557
Communications to the Editor ion. They are, however, appreciably larger than the solvation numbers obtained from nmr studies and comparisons of the results obtained by these methods must be made with the knowledge that they measure different quantities, although each can provide useful information about the behavior of closely related solventsllJ2 and ions.11+13
Nonweighted linear fitb
1.156 rt 0.028
5745
119
1.234 f 0.033 1.231 f 0.045
5556 f 93 5559 89
Weighted linear
References a n d Notes
fitb
(1) M. Della Monicaand L. Senatore, J. Phys. Chem., 74, 205 (1970). (2) H. Hartley and H. R. Raikes, Trans. Faraday Soc., 23, 393 (1927). (3) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," Butterworths, London, 1968, p 44. (4) B. E. Douglas and D. H. McDaniel, "Concepts and Models of Inorganic Chemistry" Blaisdell, New York, N. Y., 1965, p 126. (5) Reference 3, p 12. (6) Reference 3, p 125. (7) J. E. Huheey, "Inorganic Chemistry, Principles of Structure and Reactivity," Harper and Row, New York, N. Y., 1972, p 73. (8) J. H. Swinehart and H. Taube, J. Chern. Phys., 37, 1579 (1962). (9) Reference 3, p 62. (IO) N. A. Matwiyoff and H. Taube, J. Amer. Chem. Soc., 90, 2796 (1988). ( 1 1) A. Fratiello in "Inorganic Reaction Mechanisms," J. 0.Edwards, Ed., Interscience, New York, N. Y., 1972, p 57 f f . (12) A . P. Zipp, J. Phys. Chem., 77, 718 (1973). (13) P. Bruno, M. Della Monica, and E. Righetti, J. Phys. Chem., 77, 1258 (1973).
Department of Chemistry State University College Cortland, New York 13045
TABLE I: Comparison of Least-Squares Fits of Spectral D a t a For Et&-& in Heptane at 150a
Arden P. Zipp
Received October 23, 7973
Some Comments on the Calculation of Equilibrium Constants and Extinction Coefficients for 1:I Complexes Publication costs assisted by the National Science Foundation
Sir: Several authors have reported methods for using nonlinear least-squares analysis to infer formation constants, K, and optical extinction coefficients, ~ D A ,for a molecular complex, DA, existing in solution in equilibrium with free donor and acceptor mo1ecules.l Standard errors in the parameters have also been computed from the shape of the error surface (the sum of squares of absorbance deviations as a function of K and eDA) in the vicinity of the optimum ~ investigators have argued values of K and c D A . ~ C .Various that numerical optimization methods are superior to graphical or linear least-squares methods based on the Benesi-Hildebrand (BH) equation2 and related linear relations. What seems not to have been properly appreciated by many workers in the charge-transfer field is that use of any of the linear forms will yield answers virtually identical with those inferred from the more complicated nonlinear analyses if data are correctly weighted. For example, a recent paper in this journal3 includes a discussion of the relative merit of various linear forms of the BH equation (in terms of differences in correlation coefficients of the least-squares fits) and a comparison of two nonlinear fitting methods with the linear forms. In our opinion, this type of discussion is unproductive, since all of the various linear and nonlinear forms based upon the same physical
Nonlinear fit
a Data taken from the Ph.D. Dissertation of J. D. Childs, The University of Oklahoma, 1971; absorbances were measured at 250 nm. Corrections have been made for the absorbance of free donor and acceptor in all three fitting methods. Benesi-Hildebrand equation2 was used.
