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Applications and Analogies
Ron DeLorenzo Middle Georgia College Cochran, GA 31014
Mice in the Box for Zero-Order Kinetics Francisco J. Arnáiz Laboratorio de Química Inorgánica, Universidad de Burgos, 09001 Burgos, Spain; *
[email protected] The idea of zero-order kinetics frequently produces astonishment the first time it is presented to students because it is difficult to imagine a reaction as being independent of the concentration of all the species involved. This situation, when mentioned in texts, is seldom illustrated with examples, so that reactions following zero-order kinetics are sometimes considered chemical curiosities (1). To clarify the authentic meaning of zero-orders kinetics the following simple analogy can be used. Imagine a room in which there is a box with stressed mice attempting to escape. The box is not perfectly sealed so that mice can escape, although the exit is not visible to them. Also in the room are cats, hungry and efficient hunters, who have detected the existence of mice and are on the watch so that those that escape are immediately captured (see figure). In this “reaction” the reagents are mice (M) and hungry cats (C), the product is the pair formation M–C, the room is the scenario of the reaction, and the reaction is over when each cat has obtained one mouse.
In kinetics the rate of a reaction is measured under a variety of conditions to obtain as much information as possible, and care is required to interpret the results from different runs to draw coherent conclusions. Imagine a run starting with 500 mice and 5 cats and three observers examining variations in M, C, and M–C, respectively. The first observer concludes that the rate of disappearance of mice is zero ({d[M]/dt = 0) because it is not possible to detect the small change from 500 to 495 in the group of running mice (1% resolution is associated with most measurements). Of course, the correct conclusion from this observation should be: the fact that we are unable to appreciate changes in a system is not a sufficient condition to conclude that nothing is happening in it. The second observer concludes that the rate of disappearance of cats is constant ({ d[C]/dt = constant) because the time required by the fifth mouse to escape (when the mice are 496) is imperceptibly longer than that required by the first (at 500). The third observer comes to a similar conclusion: d[M–C]/dt = constant. Thus, in runs starting from a large excess of mice over cats, the above rate law transforms into { d[C]/dt = d[M–C]/dt = k[M]x [C]0 = kobs
Applying common sense—despite its subjective component—it is reasonable to admit that: 1. The higher the number of mice in the box, the higher the probability of one mouse finding the exit; that is, the rate at which mice escape depends on their number (concentration) in the box. 2. The rate of disappearance of hungry cats, the same as the absolute value of the rate of mouse–cat pairs formed, is independent of their number because they must wait for free mice in order to capture them.
Consequently, the rate law for the reaction, with respect to C and M–C, can be expressed as { d[C]/dt = d[(M–C)]/dt = k[M]x [C]0 = k[M]x The overall order of the reaction is x and the partial orders are x (unknown; however, constant for a particular run) with respect to [M] and 0 with respect to [C] (2).
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and the reaction seems to follow zero-order kinetics. It is said that the reaction has run in pseudo-zero-order conditions. It should be noted that distinct values for kobs are obtained for runs of this kind starting from different numbers of mice. In conclusion, pseudo-zero-order kinetics can eventually be established when measuring the variation in concentration of the limiting reagent in the presence of a large excess of others (in fact, overall first-order reactions involving more than one species might run under pseudo-zero-order conditions). If so, the mouse–cat analogy can be useful in shedding light on the problem, in the same way that mice inside the box are “unreactive” and become “very reactive” once they escape. This is analogous to a reagent that requires an initial transformation (isomerization, dissociation,…) to become reactive (slow step) so that the resulting species reacts immediately (fast step) as it forms. Acknowledgment To Jose A. Olivares for helpful comments. Literature Cited 1. Hindmarsh, K.; House, D. A. J. Chem. Educ. 1996, 73, 585. 2. For the terminology on kinetics see Reeve, J. C. J. Chem. Educ. 1991, 68, 728 and references therein.
Journal of Chemical Education • Vol. 76 No. 10 October 1999 • JChemEd.chem.wisc.edu
