The Kinetic Rate Law for Autocatalytic Reactions Fernando Mata-Perez and Joaquin F. Perez-Benitol Universidad Central de Barcelona, Avenida Diagonal 647, 08028 Barcelona, Spain
It is a common practice among authors of chemical kinetics textbooks to include a series of rate laws anolvine . to different kinds of chemical reactions (I).However, the case of autocatalvtic processes is often ienored despite the exten. . s k e oanlrrence of this sort of phenomenon. As a consequence uf this omissiun, there exist.; a definite lack of iniormation about the way the rate constants of this type of reaction can be derived from experimental data. In part to fill this gap in the teaching of kinetic rate laws, hut also in the belief that this topic may provide an interesting.applica-. tion of mathematical t o resolvechemical problems, a method of obtaining accurate rate constants for autocatalytic reactions will he presented in this paper. In order to illustrate the general procedure with a practical case, the autocatalvtic oxidation of dimethvlamine hv . .nermaneanate ion in aq;eous solution has been ihosen as an exampik (2).
..
The Dlfferenilal Rate Law A reaction is autocatalytic when a product-concentration vs. time plot exhibits an S-shaped profile (3);accordingly, a reaction-rate vs. time plot shows a bell-shaped profile with both an acceleration and adecay period. For that behavior to be explained, two mechanisms must he involved in the total process, one of them being responsible for the start of the reaction and the initial formation of products and the other using one of those nroducts to accelerate the reaction. These twomechanisms can be designated as noncatalytic and catalvtic, res~ectivelv.If pseudo-first-order conditions are used. the reaction rate-can b e expressed as r = kJR1 + k,[RI[Pl
(1)
where kl and kz are the rate constants corresponding to the noncatalvtic and the catalvtic mechanisms. resnectivelv. I t has bwnassumed that the noncaralytic mechanism is &st all theothers being in creat order in uneof the reactanrs (HI, excess, and that the catalytic mechanism is f i r s t i r d i r in both reactant (R) and autocatalytic product (P). This assumption allows an easy mathematical solution, besides heing, among all the possibledifferential rate laws, that of most general applicability. Equation 1can also be written as -de/dt = kle
+ k,c(c, - c)
(2)
where c represents the reactant concentration, its initial value being co, and (co - c) is the product concentration, provided that one molecule of autocatalytic product is generated when one molecule of reactant is consumed. If another stoichiometric relation holds. the nroduct concentration will be directlv proportional to ('c - c l , and the mathemiltical formof ea ?will still hr valid. It is easv tonee that nnother form of writ& the differential rate lawis ~
~
Therefore, when the ratio r/c is plotted against c a linear relationship must be obtained.
' Author to whom correspondence should be addressed.
Table 1. Concentration and Rate Values during the Reactiona
u
t (min)
10'c
3 4 5 6 7 8 9 10 11 12 13
4.814 4.766 4.712 4.648 4.578 4.492 4.401 4.299 4.186 4.057 3.918 3.768 3.601 3.429 3.247 3.054 2.855 2.657
14
15 16 17 18 19 20
'[Permsnganatel = 4.93 X to4 = 25.0 OC. 'Permanganate CReactil)nrats.
(M)~
M;
10Br(Mm i n - y
5.10 5.90 6.70 7.80 8.85 9.65 10.75 12.10 13.40 14.45 15.85 16.95 17.70 18.75 19.60 19.85
-
[dlmemylsmioel = 0.20 M: pn = 8.0: tempera-
tUrB
concentration.
In Tahle 1 the data obtained from monitoring the reaction between dimethylamine and permanganate ion in aqueous solution are given. A phosphate buffer was used to keep the DH of the svstem constant. and dimethvlamine was oresent in great excess with respect to permanganate. Hence, c represents in this case the permaneanate ion concentration. Since the rate of reaction, r, is given by the negative of the derivative of the reactant concentration with respect to time, -dc/dt, determination of the rate values a t several moments throughout the reaction requires that the corresponding derivatives be estimated. For that purpose, a simple device may be used. If three consecutive time-concentration pairs of data from Tahle 1,designated at (tl, cl), (tn, c, and (tg, cg), are fitted according to the first few terms of a Taylor series expansion what is indeed a reasonable approximation for short intervals ( x , y, and z are parameters to he calculated in every case), the rate value corresponding to the middle point will be r2 = - y - 2zt2
(5)
By substituting the three pairs of values in eq 4 and taking into consideration that the differences in time between two consecutive points from Table 1are constant. At. it is easv to deduce the following approximate expression fo; the rate of reaction at time tz:
Equation 6 provides a method, both simple and accurate, Volume 64
Number 11 November 1987
925
0
10
20
t , min Figure 2. integral rate-law plot
Figure 1. Differential rate-law plot,
+
to obtain the values of the rate of reaction from the concentration data. These rate values have also been compiled in Table 1, and we can see that they correspond to the acceleration period of the reaction. The ratio rlc has been plotted against c in Figure 1, and the linear relationship confirms that eq 3 is fulfilled. From the intercept and the slope of this straight line the values of the rate constants, kl and kz, can he derived. These results, along with the linear correlation coefficient, appear in Tahle 2.
