interpretive experiment1
edited by FRANKDEHAAN
Occidental College LOS Angeles. CA 90041
The Rigorous Evaluation of Spectrophotometric Data to Obtain an Equilibrium Constant John R. Long and Russell S. Drago The School of Chemical Sciences, University of Illinois, Urbana, IL 61801 Many chemical systems are represented by the general equation which is characterized by the equilibrium constant expression in eqn. (2),
K=- (IAB
(2)
~ACB
where a*, aB, and a m are the activities of species A, B, and AB. In dilute solutions the true thermodynamic equilibrium constant is approximated by Kc, the concentration equilibrium constant:
Using eqn. 3, most undergraduates are able t o determine the equilibrium concentrations of the different chemical species, given the equilibrium constant and the initial conditions. Most undergraduates also know how to determine the equilibrium constant given sufficient information to enable them to calculate the equilibrium concentration of A, B, and AB. Most, however, would not know how to determine the equilibrium constant and estimate the error in it from spectrophotometric data that contains experimental errors. In fact, many research chemists when confronted with this same ~roblem.introduce unnecessary approximations to simplify the mathematical analvses that often introduce errors into the values calculated. his "dry-lab" experiment describes a method that may be used to determine the "best-fit" value of the 1:1equilibrium constant to spectrophotometric data. Sufficient detail is given so that the reader will have an appreciation for proper experimental design and will be able to write a computer program to analyze the experimental data. The Experiment
Experimental Design Proper design of the experiment is important and makes the difference between good and poor data. There are a number of considerations. For ease of discussion, though, i t shall be assumed that species B is being added to species A, and that B does not absorb in the spectral region being studied, but the spectra of A and AB overlap. The first consideration is the choice of the initial concentration (i.e., formal or total concentration) of species A. The concentration range must be such that Beer's law is obeyed. Also, the initial concentration of A, denoted AT, needs to be sufficiently large so that the absorbance changes that result from the conversion of A to AB are reasonably large (i.e., >0.3 absorbance units). Second, the range of total concentrations of B, denoted BT,
needs to be varied in such n way that the percent complexation ranges from low to high values. For more details see references 1 &d 2. Third, it is convenient to prepare a series of solutions in which the total concentration of species A is the same for each solution, letting only BT v q . This will allow one to investigate for the presence of isosbestic points, vide infra (3). Analysis of Data . The analysis of the data should provide three items: a "hest-fit" value for the eauilibrium constant. the error limits on K, and a knowledge ofthe uniqueness of the fit. Unfortunatelv. manv of the techniaues commonlv used to determine the e&ilibr& constant from spectrophotometric data do not orovide one or more of the above. For examde. the Hill equation, which utilizes a logarithmic display of the data, has been shown to artificially smooth the experimental uncertainties ( 4 , 5 ) ,making the data appear better than i t really is (5).Other methods, such as the Benesi-Hildebrand treatment (6),are valid only under certain limiting conditions. This eliminates the utilization of otherwise useful and needed data resulting in a poor estimate of the true value of K and of the corresponding error in the value of K. Data Analysis
Due t o advances in computer technology it is possible to upgrade the older methods of data analysis, and use an exact expression for the equilibrium constant over.the entire range of concentration (8, 9). T o accomplish this the method of nonlinear least-squares shall be used. The method of nonlinear least-squares is based on the assumption that the best estimate for the values of the parameters (e.g., K ) are those values that minimize the sum of the squared deviations between the observed experimental data points and that calculated by the .orooosed . model. For the case of the s~ectroohotometricexperiment we want to minimize the sum of the squared deviations between the observed absorbances and that calculated by the equilibrium model. Mathematically, we want the values of the parameters that minimize ~ 2 :
Where N is the number of datapoints, Absio"is the observed is the absorbance for the ith spectral curve, and analogous calculated quantity. T o determine the Absica'c values, we shall consider the case for which species A and AB have overlapping absorbances and the extinction coefficient of AB cannot be measured directly. If A and AB are the only absorbing species in the region under study and they both obey Beer's law, the total absorbance (Abs) a t any particular wavelength for a 1cm pathlength is given by Volume 59
