USE OF FIRST-ORDER RATEEQUATION IN TREATING KINETICDATA
495
The Use of the First-Order Rate Equation in Treating Kinetic Datal
by DeLos F. DeTar and Victor M. Day Department of Chemistry and the Institute of Molecular Biophysics of the Florida State University, Tallahassee, Florida (Received August 26, 1966)
A detailed analysis is presented of the use of the first-order rate equation with data which only approximate to first order. Rather large departures from first-order kinetics may be undetectable within a single run. The relationship between the parameters of the firstorder equation and the true rate constants and factors affecting the precision of the rate constant are considered. By proper use of the apparent first-order rate constants, it is possible to get a reliable estimate of the first-order component of a reaction which does not deviate further from first order than the examples cited (e.g., 25% of second order) or of B with B present in moderate excess. the second-order rate constant for the reaction A The approach has its limitations, and for complete treatment other methods of calculation must be used which assume the correct kinetic form.
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tures from first-order kinetics, what is the relationship Few actual chemical systems yield rate data which between the parameters of the best first-order equafollow a simple reaction order.2 It is, nevertheless, tion and the exact parameters? often fruitful to use the first-order expression for data It is tedious and not very useful to work out these which are approximately first order. This is now quantitative relationships in explicit form but quite particularly convenient since good computer programs practical to use an empirical approach. We have done are available for finding parameters adjusted by the this for two classes of reaction. The first involves a least-squares criterion.’ The best ways of finding single reactant, A, and has three subdivisions: (a) parameters for the first-order equation for data which conform accurately to first order have been d i s ~ u s s e d . ~ ~first ~ plus second order: -dA/dt = klA kzA2; (b) first plus three-halves order: -dA/dt = lc~A The gross deviations are well known. If data conk’A”/’; (c) Erst plus half order: -dA/dt = klA forming to a second-order expression are fitted to the k‘A’/z. firsborder equation, the resulting compromise underCertain known reactions approximate to these estimates the amount of reactant which has disappeared various subdivisions.6 Furthermore, traditional “salt during the early part of the reaction and overestimates effects” can be approximated by such expressions. the amount in later stages. Corresponding treatment The second class involves a clean second-order reacof data for a zero-order reaction produces the contion: -dA/dt = k2AB. It is assumed that B is verse effect. present in considerable but not overwhelming excess. Of more immediate interest is definite information about reactions which deviate only slightly from first order kinetics. The following points are of practical (1) This work was supported by the United States Air Force, Office of Aerospace Research, under Grant AF-AFOSR 629-64. All comimportance. putations were performed by L S K I N ~ written by D. F. DeTar and (1) For a given precision of analysis for reactants C. E. DeTar. (2) Many general treatments are available, e.g., A. A. Frost and R. G. or products, how far must a reaction depart from first Pearson,“ Kinetics and Mechanism,” 2nd ed., John Wiley and Sons, order before this departure can be clearly demonInc., New York, N. Y., 1963. strated? (3) D.F. DeTar, J . Am. Chem. SOC.,7 8 , 3911 (1956); D. F. DeTar and A. R. Ballentine, ibid., 7 8 , 3916 (1956). (2) Assuming that a departure has been demon(4) C.J. Collins, Advan. Phys. Org. Chem., 2 , l(1963). strated or is suspected, what is the best way to treat (5) K. Noraki and P. D. Bartlett, J. Am. Chem. SOC.,68, 1686 the data so as to obtain correct values for the rate (1946); B. Barnett and W. E. Vaughan, J. Phys. Chem., 5 3 , 926, constants? More explicitly, for given known depar945 (1949).
