Research: Science and Education
Nonlinear Fitting to First-Order Kinetic Equations Ian J. McNaught † School of Chemistry, University of Sydney, Sydney, NSW 2006, Australia;
[email protected] A recent paper (1) discussed analyzing first-order kinetic data where the infinite-time measurement is unreliable or unavailable. The discussion of time-lag methods makes it clear that the derived rate constant and infinite-time reading are dependent on the particular time-lag chosen for the analysis. This undesirable feature can be avoided by using nonlinear regression techniques (2). The ready availability of easily used microcomputer programs that permit nonlinear regression (e.g., Mathcad [3], TableCurve [4 ], Mathematica, or the Solver feature in Excel [5]) makes it almost as easy to do a nonlinear regression as a linear regression. The standard deviations in the parameters are as important as the parameters themselves. TableCurve and Mathematica automatically generate standard deviations for nonlinear fits, whereas Mathcad and Excel do not. However, a simple technique has been described for using Excel to calculate the standard deviations (6 ). This approach has several advantages over unweighted linear regression and time-lag methodologies: the results are independent of the chosen time lag, the readings do not need to be equally spaced, all the readings are used, and the parameters are not biased by an inappropriate weighting of the data points (7). The table shows the results of fitting the data in (1) to the equation At = A∞ – (A∞ – A0 )exp({ kt)
(1)
where A ∞, A0 , and k are the required parameters. The values should be compared with the results of fitting to the linearized equation from (1), (with τ = 1200 s): ln(At+τ – At) = ln{(A∞ – A0)[1 – exp({ kτ)]} – kt
(2)
The significant difference between the results of the two fits illustrates the bias that is introduced by linearizing the equation. If the data are analyzed using the linearized equation but weighting the points by (At+τ – At)2 (8), then the derived parameters effectively agree with the nonlinear regression analysis. These results underline the importance of correctly weighting the data when doing regression analyses. The analysis presented here assumes that the initial data have errors only in the dependent variable and the standard † Current address: School of Human and Biomedical Sciences, University of Canberra, Canberra, ACT 2601, Australia.
Results of Fitting Eqs 1 and 2 to the Data in Ref 1 Eq
Parameters Weight (No.)
1
3
equal
2
2
equal
2
2
(At + τ – At )
aDerived
A∞
A0
A∞ – A0
104 k / s{1
1.253 ± 0.007 ± 1.246 ± 0.009 0.003 0.009a
6.35 ± 0.09
2
—
—
1.240 ± 0.003
6.53 ± 0.09
—
—
1.250 ± 0.009
6.38 ± 0.10
from A∞ and A0.
deviations for all the points are equal. Under these conditions unweighted nonlinear regression correctly weights the data, whereas unweighted linear regression (on the artificially linearized equation) does not. If the data had varied standard deviations, then weighted nonlinear regression would be required. This is simply implemented by weighting each point by 1/s(y i) 2 as discussed in the procedure described in (6 ). This can introduce an additional difficulty, as it is not unusual for the weights to depend on the parameters being determined, necessitating an iterative approach to their calculation (8). This is the case when linearizing eq 1: the weighting is given by (A∞ – At)2, depending on the desired parameter A∞. It may be noted that the three methods considered in ref 1 produce different parameters. Each yields the rate constant k, but method I gives A0 (with an assumed A∞), method II gives A∞ – A0, and method III gives A∞. Literature Cited 1. Hemalatha, M. R. K.; NoorBatcha, I. J. Chem. Educ. 1997, 74, 972. 2. Copeland, T. G. J. Chem. Educ. 1984, 61, 778. 3. Sauder, D.; Towns, M. H.; Stout, R.; Long, G.; Zielinski, T. J. J. Chem. Educ. 1997, 74, 269. 4. Boyd, D. B. Rev. Comput. Chem. 1995, 7, 336. 5. Machuca-Herrera, J. O. J. Chem. Educ. 1997, 74, 448. 6. Harris, D. C. J. Chem. Educ. 1998, 75, 119. 7. Zielinski, T. J., Allendoerfer, R. D. J. Chem. Educ. 1997, 74, 1001. 8. Sands, D. E. J. Chem. Educ. 1974, 51, 473.
JChemEd.chem.wisc.edu • Vol. 76 No. 10 October 1999 • Journal of Chemical Education
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