4044 for p-xylene indicate no ionization potential lower than also obtained a about 8.2 eV.39 This same much lower a2u ionization potential for benzene than that calculated by Newton, Boer, and Lipscomb,26and on that basis questioned the calculations. We must concur with the criticism and hence have, for instance, assumed the transition energy of the w j + 2b,, transition in p-xylene-TCNE (Table IX) to be 2.70 eV (this effectively pulls orbital w j down to -8.19 eV, within the region expected from ionization potential studies). The errors in the calculation, however, are rather serious and allow us to make no comments on the effects of increased methylation in molecular complexes. Three very recent works have been brought to our attention dealing with the problem studied here. 36,37,40 Mantione has calculated the inductive contribution to the dipole moments3’ and the energy of formation36of a rather large number of TCNE-aromatic complexes. The form of the perturbation theory used by her does not contain a charge-transfer term and is probably equivalent to (25), although with a different form of the (38) A . D. Baker, D. P. May, and D. W. Turner, J. Chem. SOC.,E, 22 (1968). (39) The original paper of Newton, Lipscomb, and Boer26 calculated an ionization potential for j7-xylene of 6.98 eV, better than the present calculation but still subject to the same criticism, The difference in calculations is due to a different choice of diagonal U,,matrix elements. (40) R. J. W. Le Fivre, D. V. Radford, and P. J. Stiles, J . Chem. Soc., B, 1297 (1968).
repulsion. Her method of evaluating electrostatic terms is much like ours except that she does not include a-quadrupole effects. In fact, comparable terms for benzene-TCNE in the Mantione work” are almost identical with those calculated here. Neglect of T quadrupoles, however, gives AE, = -4.80 kcal/mole for benzene-TCNE, as compared to the present AEf = -6.27 and measured AEf = -7.36. Neglect of a quadrupoles would seem important. Le FCvre, Radford, and Stiles40 have attempted to evaluate the amount of charge transfer in the ground state of molecular complexes by attributing that part of the measured dipole moment not accounted for by the induced moment as being due to charge transfer, and hence a measure of the amount of charge transfer in the ground state. Their conclusion, that van der Waals forces rather than charge-transfer interactions seem primarily responsible for the ground-state dipole moment, is not in disagreement with our results. We do feel, however, that the present state of experimental complex dipole moments does not warrant use of this procedure as an accurate measure of amount of charge transfer. Acknowledgment. We wish to thank Dr. P. O’D. Offenhartz for making several computer subroutines available to us and for invaluable assistance during the debugging of the programs. We also wish to thank Dr. D. R. Williams for a number of helpful discussions.
Theory of the Measurement of Weak Molecular Complexes. I. General Considerations David A. Deranleau Contribution from the Laboratorium f iir Molekularbiologie, Eidgenossisch Technische Hochshule, 8006 Zurich, Switzerland. Received September 30, 1968 Abstract: Some aspects of the measurement of formation constants and other constants for weak molecular complexes are considered from the standpoint of basic binding theory. The theoretical minimum errors in the formation constant k and the extinction coefficient B are discussed as a function of the saturation fraction of the most dilute component, and it is shown in agreement with previous work that the most accurate values of either parameter are obtained when the saturation fraction lies between 0.2 and 0.8. Outside this region, the determined values be-
come extremely uncertain, and in addition, insufficient data are available to adequately fit a given stoichiometric model. It is suggested that the criteria that the highest concentration of the excess component be not less than O.l/k are insufficient to assign a given stoichiometric model to the data, but apply instead to the smallest range of saturation fraction required to show that a line of finite limiting slope and/or intercept exists. To develop criteria by which a given stoichiometric model can be considered in 1 :1 correspondence with the phenomenological equation fitting the data, the principles of information theory are applied to the binding process. It is concluded that of the order of 75% of the saturation curve is required to show the correspondence between the equation of the model and the equation fitting the data. Also discussed are various plotting forms and the errors arising from various rearrangements of the basic binding equation.
T
he problem of the determination of formation constants and other constants for weak intermolecular complexes has received considerable attention during the past several years, and for good reason, since the proof of existence of such complexes depends largely on the obtention of unique values for the formation constant k and the extinction coefficient E . The most critical discussion is that given by Person, who points Journal of the American Chemical Society 1 91:15 1 Jury 16, 1969
out what some authors seem to have missed: that the most accurate values of the formation constant are obtained when the concentration of the complex is approximately the same as the free concentration of the most dilute component. Although Person limited his discussion to a consideration of the errors arising when the concentrations fell below or above certain limits, (1) W. B. Person, J . A m . Chem. SOC.,87,167 (1965).
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resulting in the inability to determine either k or E separately,2 the central idea has more far reaching consequences in terms of the assignment of an adequate model for the observed interaction. In order for limit criteria such as presented by Person to be applicable, it is of primary concern to determine that the model from which the limit criteria were derived is in fact the model correctly describing the experimental situation. A further complication arises when the limit criteria depend on a prior knowledge of k, a quantity determined in the experiment and, under certain conditions, admittedly incalculable because of the inability to separate the product kE. In addition to problems generated by insufficient data to fit the model properly, other factors have been postulated to account for observed or suspected anomalies in the reported values of k and E. These include deviations from Beer’s law, solvent interactions, the presence of complexes with other than 1 : 1 stoichime try,^-' and the use of more exact equations for numerical a n a l y s i ~ . ~It~is~probable that many of these are apparent anomalies arising strictly from the incorrect use of the simple Benesi-Hildebrand’O or Scott” equations evaluating k and E, especially when multiple equilibria might be a contributing factor. In view of the considerable diversity of interpretation of these anomalous results, it seems pertinent to reexamine some of these problems in greater detail. In the present paper, we will concern ourselves mainly with a simple model, reserving a discussion of a more complicated system for the following paper. Basic Relationships. The formation constant of a weak molecular complex is commonly determined from measurements in which a dilute component P, maintained at a fixed total concentration [Po], is “titrated” by the addition of a second component X with total concentration [Xu]. When a 1 : 1 complex PX is formed to the exclusion of all other complexes, the formation constant is defined by the usual mass law expression *r5
[PX] = k[P][X] = ~ ( [ P u ] [PX])([Xo] - [PX]) (1) and the entire experiment as [PX] varies from zero to [Po] can be conveniently described in terms of the saturation fraction of the dilute component. s
[PX]/[Po] = k[X]/(l
+ k[X])
0
