Research: Science and Education
Kinetic Analysis of Parallel-Consecutive First-Order Reactions with a Reversible Step: Concentration–Time Integrals Method A. E. Mucientes* and M. A. de la Peña Departamento de Química Física, Facultad de Química, Universidad de Castilla La Mancha, Avda Camilo José Cela 10, 13071 Ciudad Real, Spain; *
[email protected] French (1) and Wideqvist (2) independently described an approach for the treatment of kinetic data that makes use of the area under the concentration–time curve to obtain the rate constants. The concentration–time integral is the area (in mol s L‒1) under the curve for a concentration of a given species versus time and is therefore directly available from conventional concentration–time data. Thus, for a reactant A, the concentration–time integral, denoted θA,t, is RA, t
t
± < A> d t
The experimentally determinable concentration–time integrals method is useful for solving any kinetic equation, although it is especially applicable in solving complex kinetic equations of composite second and higher orders. The essential effect of this treatment is a reduction by one of the apparent order of the reaction. The availability of modern computer techniques for integrating experimental curves makes this method particularly attractive. The aim of this article is to show the student that the concentration–time integrals method provides an easy and accurate way to study parallel-consecutive first-order reactions with a reversible step. Throughout the text students can see that this method is a simple alternative to other methods for kinetic analysis and is therefore extremely useful in the treatment of kinetics in physical chemistry and biochemistry courses.
Consider a parallel-consecutive first-order reaction with a reversible first step. k1 k2
k3
B
k4
C
(2)
C where A is transformed into C via the reversible formation of intermediate B, and A also undergoes an independent and concurrent reaction to yield the same product C. There are numerous chemical reactions that show this parallel-consecutive first-order behavior. Examples include the reactions between the p-phenylene-bis-diazonium ion and water (3), the hydrolysis and cyclodehydration of dipeptide (4), and the decomposition of glucose (5). The differential rate equations are
390
(4)
d k3 < A > k4 dt
(5)
Integration of eqs 3 and 4 with [A] = [A]0, [B] = [C] = 0 at t = 0 gives
< A >0
© k2 k3 B1 exp B1 t
(6) B 2 B1 « k2 k3 B 2 exp B 2 t ¸º
k1 < A >0
©exp B1t exp B 2 t ¸º B 2 B1 «
(7)
The expression for [C] is easily derived using the mass balance relationship, [C] = [A]0 − [A] − [B],
< A >0 1
B 2 k4 exp B1t
B 2 B1
B k4 1 exp B 2 t
B1 B 2
(8)
where
Theoretical Analysis
d k1 < A > k2 k4 dt
(1)
0
A
d < A > k1 k3 < A> k2 dt
(3)
B1
p
q 2
B2
p
q 2
(9)
and p k1 k2 k3 k4
q ©« p 2 4 k1 k3 k2 k4 k3 k4 ¸º
1
2
(10)
The exact solutions are too complex to be used to obtain rate constants from experimental data (6) and, as a result, the steady-state and pre-equilibrium approximations are normally used to simplify the treatment of the data (7). However, despite of the complexity of the exact solutions, methods based on these have been developed to find the rate constants (8). We will now apply the concentration–time integrals method to the process depicted in eq 2. Equation 3 can be solved by integration between the limits [A]0 and [A] for a concentration of A, and 0 and t for time
t < A> t d A k k A d t k 2 ± < B > d t (11) ± < > 1 3 ±< > A 0 0 < >0
Journal of Chemical Education • Vol. 86 No. 3 March 2009 • www.JCE.DivCHED.org • © Division of Chemical Education
Research: Science and Education
< A >0
< A > k1 k3 RA,t k2 RB ,t
(12)
Differential eqs 4 and 5 can be integrated in a similar way, so that the expressions for [B] and [C] are
k1 RA, t k2 k4 RB,t
(13)
(14)
k3 RA,t k4 RB,t
Results and Discussion As an example we will use the hydrolysis of the p-phenylenebis-diazonium ion (A), which involves the reversible formation of a mono-anti-diazohydroxide (B) (3). Each of these substances yields irreversibly the p-hydroxybenzenediazonium ion (C) according to the kinetic scheme, k1
ArN2OH
k2
k4
ArOH
k3
k1 RA, t b k2 k4 RB, t b
< A >0
(17)
k3 RA,t b k4 RB,t b
RA, t b
[B]/[A]0
0.2 0.0 0
5
t
0
0
dt
RB,t b
t
0
0
±
dt
(19)
Concentration–time integrals, θA,t’ and θB,t’ at different times (Table 1) were obtained using the Excel complement Interpolation.xla, which contains a series of functions to interpolate (9). We used the function CERCHAREA, which enables the area under the curves [A]∙[A] 0 , t and [B]∙[A]0, t to be calculated from 0 to t, using cubic splines.
