THE JOURNAL OF
PHYSICAL CHEMISTRY Reaistered in
U.S. Patent Oflce @ Copyright, 1970, by the American Chemkal
Society
VOLUME 74, NUMBER 23 NOVEMBER 12, 1970
Linear Regression Models in the Study of Charge-Transfer Complexation by Robert A. LaBudde and Milton Tamres Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48104 (Received November 4, 1969)
Linear regression models are analyzed in the determination of equilibrium constants, K , and extinction coefficients, E , of charge-transfer complexes. Criteria for separating K and e from the Ke product in weak complexes are refined. For strong complexes where linear approximations lead to large error, equations are developed which permit reasonable corrections to be made from a knowledge of the experimental conditions.
Introduction I n recent years there has been a growing number of spectrophotometric studies of charge-transfer (CT) complexes in the vapor phase.1-6 The experimental techniques generally are more involved than are those in solution and, consequently, the determination of the thermodynamic and spectral characteristics of the CT complexes are subject to greater uncertainty. First, concentrations may not be as precise because very small amounts of reagents are involved in many cases, and they may have to be determined by means other than weighing. Second, because of the interest in learning about solvent effects,' many of the vapor studies to date have focused on the relatively nonvolatile acceptor molecules such as i ~ d i n e l J *and ~ ~tetra*~ cyanoethylene4 because of the many solution data already available for these acceptors. This has necessitated the vapor studies to be made a t higher temperatures, a condition less favorable for complexation. Third, while pressure cells have been used for some contact CT observations in the vapor p h a ~ e they , ~ ~have ~ not yet been employed to study more stable CT complexes. Until they are, the maximum concentration of the system in vapor studies will remain less than 10-1M , and more often closer to 1 0 - 2 M . Since vapor phase results are more directly related to theory,I0 they are more likely to be quoted, which is all the more reason for keeping the above limitations in mind. Person" has focused on the difficulty in obtaining a meaningful separation of the equilibrium con-
stant, K , and the extinction coefficient, e, when the extent of complexation is low. More recently, Deranleau12 has stated that a reasonably low per cent error in K is possible only if the component of smaller concentration is between -20 and 80% complexed, a situation which surely is not met in many of the vapor as well as some solution studies. I n this paper we reexamine these concepts and, assuming the favorable situation of only 1 : l complexation, a criterion is de(1) (a) F.T. Lang and R. L. Strong, J . Amer. Chem. Soc., 87, 2345 (1965); (b) E.I. Ginns and R. L. Strong, J . Phys. Chem., 71, 3059 (1967). (2) (a) J. Prochorow and A. Tramer, J . Chem. Phys., 44, 4545 (1966); (b) J. Prochorow, ibid., 43, 3394 (1965). (3) (a) J. M . Goodenow and M . Tamres, ibid., 43, 3393 (1965); (b) M . Tamres and J. M. Goodenow, J . Phys. Chem., 71, 1983 (1967); (c) M . Tamres. W. K. Duerksen, and J. M. Goodenow. ibid.. 72. 9e6 (1968); (d) W. K. Duerksen and M. Tamres, J . Amer. Chem: Soc., 90, 1379 (1968). (4) (a) M. Kroll and M. L. Ginter, J . Phys. Chem., 69, 3671 (1965); (b) M . Kroll, J . Amer. Chem. Sac., 90, 1097 (1968). (5) (a) S. D. Christian and J . Grundnes, Nature, 214, 1111 (1967); (b) J. Grundnes and S. D. Christian, J . Amer. Chem. SOC.,90, 2239 (1968);Acta Chem. Scand., 22, 1702 (1968). (6) A. A. Passchier and N. W. Gregory, J. Phys. Chem., 72, 2697 (1968). (7) R. 8. Mulliken, Proceedings of the Robert A. Welch Foundation Conferences on Chemical Research XI, Radiation and the Structure of Matter, Dec 4-6, 1967. (8) D.F. Evans, J . Chem. Sac., 1735 (1960). (9) E.C. Lim and V. L. Kowalski, J . Chem. Phys., 36, 1729 (1962). (10) R. S.Mulliken, J . Amer. Chem. Soc., 74, 811 (1952). (11) W.B.Person, ibid., 87, 167 (1965). (12) D.A. Deranleau, ibe'd., 91, 4044 (1969).
