EQUILIBRIUM CONSTANTS FOR
THE
FORMATION OF WEAKCOMPLEXES
3843
iance with the range suggested by Chittenden and P-N vibration in (CH3)2NPC1z. Thus, the earlier freThomas22and R4cIvor and H ~ b l e y . Chittenden ~~ and quency range s ~ g g e s t e dfor~ the ~ ~P-N ~ ~single * ~ bond ~ ~ ~ ~ Thomasz2rejected this range on the basis that many stretching vibration is correct. compounds containing P-N band have no infrared band Acknowledgment. The authors gratefully acknowlin the region 660 to 775 cm-I which they could assign to edge the financial support given this work by the Nathis normal vibration. It is quite probable that many tional Aeronautics and Space Administration through of these compounds may have weak infrared bands Grant NGR-41-002-033. arising from the P-N motion and that the Raman effect is much better suited for the determination of a char(24) R. A. McIvor and C. E. Hubley, Can. J. Chem., 37, 869 (1959). acteristic frequency for the P-N bond. The analogy (25) J . R. Durig and J. 5. DiYorio, J . Chem. Phys., 48, 4154 (1968). drawn between P-0-C and P-X-C bands is not ex(26) J. R . Durig and J. W. Clark, ibid., 50, 107 (1969). pected to lead to a characteristic frequency for the P-N (27) B. Holmstedt and L. Larsson, Acta Chem. Scand., 5, 1179 stretch since the coupling of the P-Oand O - C m o t i o n ~ ~ ~ (1951). -~~ is significantly different from the coupling found for the (28) L. Larsson, {bid., 6, 1470 (1952).
Equilibrium Constants for the Formation of Weak Complexes by Robert L. Scott Contribution No. 291%f r o m the Department of Chemistry, University of CalZfornia, Los Angeles, California (Receaved March 26, 19YO)
SO024
Publication cost assisted by the National Science Foundation
The determination of equilibrium constants for the formation of weak complexes is examined in the light of a quasi-chemical calculation for a lattice of donor, acceptor, and inert solvent molecules. It is apparent that the equilibrium constants determined from spectroscopic or other nonthermodyriamic measurements (e.g., using the Benesi-Hildebrand equation) are “sociation” constants which yield the number of “complexes” in excess of that calculated on the basis of random probabilities.
It is well known that the concentration of weak complexes in solution cannot be determined from thermodynamic measurements alone without the aid of a detailed (and unverifiable) molecular model. However, with the aid of spectroscopic or other nonthermodynamic measurements and with the additional assumptions of (a) Beer’s law behavior for all species in the solution and (b) ideal solution behavior for all molecular species, concentrations and equilibrium constants can be deduced. For measurements of visible or ultraviolet absorption, the usual procedure is to use the BenesiHildebrand equation1 or modifications Similar procedures have been used with nmr spectra in s~lution.~~’ Some of the equilibrium constants so deduced are extremely small (e.g., K , = 0.009 1. mol-’ or K , = 0.07 for benzene carbon tetrachloride in n-hexane solution*), and one may reasonably question their physical significance. In particular, it has been pointed outg-ll that in a plausible quasi-lattice model of the
+
solution, random contacts between donor and acceptor molecules lead to K , = z, the number of nearest neighbors, or KO’sof the order of 0.30 1. mol-l, larger than many of those reported. (1) H. A. Benesi and J. H. Hildebrand, J . Amer. Chem. SOC.,71,2703 (1949). (2) J. A. A. Ketelaar, C. van der Stolpe, A. Goudsmit, and W. Dzcubas, Red. Trav. Chim. Pays-Bas, 71, 1104 (1952). (3) R. L. Scott, ibid., 75, 787 (1956). (4) R. S. Drago and N. J. Rose, J. Amer. Chem. Soc., 81,6138 (1959). (5) G . Briegleb, “Elektronen-Donator-Acceptor Komplexe,” SpringerVerlag, West Berlin, 1961. (6) M . W. Hanna and A. L. Ashbaugh, J. Phys. Chem., 68, 811 (1964). (7)R. Foster and C. A. Fyfe, Trans. Faraday SOC.,61, 1626 (1965); 62, 1400 (1966). (8) R. Anderson and J. M. Prausnitz, J . Chem. Phys., 39, 1225 (1963). (9) R. L. Scott, Proceedings of the Third International Conference on Coordination Compounds, Amsterdam, 1955, p 345. (10) J. E. Prue, J . Chem. SOC.,7534 (1965). (11) J. E. Prue, Chemical Society Symposium on the Physical Chemistry of Weak Complexes, Exeter, England, April 1967. T h e Journal of Physical Chemistry, Vol. Y6,No. 26, 1971
ROBERTL, SCOTT