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~~~
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Volume YO, Number 2 February 1066
496
DELOSF. DETARAND VICTORM. DAY
We have obtained values of A as a function of time for a number of examples of both classes. In some cases we introduced a known error into these A values by use of standard tables so that the data would approximate more closely those obtained experimentally.'j The resulting data were then fitted to the first-order expression, eq. 1. For this purpose we used the com-
A = A,
+
(A0
- A,)e-'
Li
The Magnitude of the Systematic Error Figures 1-6 summarize the answers to the first point, the question of how small a departure from accurate first-order kinetics is detectable. All of these figures represent a plot of the residuals, ( A o w - A c s d vs. per cent of reaction. These are presented in control chart form. The inner pair of dashed lines represents one standard deviation, and the outer pair (where shown) represents about two standard deviations. Roughly half of the points should be within the inner pair and 95% of the points within the outer pair. If the data conform to the first-order expression, there should be no trend. Figure 1 is a typical example. It represents accurate first-order data upon which have been imposed a scalar error in A which is nominally 0.001; A0 = 1.000. Figure 2 shows what happens if the reaction is not first order but instead has a second-order component. The data in Figure 2 correspond to an ideal case. The accuracy (standard deviation) with which A can be determined was taken as 10.001, and the initial concentration was 1.000. The reaction has been followed from 0-99%. The fraction of second-order reaction of course decreases as the reaction proceeds. Initially the second-order component was 5%. Under these almost ideal conditions the deviation from firstorder kinetics shows up plainly as an S curve. Initially A disappears faster than expected for a first-order process, and t,oward the end of the reaction it disappears more slowly. Such a result is, of course, exactly that
-2.0
-- ----- - --------0
io
-3.0
(1)
puter program L S R I N ~ , which has a variety of options and permits use of any of several types of weighting required. In general, Ao, A,, and IC were all treated as disposable parameters, and the scalar error in A was minimized. The effect of minimizing the scalar error when the relative error (dA/A or d In A ) is normally distributed and the converse have a relatively minor effect in the applications under discussion, but proper weight is necessary if the highest precision is required. It is also possible to take either A . or A , or both as accurately known. The main effect of this approach is to extend the range of reaction covered by the data.
The Journal of Physieal Chemistry
* U
do
$0
sb
loo
PER CENT REACTION
Figure 1. Exact first-order data; standard deviation of A nominal 0.001, actual 0.00094. Ao = 1.000.
------- - ---
I
I
I
20
100
40 60 80 PER CENT REACTION
Figure 2. Initial reaction 5% second order, 95y0first order. Points represent nominal standard deviation of 0.001. Line is drawn through exact values. A0 = 1,000.
N
0 X V
'ij
---- - - - - - - - -0 - - -- -
2.01
I.Ok-Q
0::
_ _ _ - _ _ _ _o--Q_-
- __----_ r
0 n
0
0
--Q,,Q-0 0 ----------- ---i-2.0:
a
I-1.0-
0 00 'o
a
expected for a reaction somewhat greater than first order. If it is more difficult to analyze accurately for A, then a large amount of second-order component may escape detection. This is shown in Figure 3. Here the initial reaction is 25% second order, and A is determined with an initial accuracy of 1%. Such accuracy might be better than actually attainable from infrared spectral data, for example. (6) W. E. Deming, "The Statistical Adjustment of Data," John Wiley and Sons, Inc., New York, N. Y., 1943,Appendix. We used 10% larger values than those in the table since these gave actual errors more nearly in accord with the nominal error.
