Sherril D. Christian'
of Oslo Blindern, Oslo, Norway Universtty
I
Rapid Graphical Method for Determining Formation Constants
In a recent article in THIS JOURNAL, Ramette ( I ) has discussed some of the numerous methods which have been uscd to infer formation constants for donor-acceptor complexes from spectrophotometric data. He has provided a useful summary of conditions for obtaining meaningful data and methods of cttlculating equilibrium constants in cascs where one or more of the extinction coefficients (absorptivities) of donor, acceptor, and complex are unknown. I n addition, he has described a general computer method for calculating equilibrium constants for 1:l complex formation reactions from spectrophotometric data, properly accounting for several common complications that have plagued workers studying molecular or ionic complexes. The most popular computational procedures for calculating formation constants for molecular complexes in nonaqueous media have been the BenesiHildebrand method and several related graphical and numerical techniques (2). Critical discussions of the range of applicability and validity of these methods have been presented by many authors (3). I n particular, Person (3) has stressed the importance of utilizing suitable ranges of solute concentrations to ensure that the equilibrium concentration of the complex is comparable to that of the least concentrated reactant. The present tendency in treating spectral data to infer equilibrium constants is to rely almost entirely on computer methods. Undoubtedly this approach provides in general the best utilization of the available data and facilitates a proper statistical analysis of experimental errors and errors in derived parameters. However, the student who embarks upon an investigation of molecular complex formation may fail to appreciate the causal relation between the data he has collected in the laboratory and the values of absorptivity and equilibrium constant printed out by the computer. He may feel more personally involved in the analysis of data if he a t least has the opportunity t o display his spectral results in a form that allows direct evaluation of the complex formation constant. This paper presents a rapid graphical method for inferring formation constants and absorptivities of 1:1 complexes in dilute solution. The procedure is applicable w d e r quite general conditions, and possesses several advantages over graphical methods described prcviously. Theory
I t is assumed that solvated molecules of the complex (AD), donor (D) and acceptor (A) ark present in solution a t concentrations governed by the equilibrium 'Permanent address: The University of Oklahoma, Norman, Oklahoma 73069.
A
+
D = AD. The solution is presumed to be dilute enough so that Beer's law and Henry's law are obeyed by all three solute species individually. For simplicity it will be assumed that A and AD molecules absorb at the wavelength selected for analysis but that D does llot absorb. It is furt,her assumed tlrat the tola1 or analytical concentration of donor (el+)' is considerably greater than the equilibrium concentration of the complex (cAn). Later it will bc shown that the graphical method developed here can be employed in slightly modified form even if the last two assumptions are not justifiable. The absorbance at the chosen wavelength, A, can be expressed in terms of the ahsorptivities and equilihrium molar concentrations of acceptor and complex and the cell length as
The absorbance that would be observed if all of the acceptor molecules were present in the uncomplexed form can be written where C A ~is the total or analytical concentration of A. However, since the total concentration of A in the presence of donor molecules is equal to the sum of the concentrations of uncomplexed and complexed acceptor, the expression for Ao may be written An = a&*
+ cm)l
(2)
Equations (1) and (2) may be combined (by subtracting eqn. (2) from eqn. (1) and dividing by eqn. (2)) to give the relations
in which the equilibrium constant, K, equals em/ , (3) reduces to ( c ~ c ~ )If. CDO >> c ~ neqn.
I t should be emphasized that eqn. (3) is exactly valid ~rovidedthe laws of dilute solution and Beer's law are obeyed, whereas eqn. (4) is only approximately correct. However, under conditions prevailing in many investigations of charge-transfer and hydrogen-bonded complexes, the donor concentration is several orders of magnitude greater than CAD, and little error is introduced by substituting enofor en. The important factor Kc$/(l Kenn) appearing in eqn. (4) represents the fraction of the acceptor which is complexed at a given formal concentration of donor. This factor depends only on the single dimensionless
+