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ROBERTA. LABUDDE AND MILTON TAMRES
veloped separating K and E from the KEproduct which is an extension of Person’s. For stronger complexes, the concentrations in the vapor phase and solution are comparable, and more such studies are to be anticipated. I n comparing the results in the two phases, it should be recalled that analysis of most of the earlier solution work was based on a linear regression model for a 1: 1 complex. It is now known that this model leads to e r r ~ r . ~ I~n Jthe ~ second part of the paper, equations are given which permit the effects of a linear approximation to be estimated from the experimental conditions, and these are analyzed using known, synthetically generated data. 1:1 Complexes. The vast majority of CT complexes studied have been considered to be of 1:1 composition. Recently, the question has been raised whether only 1:1 complex formation can be assumed from studies involving a limited c ~ n c e n t r a t i o n . ~ Evidence ~~’~ has been given that even hexamethylbenzene-tetracyanoethylene forms both 1 : 1 and 2: 1 complexes,1ewith the former being dominant a t the lower concentrations. For weak complexes or a low extent of complexation, the 1: 1 species is favored and, for the purposes of this paper, it is assumed to be the only significant interaction. Then
K =
(Do -
C C)(Ao - C)
=
EA[AO- C]
+ ED[DO- C] + EGG‘
(2)
Let
OD’
= E’
- ~ A A-O EDDO = €0 - EA - ED OD
=
e’C
(3)
Combining (3) with expression 1 gives the wellknown relation
Since this relation is symmetric in A0 and Do, it is expedient analytically to make the following transformation~~’
p
=
2 2) (or
The Journal of Physical Chemistry, Vol. 74, No, $3, 1070
The assignment
4 a2@ y = -AoDo - OD’ (1 @)20D’
+
sometimes will be referred to for convenience. Weak Complexes. A linear approximation of relation (4) [or (6)] is commonly used in the statistical estimation of stability constants from spectrophotometric data, i.e.
which is of the form
Y
= a
+ b2a
(9)
from whence
and
For very weak complexes, 1/K is large; the linear term in (8) vanishes in relation to the constant term. Since K and E’ can be determined separately only to the significance of the linearity of (9), some criterion governing the reliability of (11) is important. Person1’ suggested a criterion of
Do ’v 2a 2 0.1/K (or 2aK 1 0.1)
Then (2) becomes
OD’
and
(1)
where DO and A0 are the initial concentrations of donor and acceptor, respectively, and C is the equilibrium concentration of the complex. If it is further assumed that the only absorbing species are A, D, and C in the region studied spectrophotometrically, with molar extinction coefficients EA, ED, and ec, respectively, the Beer-Lambert relation for the complexation is
OD
where p is chosen to be greater than one. Substitution in (1) and (4) gives, respectively
(12)
This is based on a standard deviation ( u ) of 1% and a confidence level of 99% (or 317). Person extended this to 1Ou by considering that very few data points are available. Since the confidence level varies with the number of data points, one refinement would be to take this into account.18 Also, the error limits in a linear regression depend on the range covered in the x-coordinate. Thus (13) N.J. Rose and R. S.Drago, J . Am. Chem. SOC.,81,6138 (1959). (14) M.Tamres, J . Phus. Chem., 65, 654 (1961). (15) D.A. Deranleau, J . Amer. Chem. Soc., 91, 4050 (1969). (16) P. J. Trotter and D. A. Yphantis, J . Phys. Chem., 74, 1399 (1970). (17) For the case when DO>> Ao, Le., the treatment of BenesiHildebrand, J . Amer. Chem. SOC.,71, 2703 (1949),then a ~li D0/2. (18) E.L. Heric, ibid., 73, 3496 (1969).
