Chemical Education Today
Letters Don’t Be Tricked by Your Integrated Rate Plot
Pitfalls of Using Integrated Rate Plots In the article “Don’t Be Tricked by Your Integrated Rate Plot” (1), Edward Urbansky describes the possible pitfalls of using integrated rate plots. I think the point the author raises is valid, and one should be careful to avoid pitfalls like this. However, as a practising kineticist, I find it most disturbing that the author attributes the error to using an integrated rate plot. In fact, it clearly arises from linearizing the integrated rate law. Urbansky claims that “treatment of reaction order necessarily requires presentation of the linear integrated rate plots”. However, the unfortunate statistical consequences of linearization are well known, and the preferable practice is to use the original (non-linear) integrated rate laws and a nonlinear least squares fitting algorithm (2). Comparing measured and fitted curves and observing the residuals of the fit will always reveal deviations that could be hidden in linearization. Most of these arguments are also clearly presented in various sections of Espenson’s textbook (2). Literature Cited 1. Urbansky, E. T. J. Chem. Educ. 2001, 78, 921. 2. Espenson, J. H. Chemical Kinetics and Reaction Mechanisms, 2nd ed.; McGraw-Hill: New York, 1995.
Department of Inorganic and Analytical Chemistry University of Debrecen H-4010 Debrecen, POB 21, Hungary
[email protected] Reaction Order Ambiguity In the article “Don’t Be Tricked by Your Integrated Rate Plot” (1), E. T. Urbansky draws attention to the dangers of determining reaction (partial) order by use of integrated rate equation plots when the reaction has been followed to insufficient extent, particularly when there may be substantial random error in concentration. I would accept the key point that reactions must be followed to substantial extent if, solely, integrated rate equation plots are to be used to determine reaction order; indeed, I would always make that point strongly myself to students in laboratory courses. Urbansky illustrates the problem with a specific numerical problem, using synthesized first order data, based upon chosen rate constant and initial reactant concentration. I would agree that use of specific data is probably sensible to get over the ambiguity problem to undergraduate students, particularly those at an early stage of higher education, but here I want to draw attention to a more general algebraic method of presenting Journal of Chemical Education
Zeroth order: C = 1 – T/2
ln(C) = ln(1 – T/2)
1/C = 1/(1 – T/2)
First order: C = 2–T
ln(C) = –T ln(2)
1/C = 2T
Second order: C = 1/(1 + T)
ln(C) = –ln(1 + T)
1/C = 1 + T
With the obvious requirement that linearity of the function of C with T will automatically imply linearity of the same function of concentration [or strictly, for the logarithmic function, concentration/(a concentration unit)] with time, one can easily see which three of the above nine equations give linear plots, namely, those with right sides linear in T. Furthermore, for the purpose of Urbansky’s paper, one can see by plotting the other six right sides against T, how nonlinear are these functions and to what value of T one should ideally follow a reaction to obtain an unambiguous order. Literature Cited 1. Urbansky, E. T. J. Chem. Educ. 2001, 78, 921–923.
Gabor Lente
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the argument. Integrated rate equations (for constant reaction volume) may be given in terms of relative reactant concentration, C (= concentration/initial concentration) and relative time, T (= time/half-life); in these forms, the equations are independent of rate constants and initial concentrations. The equations, stated here for three functions of C (not merely those giving a linear plot against T) are restricted to orders 0, 1, and 2. They are not difficult to prove from conventional integrated forms (and consequent expressions for half-life) and can be extended to non-integral orders.
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Sue Le Vent Department of Chemistry University of Manchester Institute of Science and Technology Manchester, M60 1QD, UK
[email protected] The author replies to both letters: I want to thank all of those who took the time to write to me personally following the publication of my paper (1) on the analysis of kinetic data. My primary aim in writing the article was to show a weakness in introductory and physical chemistry texts with regard to the presentation of linear integrated rate plots. I will try to address briefly the key points raised in correspondence. I want to acknowledge that the importance of gathering data from a sufficient extent of reaction is mentioned in Sam Logan’s textbook, Fundamentals of Chemical Kinetics (2). In a personal communication, Logan reaffirmed my concern: “This point [the need to follow the reaction for long enough] has frequently occurred to me in tackling kinetics problems in textbooks, for so often the data presented only go to an extent of reaction that is patently inadequate for reliable conclusions to be drawn.”
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Chemical Education Today
Letters In actual practice, a useful tool for analyzing kinetic data, as mentioned by Jim Espenson in a personal communication, is a plot of the residuals (the differences between the actual and calculated concentrations/signal) as a function of time. The trend and the magnitude in such a plot can assist in determining the reaction order. Also, by varying the starting reactant concentration by two-fold to four-fold and plotting the integrated rate expressions associated with the possible reaction orders, one can look for those expressions that yield a constant value of the rate constant (k) even as the starting concentration is varied. When k is independent of initial reactant concentration, it is likely that the correct integrated rate expression has been used. I agree with Gabor Lente that nonlinear fitting schemes can provide better results than linear ones. Again, I emphasize that my article was directed towards teachers of introductory and physical chemistry where the mathematical sophistication of the students may be inadequate to use nonlinear fitting. Moreover, I believe that the pedagogical benefits of teaching the linear fits are substantial enough that this practice will not change any time soon. It is easier to see an invalid fit when using a nonlinearized integrated rate plot of concentration (c) versus time (t), but the linearization alone cannot be blamed for the problem. In the early stages of a reaction, the order is not necessarily clear when sets of (t, c) data are fitted using first and second order functions. Whether the plots themselves are compared visually or using a mathematical criterion for goodness of fit, I am unconvinced that it will be possible to objectively advocate c = c0e᎑kt over c =
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(c0᎑1 + kt)᎑1 or vice versa. Even plotting the residual [c0 e᎑kt ᎑ (c0-1 + kt)᎑1] as a function of time may not be sufficient if less than 75% of the reaction is monitored. In my mind, no trick of data analysis replaces gathering enough data, although it may be possible to rely on fewer data when an experimental system is cooperative and well behaved. Sue Le Vent has offered some alternative functions that may be plotted to determine reaction order. Based on both Le Vent’s and Lente’s letters, it is safe to say that there is no one way to analyze kinetic data, and careful investigators will make efforts to examine their data in multiple ways when necessary to reach an accurate and sound conclusion. However, if one chooses to use a linearized integrated rate plot to determine reaction order, then one must collect data for enough of the reaction for the various forms to be unquestionably distinguishable. Literature Cited 1. Urbansky, E. T. J. Chem. Educ. 2001, 78, 921–923. 2. Logan, S. R. Fundamentals of Chemical Kinetics; Longman: Harlow, England, 1996. Edward Urbansky Joint Oil Analysis Program Technical Support Center Naval Air Station Pensacola, FL 32508-5010
[email protected] Vol. 81 No. 1 January 2004
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