The lntegral Rate Law Integration of eq 2 leads to the following rate law: We can see that a plot of the left-hand side of eq 7 vs. time must he linear and that from the intercept and the slope of this straight line the values of kl and kz can be deduced. However, obtaining the rate constants of an autocatalytic reaction by means of an integration method is not an easy task, for the values of those rate constants must first he known to make possible the plot of the left-hand side of eq 7 against time. T o resolve this situation an iterative procedure can he used. First, the left-hand side of eq 7 is fitted against time by the least-squares method with the help of some trial values of the rate constants. From the intercept and the slope of that fit new values of kl and kz are ohtained, and the fit is then repeated with them. The cycle is continued till convergence is achieved, yielding a t the end the correct values of the rate constants.
Table 2.
"
5.
Equation 7 requires that the use of an iterative procedure with two variables (kl and kz) be learned. Application of such a procedure is not frequent in chemistry. Figure 3 shows that the final values of the rate constants obtained from the iterative method are totallv inde~endentof the trid values used to start the calcula~ions.This intqral rafe law has been usrful for the kinetic studv of the autocaralvric permanganate oxidations of dimethylamine (2),trimethGlamine (4) and other amines (51,and aminoacids (6, 7). and was also proposed for the acid-catalyzed hydrolysis of esters (8).
I
I
Results Obtained from the Differential and the Integral Methods
Method
103k, (rnin-')*
Differential Integral
6.01 5.96
k2(M-' 308 309
'Noncatalytic rate constant. LCatalytic rate constant. r ~ i n e carelation ~r coefficient
928
A plot of ln[(kl kzco - klc) fc] vs. t is shown in Figure 2. In Tahle 2 the rate constants and the linear correlation coefficient obtained from the integration method are given and can he compared with those inferred from the differential method. We see that both methods lead to nearly the same rate constants, although the correlation coefficient associated with eq 7 is better than the one corresponding to eq
Journal of Chemical Education
'w0
0.99988 0.9999969
Figure 3. Convergence of the successive iterations toward the final values of the rate constants from different starting points.
Figure 5. Dependenceol the rate constants obtained hornthe Integral rate law on the initial concentrationvalue. Figure 4. Dependence of the rate constants obtained hom the differentialrate law on the initial concentration value.
Dependence on the Initial Concentration
I t can he inferred from eq 3 that the rate constant k2 derived from the differential method has no dependence whatever on the initial concentration value, hut the same cannot he stated as far as kr is concerned. In fact. k~ can he ohtained from the interrept'corresponding to eq 3 only if an accurate estimation uf thc initial conrentrarion is available. This situation has been represented in Figure 4, and we can see that a kl vs. co plot gives a straight line whose slope is -kz. On the other hand, both the kl and the kz values deduced from eq 7 depend on the co value used in the calculations, for the left-hand side of that equation includes co. The integral rate law has been applied with different co values and the results ohtained for the rate constants are shown in Figure 5. The k, vs. co wlot is aeain a straight line. the done heine now +kp (the converse ofthat o f ~ i g . i )while , the rate const& of the catalvtic mechanism decreases in a nonlinear wav as the initial conrentrution increusw. The lineur rorrelation coeffirient associated with cu 7 has been plotted in Figure 6 against the initial concentraGon. It can he appreciated that, while the correlation coefficient corresponding to the differential rate law is independent of the co value, that of the integral rate law shows a maximum when the correct co value is used and decreases as co either increases or decreases with respect to that value. I t follows from this discussion that availahilitv of an accurate value of the initial reactant concentration isan essential factor in ohtainine correct values of the rate constants of an autucutalytic reaction, although the importance or that fnctor is clearly reduced if the ditferential method id used instead of the integral rate law. Notwithstanding, since the integration method leads to a higher linear correlation coefficient (see Table 2), the rate constants obtained from eq 7
F g u e 6 Dependence01me m a r carrelatton mlfcient associated withme inegtai rate aw on the mtia concentrat on va ue.
might be more reliable than those deduced from eq 3. Literature Cited 1. Wilkinaon, F. Chemicd Kinetics and Reoction Mecharisma; Reinhold: New York. 1980: P 36. 2. Mata-Perez, F.; Perez-Benito, J. F.: hranz. A. 2.Phya. Chem. (NeueFdge) 1984 135, ill. 3. Mooro,J. W.:Pesrson.R. G.KinefiesandMechonism3rded.: Wiley: NewYork, 1981:~ 26. 4. Msts-Perez, F.: Perez-Benito. J. F. An. Quim. 1985.81A. 79. 5. Mala-Perez, F.; Perez-Benito, J. F. 2. Phys. Cham. ( N e w Folge) 1984,141,213. fi. Andres. F.J.;Ar"zabaisga,A.; Martinez, J. 1.A" Quim. 198480A.531. 7. Andres, F. J.:Arrizabalaga.A. An. Quim. 1985,8IA,431. 8. Capoilos. C.; Bioiski. 8 . H. J. Kinetic Syalems: Wiley: NewYork, 1972:p 63.
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Number 11
November 1987
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