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variable Kcn% The logarithmic form of eqn. (4) may be written
Through eqn. ( 9 , the measurable difference in absorbance, AA, is related to coo by two unknown constants, both of which can be obtained by using a curve-matching procedure. The derivative of eqn. ( 5 ) with respect to log ennmay bc written d log aA I d log c 9 = d log
(-%=)/d Km0
log eno
From eqn. (6) it can be seen that a plot of log AA versus log coo must have a slope at any given value of coo that equals the slope of a plot of log [Kcno/(l Kcon)] versus log (KcDo)corresponding to the value Kcon. Thus, if a standard plot of log [Kcoo/(l Kcoo)] versus log (KcDo)and a plot of data in the form log AA versus log cooare constructed on graphs having the same scales, the two curves will coincide when the standard curve is displaced horizontally a distance -log K and vertically a distance
+ +
log
[ ( a ~ ) A,]
The equations in (6) and the curve-matching procedure described in the next section are equally valid if the acceptor does not absorb at the selected wavelength. I n case a* = 0, eqn. (4) involves zero terms in the denominator of both sides, but an equation of the same form,
+
AA/1 = c ~ ~ a ~ n K c n o l ( Kcno) 1
+
Figure 1 shows the standard curve, KcDo/(l Kc2) plotted against Kc2 on log-log paper. Kumerical values noted along the curve indicate the percentage of acceptor molecules existing in the complexed form a t various values of Kcn! Included in the figure are points representing data reported by Iietelaar and coworkers (4) for the system dioxane (donor)-iodine (acceptor) in carbon tetrachloride at 17°.2 Each AA value is the difference between the absorbance (measured a t 451.6 m ~ of) a solution containing a given total concentration of donor and a fixed concentration of acceptor (cAO= 0.000716 A+') and the absorbance measured a t coo = 0,with the same total concentration of acceptor. The experimental values may be matched with the standard curve by a combination of vertical and horizontal displacements (but not rotation). It is convenient to trace the standard curve, and marks indicating percent complexed, onto transparent, stiff paper, and to slide the curve over the points to achieve the best fit of data. After the data have been matched, the value of K can be calculated from the value of cuo corresponding to 50y0 complexation of the accept,or; the reciprocal of this value of eno is equal to K. Using the value of K, and aA/A0 a t the 50% point, the quantity (aAD - aA)/aAmay be calculated from eqn. (4L3 The point marked (x) in Figure 1 represents the 50% point obtained by matching the standard curve with the dioxane-iodine data. The reciprocal of the 50% value
(7)
can be used in fitting data. The standard curve, log Kcno)versus log (Kcno),will match the data Kc$/(l plot if translated horizontally a distance of -log K and vertically a distance of log ( C * ~ ~ A D ) .
+
Applications
Only data corresponding to donor concentrations less than 1.1
M have been represented in t,he figure. Both Rose and Drago, and Nash ( 8 )have observed that it is not possible to fit data for this system a t donor concentrations greatly in excess of 1 M by using t,he value of the equilibrium constant obtained from data. for the dilute concentration region. a I t is interesting to note that adsorption data can be fit,ted to the Langmuir adsorption equation ( 5 ) by using the standard curve in Figure 1. Langmuir's theory of manomolecular adsorption yields the relation 0 = p/(a p) or
+
where p denotes gas pressure, 0 is the fraction of adsorbent strrface covered by adsorbed molecules, v is the volume of gas adsorbed (referred to a given pressure and temperature) and a and b are constants characteristic of the adsorbent and sdsorbent-adsorbate interaet,ions. The function ( p / a ) ! ( l p/a) appearing in the expression for s i~ equivalent. to the function plotted in Figure 1. The nwnhers indicated a t points on the standard curve now denote percentages of the adsorbent surface covered by ad8orbat.e. Once data in the form log u vesus log p have been mat,ched with the standard curve, both a and b can he determined directly from the coordinates of the 50 percent point. In general, if two measurables, y and x, and two constants, a and 8, are func.. tionally related by theequstian g(y) = ah&), the type of matching procedure described here will he applicable. The relation requires that
+
d log g(y)/d log z = d log h(pz)/d log z = d log h(gz)/d log (pz) =
Figure 1.
Stondard curve ond rpectrol doto.
714 / Journal
of Chemical Education
d8r)
and a plot of data in the form lug g(y) versus log z must match a standard plot of log h(&) verma log (Bx) by a comhinatmn of horizontal and vertical displacemenla.
