A common problem in chemical kinetics is the development of a rate law that describes the dependence of the reaction rate on the surrounding conditions such as concentrations of reacting species or temperature of the reacting media (see Chap. 1). The most direct approach to solving this problem is to measure reaction rates under systematically varied conditions and then to perform mathematical analyses on these data to determine the form of the rate law and to generate estimates of any unknown constants, or parameters, that make up the proposed rate law. Other chapters in this book provide information both for designing kinetics experiments and for selecting appropriate rate laws for a variety of geochemical reactions. In this chapter we describe the mathematical analyses — known collectively as curve fitting or regression analysis — that can be used to select a rate equation that matches a given data set, to generate estimates for any unknown parameters in the rate equation (e.g., rate constants or reaction orders), and to quantify the uncertainty associated with the estimated values for the parameters. As we traverse this entirely quantitative process we will attempt to describe the underlying, qualitative process of looking at kinetic data: plots to make, features of these plots to examine, and conceptual sketches to draw.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Acker J. G. and Bricker O. P. (1992) The influence of pH on biotite dissolution and alteration kinetics at low temperature. Geochimica et Cosmochimica Acta 56(8), 3073-3092.
Agrawal A., Ferguson W. J., Christ J. A., Bandstra J. Z., and Tratnyek P. G. (2002) Effects of carbonate species on the kinetics of dechlorination of 1,1,1-trichloroethane by zero-valent iron. Environmental Science and Technology 36, 4326-4333.
Amrhein C. and Suarez D. (1992) Some factors affecting the dissolution kinetics of anorthite at 25◦ C. Geochimica et Cosmochimica Acta 56, 1815-1826.
Bandstra J. Z. and Tratnyek P. G. (2004) Applicability of single-site rate equations for reactions on inhomogeneous surfaces. Industrial and Engineering Chemistry Research 43(7), 1615-1622.
Bandstra J. Z. and Tratnyek P. G. (2005) Central limit theorem for chemical kinetics in complex systems. Journal of Mathematical Chemistry 37(4), 409-422.
Bennett P. C. (1991) Quartz dissolution in organic-rich aqueous systems. Geochimica et Cosmochimica Acta 55, 1781-1797.
Bennett P. C., Melcer M. E., Siegel D. I., and Hassett J. P. (1988) The dissolution of quartz in dilute aqueous-solutions of organic-acids at 25◦ C. Geochimica et Cosmochimica Acta 52, 1521-1530.
Berger G., Cadore E., Schott J., and Dove P. M. (1994) Dissolution rate of quartz in lead and sodium electrolyte solutions between 25 and 300◦ C: effect of the nature of surface complexes and reaction affinity. Geochimica et Cosmochimica Acta 58, 541-551.
Bird H. A. and Milliken G. A. (1976) Estimable functions in the nonlinear model. Communications in Statistics: Theory and Methods 6, 999-1012.
Brady P. V. and Walther J. V. (1989) Controls on silicate dissolution rates in neutral and basic pH solutions at 25◦ C. Geochimica et Cosmochimica Acta 53, 2823-2830.
Brady P. V. and Walther J. V. (1990) Kinetics of quartz dissolution at low temperature. Chemical Geology 82, 253-264.
Brownlee K. A. (1965) Statistical Theory and Methodology. New York, John Wiley & Sons.
Busenberg E. and Clemency C. V. (1976) The dissolution kinetics of feldspars at 25◦ C and 1 atm CO2 partial pressure. Geochimica et Cosmochimica Acta 40, 41-49.
Cama J., Metz V., and Ganor J. (2002) The effect of pH and temperature on kaolinite dissolution rate under acidic conditions. Geochimica et Cosmochimica Acta 66 (22), 3913-3926.
Carroll-Webb S. A. and Walther J. V. (1988) A surface complexation model for the pH-dependence of corundum and kaolinite dissolution rates. Geochimica et Cosmochimica Acta 52, 2609-2623.
Casey W. H., Lasaga A. C., and Gibbs G. V. (1990) Mechanisms of silica dissolution as inferred from the kinetic isotope effect. Geochimica et Cosmochimica. Acta 54, 3369-3378.
Casey W. H., Westrich H. R., and Holdren G. R. (1991) Dissolution rates of plagioclase at pH = 2 and 3. American Mineralogist 76, 211-217.