Number 12 December 1982
1037
where r~ is the molar absorhtivity of species A, and (AH is the molar absorbtivity of the complex AD. With the appropriate substitutions ( I ) , the above equation can he reduced to where Abso equals EAAT,the absorbance that would result if there were no B present and is ohtained from the spectra of free A. By defining AA = Abs - Abso, and AE CAB - €A,and substituting into eqn. ( 6 ) ,we get Equation (7) is a very useful form as can he seen if equation (4) is rewritten by adding and subtracting Abso t o each term
Furthermore, solving eqn. (3)for [AB], we obtain ( I )
(AT+BT+~)-~(AT+BT+$ 1
[AB] =
1 2
-~ATBT (10)
2
+
where the substitutions AT = [A] [AB] a n d B = ~ [B] + [AB] have been made. This defines all the terms necessary to calculate u2 excent for K and AE.' he;; are many ways todetermine the values of K and Ar that minimize .. ~ ~.1 1. 0Weshall 1. describe herea method that combines linear least-squares and an iterative-minimum seeking routine. TIIstart, we make a guess o i the value of K; the valueof At that would minimize .x2. for this initial estimate of K , is given by
Equation (11)is obtained by setting the partial derivative of x2with respect to Ac equal to zero, and then solving for AE. From the initial guess of K we calculate the best value of AE using eqn. (ll), and the corresponding x2is then calculated using eqn. (9).The value for K is then varied, and the procedure repeated until the smallest value for x2is obtained. As the y 2 values for a euessed K eet smaller. the amount that K is v&ed for the next iteration is made s;ccessively smaller, and the iteration is stopped when the change applied to K produces a change in the value of x2 smaller than a predetermined value. The details of this are eiven with the comnuter listing, or can be found in any texthoik of numerical ankysis that describes "root" problems. The statistical parameters were calculated in accordance with the procedure presented in reference (11): conditional standard deviation in K = marginal standard deviation in K =
::-
Dnd]L'2 ~ v d c n p , ",m,
(13)
where
If AT is the same for all spectral curves then Abso is constant; if AT is not the same for all spectral curves then the calculation of via
eqn. (10) needs to take this into account. 1038
Journal of Chemical Education
By substituting in the values of AA, AT, BT for aparticular spectral curve and wavelength, AEcan be varied, *lo%, from its best fit value to calculate the value of K-1 as a function of Ar. A plot of these K-', Ar data pairs produces a nearly linear curve for each spectral curve. Plotting such a curve for each spectral curve on the same plot results in a plot of several nearly linear curves, which, if it were not for experimental errors, would all intersect at one point, the unique solution to the simultaneous equations. The utility of the plot, however, is not in the determination of K-' and Ar, hut rather in determining how well the data defines the values of K and AE. In general K-' and AEare well defined if the curves intersect in a narrow region and exhibit widely varying slopes in the region of the intersections. If instead the plot consists of curves that are nearly parallel and closely spaced in the region of the hest fit value of K-', and AE,a s m d standard deviation would result, hut the value for K and Ar would not be well defined (12).The nearly parallel n w e s result from poor experimental design; the range of percent complexation was probably too small (12).We have found (12)empirically that the ratio of marginal to conditional standard deviations provides a good indication of whether the slopes of the various lines in the above plot will vary substantially, and hence, whether the values of K and AEare well defined or not. If with an acceDtable standard deviation the ratio is greater than 12, the v&es calculated for K and AE should he considered as not beine meaningtul. If the rntk~is less than 4, and the standard deviations are small, then the resultn are meaningful. If the ratlo
(12)
[($: 2) (Dl*.
'
and N equals the number of data points. The statistical parameters given in eqns. (12) and (13)give an indication of how well the model fits the data, hut does not necessarily indicate how well the data defines the values of K and AE.For this reason a graphical method was developed to determine how "well defined" the value of K is. I t has been shown previously ( I ) that eqns. ( 3 )and (7) can be combined to yield
x2
Spectrophotomebic titration of Co(LCF3) w i a Concentrations are listed in the Table.
1-Melm in
toluene at 19.4OC.