USE O F FIRST-ORDER RATEEQUATION IN
TREATING KINETICDATA
Figure 4 shows a reaction which is first plus threehalves order with the three-halves order component 10% initially. As in Figure 2 , the accuracy of determining A (scalar error) is 0.1% of its initial value. Figure 5 shows a reaction which is first plus half order. The trend of the deviation in Figure 5 is the converse of the trend in those cases where the reaction has a component higher than first order. It should be noted that any reaction which is less than first order comes to an abrupt stop at some finite time. Figure 6 shows the result of following a reaction over a more restricted range. This is often forced upon the experimenter because of side reactions. Considerably larger extents of nonfirst-order component will readily escape detection. The deviation for a second-order reaction between A and B gives curves which look very much like those for the case in which A disappears by both first- and second-order steps as in Figure 3. To define the situation more precisely, we note that there are two contributions to the observed standard deviation of A , errors arising from the analysis for A and systematic errors from the nonfirst-order part of the reaction. The reliable detection of departures from first-order kinetics depends upon having the systematic error relatively large compared with the analytical error. While it is possible to develop statistical criteria to test for systematic error, unless the systematic error is large enough to show up plainly as in Figures 4 and 5 , such tests are not of much practical use. Anyone with experience in kinetics knows that the most reliable way to determine reaction orders is to study the behavior of ICobsd, preferably the initial rate, as initial concentrations are varied. It is, however, most useful t o know how the two types of error interact t o produce an over-all error pattern. We have, therefore, used exact values of the variable, A , corresponding to known kinetics in order to determine the contribution of the nonfirst-order component. The situation is complicated by the fact that these deviations are a complex function of the extent of reaction covered by the data. As a reference point we take the ideal case in which the reaction has been followed from 0 to 99%. Representative values are then presented for other extents of reaction. The results are summarized in Table I for the first class of reaction and in Table I1 for the A-B second-order class. It is seen that for a first- plus second-order reaction, the systematic part of the error in A varies from 0.0012 for A . = 1.000 with 501, of second-order component to 0.0065 for 25% of secondorder component. It turns out, however, that, if the
497
i
n
-u
= l -3.0
20
40 60 80 PER C E N T REACTION
100
Figure 4. Initial reaction 10% three-halves order, 90% first order. Points represent nominal standard deviation of 0.001. Line is drawn through exact values. A0 = 1.000.
a 3.ob3.5
I
I
I
20
40 60 Per cent Reaction
80
100
Figure 5. Initial reaction 5% half order, 95% first order. Points represent nominal standard deviation of 0.001. Line is drawn through exact values. A0 = 1.000. 3.0 R
---_---------,,o ,o- ----0-om- - - - -- --
2,0
X
0 0
I
B
%
a
O ‘ o y - \ 0
------- 75-------2.o= - - - _ _ _ - _ - - - _ _ - _ -1.ok
-3.0
I
I
I
I
nominal analytical error in A is 0.01, for A . = 1.000, then the calculated standard deviation for A is 0.0080.01 for all examples in Table I. See footnote b. Table I1 shows the corresponding data for the secondorder reaction. Roughly speaking, a reaction following first- plus second-order kinetics with a 25% secondorder component is equivalent in behavior to a true second-order reaction in which the initial A :B ratio is 1:4. The effect of using points that cover various extents of reaction is, in general, cpiite complex. The exVolume 70,Number 3 February 1966
498
DELOSF. DETARAND VICTOR111. DAY
~
Table I: Effect of Nonfirst-Order Component on Error in Variable and on the Apparent Rate Constant Extent followed, %
Error in variable from nonfirst-ordera
Obsd.a rate const. x 10'
10% 2 10% 2 10% 2 10% 2
0-99 5-9 1 5-72 5-53
0.24 0.12 0.04 0.01
4.85 4.95 5.09 5.20
5% 2
0-99
0.12
4.93
25% 2 25% 2 25% 2
0-99 0-92 0-75
0.65 0.32 0.11
4.62 4.85 5.20
5% 10% 10% 25%
0-99 0-99 0-90 0-99
0.08 0.16 0.06 0.41
4.96 4.93 5.00 4.82
0-99 0-98 0-96
0.16 0.21 0.45
5.03 5.00 4.95
Reaction type
"2 "2
5% '/2 10% ' / z 25% ' / z
This table refers to data in which the scalar error in the variable is normally distributed and has been minimized. The systematic error due to the nonfirst-order component is treated as though it were a random error and has been tabulated as the percentage of the initial value. b T h e true initial k in every case was 5.00 X 10-3. The observed or apparent k depends both upon the per cent of nonfirst-order component and upon the extent of reaction covered. Taking the nominal standard deviation of the variable as 1% of its initial value, the standard (2% of k ) for all k values in this deviation of k was 0.1 x table, The error arising from the systematic deviation (column 3) has practically no effect a t the levels present here. Why this is so, can be seen from an example. Thus, the over-all error for line 6 is nominally (0.652 l.Oz)'/a = 1.2, and there is a considerably larger random variability in the standard deviation from its nominal value than the difference between 1 and 1.2.