15
t / min
20
25
30
RA,t b [A]0 ź [A] ź ä 0.143 ź 0.0746 min 1 [A]0 RB,t b RB,t b R 2 ä 0.9998
0.6
0.4
0.2
0.0
ź0.2
0
1
2
3
RA,t b
4
5
6
RB,t b Figure 2. Plot of ([A]0 – [A])∙[(A]0 θB, t′) versus θA, t′/θB, t′.
Table 1. Relative Concentrations of A, B, and C and Concentration–Time Integrals at Different Times Time/min [A]/[A]0
±
10
Figure 1. Relative concentrations of A ( ), B ( ), and C ( ) versus time (Data from ref 3).
(18)
where
0.4
(15)
where Ar represents p-N2+C6H4. All rate constants are pseudofirst-order rate constants that depend on the acidity. Plots of the observed relative concentrations of A, B, and C versus time at [H+] = 8.71 × 10‒4 are given in Figure 1. Because the concentration data are expressed as relative concentrations with respect to [A]0, eqs 12–14 can be rewritten as < A >0 < A > k1 k3 RA, t b k2 RB, t b (16) < A >0
< A >0
[C]/[A]0
0.6
0.8
ArOH
[A]/[A]0
0.8
minź1
ArN2
1.0
[A]0 ź [A] [A]0 RB,t b
1.2
Relative Concentration
Using the concentration–time integral concept, defined in eq 1, the integrated rate expression thus has the form
[B]/[A]0
[C]/[A]0
θA,t ′/min θB,t ′/min
0
1
0
0
−
−
1.4
0.89
0.13
0.05
1.35
0.10
3.6
0.62
0.25
0.12
2.96
0.54
6.9
0.44
0.35
0.21
4.66
1.54
10.2
0.35
0.37
0.28
5.98
2.75
13.3
0.29
0.36
0.35
6.97
3.87
16.4
0.26
0.33
0.41
7.82
4.94
19.8
0.23
0.30
0.47
8.64
5.99
22.6
0.20
0.27
0.52
9.26
6.81
25.7
0.17
0.24
0.58
9.82
7.60
28.6
0.15
0.22
0.63
10.28
8.27
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 86 No. 3 March 2009 • Journal of Chemical Education
391
Research: Science and Education RB,t b [B] ź ä ź0.105 á 0.107 min 1 [A]0 RA,t b RA,t b
0.12
[C] [A]0 RA,t b
minź1
0.08 0.06 0.04
[C] ä [A]0 RA,t b
0.0314
RB,t b RA,t b
á 0.0337 min
ź1
R 2 ä 0.9854
0.06
minź1
R 2 ä 0.9954
0.10
[B] [A]0 RA,t b
0.07
0.05
0.04
0.02 0.00
0.03 0.0
0.2
0.4
RB,t b
0.6
0.8
0.0
1.0
0.2
0.4
RB,t b
0.6
1.0
RA,t b
RA,t b Figure 3. Plot of [B]∙[A]0 θA,t′ versus θB, t′/θA,t′.
Figure 4. Plot of [C]∙[A]0 θA, t′ versus θB, t′/θA, t′.
