Research: Science and Education
Reaction Order Ambiguity in Integrated Rate Plots Joe Lee School of Chemistry, The University of Manchester, Manchester, M60 1QD, United Kingdom;
[email protected] This article addresses an important problem in reaction kinetics that is often presented as a simple exercise in deciding whether a given graphical plot is linear and then, if the plot is linear, assigning an unambiguous reaction order. The reality is much more complicated since a simple failure to establish sufficient experimental points can lead to uncertainty, or worse, to an incorrect assignment of the order. Graphical procedures based on so-called integrated rate equations form a well-established methodology, described in most physical chemistry texts (e.g., 1–3) and in general texts on chemical kinetics (e.g., 4–6), for processing reaction kinetics data for the purpose of determining (partial) orders and thence (pseudo) rate constants. For particular postulated orders, corresponding functions of reactant concentration plotted against reaction time should yield a straight line if that order is to be confirmed. However, allowing for experimental error in the measurement of concentration (or some quantity linearly related to concentration) at various reaction times, it may be difficult to decide whether it is legitimate to draw a straight line “through” the experimental points or whether a curve ought to be more appropriate. The problem with integrated rate equation methodology is that when a concentration function appropriate to a particular order, but inappropriate to the true order, is plotted against time and the progress of the reaction has been monitored over an insufficient fraction of reaction, one can always construct a straight line within say the error bars of the function; this is particularly so when the function corresponds to an order close to (say within one of ) the true value (7). Students are often given rough rules to avoid reaction order ambiguity, for example, follow the reaction to at least 75% completion (2 half-lives). The purpose of the present article is to put this kind of rule on a firmer theoretical basis; this article is in fact an expansion of an brief letter to this Journal (8). Dimensionless Forms of Simple Rate Equations In chemical reactions (at constant temperature and absolutely or essentially constant volume) it is well-known (e.g., 1–6) that the rate of concentration change of a particular reactant A with time t, d[A]/dt, (= rate of reaction × stoichiometric number of A) is frequently directly proportional to a simple power—the order—of the concentration of A, [A]n. If other reactants are involved, d[A]/dt may also be dependent on the concentrations of these; in some cases it may also depend on product concentrations. However, if this additional dependence exists, by the standard practices of maintaining (i) other reactants (and possibly products) in substantial excess and/or (ii) reactant concentrations in their stoichiometric proportions, appropriate when reactant concentrations also follow power laws,
d[A]∙dt may be expressed solely in terms of [A], that is,
d n k dt
(1)
where n is (partial) order and k is a (positive) proportionality constant. Orders are commonly non-negative integral (0, 1, 2, or 3), but half-integral values (for multistage reactions) are not uncommon. For the purpose of this article, we shall consider orders of only 0, 1, and 2 to illustrate the arguments. The integrated rate equations for these orders and corresponding equations for first half-lives t½ (when [A] is half the initial concentration of A, ½[A]0) are well-known (e.g., 1–6 ). These are given in equation sets 2–4; each set also includes an equation obtained by eliminating k: Ordder 0: k t 0
k t1 2 t
t1 2
1
2
0
k t1 2 ln 2 t t12
0
ln
k t12 t t1 2
(3)
ln 2
1 A < > 1 0
0 1 A < >0
Order 2: kt
(2)
0
2 1
Order 1: k t ln
0
(4)
1
By representing [A]∙[A]0 by C and t∙t½ by T, we then convert the final equation in each of sets 2–4 to the following dimensionless forms, respectively, for orders 0, 1, and 2:
C 1
T 2
ln C T ln 2
1 1 T C
(5)
(6) (7)
For these orders, eqs 5–7 indicate that C, ln C, and 1∙C are linearly related to T for orders 0, 1, and 2, respectively. Since C is proportional to [A] and T is proportional to t, this linearity must correspond to linear relationships of [A], ln([A]∙concentration unit), and 1∙[A] to t for these orders.