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amples in Tables I and I1 all start with 0 or 5% of reaction and go to various extents. The effect of ignoring part of the data is to reduce the extent to which the data deviate from first-order kinetics. The first four entries in Table I and the latter part of Table I1 show the magnitude of the effect. For a trend in the residuals to be clearly discernible, it is necessary that the standard deviation of the analysis for the variable be smaller than the systematic error for the variable given in Tables I and 11.
Factors Affecting the Standard Deviation
of the Rate Constant The standard deviation of the rate constant is determined by the standard deviation of the observed variable and by a propagation of error term. This The JoUTnd of Physical Chemistry
Table 11: Relationship between Parameters of First-Order Equation and True Second-Order Values (Bo 0.1000, k = 5.00 X 10-8)
lO'Ao
5.0 10.0 20.0 25.0 5.0 10.0 20.0 25.0 4.0 10.0 20.0
% reaction
0-99 0-99 0-99 0-99 0-51 0-52 0-53 0-54 1-40 1-41 1-43
Std. dev. in A due to 2nd ordera
0.11 0.23 0.46 0.58
... 0.01 0.04 0.03 0.00 0.00 0.00
10'kobadb
4.93 4.86 4.74 4.68 5.12 5.23 5.48 5.62 5.10 5.31 5.63
Tabulated as per cent of A Oalthough scalar error in A was minimized. True initial kBo was 5.00 X lo-' for all runs.
term is obtained from the reciprocal matrix of the normal equations.? By concentrating on the contribution from the propagation of error term, it is possible to form conclusions which concern a property of the first-order equation itself. These are independent of departures from first-order kinetics. However, the way in which the standard deviation of the variable enters depends on how the error is treated. If the normally distributed error is scalar in the variable, then an analytical accuracy in which the standard deviation is 1% of the initial value gives one over-all result. On the other hand, if the normally distributed error is the relative error in the variable, then an analytical accuracy in which the standard deviation is 1% of the variable gives a different overall result, for in effect the analysis is much more accurate. A normally distributed error in per cent T gives still another result which depends further upon the specific per cent T values covered. While it is desirable for best results to minimize the proper error, this is a weighting problem which has only indirect bearing on the above point. Data usually do not conform rigorously to a particular form of weighting. The effect of calculating by an improper procedure is to give somewhat poorer estimates of k but not the gross differences which appear in Tables I11 and IV. As experience with a given technique accumulates, each calculation produces an estimate of the error in the variable appropriate to the mode of calculation used (ie., which error is minimized), and it (7) See ref. 6, p. 167.
USE OF FIRST-ORDER RATEEQUATION IN TREATING KINETICDATA
is this error estimate which is to be used to get correct error estimates of the parameters. To illustrate these factors, representative results are shown in Tables 1.11 and IV. Table I11 refers to the special case in which product formation is observed spectrophotometrically and the error in per cent T is minimized. The initial T was 9575, and the final T was 3.5%. For purpose of tabulating, it was assumed that the standard deviation of T was 1%. The standard deviation of k is directly proportional and can readily be calculated for any other assumed error. Table 111: Relationship between Error in Rate Constant and Extent of Reaction Covered“ Reaction points, % reaction
SM. dev. of k, %b
50, 75, 87 1, 75, 94 30, 60, 90 30, 50, 5’0 20, 50, SO 10, 50, 90 5, 50, 95 1, 50, 99 2-97, “best”
97 31 30 56 28 19 16 14 5
” Except for the last run which included 20 points, all runs Based on 1%scalar accuracy in per cent involved just three. T measurements: 16 = 97%; T, = 3.5%.
Table N : Relationship between Error in Rate Constant and Extent of Reaction Covered % reaction covered”
0-99 (22) 5-91 (18) 5-72 (14) 5-53 (10) 0-98 (22)
Std. dev. o f k %b
1.Qb 3.5b
8.4b 22b 0.3c
” Number of evenly spaced points taken is given in parentheses. Based on scalar error in the variable of 1% of the initial value. Scalar error minimized. Based on 1% relative error in the variable; relative error minimized.