CHARGE-TRANSFER COMPLEXATION the range of the variable 201 should be considered and not just its magnitude. Several statistical estimators can be used to test if the data are linearly dependent on a as given by (9). One choice might be the correlation coefficient r; an equivalent choice might be the student-t value for the slope b. The evaluation of K and E‘ is not considered acceptable if the statistic r is not significantly greater than zero at the lOO(1 - y)% level, where y is the maximum acceptable probability that the observed dependence of Y on 2a is spurious. I n this paper y will be taken as 0.05 ( i e . , the 95% confidence level). The test can be applied in reverse fashion by asking, “For which minimum value of K is significance still possible at the 95% level?” Suppose the variable Y has 95% confidence intervals Y AY. Then let
+
AY 17’-
Y
which is the convenient measure of the precision of the data. If, as often is the case in spectrophotometric studies the errors in Aol Do, and cell length can be negle~ted,’~ then q = AOD’/OD’, which is the same as Deranleau’s’2 q = AAbs/Abs. This is equated in his treatment with As/s, where s is the fraction complexed of the component present in smaller concentration. For a fixed As, q varies over the entire range of complexation. It is this variation which leads to the conclusion that for As = 0.01 the error in E rises rapidly when complexation is less than 20%. Similarly, errors in K rise rapidly outside the range 20-8001, complexation, as is evidenced from eq l when Do>> A,; ie., large errors in K occur when C is very small or very large, and the smallest error occurs when C = A o / 2 . However, this applies only to the situation where the same experimental procedure is used over the entire range of complexation. It should be possible to have a small 17 for a very low extent of complexation by using a more sensitive method to detect differences. Spectrophotometrically, this is achieved by varying the cell length, so that q remains essentially constant over the entire range of measurements. If i l % error limits on Y are taken over the full range of 2a and these limits correspond to a 95% confidence level, then q = 0.01.20 It follows from the definition of the correlation coefficient, if L is the range of the variable 201, only those values of K which satisfy
can have regressions significant at the lOO(1 - y)% level, where r, is the appropriate critical value of the correlation constant for N data points and t, is the appropriate value of the (one-sided) student-t statistic. Equality in (13) occurs when half of the data
4011 points are at each end point of the range L and corresponds to the minimum value of K L for which lOO(1 y)% significance may still be attained. Similarly, the inequality
governs those K L which do not have lOO(1 - y)% significance. Equality in (14) occurs when the least favorable selection of N data points in the range L is made and corresponds t o the maximum value of K L for which a significance of lOO(1 - y)% may still not be obtained in the regression. Since the Person criterion (12) governs those experiments which must satisfy the conditions for acceptance; the refinement of his criterion is therefore (14). Values of (KL),i, and (KL)max from (13) and (14), respectively, for different sample sizes for y = 0.05 and q = 0.01 are given in Table I. For N = 4, the Table I : h1inimum and Maximum Critical Values of [ K L ]a t the 95% Significance Level“ N
4 5 6 7 8 9 10 11 12 13 14 15 20 100 m
[KLImin x 10%
IICLlmax x 102
2.06 1.36 1.07 0,901 0.793 0.716 0.658 0.611 0.573 0.541 0.514 0.491 0.410 0.167 0.000
2.92 2.15 1.85 1.69 1.59 1.52 1.47 1.43 1.40 1.38 1.36 1.35 1.29 1.18 1.16
0 For 9 = 0.01 ; this corresponds to approximately i :1% error in Y . For an error of . t p % in Y , the entries in this table should be multiplied by p .
value would be close to that of 0.1 given by Person if a 99% confidence level had been used. For illustration, suppose a ten-point sample is taken with an experimental range L = 0.01, as may be the case in many vapor phase studies, and separation a t the 95% significance level is desired. If K < 0.658, even the most optimal design of the experiment will not achieve separation (19) For inclusion of the various sources of error see W. Wentworth, W. Hirsch, and E. Chen, J . Phys. Chem., 7 1 , 218 (1967); also P. R. Hammond, J . Chem. Soc., 479 (1964). (20) If the error limits correspond to a different confidence level, this value should be multiplied by the ratio of the length of a 95% confidence interval to that of the given confidence in terms of standard deviations, e.g., if i1% error limits are for the 99% confidence level, 7 j = 2 u / 3 u X 0.01 = 0.067.
The Journal of Ph&al
Chemistry, Vol. ‘7q3No. $3, 1970
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ROBERT A. LABUDDE AND MILTON TAMRES
with the desired level of significance. If K 2 1.47,95% significance will be obtained no matter how the points are chosen within the range. Conversely, if it is known after the fact that significance a t the 95% level is not obtained, then it can be stated with regard to the true value of K that it must be less than 1.47. A point which should be kept in mind is that the use of the values in Table I implies the choice of 7 = 0.01, I n agreement with Deranleau,12 a larger r makes it more difficult to determine small K. For some of the vapor phase studies, 7 may be appreciably larger. Similar inequalities hold for the error in the slope b.