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Unambiguous Order Resolution Equations 5–7 can be extended to include all three functions of C present in these equations not just in the one order as given but for each of the three orders, as follows: For order 0:
C 1
T 2
ln C ln 1
T 2
dent of [A]. As shown in the Appendix, the consequence of this is that σ(ln C) increases substantially with increasing magnitude of ln C. Another extreme possibility, which will be used as illustration here, will be that σ([A]) is proportional to [A]. For this case, the Appendix shows that σ(ln C) is a constant, κ. A 95% confidence interval for ln C would then be ln C ± 1.96κ. As a specific example of this, let 1.96κ = 0.1, a reasonable typical value
(8)
1 2 C 2 T
0.0
0 ź0.5
For order 2:
C 2T ln C T ln 2 1 2T C C
(9)
2 1
ź1.5
ź2.0
1 1 T
ln C ln 1 T
ź1.0
ln C
For order 1:
2
0
ź2.5 0.0
0.2
0.4
0.6
The key point now is that, for each order, C, ln C, or 1∙C can be
1.0
1.2
1.4
1.6
1.8
T
(10)
1 1 T C
0.8
Figure 1. Plots of ln C [variously represented by ln(1 − T/2), –T ln 2 and –ln(1 + T ) for reaction orders 0, 1, and 2] against T as indicated. The zero-order plot descends to –∞ at T = 2.
represented as its equivalent function of T, so enabling function plots of C, ln C, or 1∙C against T.
142
0.0
ź0.5
ln C
It must be emphasized that these are universal and theoretical plots—universal in the sense that they are independent of [A]0 and k—devised merely to assess the ease with which an incorrect decision can be reached about the order of a process using traditional integrated rate law plots {[A], ln([A]∙concentration unit), or 1∙[A] against t}. There is no question that experimental plots of C, ln C, or 1∙C against T (or t) are being advocated; in any case, absent knowledge of t½ would prevent the conversion of experimental t values to T values. To exemplify the key point above, let us consider multiple plots of just one of these functions, viz. ln C—in the form of its three equivalent functions of T—against T. This is shown in Figure 1. The order 1 function plot (‒T ln 2) is, of course, the straight line; the order 0 function plot [ln(1 − ½T)] is the steeply descending curve; the order 2 plot [‒ln(1 + T)] is the remaining curve. The functions coincide at T = 1 where all three functions equal ‒ln 2. The question now is to what value of T should one follow a reaction to be sure that the line is or is not straight bearing in mind that the experimental values of [A] and therefore ln C will be subject to random error. One can reasonably assume that t values are essentially error-free but, of course, uncertainty of [A] will result in a corresponding uncertainty of t½ and therefore of T. Depending upon the technique for monitoring [A], there are many possibilities for the dependency of σ(ln C)—standard deviation of ln C—upon the true value of ln C. One extreme possibility would be that σ([A]) is a constant, that is, indepen-
ź1.0
ź1.5
ź2.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
T Figure 2. Plot of ln C = ln(1 − T/2) against T (thick solid curved line) for order = 0 to which is added 95% confidence curves (thinner lines) for an assumed constant standard deviation of ln C. The confidence range is taken as ±0.1 regarded as a reasonably typical value. Also included are (i)—as dashed lines—the consequent confidence limits for T = 1, and (ii)—as a thick straight line—the longest straight line entrapped between the 95% confidence curves and starting at the invariant (T = 0, ln C ). This line stays within the limits up to T ≈ 1.33, indicating that one should follow a zeroth-order reaction beyond this value to ensure that it is not indicated to be first order.
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Research: Science and Education
(but readers are of course at liberty to choose a different value), and then ask the question whether an incorrect order could be assigned to a process based on experimental data. To do this, 95% confidence limits have been separately attached (as thinner lines) to the individual lines of Figure 1, and shown in Figure 2 (order 0), Figure 3 (order 1), and Figure 4 (order 2); (these figures also show as dashed lines the consequential errors in T
(a) Figure 2 also includes a limiting straight line starting at the invariant—see the Appendix—fixed point (T = 0, ln C = 0) and passing through at least the extremes of the confidence limits as far as T = 1.33. This suggests that one should follow a truly zeroth-order reaction beyond 1.33 true half-lives (60% completion) to be sure that the reaction is not first-order. (b) Figure 4 similarly includes a limiting straight line and in this case suggests that one should follow a truly secondorder reaction beyond 2.00 half-lives (75% completion) to be sure that the reaction is not first-order.
0.0
ź0.5
ln C
at its value of 1). We will now ask the separate questions: Could a reaction truly of (a) zeroth or (b) second order be mistakenly interpreted as being of first order?
ź1.0
ź1.5
ź2.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
T Figure 3. Plot of ln C = –T ln 2 against T (thick solid line) for order = 1 to which is added 95% confidence curves (thinner lines) for an assumed constant standard deviation of ln C. The confidence range is taken as ±0.1 regarded as a reasonably typical value. Also included are—as dashed lines—the consequent confidence limits for T = 1.