Charnes A., Frome E. L., and Yu P. L. (1976) The equivalence of generalized least squares and maximum likelihood estimates in the exponential family. Journal of the American Statistical Association 71(353), 169-171.
Chen G. and Balakrishnan N. (1995) A general purpose approximate goodness-of-fit test. Journal of Quality Technology 27(2), 154-161.
Chou L. and Wollast R. (1984) Study of the weathering of albite at room temperature and pressure with a fluidized bed reactor. Geochimica et Cosmochimica Acta 48 (11),2205-2217.
Chou L. and Wollast R. (1985) Steady-state kinetics and dissolution mechanism of albite. American Journal of Science 285, 963-993.
Cygan R. T., Casey W. H., Boslough M. B., Westrich H. R., Carr M. J., and Holdren G. R. (1989) Dissolution kinetics of experimentally shocked silicate minerals. Chemical Geology 78, 229-244.
D’Agostino R. B. and Stephens M. A. (1986) Goodness-of-Fit Techniques. New York, Marcel Dekker.
Dennis J. E., Jr., Gay D. M., and Welsch R. E. (1981) An adaptive nonlinear least-squares algorithm. ACM Transactions on Mathematical Software 7, 348-368.
Dennis J. E., Jr., Gay D. M., and Welsch R. E. (1981) Algorithm 573. NL2SQL-An adaptive nonlinear least-squares algorithm. ACM Transactions on Mathematical Software 7, 369-383.
Dennis J. E., Jr. and Schnabel R. B. (1996) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. New Jersey, Prentice-Hall.
Draper N. R. and Smith H. (1966) Applied Regression Analysis. New York, John Wiley & Sons.
Eggleston C. M., Hochella M. F., Jr., and Parks G. A. (1989) Sample preparation and aging effects on the dissolution rate and surface composition of diopside. Geochimica et Cosmochimica Acta 53, 797-804.
Frogner P. and Schweda P. (1998) Hornblende dissolution kinetics at 25◦ C. Chemical Geology 151, 169-179.
Ganor J., Mogollon J. L., and Lasaga A. C. (1995) The effect of pH on kaolinite dissolution rates and on activation energy. Geochimica et Cosmochimica Acta 59 (6),1037-1052.
Gillespie D. T. (2000) The chemical Langevin equation. Journal of Chemical Physics 113(1), 297-306.
Gislason S. R. and Eugster H. P. (1987) Meteoric water-basalt interactions. I: A laboratory study. Geochimica et Cosmochimica Acta 51, 2827-2840.
Golubev S. V., Pokrovsky O. S., and Schott J. (2005) Experimental determination of the effect of dissolved CO2 on te dissolution kinetics of Mg and Ca silicates at 25◦ C. Chemical Geology 217, 227-238.
Green P. J. (1984) Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives. Journal of the Royal Statistical Society B 46(2), 149-192.
Guidry M. W. and Mackenzie F. T. (2003) Experimental study of igneous and sedi-mentary apatite dissolution - Control of pH, distance from equilibrium, and tem-perature on dissolution rates. Geochimica et Cosmochimica Acta 67(16), 2949-2963.
Hartley H. O. (1961) The modified Gauss-Newton method for the fitting of nonlinear regression functions by least squares. Technometrics 3, 269-280.
Hodson M. E. (2003). The influence of Fe-rich coatings on the dissolution of anorthite at pH 2.6. Geochimica et Cosmochimica Acta 67(18), 3355-3363.
Holdren G. R. and Speyer P. M. (1987) Reaction-rate surface area relationships during the early states of weathering; II, Data on eight additional feldspars. Geochimica et Cosmochimica Acta 51(9), 2311-2318.
Hougen O. A. and Watson K. M. (1943) Solid catalysts and reaction rates. Industrial and Engineering Chemistry 35, 529.
House W. A. and Orr D. R. (1992) Investigation of the pH dependence of the kinetics of quartz dissolution at 25◦ C. Journal of Chemical Society Faraday Transactions 88,233-241.
Huber-Carol C., Balakrishnan N., Nikulin M. S., and Mesbah M. (2002) Goodnessof-Fit Tests and Model Validity. Boston, Birkhauser.
