium Constants for Weakly Bound Chargetransfer Complexes Ramona Zaini, Andrew C. Orcutt and Bradley R. Arnold* Department of Chemistry and Biochemistry, University of Maryland, Baltimore County, Baltimore, MD, USA Received 4 September 1998; accepted 18 January 1999
ABSTRACT
reactive partner acts as an electron acceptor while the other as an electron donor. These complexes are typically characterized by the appearance of a weak electronic absorption that is not present in the spectra of the individual reaction partners. Mulliken theory explains the appearance of this new absorption band as an interaction between the highest occupied molecular orbital (HOMO) of the donor and the lowest unoccupied molecular orbital (LUMO) of the acceptor (9-12). Many relationships showing the interdependence of ionization potential, electron affinity, redox potentials and calculated HOMO-LUMO energies to the formation constants and absorption spectra have been published (13-20). The accurately determined equilibrium constants for complex formation (KCT) are required to probe fully the applicability of the Mulliken model to these complexes. A quick glance at the literature leads to Benesi-Hildebrand (BH) analysis and the seminal report of the determination of the equilibrium constant for the formation of an iodine/benzene CT complex (21). The determination was based on a simple one-to-one donor/acceptor complex being formed according to a model such as shown in Eq. 1.
The evaluation of the equilibrium constants for chargetransfer complex formation has been of interest for five decades. During this time, absorption spectroscopy using Benesi-Hildebrand, or related methods, has been used to obtain the equilibrium constants. These methods require relatively high concentrations of donor or acceptor to be present in solution when weakly bound complexes are studied, conditions that lead to the formation of higher order complexes and inconsistent determinations of these constants. A new method is presented that allows weakly bound charge-transfercomplexes to be studied under low concentration conditions and the equilibrium constants to be determined accurately for the first time. Using this method, the equilibrium constant for the formation of 1,2,4,5-tetracyanobenzene/pentamethylbenzenechargetransfer complex was found to be KCT = 6.8 0.3 M-l with an extinction coefficient at 400 nm of eCT= 150 30 cm-I M - l .
*
INTRODUCTION
A+B=AB.
For many types of reaction the formation of a discreet complex ensues immediately prior to commencement of the reaction. These intimate complexes, or contact pairs, are distinguishable from the individual reaction partners by relatively weak noncovalent interactions, yet they play an important role in many chemical systems. One example is the folding and assembly of macromolecular systems where the tertiary structures adopted depend largely on these rather minor interactions. The measurement of the thermodynamics of these weak interactions has been an area of interest for the last decade as a first step toward understanding, and potentially controlling, the problem of macromolecular assembly (1-8). An important class of these interactions leads to the formation of a charge-transfer (CT)t complex in which one
The equilibrium expression for this model is
Assuming the initial concentration of B, [B], is present in vast excess of A, and considering the mass balance expressions for A and B it can be shown that [B] [B], under all conditions. This must be true because the complex concentration, [AB], can never be greater than [A],. If it is further assumed that only the complex absorbs at the monitoring wavelength, optical density (OD) = eCTb[AB]and that eCT does not change with changing concentration of B, it is possible to arrive at the BH Eq. 3.
-
1/OD = l/(bK~c-[A]o[B]o)
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*To whom correspondence should be addressed at: Department of Chemistry and Biochemistry, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA. Fax: 410-455-2608; e-mail:
[email protected] tAbhrevations: BH, Benesi-Hildebrand analysis; CT, charge transfer: HOMO, highest occupied molecular orbital; LUMO, lowest unoccupied molecular orbital; OD, optical density; PMB. pentamethylbenzene: SVD, singular value decomposition; TCNB, 1,2,4,5-tetracyanobenzene. 0 1999 American Society for Photobiology 003 1-8655/99
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(1)
+
l/(b€,--[AIo).