’
I n order to show clearly the effect of including various extents of reaction, each “run” utilized just three points. As a standard, the “best” run used 20 points with optimum distribution (at eighth-life intervals). Of course, three points provide no error estimate for the variable, but they do provide the propagation of error term, and the error estimate for the variable was supplied independently.
499
The pronounced falloff in the standard deviation of
k as the per cent of reaction covered is reduced from 8 half lives to 7 to 6 to 1 is shown in Table IV. The ratio of 6 in the standard deviation for the first and last lines in Table IV gives an idea of the improvement in analytical accuracy involved in having a 1% relative error in the variable rather than a scalar error in the variable which is 1% of the initial value. There is not too much difference between the 1% scalar error and the 1% error in per cent T (last line in Table 111), particularly in view of the fact that the final stages of the reaction in Table I11 are subject to very large relative error owing to the particular range of T values used. It is possible to get highly precise estimates of a firstorder rate constant, but extreme precautions are essential if better than about 1% accuracy is to be achieved. The precision of measuring the values of the variable within a run is clearly only part of the story. Relationship between Apparent First-Order Rate Constant and True Rate Constants In actual systems departures from first-order kinetics may relate to no simple analytical expression. It is necessary to consider limiting cases, however, in order to show how typical departures can be treated. Tables I and I1 show that the apparent first-order rate constant obtained for a given set of data depends systematically on the extent of reaction covered. This is to be expected since the higher order terms contribute more heavily to the initial part. I n general, the rate constants are lower than the true value if the reaction is followed over 7 or 8 half lives and higher than the true value if followed to a considerably smaller extent of reaction. In general, with proper procedures, it is possible to get a reliable (2-4%) estimate of the first-order rate constant of a reaction of first plus some other order and of the second-order rate constant, in the A-B case. However, there seems to be no elegant way to obtain a reliable estimate of the nonfirst-order terms from the apparent first-order rate constants alone. The correct procedure for obtaining the rate constants is a small but essential variant on the standard way: the apparent first-order rate constant is plotted as a function of A0 for a series of values of Ao. For a reaction of first plus second order the abscissa involves Ao, for first plus three-halves order it involves and for first plus half order it involves A0-l”. For the A-B second order reaction, a plot of k/Bo US. Ao/Bo gives a straight line which gives the correct value of k (BO= 1)where Ao/Bo = 0. Volume 70,Number 9 February 1966
500
The key point is that in making these plots it is essential that all data refer to the same extent of reaction within a few per cent. It is obvious from Table I1 that it would be futile to attempt to combine data covering 5-99% reaction with data covering 5-50%. However, the 5-50010 data are not incompatible with the 0-40% data, and some leeway is therefore available. The optimum extent of reaction to use in applying the first-order equation is not immediately obvious in view of the fact that the standard deviation of k must go through a minimum a t some extent of reaction. Fortunately, this point needs t o be answered only in rather general terms. In most cases it is advantageous to cover as large a fraction of reaction as possible. The effect is shown by comparing the first
T h e Journal of Physical Chemistry
DELOSF. DETARAND VICTORM. DAY
four entries in Table I with the entries in Table IV. If the accuracy in determining the variable is 0.1% of its initial value, then the standard deviation of k is 0.24 X 1.9 = 0.46% for &99% reaction, 0.17 X 3.5 = 0.6% for 5-91%, and about 0.1 X 8.4 = 0.8% for 5 7 2 % reaction. In the first two cases, the error in k arises primarily from systematic error while in the latter case it arises from the analytical error. The larger the analytical error, the greater the advantage in covering as great an extent of reaction as possible. In conclusion, the proper application of the firstorder equation to reactions deviating only moderately from first-order kinetics can be a very powerful tool in analyzing sets of data and in obtaining the correct rate constants. The approach has its limitations, and for complete treatment other methods of calculation must be used which assume the correct kinetic form.