3844 Of course, many complexes require a special orientation of donor and acceptor with respect to each other, a condition which only a small fraction of the random contacts would satisfy. This would seem to resolve the problem, except that Orgel and i\ilulliken’2have shown that if there are several 1 : 1 complexes, each with equilibrium constant K i and absorptivity ci, the “observed” K and e are the appropriate averages ZiKi and ZiKiei/ ZiKi, Thus the contact “complexes,” even if they do not absorb light ( e = 0) and contribute nothing to Eobsd, would seem to contribute substantially to K. A simple quasi-chemical calculation for a lattice of donor, acceptor, and inert solvent molecules (D, A, and S) can help to illuminate this problem. We assume a lattice of coordination number z, in which the site fractions of donor, acceptor, and solvent are X D , XA, X S , respectively, and further that all pair interaction energies are the same except for a particular one of the x “faces” of D with a similar particular “face” of A, which differs from the others by an energy w and has an absorptivity E. (This assumption is, of course, only for simplification; in any actual system, each interaction will have a somewhat different energy. However, even for “weak” complexes, it seems reasonablephysically and geometrically-to assume that one interaction is somewhat stronger than all the others.) I n a lattice of N sites, the number of random contact pairs DA will be N X D X A X , of which (in the random mixing approximation) only a fraction 1/x2 will have the “complex” orientation. If we introduce a factor t oto allow for the energetic preference for complex formation, the number of “complexes” will be N X D X A V C / Z , in excess of the random value (VC = 1). Solution of the quasi-chemical equations for this model yields (see Appendix) an equation for ?IC XAXD(K’
- 1)vc’ - [(x,
+
XD)(K
- 1)
+
X]XZVC
112x2
=
0
+ (1)
with w the usual interchange energy. where K = Where X D >> X A = 0, an essential approximation in the Benesi-Hildebrand theory, eq 1 simplifies to 17c =
HZ
(x -
1)xD
+
The number of complexes is then
N c = NXAXDW~
x
NHXAXD
_____(x - 1)sD
+z
(2)
and the absorption A per length 1 is (3)
In the Benesi-Hildebrand treatment when XAZ/Ais ~ plotted us. l / x D one interprets the intercept as 1 / and the slope as 1/Ke where K is the mole fraction equilibrium constant. In the equivalent Scott modification The Journal of Physical Chemistry, Vol. 76, No. $6,1971
XAXDZ/A is plotted against ~ / X Dto yield an intercept 1/Ks and a slope 1 / ~ . If we so transform eq 3
-- -z
X A X ~
+
(H
A
-
~)XD
HE
(4)
It is evident from eq 4 that the usual procedure yields,
Thus KobsdEobsd = KE, but Kobsd < K while Eobsd > e. Since l/z is the entropy prefactor, it is evident that the “observed” constant is just the excess over the random. It is easy to generalize the treatment to include several 1:1 complexes, for which case the earlier Orgel-Mulliken formulation is replaced by Kobsd = C ( K i i
- Ki,random)
(7)
CKdet Eobsd
i
C(Ki i
- Ki,random)
(8)
where, in the lattice of coordination number z, EiKi,randorn iS just 2 . Consequently, it is evident that very small values of Kobsd can be physically significant if the uncertainties arising from experimental errors13can be minimized. Substantially these same equations were proposed earlier by Carter, Murrell, and Rosch,14who analyzed the competition between solvent and donor molecules for the acceptor in the reaction
A*Sz
+D
----t
D*A.S,-,
+ yS
Their paper has not received the attention which it deserves because it seems to require a specific interaction (“solvation”) between solvent and acceptor. Actually the solvent molecule occupies space (e.g., a lattice site); it must be included in the bookkeeping because it is there. Some years ago, Guggenheim15 suggested a distinction between a degree of “association” and a degree of 1L sociation,” the latter being the excess over that based upon probabilities in a random mixture. The BenesiHildebrand Kobsd is clearly a “sociation” constant; only when one is suficiently confident of the validity of a theoretical model for calculating Krandom(or alternatively the “true” absorptivity of the complex) can one hope to evaluate the “association constant” K. For strong complexes ( K >> Krandom) the relative am(12) L. E.Orgel and R. S. Mulliken, J. Amer. Chem. Soc., 79, 4839 (1957). (13) W.B.Person, ibid., 87, 167 (1965). (14) 8. Carter, J. N. Murrell, and E. J. Rosch, J. Chem. Soc., 2048 (1965). (15) E.A. Guggenheim, Trans. Faraday Soc., 56, 1159 (1960).
