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1458 Journal of Chemical Education • Vol. 76 No. 10 October 1999 • JChemEd.chem.wisc.edu

The idea of zero-order kinetics frequently produces as-tonishment the first time it is presented to students becauseit is difficult to imagine a reaction as being independent ofthe 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 sometimesconsidered chemical curiosities (1).

To clarify the authentic meaning of zero-orders kineticsthe following simple analogy can be used.

Imagine a room in which there is a box with stressedmice attempting to escape. The box is not perfectly sealed sothat mice can escape, although the exit is not visible to them.Also in the room are cats, hungry and efficient hunters, whohave detected the existence of mice and are on the watch sothat 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 thescenario of the reaction, and the reaction is over when eachcat has obtained one mouse.

Applying common sense—despite its subjective compo-nent—it is reasonable to admit that:

1. The higher the number of mice in the box, the higherthe 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 asthe absolute value of the rate of mouse–cat pairsformed, is independent of their number because theymust wait for free mice in order to capture them.

Consequently, the rate law for the reaction, with respect toC 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 ordersare x (unknown; however, constant for a particular run) withrespect to [M] and 0 with respect to [C] (2).

In kinetics the rate of a reaction is measured under avariety of conditions to obtain as much information as possible,and care is required to interpret the results from different runsto draw coherent conclusions. Imagine a run starting with 500mice and 5 cats and three observers examining variations inM, C, and M–C, respectively.

The first observer concludes that the rate of disappearanceof mice is zero ({d[M]/dt = 0) because it is not possible to detectthe small change from 500 to 495 in the group of runningmice (1% resolution is associated with most measurements).Of course, the correct conclusion from this observation shouldbe: the fact that we are unable to appreciate changes in asystem is not a sufficient condition to conclude that nothingis happening in it.

The second observer concludes that the rate of disap-pearance of cats is constant ({d[C]/dt = constant) becausethe time required by the fifth mouse to escape (when the miceare 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 overcats, the above rate law transforms into

{d[C]/dt = d[M–C]/dt = k[M]x [C]0 = kobs

and the reaction seems to follow zero-order kinetics. It is saidthat the reaction has run in pseudo-zero-order conditions. Itshould be noted that distinct values for kobs are obtained forruns of this kind starting from different numbers of mice.

In conclusion, pseudo-zero-order kinetics can eventuallybe established when measuring the variation in concentrationof the limiting reagent in the presence of a large excess ofothers (in fact, overall first-order reactions involving more thanone species might run under pseudo-zero-order conditions).If so, the mouse–cat analogy can be useful in shedding lighton the problem, in the same way that mice inside the boxare “unreactive” and become “very reactive” once they escape.This is analogous to a reagent that requires an initial trans-formation (isomerization, dissociation,…) to become reac-tive (slow step) so that the resulting species reacts immedi-ately (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.

Mice in the Box for Zero-Order Kinetics

Francisco J. ArnáizLaboratorio de Química Inorgánica, Universidad de Burgos, 09001 Burgos, Spain; *farnaiz@cid.cid.ubu.es

Applications and Analogiesedited by

Ron DeLorenzoMiddle Georgia College

Cochran, GA 31014