(3)
A plot of 1/OD versus l/[B], should be linear with a slope of l/bKe,,[A], and y intercept of l/br,,[A],. It is assumed that b = 1 cm for the remainder of this report. The value of K,, is determined by the ratio of the y intercept to the slope or obtained graphically from the x intercept. The component to be held constant, A, must be lower in concentration than B and the ratio [B]d[A], is usually at least 10-20. There are several relationships that have been derived
443
444 Ramona Zaini eta/. from Eq. 3, such as the Scott equation (22)or the FosterHammick-Wardley equation ( 2 3 ) .These relationships do not differ from BH in substance and correctly weighted data will yield the same answer in each case. An alternative method (24)developed by Rose and Drago uses assumed values of KcT and eCT to define the OD measured at one wavelength for a solution of A and B. Subsequent measurements are taken for different concentrations of A and/or B and KcT and ecT are determined by solving a series of linear equations. The accuracy of the determination depends on a judicious choice of the concentrations of A and B. With increased access to computer power has come the ability to study entire sets of spectra simultaneously. These sets of spectra constitute a matrix of data with, in this case, OD measured as a function of wavelength and concentration of a titrant. The data matrix is broken down using singular value decomposition CSVD) into two sets of eigenvectors, one set corresponding to the observed spectral changes the other set to the evolution of each spectral change vector as a function of concentration (25-27).Careful examination of the cigenvectors allows the determination of the minimum number of species required to reconstruct the data accurately. This method has tremendous advantages over single wavelength determinations because averaging occurs over many wavelengths and no assumptions about the relative concentrations of components are needed. Unfortunately. a considerable variation in the measured values of the equilibrium constants using these procedures occurs. particularly for weakly bound complexes (3,13,15,2830).In spite of what must be perceived as a problem, reports using these methods continue to appear. This report compares the results obtained using BH and global analysis methods to determine K,, and the complete absorption spectrum for 1,2,4,5-tetracyanobenzene-pentamethylbenzene (TCNB/PMB) CT complex. Potential errors in these analyses are highlighted and a better method to determine K,.., and eCT is presented.
MATERIALS AND METHODS Marrriuls. The sample o f TCNB. purchased from Aldrich Chemical Company, was puritied by passing it twice through silica gel with dichloromethane a s the elution solvent followed by recrystallization twice from chloroform. Pentaniethylbenzene was purchased from Aldrich Chemical Company and purified by passing it through aluminum oxide with dichloroethane as the elution solvent followed by recrystallization from ethanol. In both cases. baseline absorbance was monitored and purification was continued until no further improvements were observed. Mr4wrl.s. The absorption spectra of the solutions were measured at 75°C with a Becknian spectrophotometer model DU-640. The temperature was kept constant bq circulating temperature-controlled water through the cell compartment using a VWR Scientific circulator. Solutions containing acceptor and donor were prepared immediately prior to measurement. For BH analysis the concentration of the acceptor was typically 5 .? 10 M . A 0.5 M stock solution of PMB was prepared and added in successive volumes to the TC N B stock solution ( 2 mL.) in a 1 cm quartz cell. In some cases a 10 cm quartz cell was used. All absorbances were corrected for the effect of dilution. When appropriate. individual samples were made by dilution of I0 M stock solutions of acceptor and donor. The global analysis program used was Specfit version 2.10, ohrained from Spectrum Software Associates, Chapel Hill, NC.
0.9 0.8
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0.3
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1
02 01
00 295
345
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Wavelength (nm)
Figure 1. Plot of the spectra obtained by adding P M B (0.235-0.910 M ) to a solution of TCNB (3.2 X 10 M ) in 1.2-dichloroethane at 25°C. Th e spectra are corrected for dilution. Inset: Benesi-Hildebrand plots obtained using data between 350 and 445 nm. The value of the apparcnt KCT = 0.31 M-I.
