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Textbook Deficiencies: Ambiguities in Chemical Kinetics Rates and Rate Constants Keith T. Quisenberry and Joel Tellinghuisen* Department of Chemistry, Vanderbilt University, Nashville, TN 37235; *
[email protected] In chemical kinetics, one determines reaction rate laws and rate constants by following the temporal changes in the concentrations or pressures of reactants or products. For example, two gas-phase reactions that feature prominently in most general and physical chemistry textbooks are 2 N2O5(g)
4 NO2(g) + O2(g)
Similarly, for the reaction in eq 2
2 NO(g) + O2(g)
(2)
The reactants and products are consumed and generated with rates proportional to their stoichiometry coefficients. Thus in the reaction in eq 1, consumption of 2 mol of N2O5 leads to production of 4 mol of NO2 and 1 mol of O2, so at any time the absolute rates of change of these three are in the ratio 2:4:1. Similarly, in the reaction in eq 2, the absolute rates of change of NO2, NO, and O2 are in the ratio 2:2:1.1 To define reaction rates and rate constants unambiguously with respect to these different physical rates, it is necessary to adopt a convention. As recommended by the IUPAC, for the generic reaction, bB + cC
1 d [B ] 1 d[C ] 1 d[D] 1 d[ E ] = − = = (4) c dt d dt e dt b dt
where the rate is a product of the rate constant k and some function of the concentrations of reactants and products. This function must be determined experimentally and constitutes the rate law. For the reactions in eqs 1 and 2, the rate laws have been found to be rate1 = k1 [N2 O5]
(5)
and
rate 2 = k2 [NO2]
2
(6)
The convention is often stated in the form of eq 4 in physical chemistry texts. In general chemistry, it is common practice to introduce the rate concept with finite differences in place of differentials, so ∆’s appear in place of the d ’s. Accordingly, for the reaction in eq 1, the rate law appears as ∆[ N2 O5 ] = −2 k1 [ N2O5] ∆t
(7a)
2NO2(g) + 1 2 O2(g)
(1´)
NO(g) + 1 2 O2(g)
(2´)
and NO2(g)
the rate law expressions change to, for example,
∆[ N2 O5 ] = − k1′ [ N2 O5] ∆t
(7a´)
∆[ NO2 ] 2 = − k2′ [NO2] ∆t
(8a´)
and
The quantities on the left hand sides of eqs 7 and 8 are physical rates that cannot depend upon how we choose to write the reactions. On the other hand, the reaction rates and rate constants constitute conventional quantities, since their values are tied to the written reaction through the convention of eq 4. Thus, k1´ = 2k1 and k2´ = 2k2; and hence rate1´ = 2 rate1 and rate2´ = 2 rate2. Clearly, the reaction rate and rate constant are ambiguous without explicit reference to the written reaction.2 Probably all texts correctly recognize that reactants and products disappear and appear with rates proportional to their stoichiometry numbers, and most seem to correctly state the convention of eq 4 (in finite difference form for general chemistry). However, it has come to our attention that many general chemistry texts manage to “snatch defeat from the jaws of victory” in the leap to the integrated rate laws that follow from eqs 7 and 8 and other cases where the stoichiometry numbers are not unity. This may be because they seldom actually solve the differential forms of these equations, rather just state the results for first- and second-order reactions, usually in the generic forms,
[A]
= [ A ]0 e −k t
(9)
and
or ∆[ NO2 ] = 4 k1 [ N 2O5 ] ∆t
510
(8b)
N2O5(g)
the convention consists in defining the reaction rate as rate = −
∆[ O2 ] 2 = k2 [NO2 ] ∆t On the other hand, if these reactions are rewritten as
(3)
dD + eE
(8a)
and
(1)
and 2 NO2(g)
∆[ NO2 ] 2 = −2 k2 [NO2 ] ∆t
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1 (7b)
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[A ] •
=
1
[ A ]0
+ kt
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(10)
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respectively. These forms are correct for reactions written A → products, which means that eq 9 is correct for most firstorder processes, which usually are written this way. However, for the reaction in eq 1, the correct integrated expression is
[ N2 O5 ]
= [ N2 O5 ] e
−2 k1 t
and since second-order processes are almost always written in the form 2A → products (e.g., eq 2), eq 10 is wrong for these. Instead it should read 1 + 2 kt [ A ]0
(10´)
