H. N. Wachter' ond V. Fried Brooklyn College of the City University of New York Brooklyn, 11210
Use of the Rose-Drago Method for Evaluation of Complex Formation Constants
The calculation of association constants and structural parameters of weak molecular complexes, a subject of increasing importance (1-3), has been examined by two authors in this Journal (4, 5). While several methods are available for obtaining this information (6-9), evidence bas been steadily mounting against the use of the BenesiHildebrand equation (9-12) for treatment of spectrophotometric data. In recent years there is a rapidly developing trend towards the use of the nmr spectrometer in the study of donor-acceptor interactions (6, 7, 12-15). Some researchers have applied the Rose-Drago method, originally derived for spectrophotometric measurements, to nmr studies, without giving a complete and simple justification for their procedure. Bolles and Drago (16) present a procedure employing coupling constants, which are often not easy to determine. Goldstein, et al., (17) give a better equation, hut the inexperienced researcher may have significant difficulty applying it. Mullins (18) may treat the problem more thoroughly, but his work is contained in a relatively inaccessible form. This paper fully derives the nmr version of the Rose-Drago method (in simple notation), and details its general application to weak 1:l molecular complexes in dilute solution. We also provide the reader with the results of a sample system for examination.
If the solvent interaction is ignored (11-12) and if all the species in the dilute solution behave ideally, then, for a 1:l donor-acceptor complex, the equilibrium constant may be expressed as
where Co and CA are the initial analytical concentrations of donor and acceptor, respectively, and CADis the equilihrium concentration of complex. Expanding eqn. (2), we obtain
When the donor molecule nmr peaks are well removed from the acceptor peaks in the mixture, one can write
where 6 is the observed nmr shift of a specific acceptor proton (or an equivalent set) in the equilibrium solution, 6~ is the shift of the free acceptor proton and 6 ~ is 0 the shift of the pure complex (not observable); the P s are the relative nuclear populations, which are defined by the expressions
The NMR Rose-Drago Equation
It is assumed that the donor (D) and acceptor (A) molecules will react in a dilute solution, to form a complex according to the reaction A + D Z A D (1) 'Undergraduate NSF Research participant from SUNY Stony Brook.
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Joornal of
Chemical Education
PA =
cA c,O + C*"
(6)
where CnO is the concentration of the uncomplexed acceptor a t equilibrium. Since the donor is of no consequence here (i.e., only the acceptor proton is under study), it is obvious that
Substitution of eqn. (7) into eqn. (4) results in 6 = PAD(~AD - 6*)
+ 6~
(8)
Solving for PAD and comparing with eqns. (5) and (61,i t follows that
Equating the last two equations for P:\n yields 6 =
+
+
c , o ( 6 ~ c - 6 ~ ) 6,
(10)
iio
If the complex stoichiometry is 1:1, i t is clear that CAnand consequently
CAO
+
CAD 6 = -(6
c,
AD
- 6), + 6,
C .
=
(11)
Solving eqn. (11) for C 4 n , we obtain Rose-Drago plot of the system p-xyiene-nitrobenzene
0 the where we define A = 6~ - 6 and l o = 6~ - 6 ~ (i.e., structural parameter). A/& is called the saturation fraction (9), and is equal to P\". Insertion of the above expression for C m into the expanded equilibrium constant, eqn. (3), gives
After rearrangement, the result is
the mean are rejected, and a new average value is computed. The results of the process are illustrated by the figure. It is found that K = 0.288 ,018 and lo= 39.0 0.8 Hz for the nitrobenzene-para-xylene system. A Benesi-Hildebrand treatment of the same complex yielded K = 0.272, A. = 38.8 Hz (21). Since the solution is dilute, the agreement is expected. Equally good results were obtained for. five other nitrobenzene complexes. For help in determining what criteria should he applied in order to establish the existence of a molecular complex, the reader should consult (1, 9, 12, 20-22).
*
*
Acknowledgment
From this relation, both K and Ao can he rapidly ohtained by the graphical method described below. Equation (13) can be easily transformed into eqn. (3) of reference (17), a more complicated hut equivalent version. As expected, the RD expression is more rigorous than that of Benesi and Hildehrand (19); the former reduces to the latter only for LC., = 0. Application of the NMR RD Equation
The calculation of K and A. is done in the following manner. For each donor concentration of a particular system, a graph of K-1 versus An is constructed. The values of CA (constant), CD, and A are known. The h e 1s formed by arbitrarily varying An between 10-100 (the limits vary depending upon the strength of the complex), and computing the corresponding value of K-' from eqn. (14). All plots for any one donor are drawn on the same set of axes. Although these lines are theoretically expected to meet a t one point, a closely packed set of intersections is obtained in practice. Thus one must take the mean of all crossings. Following the procedure of Rose and Drago (13). intersections differing by more than 2.5 standard deviations from
The financial support of the project by an NSF Summer Research Grant GY-10689 is appreciated. Literature Cited 111 Sl4ko.F.. D r a m R..J Amsr. Chem. S o e . 94.6516(19721. (21 ScofL.R..J Phvr Chm~,76.3843119711. (31 Foster. R. (Editor). "Molecular Complex&." Paul Elek Ltd.. London. 1979. I41 Christian. S..J CHEM. EDUC..IS.713 119681. (5) Ramette. R.. J.CHEM.EDUC.. 11.647 11967). . and w e , c..~ m e N r M R Spmrmrc., 4. l(1969). (61 ~ o s t e rR.. 171 Andrewe. L.. and Keeier. H.."Molecular Complexes in Oreanic Chemistn." H o l ~ d ~ n ~ D sSan y . Francisco. 1961. (81 Hanna. M..and Rapo,D..J. Amm Chem Sor. 91,2M(19721. (91 Derenleau. D., .I.A m w Chem. Snc, 91,4044 119691. (LO) Christian. S . J A m s r Chem. Soc.. 91,6861 (19721. I111 Caner, S . J C h m Sac.. lAl.404 (19681. (121 O r a m R., J. Amor Chem S o r , 91.84.90(19721. (131 k e . N . , andDraso. R . J . Am=. Chsm. S o r . R1.6138l19191. Sprinprr~Vedae.Berlin. (I41 Brkglieb. G.. "Electronen-Damtor-A~ceptor~K~mi)!eze." 1964. 1151 Trotter. P..andHsnna. M.. J. A m w C h e m Soc.. 8R.3724(10661. 1161 Ro1ler.T. andD~ago.R . J . Amer Chem Soc.. 88.3921 119661. I l i l Goldstein. M..etal., J. Chem Sac. IB). 3ZL (19701. (18) Mullins. C. 8.. P h D . Thesis. Univerrit~ofLondon. 1868. (191 Benesi. H..and Hildebrand.J..J. A m c Chem Sor 71.2703(19d3l (20) Hanna. M.. etal..J. Amsr. Chem Soc. 91.4035 119691. 1211 Wachler, H., andFlied.V., no" puhlishoddata. 1221 Penon. W . J . Amsr. C h r m Soc. 87,165 119651.
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Volume 51, Number 72,December 1974 / 799