Therefore, the greater the range L of the points, the smaller the error in the slope. Also, if additional points are chosen near the extremes, the error in the slope will decrease as N-'Ia; if near the center, more points may not affect the error in the slope. Strong Complexes. The linear approximation (8) to determine K and E is known to produce error for strong however, to estimate the c ~ m p l e x e s . ~It * ~is~possible, ~ effect of the approximation. Equation 6 may be rewritten 1 2a -c y = -j + KE E E'
-
Assuming for the moment that there is no error in the observations, a linear regression of (15) will then correspond to a linear regression of
C =d
+ 2ea
(16)
over the point set, and
i.e., the last term is an "average" correction for the term omitted from the full eq 4. Rearranging K* E*+ODIK* K E* -Iv
Therefore, the fractional error in K can be approximated quite simply from usually readily available data by using the relation K* - K rv OD' -K* fractional error in K = (22) K - E* This expression is in a more convenient form for spectrophotometric studies than a similar one given by Deranleau.12t21 It differs from Deranleau's with regard to sign; his fractional error is negative whereas that in (22) is positive since all terms on the right-hand side are positive. The difference is due to retaining 2a = Do A , in this paper, whereas Deranleau used 2a 'v D0.12,17 If, as is common, C increases with a (ie., e > 0 ) )then (22) will overestimate the fractional error in K . Improved estimates of K and e' can be obtained by estimating d and e directly from the experimental data. This can be done by approximating C from the linear terms of its Taylor series expansion (after solving for C in eq 5). Letting a, and P, be the mean values of a and p, respectively, it can be shown that the following expressions ford and e are obtained for the two cases. Case 1. The species of lower concentration is held constant (a common procedure for CT studies in solution).
+
d= (17)
2 am2Pm Sm(1 Pm)'
+
[
aml X
am
+ %*
If K* and E* are the apparent values obtained from relations in (1 1), then
or
Case 2 . /3 varies independently of a (ie., A . and Do vary independently).
If p is a monotonically increasing function of
a,then d
-
and e are both positive, although generally quite small. Therefore E*will be larger than E' by the small factor 1/(1 - e), Also, K* will be larger than the true constant by the factor (1 - e)/(l - Kd), and for large K it is possible for K* to be negative. For small e, C does not vary much, and
OD' C%-. E*
where
E*
(20)
E'
OD' is the average optical density.
(21) Deranleau's expression can be rewritten as
K'
-K
-E--
Then
The Journal of Physical Chemistry, Vol. 74, No. 28, 1070
K
OD'
e*Do
4013
CHARGE-TRANSFER COMPLEXATION
where, in both cases
8,
= (a,
-t
&) x
Some general points can be made for cases 1 and 2. Clearly e > 0 for both cases. When case 2 holds, d < 0 and e is maximized. When case 1 holds, e is minimized; and for Pm moderately large and a, > 1/2K*, d > 0 and e vanishes. It should be noted that only estimates a, and @, are needed t o calculate d and e, and hence t o calculate corrected values of K and E'.
K =
n .-
1-e
+ K*d'
*
E'
=
e*(1 - e)
(28)
If desired, an iterative procedure can be used for further refinement. These relations were examined using synthetic data. Seventeen representative samples were chosen for varying magnitudes K and E'. The value of K , E', ampm,and OD' for each sample is listed in Table 11. Effort was made to keep the optical densities of the samples within typical experimental range. All samples correspond t o sets of eight concentrations, except for number 2 which consists of only six concentrations. I n order to isolate the systematic trend with the value of K , samples number 1-16 were chosen t o obey case 1 ; sample number 17 was included as an example of case 2. Values of K* and E * were calculated from eq 8, and d and e were calculated from a regression of eq 16. These may be compared with d and e (called d* and e* in Table 11) calculated from (23) and (24) for numbers 1-16, and from (25) and (26) for number 17. The values of d* and e* listed are the final values of four successive iterations of the equations, and the final corrected values of K* from (28) are given for comparison. The error in K* was usually of the order of that predicted by (22). As can be seen from samples 1-14, the systematic error in K* increases rapidly as K increases, the error being less than 1% for K < 100. The error in E* is small when case 1 holds. The calculated values of d* and e* from case 1 give good estimates of d and e. In samples 5 and 15 and in 8 and 16, the only difference within the pairs is in the magnitude of am and A smaller error for E* in samples 15 and 16 is obtained, in accordance with eq 24. I n samples 13 and 14, for very large K , d becomes greater than 1/K
V d
I
@me
T h e Journal of Physical Chemistry, Vol. 74, No. I S , 1070
ALANR. h/IONAHAN AND DANIEL F. BLOSSEY
4014 and K* becomes negative. The corrected values of K* for samples 1-16 are quite good. The much poorer result for sample 17 using eq 25 and 26 is due to the large variations in a and P which makes the Taylor series approximation more in error.