0.0
Literature Cited 1. Atkins, P.; de Paula, J. Atkins’ Physical Chemistry, 7th ed.; Oxford University Press: Oxford, 2002; pp 871–875.
ln C
ź0.5
2. Moore, W. J. Physical Chemistry; 5th ed.; Longman/Prentice-Hall: Eaglewood Cliffs, NJ, 1974; pp 333–338.
ź1.0
ź1.5
[The general formula for N half-lives is percent completion = 100 (1−2−N).] One would then conclude that, with the experimental errors indicated above, if one can assume an order between 0 and 2, one should follow the reaction to beyond 75% completion to ascertain whether the reaction is first-order. If the reaction is truly first-order, one can see from Figure 3 that (assuming all experimental points lie within the 95% limits) a straight line could always be drawn through the points, but that the larger the maximum T (the number of half-lives) the closer will the slope of the line to the true slope of ‒ln 2. The true slope could in fact be recreated by the formation of new T values by multiplying the correct T by a factor, f, close to 1, equivalent to dividing the true t½ by f and therefore multiplying the true k by f. To ascertain whether the reaction is zeroth-order or is second-order, one should repeat the procedure described above but now using graphs, not presented here, of the three equivalent T functions of, respectively, 1∙C or C against T. The creation of these graphs and typical analyses based upon them will be left to the reader.
3. Levine, I. N. Physical Chemistry, 4th ed.; McGraw-Hill, New York, 1995; pp 498–501. 0.0
0.5
1.0
1.5
2.0
2.5
3.0
T Figure 4. Plot of ln C against T (thick solid curved line) for order = 2 to which is added 95% confidence curves (thinner lines) for an assumed constant standard deviation of ln C. The confidence range is taken as ±0.1 regarded as a reasonably typical value. Also included are (i)—as dashed lines—the consequent confidence limits for T = 1, and (ii)—as a thick straight line—the longest straight line entrapped between the 95% confidence curves and starting at the invariant (T = 0, ln C ). This line stays within the limits up to T ≈ 2.00, indicating that one should follow a second-order reaction beyond this value to ensure that it is not indicated to be first order.
4. Laidler, K. Chemical Kinetics; McGraw-Hill: New York, 1950; p 13. 5. Logan, S. R. Fundamentals of Chemical Kinetics; Longman: Harlow, 1996. 6. Espenson, J. H. Chemical Kinetics and Reaction Mechanisms, 2nd ed; McGraw-Hill: New York, 1995. 7. Urbansky, E. D. J. Chem. Educ. 2001, 78, 921–923. 8. Le Vent, S. J. Chem. Educ. 2004, 81, 32. 9. Davies, O. L.; Goldsmith, P. L. Statistical Methods in Research and Production, 4th revised ed.; Oliver & Boyd: Edinburgh, 1972; p 54.
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Research: Science and Education
Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2008/Jan/abs141.html
(a) If σ([A]) is essentially constant and therefore equal to σ([A]0), eq (A1) can be written
Abstract and keywords Full text (PDF) Links to cited JCE articles
T 2( ln C ) T 2 ln
T 0
0
2
0 v
1
2
2
T 2
2
2
1
0
T 0
0
2
0 v0
vln
0
T 2
T
T 0
0
T 2 0
T 2 0
2
Two possibilities will now be considered. Before this, however, one must consider the special case at t = T = 0. Here [A] = [A]0, ln([A]∙[A]0) = ln 1 = 0 and this is errorless.
1 C2 1
C2
2
T 0
0
2
1
(A2)
1
becomes more negative, for example: 1 C2
for ln C = –1, C = 0.368, and (A1)
0
σ(ln C) then increases as C decreases and therefore as ln C
for ln C = 0, C = 1, and
0
vln
144
2
T(ln nC)
Using standard propagation of errors formulas (for independently measurable quantities) (e.g., 9),
T 0
0
Appendix
T 2( ln C )
1 1 C2
2 1. 4
1 2.9 .
(b) If σ([A]) to be essentially proportional to [A]—this is equivalent to σ([A])∙[A] = σ(ln([A]∙concentration unit) being constant—eq (A1) reduces to T(ln nC)
2
T 0
constant, L (say) 0
A well-established illustration of this is the more-or-less constant error of a pH {≈ ‒log10([H+]∙mol dm‒3)} measurement; likewise pX by use of ion-selective electrodes where X represents an inorganic ion.
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