The four rate constants in eq 15 can be obtained from equations 16 and 17. As independent variables θB, t ’∙θA, t ’ or θA, t ’∙θB, t ’ may be used and as dependent variables the relative concentrations divided by θA,t’ or θB,t’ can be used, respectively. The rate constant k2 and the sum of constants (k1 + k3) were obtained by fitting the data ([A]0 − [A])∙[A]0 θB,t’, θA,t ’∙θB,t’ to eq 16 using the linear least-squares method. The slope gives 0.1426 min‒1 = (k1 + k3) and the y intercept ‒0.0746 min ‒1 = ‒k 2, respectively (r 2 = 0.9998) (Figure 2). The plot of [B]∙[A]0 θA,t ’ versus θB,t ’∙θA,t ’ was linear (r 2 = 0.9935) (Figure 3). The slope gives ‒0.105 min‒1 = ‒(k2 + k4) and the y intercept k1 = 0.107 min‒1. From these results the values of k3 and k4 were obtained and these were found to be 0.0353 min‒1 and 0.0300 min‒1, respectively. The rate constants k3 and k4 can also be obtained by fitting the data [C]∙[A]0θA,t ’, θB,t ’∙θA,t ’ to eq 18 using the linear least-squares method. The slope gives 0.0314 min‒1 = k4 and the intercept 0.0337 min‒1 = k3 (Figure 4). The values of these constants are very close to those obtained above. Students should appreciate the validity of this method by seeing the good agreement between the calculated values of [A]∙[A]0, [B]∙[A]0, and [C]∙[A]0 (solid lines) and the observed concentrations, as shown in Figure 1. The calculated values were obtained by substituting k1 = 0.107 min‒1, k2 = 0.0746 min‒1, k3 = 0.0345 min‒1, and k4 = 0.0307 min‒1 into eqs 6–8. The latter two constants are mean values of the above data. The concentration–time data can be fitted directly to eqs 6–8 (integration method) by using the non-linear least-squares method (10), but in this case the fitting produces multiple solutions due to the presence of multiple equivalent minima. Each constant is initially estimated, so to obtain accurate values of the rate constants it is necessary to use accurate starting values. Students may, therefore, observe that the application of the concentration–time integrals method to eq 15 allows one to obtain the rate constants, using the linear least-squares method, more easily than in the direct fitting. The method is, moreover, fast and accurate.
Conclusion
392
0.8
We have used the method of experimentally determinable concentration–time integrals to study parallel-consecutive firstorder reactions with a reversible step. This method avoids the use of the steady-state and pre-equilibrium approximations. It permits the fast and accurate determination of rate constants and is therefore a simple alternative to other methods for kinetic analysis. Literature Cited 1. French, D. J. Am .Chem. Soc. 1950, 72, 4806–4807. 2. Wideqvist, S. Acta Chem. Scan. 1950, 4, 1216. 3. Lewis, E. S.; Johnson, M. D. J. Am. Chem. Soc. 1960, 82, 5399–5407. 4. Faisal, M.; Sato, N.; Quitain, A. T.; Daimon, H.; Fujie, K. Ind. Eng. Chem. Res. 2005, 44, 5472–5477. 5. Xiang, Q.; Lee, Y. Y.; Torget, R. W. Appl. Biochem. Biotechnol. 2004, 115, 1127–1138. 6. Espenson, J. H. Chemical Kinetics and Reaction Mechanisms; McGraw-Hill: New York, 1981; pp 71–72. 7. Viossat, V.; Ben-Aim, I. J. Chem. Educ. 1993, 70, 732–738. 8. Andraos, J. J. Chem. Educ. 1999, 76, 1578–1583. 9. Interpolation.xla. http://personales.gestion.unican.es/martinji/ Interpolation.htm (accessed Oct 2008). 10. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969.
Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2009/Mar/abs390.html Abstract and keywords Full text (PDF) with links to cited URLs and JCE articles Supplement
Relative concentration and concentration–time data
Data used to determine the rate constants
Journal of Chemical Education • Vol. 86 No. 3 March 2009 • www.JCE.DivCHED.org • © Division of Chemical Education