Huertas F. J., Chou L., and Wollast R. (1999) Mechanism of kaolinite at room temperature and pressure Part II: Kinetic study. Geochimica et Cosmochimica Acta 63 (19/20), 3261-3275.
Huet S., Bouvier A., Poursat M.-A., and Jolivet E. (2004) Statistical Tools for Nonlinear Regression. New York, Springer.
Hughes-Hallett D., Gleason A. M., Flath D. E., Gordon S. P., Lomen D. O., Love-lock D., McCallum W. G., Osgood B. G., Pasquale A., Tecosky-Feldman J., Thrash J. B., Thrash K. R., and Tucker T. W. (1994) Calculus. New York, John Wiley & Sons, Inc.
Ishida K. (1966) Stochastic approach to nonequilibrium thermodynamics of firstorder chemical reactions. Journal of Physical Chemistry 70(12), 3807-3811.
Kalinowski B. E. and Schweda P. (1996) Kinetics of muscovite, phlogopite, and biotite dissolution and alteration at pH 1-4, room temperature. Geochimica et Cosmochimica Acta 60(3), 367-385.
Knauss K. G., Nguyen S. N., and Weed H. C. (1993) Diopside dissolution kinetics as a function of pH, CO2 , temperature, and time. Geochimica et Cosmochimica Acta 57, 285-294.
Knauss K. G. and Wolery T. J. (1986) Dependence of albite dissolution kinetics on pH and time at 25◦ C and 70◦ C. Geochimica et Cosmochimica Acta 50(11), 2481-2497.
Langmuir I. (1918) The adsorption of gases on plane surfaces of glass, mica, and platinum. Journal of the American Chemical Society 40, 1361.
Legates D. R. and McCabe G. J. J. (1999) Evaluation of the use of “goodness-of-fit” measures in hydrologic and hydroclimatic model validation. Water Resources Research 35(1), 233-241.
Levenberg K. (1944) A method for the solution of certain problems in least squares. Quarterly of Applied Mathematics 2, 164-168.
Marin-Lof T. (1974) The notion of redundancy and its use as a quantitative measure of the discrepancy between a statistical hypothesis and a set of observation data. Scandanavian Journal of Statistics 1, 3-18.
Marquardt D. (1963) An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics 11, 431-441.
Martin B. R. (1971) Statistics for Physicists. London, Academic Press.
McQuarrie D. A. (1967) Stochastic approach to chemical kinetics. Journal of Applied Probability 4, 413-478.
Miller D. C. (1984) Reducing transformation bias in curve fitting. The American Statistician 38(2), 124-126.
Moore D. S. (2001) Statistics: Concepts and Controversies. New York, W. H. Free-man.
Moore J. W. and Pearson R. G. (1981) Kinetics and Mechanism. New York, John Wiley & Sons.
Oelkers E. H. (2001) An experimental study of forsterite dissolution rates as a function of temperature and aqueous Mg and Si concentrations. Chemical Geology 175 (3-4), 485-494.
Oelkers E. H. and Gislason S. R. (2001) The mechanism, rates and consequences of basaltic glass dissolution: I. An experimental study of the dissolution rates of basaltic glass as a function of aqueous Al, Si and oxalic acid concentration at 25◦ C and pH = 3 and 11. Geochimica et Cosmochimica Acta 65(21), 3671-3681.
Pokrovsky O. S. and Schott J. (2000) Kinetics and mechanism of forsterite disso-lution at 25◦ C and pH from 1 to 12. Geochimica et Cosmochimica Acta 64(19), 3313-3325.
Press W. H., Teukolsky S. A., Vetterling W. T., and Flannery B. P. (1992) Numerical Recipies in C. Cambridge University Press.
Rose N. M. (1991) Dissolution rates of prehnite, epidote, and albite. Geochimica et Cosmochimica Acta 55(11), 3273-3286.
Rosso J. J. and Rimstidt J. D. (2000) A high resolution study of forsterite dissolution rates. Geochimica et Cosmochimica Acta 64(5), 797-811.
Schweeda P. (1989) Kinetics of alkali feldspar dissolution at low temperature. In Proceedings of the Sixth International Symposium on Water/Rock Interaction (ed. D. L. Miles), A. A. Balkema, Lisse, The Netherlands, pp. 609-612.
Seber G. A. F. and Wild C. J. (2003) Nonlinear Regression. New Jersey, John Wiley & Sons.
