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JChemEd.chem.wisc.edu • Vol. 78 No. 7 July 2001 • Journal of Chemical Education 921

Finding Reaction Order

Most students are introduced to chemical kinetics in thesecond semester of the general chemistry course. Generally, twoways of determining reaction order are presented: comparinginitial rates while varying reactant concentration, and plottingintegrated rate expressions (1–6 ). Most reactions encounteredin general chemistry are either first or second order. While anumber of complications and special cases occur, conditionscan be adjusted to make many reactions exhibit pseudo-first-or pseudo-second-order behavior. Comparison of initial ratedata is most commonly used in kinetics for determiningreaction order—as opposed to relying on the linearity ofintegrated rate plots. There are several reasons for this: thereaction may be too slow to follow for very long, side or sub-sequent reactions may take place, a reactant originally in excessmay be consumed to the point where its concentration is nolonger constant, a measured signal may increase or decreaseuntil it is outside the dynamic range of the instrument, etc.

Treatment of reaction order necessarily requires presen-tation of the linear integrated rate plots: ln [A] vs t (first or-der) and 1/[A] vs t (second order). Graphs show [A], ln [A],and 1/[A] as a function of time, and the student is advisedthat the linear plot tells the reaction order. This is technicallycorrect, but only one of the undergraduate texts consultedgave any cautionary statements about using this approach toanalyze kinetic data (1–8). In practice, integrated rate plotsare generally used for determining the observed rate constantonce the order is known. If the integrated rate plot is to beused to find reaction order, the student should be warned of itslimitations. When considering any system not well studied,comparison of initial rate data while changing reactant con-centration is the preferred strategy. However, integrated rateplots can be used with some confidence to verify agreementwith previous studies.

Analyzing Kinetic Data

For the sake of this discussion, a set of first-order kineticdata (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 1ashows typical plots of ln [A] and 1/[A], as would be found

in most introductory texts. It represents 7 half-lives of thereaction, so that [A] is now less than 1% of its initial value.Over this time domain, it is clear that the reaction exhibitsfirst-order kinetics.

It is not always possible to obtain 3–7 half-lives’ worthof data. Figure 1b shows the expanded view of 3 half-lives’worth of data. Suppose we have only 2 half-lives’ worth of

Don’t Be Tricked by Your Integrated Rate Plot!†

Edward Todd UrbanskyNational Risk Management Research Laboratory, Water Supply and Water Resources Division, U.S. EnvironmentalProtection Agency, 26 West Martin Luther King Drive, Cincinnati, OH 45268-0001; urbansky.edward@epa.gov

†The U.S. Environmental Protection Agency, through its Office ofResearch and Development, funded the work described here. It hasbeen subjected to agency review and approved for publication.This work was authored by a United States government employeeas part of that person’s official duties. In view of Section 105 of theCopyright Act (17 USC §105) the work is not subject to U.S. copyrightprotection.

Figure 1. (a) First-order and second-order integrated rate plots for acontrived 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 abscissaspans 7 half-lives. (b) Expanded view of the first 3 half-lives of thereaction; the vertical bar divides the first two half-lives from thethird. During the first 2 half-lives (0 < t ≤ 10 s), it is not possible tovisually distinguish whether the reaction is first or second order,even when the data fit a first-order equation perfectly. Only in thethird half-life (10 < t ≤ 15 s), does it become clear that the second-order fit fails.

ln [A

]

Time / s

1/[A

]

1/ [A]

ln [A]

0−8

−7

−6

−5

−4

−3

−2

5 10 15 20 25 30 350

200

400

600

800

1000

1200

1/ [A]

ln [A]

ln [A

]

−5.0

−4.5

−4.0

−3.5

−3.0

−2.5

−2.0

Time / s0 2 4 6 8 10 12 14 16

1/[A

]

10

20

30

40

50

60

70

a

b

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922 Journal of Chemical Education • Vol. 78 No. 7 July 2001 • JChemEd.chem.wisc.edu

data (75% of the reaction). In this region of Figure 1b (beforethe vertical bar), the order cannot be discerned from the in-tegrated rate plots. When the first 10 points generated by afirst-order reaction model are analyzed using a second-orderequation, the fit is surprisingly good—albeit wrong—with aregression correlation constant of .9645 and a second-orderrate constant of 2.9 ± 0.2 M�1 s�1 (standard error of <10%).Only when the third half-life region (after the vertical bar) isexamined does the deviation from linearity become visible.

This example represents an idealized case. Real kinetic dataoften have ±10% variation due to instrumental or practicallimitations. In Figure 2, a random error of up to ±10% has beenintroduced in the calculated concentration using the spread-sheet’s random number generator. The first-order analysis hasa regression correlation coefficient (R2) of .9845 with a rateconstant k = 0.137 ± 0.006 s–1. The second order analysishas R2 = .9485 and k = 2.9 ± 0.2 M�1 s�1. Without the addi-tional data in the third half-life of the reaction, it would bedifficult to argue in favor of the first-order analysis even withits more favorable statistical parameters if these data had beenobtained from a real experiment.

