In the Classroom
Don’t Be Tricked by Your Integrated Rate Plot!† Edward Todd Urbansky National Risk Management Research Laboratory, Water Supply and Water Resources Division, U.S. Environmental Protection Agency, 26 West Martin Luther King Drive, Cincinnati, OH 45268-0001;
[email protected] Analyzing Kinetic Data For the sake of this discussion, a set of first-order kinetic data (t, [A]) was generated using a rate constant of k = 0.1386 s᎑1 (so that t1/2 = 5 s) and an initial reactant concentration [A]0 = 0.10 M. Reactant concentration was calculated every second. It is assumed that the reaction goes to completion; that is, [A]∞ = 0. This represents a straightforward case. Figure 1a shows typical plots of ln [A] and 1/[A], as would be found † The U.S. Environmental Protection Agency, through its Office of Research and Development, funded the work described here. It has been subjected to agency review and approved for publication. This work was authored by a United States government employee as part of that person’s official duties. In view of Section 105 of the Copyright Act (17 USC §105) the work is not subject to U.S. copyright protection.
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Most students are introduced to chemical kinetics in the second semester of the general chemistry course. Generally, two ways of determining reaction order are presented: comparing initial rates while varying reactant concentration, and plotting integrated rate expressions (1–6 ). Most reactions encountered in general chemistry are either first or second order. While a number of complications and special cases occur, conditions can be adjusted to make many reactions exhibit pseudo-firstor pseudo-second-order behavior. Comparison of initial rate data is most commonly used in kinetics for determining reaction order—as opposed to relying on the linearity of integrated rate plots. There are several reasons for this: the reaction may be too slow to follow for very long, side or subsequent reactions may take place, a reactant originally in excess may be consumed to the point where its concentration is no longer constant, a measured signal may increase or decrease until it is outside the dynamic range of the instrument, etc. Treatment of reaction order necessarily requires presentation of the linear integrated rate plots: ln [A] vs t (first order) and 1/[A] vs t (second order). Graphs show [A], ln [A], and 1/[A] as a function of time, and the student is advised that the linear plot tells the reaction order. This is technically correct, but only one of the undergraduate texts consulted gave any cautionary statements about using this approach to analyze kinetic data (1–8). In practice, integrated rate plots are generally used for determining the observed rate constant once the order is known. If the integrated rate plot is to be used to find reaction order, the student should be warned of its limitations. When considering any system not well studied, comparison of initial rate data while changing reactant concentration is the preferred strategy. However, integrated rate plots can be used with some confidence to verify agreement with previous studies.
in most introductory texts. It represents 7 half-lives of the reaction, so that [A] is now less than 1% of its initial value. Over this time domain, it is clear that the reaction exhibits first-order kinetics. It is not always possible to obtain 3–7 half-lives’ worth of data. Figure 1b shows the expanded view of 3 half-lives’ worth of data. Suppose we have only 2 half-lives’ worth of
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Time / s Figure 1. (a) First-order and second-order integrated rate plots for a contrived data set derived from a first-order reaction (A → products) with k = 0.139 s᎑1 (t1/2 = 5 s) and [A]0 = 0.10 M. The abscissa spans 7 half-lives. (b) Expanded view of the first 3 half-lives of the reaction; the vertical bar divides the first two half-lives from the third. During the first 2 half-lives (0 < t ≤ 10 s), it is not possible to visually distinguish whether the reaction is first or second order, even when the data fit a first-order equation perfectly. Only in the third half-life (10 < t ≤ 15 s), does it become clear that the secondorder fit fails.
JChemEd.chem.wisc.edu • Vol. 78 No. 7 July 2001 • Journal of Chemical Education
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In the Classroom
Conclusion Teaching chemistry involves a balance between presenting enough information to be useful but not so much as to be overwhelming. Consequently, clarification will continually be necessary when student learning must be applied to actual
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data (75% of the reaction). In this region of Figure 1b (before the vertical bar), the order cannot be discerned from the integrated rate plots. When the first 10 points generated by a first-order reaction model are analyzed using a second-order equation, the fit is surprisingly good—albeit wrong—with a regression correlation constant of .9645 and a second-order rate constant of 2.9 ± 0.2 M ᎑1 s᎑1 (standard error of