Stewart W. E., Shon Y., and Box G. E. P. (1998) Discrimination and goodness-of-fit of multiresponse mechanistic models. AIChE Journal 44(6), 1404-1412.
Stillings L. L. and Brantley S. L. (1995) Feldspar Dissolution at 25◦ C and pH 3 - Reaction stoichiometry and the effect of cations. Geochimica et Cosmochimica Acta 59(8), 1483-1496.
Sverdrup H. U. (1990) The kinetics of base cation release due to chemical weathering. Lund University Press, pp. 246.
Swoboda-Colberg N. G. and Drever J. I. (1993) Mineral dissolution rates in plotscale field and laboratory experiments. Chemical Geology 105, 51-69.
Temkin M. I. (1979) The kinetics of some industrial heterogeneous catalytic reactions. Advances in Catalysis 28, 173.
Tester J. W., Worley W. G., Robinson B. A., Grigsby C. O., and Feerer J. L. (1994) Correlating quartz dissolution kinetics in pure water from 25◦ C to 625◦ C. Geochimica et Cosmochimica Acta 58, 2407-2420.
Tiley P. F. (1985) The misuse of correlation coefficients. Chemistry in Britain 21(2), 162-163.
Valsami-Jones E., Ragnarsdottir K. V., Putnis A., Bosbach D., Kemp A. J., and Cressey G. (1998) The dissolution of apatite in the presence of aqueous metal cations at pH 2-7. Chemical Geology 151(1), 215-233.
van Hess P. A. W., Lundstr öm U. S., and M örth C.-M. (2002) Dissolution of microcline and labradorite in a forest O horizon extract: the effect of naturally occurring organic acids. Chemical Geology 189, 199-211.
Welch S. A., Taunton A. E., and Banfield J. F. (2002) Effect of microorganisms and microbial metabolites on apatite dissolution. Geomicrobiology Journal 19(3), 343-367.
Wieland W. and Stumm W. (1992) Dissolution kinetics of kaolinite in acidic aque-ous solutions at 25◦ C. Geochimica et Cosmochimica Acta 56, 3339-3355.
Willmott C. J., Ackleson S. G., Davis R. E., Reddema J. J., Klink K. M., Legates D. R., O’Donnell J., and Rowe C. M. (1985) Statistics for the evaluation and comparison of models. Journal of Geophysical Research 90(C5), 8995-9005.
Wogelius R. A. and Walther J. V. (1992) Olivine dissolution kinetics at near-surface conditions. Chemical Geology 97(1-2), 101-112.
Wollast R. and Chou L. (1986) Processes, rate, and proton consumption by silicate weathering. Transactions of the 13th Congress of the International Society of Soil Science, pp. 127-136.
Wollast R. and Chou L. (1988) Rate control of weathering of silicate minerals at room temperature and pressure. In Physical and Chemical Weathering in Geo-chemical Cycles (ed. A. Lerman and M. Maybeck). Amsterdam, Kluwer Academic, pp. 11-32.
Zepp R. G. and Wolfe N. L. (1987) Abiotic transformation of organic chemicals at the particle-water interface. In Aquatic Surface Chemistry: Chemical Processes at the Particle-Water Interface (ed. W. Stumm). New York, John Wiley & Sons, pp. 423-455.
Zhang H. (1990) Factors determining the rate and stoichiometry of hornblende dissolution. Ph.D. thesis, University of Minnesota.
Zhang H. and Bloom P. R. (1999) The pH dependence of hornblende dissolution. Soil Science 164(9), 624-632.
Zhang H., Bloom P. R., Nater E. A., and Erich M. S. (1996) Rates and stoichiometry of hornblende dissolution over 115 days of laboratory weathering at pH 3.6-4.0 and 25◦ C in 0.01M lithium acetate. Geochimica Cosmochimica Acta 60, 941-950.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Bandstra, J., Brantley, S. (2008). Data Fitting Techniques with Applications to Mineral Dissolution Kinetics. In: Brantley, S., Kubicki, J., White, A. (eds) Kinetics of Water-Rock Interaction. Springer, New York, NY. https://doi.org/10.1007/978-0-387-73563-4_6
Download citation
DOI: https://doi.org/10.1007/978-0-387-73563-4_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-73562-7
Online ISBN: 978-0-387-73563-4
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)