Injection
(molesL-')
e, (molesL-'I
(X = 525 nm)
AA
free A
0.000212 0.000212 0.000212 0.000212 0.000212 0.000212
0 0.000515 0.00103 0.00206 0.0041 1 0.00923
0.859 0.745 0.676 0.609 0.544 0.494
0.1 14 0.183 0.250 0.315 0.365
1 2 3 4 5
A,
AbSwbance
-
is between 4 and 12 and results should he viewed a s being tentative. Interpretation of Spectra In this experiment you will determine the equilihrium constant and Ar for the binding of t h e Lewis hase l-methylimidazole t o t h e Lewis acid cohalt(II)4-trifluoromethyl-ophenylene-4,6 methoxysalicylideniminate, Co(LCF3), i n toluene solution. T h e spectra of the "titration" of Co(LCF3) with l-methylimidazole are shown (13) in t h e figure. T h e spectrum of t h e pure acid is labeled with an "0." T h e formal concentrations of Co(LCF3) and l-methylimidazole corresnondine t o t h e snectral curves are eiven in the Table. You siould 11: fnmll~ar'w~th the m a t e n d in reference (.?a,, sections 4-6 and &7. and reference (.I .) before contintrina. Perform the analysis a s follows: (1) Determine the positions of the isosbestic points.
Answer: 602,590,582, and 477 nm. (2) What is the significance of the presence of an isosbestic nnint? F " . . . . .
of a third absorbing species,the presence of isosbesticpoints at other wavelengths, and the lack of any crossing of the spectral curves in this region indicates that the extinction coefficientsare comparable hut unequal in this region. More specifically,CA > at wavelengths just above and below 560 nm. (5) At what wavelength should the absorhancesbe read to analp for the equilibrium constant and why? Answer: At about 525 nm. Since the ahsorhance changes are largest here, the most accnrate data will he obtained here. (6) Measure Abso and the Abs; values from the figure at X = 525 nm, then calculate the AAbsiobs values. Answer: See the Table. (7) Determine the best fit values of K and Ar Answer: K = 8.3 X 10%.Ac = 1.94 X lo3 (8) Determine the standid deviations. Answer:C.S.D. ofK = 13.2,M.S.D. = 25.8 (9) Make a plot of K-' versus Ac. What do the plot and the M.S.D.1C.S.D. ratios indicate about the determined values for K and At? (10) In a calorimetric experiment, the heat evolved, H', upon mixing an acid and a hase depend upon the concentrations of the reagents, K, and AHo. (a) Describe the analogous quantities in the calorimetric and spedrophotometrie systems. Answer: AT a H', c A H e , K d = K , (h) Derive an equation for calculating K and AHofrom H'measurements. Answer: See reference (8).
Answer: Isosbestic points arise when two or more absorbing species have equal molar extinction coefficientsat the same &welength, and when the sum of the concentrations of the ahove species is held constant. Since the chances of mare than two species having equal extinction coefficients at the same wavelength is small, the presence of an isosbestic point is usually indicative of the presence of only two absorbing spe~ea.~Thus, from the fw, T h e figure is available in a 8% X 11 copy, upon request. we would conclude that there are only two absorbing species, Calculator keystroke (TI-59) listings are a ~ a i l a h l e . ~ Co(LCF3)and Co(LCF+l-MeIm (i.e., there isno Co(LCFz).(l-MeIm)a in equilibrium with the above two species, and l-MeIm is colorless in this spectral region). (3) What is the significance of the presence of more than one Literature CHed isnshestic mint?. I l l Roe. N. L a n d Draso.R. S.. J . Amer Chsm. Soc,81.6138 (1959). Aniwcr: There is no particular significanc~. Thir unly indi21 nemn1em.D A . J >me, Cnem IK. ¶ I . I U 1 11M8,. cates that [he two ahsorhing species haw equal extinction , I , is, ucago. H S "Ph,a,rslMcthodr,nChcm,nry." W R SsundcnCo.. I'hilada~hla. PA11%7:,. Ibl hlwer. K C and Dradl.R 3 ,lnord ( ' h e m . 11.?010t197L coefficientsat numerous wavelengths. t4l Andcrun.S.and H'rlcl T. . n m r h e m . 1.1912 ,19651 (4) In the figure, what does the lack of an isosbestic point at X = lil Cu.dry H..and I h # .R S .I Anwr Chem S.I . 95, MA5 1197.11 and reference. 560 nm indicate? ".... Answer: Although near misses in the formation of an isosbestic 161 Beneai. H. A,. and Hildehrand. J. H..J A m . C k m . Soc., 71,2703 (1949) point frequently indicate experimentalerrors or the presence . J
~
~
.....
-
Occasionally an isosbestic point will occur when more than three species are present. The interested reader is referred to reference
therein. (10) Johnson, L., and Riesa, R., "Numerid Analysis,"Addlaon-Wdw Pub., Reading, M A ,
3(@.
TI49 listing available through Texas Instrument PPX-59 catalog, No. 418103G. Lubbock. TX.
Volume 59
Number 12
December 1982
1039