One of the main reasons for giving students the integrated rate expressions of eqs 9 and 10 is the desire to acquaint them with the concept of fractional life, especially half-life. Thus, for example, by noting that t1/2 is the time needed for [A] to fall to 1/2[A]0, one readily obtains from eqs 9 and 10 the commonly given expressions, t 1st = 1 2
ln 2 k
(12a)
1 k [ A ]0
(12b)
and t 12nd = 2
But these expressions are not correct for the reactions in eqs 1 and 2, where the correct versions are n2 ln 2 k1
(13a)
1 2 k2 [NO2]0
(13b)
t1 = 2
and t1 = 2
respectively. In going from eqs 12 to 13, it is not the half-life that is changing but the rate constant. Like the rates that appear on the left-hand sides of eqs 7 and 8, the half-life is a physical property of the system and cannot depend upon how the reaction is written. The foregoing results can be summarized in more compact form by using stoichiometry numbers νi and labeling all reactants and products as Ai. Then the balanced chemical reaction reads
∑ νi Ai
= 0
(14)
where νi is negative for reactants and positive for products; and eq 4 becomes
rate =
1 d[Ai ] νi dt
(15)
In the simplest case, where a single reactant A (unsubscripted) determines the rate law, let us take a = ᎑νA. Then the firstand second-order rate laws of eqs 7 and 8 read ∆[ A ] ∆[ A ] 2 = −ak1st [A ] and = −ak 2 nd [A ] (16) ∆t ∆t
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Accordingly, the integrated rate laws become
[A]
= [ A ]0 exp ( −a k1st t ) and
(11)
0
1 = [A]
•
1
[A]
=
1
[ A ]0
+ ak 2 nd t (17)
and the half-lives are t 1st = 1 2
1 ln 2 and t 12nd = 1 st 2nd 2 ak ak [ A]0
(18)
We have examined a dozen general chemistry texts of the sort widely used in the instruction of science and premedical students (1–12) and have concluded that only two of these have all aspects of this problem essentially correct— those by Whitten et al. (1) and Oxtoby et al. (2).3 And even these fail to emphasize that while the half-life is a physical property of the system, the rate constant and its relation to the half-life depend upon how the reaction is written. With respect to the N2O5 decomposition, Oxtoby commits a minor flaw by writing the reaction as shown in eq 1´ and then two pages later (in Example 14-5) asking students to find k for “the first-order decomposition of N2O5(g)” without actually stating the reaction. Presumably it is to be taken as eq 1´ above; but the authors have actually adopted for the sake of the example a half-life that is incorrect, as the k it yields at 25 ⬚C agrees with the literature value for k1, not k1´ (13). Of the other texts, several either fail to state the convention of eq 4 or give it and then announce that it will be ignored! In addition to the examples given in eqs 1 and 2, errors occur in connection with the discussion of the firstorder decomposition of H2O2(aq) and the (pseudo) secondorder recombination of I atoms. The texts by Petrucci et al. (6) and Brown et al. (8) bypass the problem by assiduously sticking to coefficients of unity in their use of the integrated rate laws and half-lives. In the case of Brown, this appears to be a conscious attempt to correct this problem in earlier editions; and Petrucci et al. slip up in their Table 15.4 by listing eq 1 but then giving a half-life and k that are correctly related for eq 1´, not 1. As was already noted, physical chemistry texts state the convention in differential form; but most make some of the same errors of omission and commission in applications. The topic is treated very clearly by Levine (14), through his Problem 17.5 (17.1 in the 4th edit.), in which the implications of eqs 1 and 1´ are compared directly. Levine also provides a useful simplification of eqs 16–18, by defining kA = ak. As was noted earlier, in general chemistry texts the problem arises in part because the integrated rate expressions of eqs 9 and 10 are just stated, not derived. Since texts like those we examined are used in courses for science, engineering, and premedical majors, there seems little reason not to include the derivations. In their differential forms, eqs 7 and 8 constitute the simplest kind of differential equations, solvable by separating variables and integrating. Such equations are introduced to calculus students in their first encounter with integration. This usually comes before they study kinetics in chemistry, which is typically in their second semester. The sidebox approach used by some authors is a reasonable compromise and requires little page space.