Acknowledgment. This work was supported in part by a grant from the National Science Foundation, NSF GP-6429. One of the authors (R. A,' L.) expresses his appreciation for two SSF Undergraduate Research Participation Awards.
The Aggregation of Arylazonaphthols. I. Dimerization of Bonadur Red in Aqueous and Methanolic Systems by Alan R. Monahan* and Daniel F. Blossey Research Laboratories, Xerox Corporation, Xerox Square, Rochester, New Y O T ~l4%03 (Received April 17, 1970)
The absorption spectra of the sodium salt of Bonadur Red in aqueous and methanolic solutions were analyzed in terms of monomer-dimer equilibria in the concentration range to mol 1.-'. The dimerization constants, K,, = cd/cm2, are (8.00 2.73) X IOa and (1.25 0.01) X lo4 1. mol-' for the methanol and water systems, respectively. The absorption spectra of the pure monomer and pure dimer in the two solvents were calculated. The splitting and relative strengths of the dimer bands were used along with the strength of the monomer band to calculate the relative orientations of the two dye molecules in the dimer. Assuming a parallel plane dimer configuration,the twist angle between the dye molecules was calculated to be 43 f 10" for the dye-methanol system and 63 1" for the dye-water system. The spacing between the molecules was calculated to be 4.6 =k 0.2 %I in methanol and 4.3 cfr 0.0 A in water. In addition, for the dye-methanol solution, the dimer splitting was 2700 200 cm-l with the midpoint between the H- and J-dimer bands redshifted by 200 100 cm-l from the monomer peak. For the dye-water solution, the splitting was 2200 100 cm-l with the midpoint red-shifted by 800 i 100 cm-l from the monomer peak. These findings are consistent with a greater degree of overlap of the electron wave functions between the molecules in the dimer in water than in methanol. The general similaritiesbetween the dimerizationin aqueous and methanolic systems led to the suggestion that hydrogen bonding may be responsible for aggregation. This was partially supported by the exothermic heats of dimerization of 5.7 1.0 kcal (mol dimer)-' for both solutions.
*
*
*
* *
*
*
Introduction It is well documented in the chemical literature that the dimerization of dyes and formation of higher order aggregates in solution and the solid state is responsible for dramatic color changes,' enhanced photosensitizing properties,2 etc. Azo compounds were chosen for this investigation because they form the largest class of organic pigments, and their detailed spectroscopy has not been studied in solvents capable of promoting aggregation. Non-Beer's law behavior for the solution spectra of organic dyes is generally attributed to aggregation of the dye molecules. Aggregation of dye molecules, as measured by deviations from ideality, has been found in limited classes of solvents for concentration ranges typically of the order of to M . T o form the simplest aggregate, the dimer, the dye-dye interaction must be strong enough to overcome any other forces which would favor solvation OF the monomer. Thus nonpolar solvents are presumably to be preferred over The Journal of Physical Chemistry, Vol. 7 4 , No. 03, 1970
polar solvents because, the lower the dielectric constant of the solvent, the less the screening of the dye-dye interaction by the ~ o l v e n t . ~I~n *actuality, water forms the system from which the largest body of information has been accumulated concerning dye aggregation processes. Because water does not have as low a dielectric constant as nonpolar solvents but still is a host for dye aggregations, it has been said to be a n o m o l ~ u s . ~ Aggregation has been observed in alcoholic systems,6
* To whom correspondence should be addressed. (1) (a) E.S. Emerson, M. A. Conlin, A. E. Rosenoff, K. S. Norland, H. Rodriguez, D. Chin, and G. R. Bird, J. P h y s . Chem., 71, 2396 (1967): (b) W.West and S. Pearce, ibid., 69, 1894 (1965). (2) G.R. Bird, B. Zuckerman, and A. E. Ames, Photochem. Photobiol., 8, 393 (1968). (3) K. Sauer, J. R. Lindsay Smith, and A. J. Schulta, J . Amer. Chem. SOC.,88,2681 (1966). (4) R. B. McKay and P. J. Hilson, Trans. Faraday SOC.,63, 777 (1967). (6) M. J. Blandamer, M. C. R. Symons, and M. J. Wooten, ibid., 63, 2337 (1967).