RESULTS AND DISCUSSION BH analysis The spectra obtained for the formation of TCNBPMB CT complex are shown in Fig. 1. The BH plots of 1/OD versus l/[PMB] obtained from these spectra are shown in the inset. These plots were obtained by taking data at several wavelengths between 345 nm and 445 nm across the CT absorption band. It would be a clear indication that a problem existed if the plots were curved or if different x intercepts were observed with each wavelength. Because no systematic deviations occur it is usually assumed that KCT and eCTare correctly determined. In this case KCT = 0.31 M and eCT = 3540 cm-’ M-I at 400 nm. Can these values be believed? A critical consideration in these measurements is the concentration range tested. Percent complexation (S) is defined as S = 100[AB]/[N], where [N], is the concentration of the limiting reagent. It is generally accepted that most of the complexation curve is required to measure accurately the equilibrium constant, and the majority of data points must fall in the 20-80% complexation range (28-30). This must be true because inherent in the assumption that B is present in large excess of A, and therefore that [B] - [B],, is that [A] is nor in large excess with respect to LAB]. For weakly bound complexes this usually requires high concentrations of either acceptor or donor to be present in solution. Unfortunately, it is not possible to work at such high concentrations and avoid 2:1 or 1 : 2 complexes. This is the conundrum researchers must face when studying weakly bound complexes. When the concentrations are increased to allow measurements at high S the model used may not be valid because 1:2 complex formation becomes significant. But at low concentrations, S may be too small and the analysis itself may not be valid. At very low total concentration (C, = [A], + [B],) it is almost as reasonable to assume [A] [A], than it is to assume [B] [B], regardless of their relative values. Now Eq. 4 holds and only the product KeCTcan be obtained. The intercept is zero.
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OD,drnl= KEC.rLAIo[Blo.
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(4)
In this case a single parameter, the slope, is used to describe
Photochemistry and Photobiology, 1999, 69(4) 445
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20
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300
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420
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Wavelength (nm)
Figure 2. Spectra of TCNB and TCNBPMB complex obtained from global analysis from data shown in Fig 1 and assuming K,, = 0 33 M (qee text)
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the data completely. Because this equation must hold at infinite dilution we call this the ideal OD and experiments carried out such that Eq. 4 is valid are under ideal conditions. Notice that ideal refers to ideal solution behavior and not to ideal case scenario for determining KCT. On the bright side, in order to determine the product accurately, K k , , minimal curvature in an OD versus [B], plot is required, a situation that is usually the case for weakly bound complexes. The only thing we can accurately determine is the product Kern. The best way to do this is calculate AOD/A[B], for each data point ( i e . obtain the slope between each OD point and the origin when plotted against the concentration). Plotting these values versus [B], and extrapolating to infinite dilution allows the best estimate of Kk- to be determined. For the TCNBPMB complex in 12-dichloroethane at 25"C, analysis of all of the available data, including data obtained with very low total concentrations with a 10 cm path length cell, gives an average value of KeCT= 1024 -C 20 cm-I M-?.
Global analysis If the data shown in Fig. 1 are submitted to SVD using a global analysis routine, two major and one minor spectral eigenvectors are obtained. The two major vectors account for greater than 90% of the observed spectral variation while the minor vector contributes an additional -5%. The major vectors appear to correspond to depletion of TCNB and formation of CT complex, while the minor vector is associated with a shift in the TCNB spectrum. This shift is due to changing solvent polarity with added PMB because titrating hexane into a sample of TCNB in 1,2-dichloroethane produces a single vector that is identical to the minor vector. A least-squares fit of the data, assuming a 1:l complexation model, as in Eq. 1, results in reasonable agreement with the BH estimated K,, and eCTvalues, specifically KrT = 0.33 M-' and eCr = 3000 cm-1 M-I at 400 nm and 25°C in 1,2dichloroethane. In addition, the component spectra are also predicted and are shown in Fig. 2. It is clear from the predicted spectrum of the complex that the values of KCT and eCTobtained from both the BH and global analysis techiques must be wrong. The appearance of negative extinction coefficients is due to underestimation of KCT.If the value of KCT is fixed such that a smooth absorption spectrum of the complex is obtained, free from negative absorbances, the minimum allowed value of KCTis determined, K M I N= 3.5