of coois 1.19 l/mole, which compares favorably with the values K = 1.05 l/mole reported originally by Iietelaar, et al., 1.14 0.03 l/mole calculated from the same data by Rose and Drago, and 1.18 + 0.04 l/mole calculated by Sash. The absorptivity for the complex calculated by the technique described here is 950 hole-' cm-', compared to values of 991,979 =k 10, and 957 5 given by Iietelaar, et al., Rose and Drago, and Nash, respectively. Uncertainties in the values of K and a m , estimated from the reproducibility of the curve matching procedure, seem to be roughly equal to those reported by Rose and Drago, and Nash. I n case the donor absorbs at the selected wavelength, absorbances should be corrected by subtracting c#aDl from each meawred value. If coo is not considerably larger than CAD, n simple iteration procedure may be used to obtain a correct valne of K, provided cAOis known. (Note that the acceptor coriccntration need not be known in the procedure described above for calculating K, although CAO must be constant.) To correct for the concentration of donor that is complexed, the data (AA versus coo) are first matched with the standard curve to obtain the approximate values of percent acceptor complexed at each point. Next, the value (cao X fraction complexed) is subtracted from each value of coo to obtain an estimate of the concentration of uncomplexed donor, CD. The data are replotted on the same graph as AA versus these estimated values of CD. Somewhat larger values of the percentages complexed are obtained, and the new percentages may be used to obtain improved estimates of co. The iteration procedure is illustrated in Figure 2, in which
*
*
ues obtained in the first correction step. The t,hird set of points represents final values ohtained after two additional iterations-the second and third iterations produce nearly the same values of corrected iodine concentration, and further iterations result in no change in the locat,ion of t,he points. The solid line in Figure 2 corresponds to the best fit of the corrected data with the curve-matching procedure. A value of K = 13,150 I/mole is obtained, compared to the value 13,000 l/mole reported by Augdahl, Grundnes, and Iilaboe. Several features of the method described here recommend it for use in treating data ohtained from studies of complex cquilibria. First, the procednrc is very rapid-a set of 10 or 12 absorbance versus concentration values may bc analyacd to obtain n value of K within 2 or 3 min. Second, the observed absorbance values are plotted directly against concentration, rather than in combined forms involving, for example, products, quotient,^, or reciprocals of the measurahles. Hence, it is easy to assess the significance of deviahns of observed valucs from the theoretical curve. I t is also easy to see if Person's criterion (3-that e, and cAoshould be of comparablc magnitude-is met by a given set of data. Values of K are obtained direct,ly from the matching procedure, arid not as some function of the slope or intercept of a linear plot. A rough indication of thc uncertainty of K and aao values is provided by the agreement or lack of agreement of values obtained in successive attempt,s to fit the same set of data with the standard curve. Finally, the graphical method may serve as a convenient point of departure for more sophisticated numerical treatments of data in which the presence of additional complexes or deviat,ions from the laws of dilute solution are assumcd to p1a.y an important role. The author is iudebted to Mr. .Just Grundnes for numerous helpful discussions. He gratefully acknowledges a grant in support of this research from the Royal Norwegian Council for Scientific and Industrial Rcsearch. Enlarged copies of the standard curve in Figure 1, suitable for matching spectral data, will be provided on request. Literature Cited (1) RAMEWE, R. W., J. CHEM. EDUC., 11,647 (196i). J. H., 1.Am. Chem. Soc., (2) BENESI,H. A,, .IND HII.DERB.\ND, 71, 2703 (1949); SCOTT, R. L., REC.trav, chin&., 75, 787 (19.56); K E T E L A .J. ~~ A., A,, v.\N DE STOLPE,C., GOUDSUIT, A,, nNo Ilzcunns, W., Rec. trav. ckim., 71, 1104 110.i2); Itos~,N. J., A N D ]IRAGO, R . R., I.Am. Ch,em. Suc., 81, G., i'Elektmnen-I)~nator-Aerep6138 (19.59); BRIEGLED, tor-Kornplexe," Springer-Verlap, Berlin, 1961; N.~sa, C. P., J . P h p . Chem., 64, 850 (1960); TROTTER, P. J., .\so HANNA, M. Mr., J . Am. Chem. Soc., 88,3724 (1966); WENT-
Figure 2.
lll~lrtrotionof iteration procedure.
are plotted data of Augdahl, Grundnes, and Rlaboe (6) for the system triphenylarsinc (D)-iodine (A) in dichloromethane at 29.0°C. These data were obtained by holding the concentration of donor constant at 5.69 X 10-3 M and varying the concentration of the acceptor; hence the role of donor and acceptor are reversed from that assumed in the previous discussion. Neither the donor nor the acceptor absorbs at the wavelength chosen for measurements (316 mp). The points lying furthest to the right represent the original data, and the points immediately to the left of these are val-
woam, W. E., H m s c ~W., , A N D CHEN, E., J . Phtls. Chem. 71, 218 (1967); GRUNDNES J., A X D CHRISTIAN, 6 . D., J. Am. Chem. Soe., 90, 2239 (1968); and references cited therein. (3) ORGEI,, L. E., AND MULLII(EN,II. S., .J. Am. C h m . Sac., 7 9 , 4839 (1957); TAM RE^, & J. I. Phys. , CI~ern.,65, 654 (1961); HAMMOND, P. R., J. Chem. Sac., 479 (1964); C ~ R T E H S.,, M~RRELL, J. N., m n Rosc~,E. J., (J.Chem. Soe., 2048 N R., , J . Am. Chem. Soe., 87, 167 (1965). (196.5); P E ~ E OW. See also references (8). (4) KETELAAR, J. A. A,, Y I N DE STOLPE, C.,.\ND GERS\~.\NN, H. Ii., Ree. bae. chim., 70,499 (1951). (5) I,ANGMUIR, I., J. Am. Chem. Soc., 38,2221 (1916). E., GRUNDNES, J., A N D KLAROE,P., I m q . Ch,em., (6) AUGDAHL, 4,1475 (1965).
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