In addition to indeterminate error, many “pseudo-first-order” experiments are conducted under less than rigorousconditions. Small changes in pH or reactant concentrationare often neglected because of cost or experimental difficulties(e.g., the absorbance is too high or the reaction is too fastwhen a higher concentration of the excess reactant is used).Extremely slow (weeks) and extremely fast (microseconds)reactions are particularly susceptible to these limitations.Consequently, a minimum of 4–5 half-lives is required to usean integrated rate plot to determine reaction order. Even then,caution must be exercised because more complicated systemsof one order can mimic simpler ones of another order.

Two graduate-level kinetics texts (9, 10) mention theimportance of gathering enough data before constructing anintegrated rate plot. A third (11) does not discuss the extent ofreaction required for using integrated rate plots specifically,but nonetheless warns against using them alone to determineorder. The limits of the integrated rate plot approach are wellknown to kineticists, but appear to be omitted from generaland physical chemistry texts. Because they neither emphasizethe extent of reaction needed to see curvature nor showexpanded views near time zero, general chemistry books cangive students the impression that they can determine reactionorder by graphing a few points and drawing a straight line.

General and physical chemistry classes and their text-books can be the only exposure to kinetics that some chemistsand engineers get; therefore, it is important for these treatmentsto be thorough. In fields that rely on kinetic data (e.g., for-mation and decay of disinfection by-products in drinkingwater systems), such examples of poor kinetic analysis canbe found—and have led to rejection of several manuscriptsby this reviewer.

Conclusion

Teaching chemistry involves a balance between presentingenough information to be useful but not so much as to beoverwhelming. Consequently, clarification will continually benecessary when student learning must be applied to actual

1/[A]

ln [A]

ln [A

]

−5.0

−4.5

−4.0

−3.5

−3.0

−2.5

−2.0

Time / s0 2 4 6 8 10 12 14 16

1/[A

]

10

20

30

40

50

60

70

Figure 2. Expanded view (same domain and range as Fig. 1b) ofthe first 3 half-lives when a random error of up to ±10% is incorpo-rated into the concentration values calculated for Figure 1. Thislevel of error is not uncommon in kinetics studies and makes it moredifficult to discern order from an integrated rate plot. With only 2half-lives’ worth of data (0 ≤ t ≤ 10 s), it is not possible to deter-mine the order using an integrated rate plot. The least squareregression lines are based on the first 10 s only (up to the verticalbar). If it is not possible to follow more of the reaction, anothertechnique—such as comparison of initial rates—must be used.

practice in the field. Not to be neglected in the process ischemical common sense: reactions with a second-orderdependence on one reactant are somewhat rare. However,students by definition do not have the requisite experienceto construct such generalizations. Analyzing kinetic data isoften challenging and is made even more challenging byincomplete understanding. Limitations of theoretical modelsmust be recognized, and care must be exercised in applyingthese models to real systems. Integrated rate plots are noexception. Whenever there is a need to determine reactionorder, the best approach is to use initial rates and vary reactantconcentration over a range of 5- to 10-fold or even more ifpossible. Once reaction order is known, integrated rate plotsare readily used to determine rate constants.

Literature Cited

1. Zumdahl, S. S. Chemistry, 4th ed.; Houghton Mifflin: Boston,1997; Chapter 12.

2. Bodner, G. M.; Pardue, H. L. Chemistry: An ExperimentalScience; Wiley: New York, 1995; Chapter 22.

3. Chang, R. Chemistry, 6th ed.; WCB McGraw-Hill: Boston,1998; Chapter 13.

4. Brown, T. L.; LeMay, H. E. Jr.; Bursten, B. E. Chemistry: TheCentral Science, 7th ed.; Prentice-Hall: Upper Saddle River,NJ, 1997; Chapter 14.

5. Brady, J. E.; Russell, J. W.; Holum, J. R. Chemistry: Matterand Its Changes, 3rd ed.; Wiley: New York, 2000; Chapter 13.

6. Olmsted, J. III; Williams, G. M. Chemistry: The MolecularScience, 2nd ed.; Wm. C. Brown: Dubuque, IA, 1997; p 688.

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7. Atkins, P. W. Physical Chemistry, 6th ed.; Freeman: New York,1998; Chapter 25.

8. Barrow, G. M. Physical Chemistry, 6th ed.; McGraw-Hill: NewYork, 1996; Chapter 15.

9. Connors, K. A. Chemical Kinetics: The Study of Reaction Rates

in Solution; VCH: New York, 1990; p 25.10. Espenson, J. H. Chemical Kinetics and Reaction Mechanisms,

2nd ed.; McGraw-Hill: New York, 1995; pp 31–32.11. Laidler, K. J. Chemical Kinetics, 3rd ed.; HarperCollins: New

York, 1987; p 28.