A
0
c
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IPMBl (M)
Figure 3. Plot of the apparent KCT obtained as a function of the maximum concentration of PMB used The K,, values were obtained using global analysis with TCNB ( 5 X 10 M ) and the PMB concentration varied between 0 001 and 0 74 M in 1.3-dichloroethane at 25°C
M-I. This represents the smallest possible value that K,, can assume that also predicts reasonable spectra for the components. Obviously there is a serious problem with the BH and global analysis determinations. Presumably the 1:2 complex absorption spectrum is a linear combination of the absorption spectrum of the free TCNB and the 1:l complex because two major colored species are required to fit the data, and only two if the shift is ignored. Reports of 1:2 CT complex spectra have appeared and they are usually similar in shape to the related 1:1 complexes although generally the 1:2 complexes have higher extinction coefficients (3 1.32). In light of this finding, increasing the complexity of the model to allow the formation of 1:2 complexes is futile. The second equilibrium constant KCTI:?and the spectrum of the 1:2 complex cannot be found because the problem is now underdetermined. As expected, wildly variable results are obtained by assuming the more sophisticated model and no useful values can be determined. In a similar data set the donor concentration was varied between and lo-' M. Submitting these data to global analysis allows the values of KCT to be determined as above. In each subsequent determination, the spectrum corresponding to the highest donor concentration was deleted and the remaining data reanalyzed to obtain a different estimate of KCT.The values of KCT determined are plotted versus the highest concentration of donor used in each determination in Fig. 3. The plot shows distinct variation in KrT with the total concentration of donor used. At lower concentrations, as the number of experimental spectra becomes too small, the KCT value becomes unstable, in this determination increasing. A plateau occurs at KCT = 6 M-l for concentrations of M . At higher concentrations of donor the KCTvalue decreases remarkably until it agrees with the value determined previously using BH analysis. The plateau value of KCTis much greater than the value obtained using BH analysis and is consistent with the KMINvalue as described above. One might assume that the plateau value is in fact the true KCTalthough the determination is subject to considerable error. What is evident from this plot is that higher order complexes must influence the determinations at concentrations higher than about lo-' M . To check this finding, the stoichometry of the complex
446 Ramona Zaini et a/. 0.03
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Figure 4. Job's plot for TCNB/PMB complex formation observed at 400 nm The total concentration used was 1 05 X 10 M in 1.2dichloroethane at 25°C
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y' (M')
Figure 5. Quadratic plot of the OD of TCNBPMB complex as a function of the square of the concentration. The concentrations of TCNB and PMB were equal and the data were recorded at 400 nm in 1.2-dichloroethane at 25°C. Fitted line (solid) is obtained according to Eq. 5 with KCT = 6.8 M and €,-- = 150 cm-l M - ' (assuming a 1 cm cell). The ideal line (broken) is also included for KeCT = 1024 cm- I M (see text).
'
was determined using the method of constant variations, or Job's analysis ( 3 3 ) . The solubility of TCNB in 1.2-dichloroethane is limited to -2-3 X lo-' M, therefore the maximum concentration used for these studies was 1 X lo-' M . The Job's plot obtained for TCNBPMB is shown in Fig. 4. This plot is completely symmetric about the maximum of X = 0.5, a strong indication that at total concentrations of lo-' M and lower only 1:l complexes are important. Unfortunately, due to the limited solubility of TCNB, nothing can be said about higher total concentrations.
Rationale behind a new method The problem still remains of how to work at low concentrations and yet have high percent complexation but not allow 1:2 complexes to be formed competitively. At first this looks like an impossible puzzle. Its solution depends on the realization that it is the absolute deviation from ideal behavior that is required and not the relative deviation as has been assumed. The maximum absolute deviation from ideal behavior will occur under conditions where the highest possible concentration of complex is formed for a given total concentration. According to the Job's plot shown in Fig. 4 this occurs at equal mole fraction of acceptor and donor. This is contrasted with BH conditions where large differences between the mole fractions of acceptor and donor are required. This suggests that when equal concentrations of acceptor and donor are used, [A],, = [B],) = Y, the absolute deviation from ideal behavior will be maximized. Under these conditions the predicted OD is given by Eq. 5 .