and mathematical model must give consistent results if proper statistical methods are employed. To illustrate this point, we consider using the BH equation in the form
where A is the absorbance per unit path length, and where we assume that the total donor concentration ([D]) is so much greater than the total acceptor concentration ([A]) that the concentration of free donor (CD) is practically equal to [D]. We also ignore the effects of absorbance of donor and acceptor at the wavelength chosen for analysis, although these absorbances can be taken into account with little additional difficulty. The principle of weighted linear least squares4 requires that we seek a minimum in the function
where the summation extends over all sets of measured values of At, [A],, and [D]l. Standard methods are readily applied to obtain least-squares values of K and cDA and estimates of their standard deviations, uK and utDA,,provided the W[ values can be chosen properly. If the weights are chosen injudiciously (for example, if W, is by default taken to be unity for all points) unreliable answers may be obtained with the BH equation; criticism in the literature of the uncritical use of the BH equation in this way is quite a p p r ~ p r i a t e .However, ~ the calculation of weights is straightforward if reasonable estimates of the uncertainties in absorbances and concentrations can be made. Decisions regarding these errors should not be made thoughtlessly since the calculated parameters can depend strongly on the weights, especially when the data do not permit a precise determination of K and eDA. Let us consider the case in which it is reasonable to assume that the absorbances are subject to equal absolute error at all points and that the donor and acceptor concentrations are known exactly. (These are the assumptions usually made in nonlinear analyses of spectral data.) The dependent variable (Yt = [A]t/A!) in the linear fit of Yt us. 1/[DILwill then have the weight
where U A is the (constant) error in absorbance.6 Using this weighting scheme, best values of K and tDA, and estimates of standard deviations in C D * - ~and (Ke~*)-l result directly from the standard linear least-squares analysis. Simple propagation of errors formulas than yield UK and DA
a
The Journal of Physical Chemistry, Vol. 78, No. 5, 7974
550
Table I compares results of nonweighted and weighted linear least-squares treatments of a set of spectral data. The weighted linear analysis gives results which are quite similar to those obtained by using the nonlinear leastsquares method of Grundnes and Christian.lc The latter method uses Sillen's criteria' for determining UK and ccDA from the shape of the error surface near K o p t l m u m , 6DAoptimum. Because of the near equivalence of the weighted linear and nonlinear methods, it is a matter of preference which analysis to use. However, the simplicity of the weighted linear method should make it the preferable technique in many cases. Obviously, any of the linear forms of the BH-type should give the same values of K and CDA and their standard deviations. The weighted linear least-squares method can also be applied with iteration if the complex concentration is not small compared to [ D ] . In this case, the term ~/KcDA[D] is replaced by l / K c D A C D , and an approximate value of K is used to estimate the amount of donor in the complexed form. Several passes through the linear least-squares program will ordinarily lead to convergent values of K and cDA, which agree with K o P t l m u m and ~DAOpt1"'"m inferred from the nonlinear method.
Acknowledgments. Acknowledgment is made to the National Science Foundation (Grant No. G P - 3 3 5 1 9 X ) for
The Journal of Physical Chemistry, Vol. 78, No. 5, 1974
Communications to t h e Editor partial support of this work. Edwin H. Lane wishes to express his appreciation to the National Science Foundation for a graduate fellowship.
References and Notes (1) (a) K. Conrow, G. D. Johnson, and R. E. Bowen, J. Amer. Chem. SOC., 86, 1025 (1964); (b) W. C. Coburn, Jr., and E. Grunwald, ibid., 80, 1318, 1322 (1958); (c) J. Grundnes and S. D. Christian,ibid., 90, 2239 (1968); (d) D. R. Rosseinsky and H. Kellawi, J. Chem. SOC.A, 1207 (1969). (2) H. A. Benesi and J. H. Hildebrand, J. Amer. Chem. SOC., 71, 2703 (1949). (3) P. G. Farrell and P. N. NgB, J. Phys. Chem., 77, 2545 (1973). (4) W. E. Deming. "Statistical Adjustment of Data," Wiley, New York, N. Y., 1943. (5) R. L. Scott, Recl. Trav. Chim. Pays-Bas, 75, 787 (1956); D. A. Deranleau, J. Amer. Chem. SOC.,91,4044 (1969). (6) In order to allow for variable absorbance and concentration errors, W Lis related to the error in the ith residual (res,) by '
where
r .
1
1 res, = I H J , - L - A, KcIIACDI, en,\
(7) L. G ,Sillen, Acta Chem. Scand., 18, 1085 (1964), and references therein.
Department of Chemistry The University of Oklahoma Norman, Oklahoma 73069
Sherril D. Christian* Edwin H. Lane Frank Garland
Received November 6, 7973
