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A Straightforward Method to Determine Equilibrium Constants from Spectrophotometric Data
J. Chem. Educ. 2000.77:927. Downloaded from pubs.acs.org by IOWA STATE UNIV on 01/20/19. For personal use only.
E. Keszei,* M. G. Takács, and B. Vizkeleti Department of Physical Chemistry, Eötvös Loránd University, H-1518 Budapest 112, Hungary; *
[email protected] Spectrophotometry provides useful information concerning equilibrium concentrations of reaction mixtures, and it has been used to infer equilibrium constants for many decades. As the classical methodology was developed on the basis of linearized model functions describing equilibrium, the linearized models keep surviving in textbooks, despite the availability of relatively easy-to-use software packages that allow the direct treatment of the essentially nonlinear physical models. In this paper and the related online publication we try to demonstrate that the nonlinear evaluation of spectrophotometric equilibrium measurements has become simple enough to use even in undergraduate physical chemistry courses. Learning to use a software package capable of doing nonlinear parameter estimation is not substantially more difficult than learning to use an appropriate simplification to linearize the underlying physicochemical model. The Model Function Let us consider here the prototype donor–acceptor reaction that has been treated in this Journal before, using the same notation as in the previous papers (1, 2): D+X
DX
(1)
The equilibrium constant of this reaction can be written as
K=
f DX ⋅ DX D ⋅ X fD ⋅ fX
(2)
where brackets stand for equilibrium concentrations, and the symbols ƒ indicate the corresponding activity coefficients. As activity coefficients are usually determined independently of the spectrophotometric experiment, it is sufficient to determine a “simplified version” of K, the equilibrium quotient:
Q=
DX D ⋅ X
λ
A = εXλc X + εDXλ cDX + εXλ + εDλ – εDXλ ⋅ 2 1 + Q c X ± 1 + 2Q c X + 2c DX + Q c X2 2Q
λ
A = εDλ D + εXλ X + εDXλ DX
(4)
Solving eq 3 for the case of mixing the reagent with initial concentration cD and cX and substituting the result in eq 4 we get λ
2
(5)
(6)
In eqs 5 and 6 A λ/ is the dependent variable, as a function of the independent variables c X and cD, or c X and cDX. While A λ/ is a linear function of ε λ parameters, it is a nonlinear function of the parameter Q; hence the necessity for a nonlinear parameter estimation. If we use nonlinear model functions 5 or 6 to estimate Q and the ελ values, we can also put the evaluation of the ε λ values from a band profile function (e.g., Gaussian or Lorentzian) into the model. The band profiles are also nonlinear functions of the band parameters (e.g., the absorption maximum or the spectral bandwidth), which are easily treated by a nonlinear estimation algorithm but not the linearized methods. This not only helps to extend the measurement to practically any reasonable wavelength—increasing the precision of the estimated parameter Q—but enables the determination of the complete spectrum of that component of an equilibrium mixture which cannot be prepared in a onecomponent solution. The statistical details of nonlinear parameter estimation can be found in standard textbooks (3–5). However, to use recent commercial software, it is not necessary to know the numerical details of the estimation algorithm. To take the example of the largely used method of least squares, it is enough to know that it minimizes the sum
(3)
This is the quantity we would like to infer from equilibrium spectrophotometric measurements. The measured absorbance for the above mixture at a wavelength λ can be written in terms of the optical path length and the molar absorptivities ελ according to Beer’s law:
A = εXλc X + εDλc D – εXλ + εDλ – εDXλ ⋅ 2 1 + Q c D + c X ± 1 + 2Q c D + c X + Q c D – c X 2Q
In case of mixing the product DX with one of the reagents (say, X), the corresponding expression becomes
m
S = Σ w i A obsd,i – A calcd,i
2
i=1
(7)
called the weighted residual sum of squares. All the user has to do is to write an appropriate model function and give a reasonable initial guess of the parameters to be estimated. The application program calculates the estimated mean values, their error, and some additional statistical parameters (usually described in the help or tutorial of the application). Application In the online material we discuss two detailed examples of nonlinear least squares equilibrium quotient estimation.W Here we show the simultaneous fit of eq 5 to 18 data points measured at two different wavelengths for the complex-forming reaction between I2 and DMSO in CCl 4 solution. As can be
JChemEd.chem.wisc.edu • Vol. 77 No. 7 July 2000 • Journal of Chemical Education
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Figure 1. Plot of the fit of eq 5 to the experimental data points for the I2–DMSO charge-transfer complex formation measured at 446 and 518 nm. The initial I2 concentration is 0.001 M throughout.
Figure 2. Plot of the fit of eq 5 for one of the nine mixtures, that which contains 0.04 M DMSO, with the contribution of the two absorbing species I2 and the charge-transfer complex.
seen in Figure 1, the fit is excellent, which supports a reliable estimated value of Q = 9.7. We also show part of the results of a more detailed measurement, in which nine different mixtures of I2 and DMSO were measured at 106 wavelengths. Model 5 was used together with a Gaussian band shape for I2 and a combined Gaussian–Lorentzian for the charge transfer complex. (DMSO does not absorb in this range.) In Figure 2, we can see a very good fit in the entire concentration and wavelength region. The value of Q = 11.8 is obtained for these data, with much better precision due to the large number of measured points, and with greater reliability due to the greater concentration and wavelength range covered.
Peter Holpár and András Somogyi, and the kind technical advice of the Microcal staff.
Acknowledgments E. Keszei is a beneficent of the grant OTKA T019396 from the Hungarian Research Fund. The authors acknowledge assistance in the preparation of the online version by
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Supplemental Material
Supplemental material for this article is available in this issue of JCE Online. Literature Cited 1. Ramette, R. W. J. Chem. Educ. 1967, 44, 647. 2. Collins, M. J. J. Chem. Educ. 1986, 63, 457. 3. Valkó, P.; Vajda, S. Advanced Scientific Computing in BASIC with Applications in Chemistry, Biology and Pharmacology; Elsevier: Amsterdam, 1987. 4. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes, The Art of Scientific Computing; Cambridge University Press: Cambridge, 1986. 5. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969.
Journal of Chemical Education • Vol. 77 No. 7 July 2000 • JChemEd.chem.wisc.edu