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One of us (JT) first encountered this ambiguity many years ago, while trying to understand and use literature results for the classic termolecular recombination of I atoms in the gas phase, I + I + M → I2 + M, which has added potential for confusion in usually being written “I + I” instead of “2I.” Chemistry texts should make it clear that (i) the rate and rate constant for a reaction are ambiguous without an explicit statement of the reaction for which they are meant to apply and (ii) while half-lives are physical properties of the chemical system in question, rate constants are not. Also, general chemistry students intending careers in science or medicine need no longer be sheltered from the modest amount of integral calculus needed to understand the derivation of the integrated rate laws for the simplest zeroth-, first-, and second-order reactions. Notes 1. This strict conservation can be violated at times during the course of the reaction if there are intermediates that can build up in significant quantities. 2. The use of noninteger stoichiometry coefficients is often avoided in kinetics, in recognition of the aim of understanding the reaction on the molecular level. But the problem still arises through the writing of reactions like those in eqs 1 and 2 as A → products. 3. We have exercised no “preselection” here: These were texts that were readily available to us in the form of desk copies from our shelves and from those of our colleagues and were the only ones we examined. All 12 fall under the category, introductory courses (mainly for “majors”), on the online JCE ChemEd Resource Shelf, http://www.jce.divched.org/JCEWWW/Features/CERS/ (accessed Dec 2005); the authors represent about a third of those whose books are listed there.
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Literature Cited 1. Whitten, K. W.; Davis, R. E.; Peck, M. L. General Chemistry, 6th ed.; Saunders/Harcourt: Orlando, FL, 2000; Chapter 16. 2. Oxtoby, D. W.; Freeman, W. A.; Block, T. F. Chemistry: The Science of Change, 4th ed.; Brooks/Cole–Thomson Learning: Pacific Grove, CA, 2003; Chapter 14. 3. McMurry, J.; Fay, R. C. Chemistry, 4th ed.; Prentice Hall: Upper Saddle River, NJ, 2004; Chapter 12. 4. Atkins, P.; Jones, L. Chemical Principles: The Quest for Insight, 3rd ed.; W. H. Freeman: New York, 2005; Chapter 13. 5. Chang, R.; Cruickshank, B. Chemistry, 8th ed.; McGraw–Hill: Boston, 2005; Chapter 13. 6. Petrucci, R. H.; Harwood, W. S.; Herring, F. G. General Chemistry: Principles and Modern Applications, 8th ed.; Prentice Hall: Upper Saddle River, NJ, 2002; Chapter 15. 7. Chemistry: A Project of the American Chemical Society; Bell, J., Ed.; W. H. Freeman: New York, 2005; Chapter 11. 8. Brown, T. L.; Lemay, H. E., Jr.; Bursten, B. E.; Burdge, J. R. Chemistry: The Central Science, 9th ed.; Prentice Hall: Upper Saddle River, NJ, 2003; Chapter 14. 9. Zumdahl, S. S.; Zumdahl, S. A. Chemistry, 6th ed.; HoughtonMifflin: Boston, MA, 2003; Chapter 12. 10. Silberberg, M. S. Chemistry: The Molecular Nature of Matter and Change, 3rd ed.; McGraw–Hill: Boston, 2003; Chapter 16. 11. Brady, J. E.; Senese, F. Chemistry: Matter and Its Changes, 4th ed.; Wiley, Hoboken, NJ, 2004; Chapter 15. 12. Kotz, J. C.; Treichel, P., Jr. Chemistry and Chemical Reactivity, 3rd ed.; Saunders/Harcourt Brace; Fort Worth, TX, 1996; Chapter 15. 13. Johnston, H. S.; Tao, Y.-S. J. Am. Chem. Soc. 1951, 73, 2948–2949. 14. Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill, Boston, 2002; Chapter 17.
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