A plot of the OD at 400 nm of TCNBPMB solutions against Y' for concentration varying between lo-' and M is shown in Fig. 5. As was the case with the BH plots, curvature in this plot is needed for the two parameters, KcT and eCT,to be determined independently. As can be seen, only the most careful measurements will result in well-defined curvature in these plots. At first it might appear that the desired quantities cannot be determined from these experiments, and indeed from this plot alone they cannot. As outlined above, BH-type plots for
weakly bound systems can be used to determine the product KeCT accurately. This product defines the ideal behavior line that is included on Fig. 5. Because KcT and eCTare both constants, the product KeCT must also be a constant for the system under study. The data are then required to conform to the product KECT = 1024 cm-I M *, as was determined previously. With this additional constraint KCTcan now be accurately determined. Once K,, is obtained eCT is also fixed. Analysis of the data in Fig. 5 leads to KcT = 6.8 2 0.3 M - ' and eCT = 150 2 30 cm-I M - ' , and the predicted line based on these values is included on the figure. These values are in reasonable agreement with those obtained from the plateau region using global analysis and are consistent with the limiting KMTNvalue as defined earlier. However, they are a far cry from the BH estimates of K,., = 0.32 M-I and eCT= 3120 cm-' M - I . If the BH estimates are used in Eq. 5, a line similar to the ideal OD is obtained, leaving no doubt the BH estimates are not correct. Why are these estimates so different? Benesi-Hildebrand analysis requires (1) the largest possible complexation range to be sampled and (2) the system under study must behave according to a model such as Eq. 1. The conditions needed to obtain high percent complexation for weakly bound complexes also favor 1:2 complex formation. This invalidates the determination because the model used in the BH analysis is no longer applicable. At low concentrations of acceptor and donor the analysis itself is not applicable and poor determinations result. Now that KCT and eCTfor the 1:1 complex have been determined, it is appropriate to comment on what is required of the 1:2 complex in terms of KCT1:2 and E ~ if the ~ 1:l ~ values are correct. The shapes of the 1:2 complex absorption spectra have been reported to be similar to those of the related 1:1 complexes but the extinction coefficients in the CT values are band are 2-6 times larger than the 1:1. The KCTI:> reported to be between 2 and 10 times smaller than the KCT values of the 1:l complexes. For Fig. 1 values for KCTI12 = 2 M and eCT= 1000 cm-I M-' are required to reproduce the data. These values are subject to considerable error and in light of the fact that extremely high concentrations are used such that deviations from ideal behavior, specific sol-
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Photochemistry and Photobiology, 1999, 69(4) 447 vation (13) and contact CT (13) may all influence the determinations, it is not reasonable to place much weight on such determinations. Nevertheless, these values are in reasonable agreement with what would be expected based on previous reports for 1:2 complexes (31,32).
CONCLUSION Straight BH plots and consistent values of KCTobtained at different wavelengths across the CT band are not sufficient tests of the accuracy of the determined constants. Reasonable component spectra over the entire absorption range, including overlapping regions, are also required. This finding casts considerable doubt on many of the published equilibrium constants for weakly bound CT complexes. On the other hand, the method described above allows KC- for weak complexes to be determined at relatively low concentrations where 1:2 complex formation is not significant. Percent complexation is not the major consideration in these measurements although detectable deviation from ideal behavior is required. Data obtained using BH conditions in conjunction with data obtained at equal concentrations of acceptor and donor should be used to determine the independent values of KCTand eCT.These conditions combine to allow the most accurate determination of very small equilibrium constants. The application of these measurements to other types of complexes is currently being investigated. Acknowledgements-The support of this work by University of Maryland, Baltimore County and the American Cancer Society, Maryland Division is gratefully acknowledged.
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