The author replies to Lente and Le Vent.

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.”

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 = (c₀^–1 + kt)^–1 or vice versa. Even plotting the residual [c₀e^–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.