Semianalytical Methods To Determine First-Order Rate Constant Distributions † Luuk K. Koopal,* ,‡ Maarten M. Nederlof, § Willem H. Van Riemsdijk, | and Peter A. Barneveld ‡ Department of Physical and Colloid Chemistry and Department of Soil Science and Plant Nutrition, Wageningen Agricultural University, NL-6703 HB Wageningen, The Netherlands Received November 20, 1995. In Final Form: July 23, 1996 X Semianalytical methods for the determination of first-order rate constant distributions are discussed. In general the expression for the overall decay function is an integral equation based on the local decay function and the distribution function. The analytical expressions of the distribution function obtained by inversion of the integral equation are a series of derivatives of the overall decay function. The higher the order of the approximation, the more derivatives are required. The most common method is based on the Laplace transform technique; newly derived are the coefficients for the third-order method. The second method is due to Schwarzl and Staverman, it provides a general scheme to find a distribution function from an integral equation and has been used here to obtain an expresssion for the rate constant distribution. The advantage of this method is that it provides a weighting function that maps the true distribution into its approximation; i.e., it visualizes the quality of the approximation. The third method is newly developed. It uses an approximation of the local decay function to solve the integral equation for the distribution function. The local decay function approximation or LODA method provides a physical interpretation of the approximation involved in obtaining the distribution by showing the function that approximates the true local decay function. The three methods are compared on the basis of two synthetic data sets, one composed of exact data and one of nonexact data. For nonexact data presmoothing of the data is required. For this purpose a smoothing spline technique is applied in which the smoothing parameter is obtained objectively by generalized cross validation in combination with physical constraints. It is shown that the newly developed LODA-G2 method, which needs the first and second derivative of the overall decay function, combines a good resolution with a relatively low sensitivity to experimental error. Introduction Many processes both in industrial practice and in natural environments are affected by the heterogeneity of the substrate. In order to describe the effect of the heterogeneity on a process, information on the hetero- geneity is required. In the case of adsorption or binding studies, such information can be obtained by studying the sorption process with special probes followed by an analysis of the results for heterogeneity. Methods to investigate the heterogeneity of binding reactions have been developed for both equilibrium 1-8 and kinetic conditions. 9-14 Very recently Cerefolini and Re 15 have indicated the similarities between the two types of analyses. The present paper focuses on the heterogeneity analysis of kinetic data. For complicated systems the overall reaction rate is determined by a series reactions that each have their own order and rate constant. To obtain the rate constant distribution from the overall reaction rate, the order of the reactions has to be known. With complicated reactions the reaction order is often unknown, and an assumption has to be made with respect to the order; generally a first- or pseudo-first order reaction is assumed. Under the assumption of a series of first-order reactions, the overall decay function is essentially a multiexponential decay. Such a decay function appears in a large variety of studies. 16,17 In principle, the rate constant distribution function can now be obtained without further assumptions about the rate constant distribution itself. Yet the determination of the distribution is not trivial because small variations in the overall reaction rate due to experimental error may lead to large spurious variations in the distribution function. 17,18 In order to avoid these problems, special numerical techniques have to be used. References 9, 16, 17, and 19 can be consulted for discussion and examples. Three advanced numerical techniques, based on regularized least squares to find an optimal * Corresponding author: e-mail address, koopal@fenk.wau.nl. † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovankia, September 4-10, 1995. ‡ Department of Physical and Colloid Chemistry. § Present address: KIWA NV Research and Consultancy, P.O. Box 1072, 3430 BB Nieuwegein, The Netherlands. | Department of Soil Science and Plant Nutrition. X Abstract published in Advance ACS Abstracts, February 15, 1997. (1) Koopal, L. K.; Vos, C. H. W. Langmuir 1993, 9, 2593-2605. (2) LumWan, J. A.; White, L. R. J. Chem. Soc., Faraday Trans. 1991, 87, 3051-3062. (3) Von Szombathely, M.; Brau ¨ er, P.; Jaroniec, M. J. Comput. Chem. 1992, 13, 17-32. (4) Jagiello, J.; Schwarz, J. A. J. Colloid Interface Sci. 1991, 146, 415-424. (5) Jagiello, J. Langmuir 1994, 10, 2778-2785. (6) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L. K. J. Colloid Interface Sci. 1990, 135, 410-426. (7) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L. K. Environ. Sci. Technol. 1992, 26, 763-771. (8) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L. K. Environ. Sci. Technol. 1994, 28, 1037-1047. (9) Stanley, B. J.; Topper, K.; Marshall, D. B. Anal. Chim. Acta 1994, 287, 25-34. (10) Shuman, M. S.; Collins, B. J.; Fitzgerald, P. J.; Olson, D. J. In Aquatic and terrestrial humic materials; Christman, R. F., Gjessing, E. T., Eds.; Ann Arbor Science Publishers: Ann Arbor, MI, 1983; pp 349- 370. (11) Olson, D. L.; Shuman, M. S. Anal. Chem. 1983, 55, 1103-1107. (12) Olson, D. L.; Shuman, M. S. Geochim. Cosmochim. Acta 1985, 49, 1371. (13) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L. K. Environ. Sci. Technol. 1994, 28, 1048-1053. (14) Lavigne, J. A.; Langford, C. H.; Mak, M. K. S. Anal. Chem. 1987, 59, 2616-2620. (15) Cerefolini, G. F.; Re, N. J. Colloid Interface Sci. 1995, 174, 428- 440. (16) Zimmermann, K.; Delaye, M.; Licinio, P. J. Chem. Phys. 1985, 82, 2228-2235. (17) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 213-227. (18) Noble, B. The state of the art in numerical analysis; Academic Press: London, 1979; pp 915-966. (19) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 229-242. 961 Langmuir 1997, 13, 961-969 S0743-7463(95)01054-7 CCC: $14.00 © 1997 American Chemical Society