EQUILIBRIUM CONSTANTS FOR THE FORMATION OF WEAKCOMPLEXES
3845
biguity is small, but for weak complexes it can be great. It seems probable that experimental measurements never yield “association constants” but only “sociation” constants. It is interesting to consider the evaluation of the heat of formation AHf of the complex, Le., the parameter w in the quasi-chemical treatment. Two alternative methods are available. (a) The intercept obtained when XAXDIIAis plotted against X D is XIHE, or 1/Kobsd eobsd in the Benesi-Hildebrand terminology. If one assumes16that E is independent of temperature, then when the logarithm of this intercept is plotted against 1/T, the slope is w/k and can be interpreted, presumably correctly, as AHr/R. (b) On the other hand, if Kobsd is regarded as a true equilibrium constant, a plot of In Kobsd vs. 1/T yields a more complicated curve rather than a straight line, and the slope at a particular temperature is
___ A H o b s d - -d In Kobsd - -d In (e-”’kT - 11 R d(l/T) d(l/T)
When is large by comparison with unity, this leads to w/k,but for weak complexes, this AHobsdd l be larger than the “true” AHr. For example, for AHf = -2RT ( - 5 kJ mol-’ at room temperature, the order of magnitude of AHf for the benzene-iodinc complex), e2/(e2 - 1) = 1.16, leading to a A H o b s d 16% high. For still weaker complexes, this procedure would yield even less reliable enthalpies of formation. Comparisons with experimental data from the literature can be made (and were made by Carter, et a l l 4 ) , but for this highly oversimplified model-a lattice of molecules of exactly equal size, all interactions but one equal- they arc not likely t o be very meaningful. The purpose of this paper is t o clarify the problem with a simple mode1.l’ Acknowledgments. I wish to thank Dr. J. Prue and Professor J. S. Rowlinson for helpful discussions of this problem. This work was supported in part by a grant from the National Science Foundation,
Appendix : The Quasi-Chemical Calculation We assume a lattice of coordination number x and N sites, occupied by NXDdonor molecules, N X Aacceptor molecules, and N X Ssolvent molecules. All energies of interaction are equal cxccpt that between one unique “face” of D (designated as D’) and one unique “face” of A (designated as A’); this D’A’ interaction is the “complex” (designated as C) and differs from the others by an energy w and has an absorptivity E. The N z / 2 nearwt-neighbor pairs are of 15 types: SS, SD, SD’, SA, SA’, DD, DD’, DA, DA’, D’D’, D‘A, D’A’ =
(16) T. M . Cromwell and R. L. Scott, J . Amer. Chem.SOC., 20,4090 (1948).
(17) After this manuscript was substantially complete, I learned from Professor N. S. Bayliss that he had proposed a similar idea some years ago in a paper which has never been published.
The Journal of Physical Chemistry, Vol. 76,No. 86, 1971
