J. Phys. Chem. 1993,97, 5321-5326
5321
Reversible, Mixed First- and Second-Order and Autocatalytic Reactions as Particular Cases of a Single Kinetic Rate Law D. Lavabre, V. Pimienta, G. Levy, and J. C. Micbead Luboratoire des IMRCP, URA uu CNRS 470, Universitb Paul Sabatier F-31062 Toulouse, France Received: December 29, 1992; In Final Form: February 18, 1993
Seven general kinetic schemes encompassing reversible, mixed first- and second-order and autocatalytic reactions were shown to be governed by a single integrated rate law involving two parameters a and m: Y(t) = (1 m)/(exp(ar) - m),where Y(t)is the normalized amplitude of the kinetics. This mathematical expression is derived by integration of a single variable quadratic differential equation. The parameter m characterizes the geometrical shape of the curve and is independent of the scales of both signal amplitude and process duration. The value of this parameter (--EO < m < 1) provides information on the nature of the reaction mechanism. Parameter a represents an overall rate constant. Experimental examples and numerical simulations illustrate the properties of this mathematical expression, especially for the description of the kinetics of many common chemical reactions.
Introduction
I
A large number of chemical systems have a relatively simple reaction mechanism. Using one or more conservationequations, the number of independent variables in such systems can often be reduced so that they can be described in terms of a single kinetic equation. Among systems of this type, we describe here seven general kinetic schemes whose rate equation in a batch reactor can be expressed as a single-variablequadratic equation:
r - .
\
\
/'
+
D = -dY/dt = a Y byz (1) where Y is a dimensionless variable that is characteristic of the course of the reaction; t is time. a and b are parameters whose values depend on the particular kinetic scheme under consideration. Y lies between 1 (for f = 0) and 0 (for t m); Y is thus proportional to the extent of departure from equilibrium.' In the present paper, we establish the integrated function corresponding to the quadratic rate eq 1 and show that it applies in turn to the seven kinetic schemes described below.
-
Properties and Integration of a Quadratic Rate Equation To illustrate the relative influence of the parameters a and b, Figure 1 shows the parabolic relationship between rate u and the progress of the reaction represented here by (1 - Y). In the neighborhood of the equilibrium Y -,0, the quadratic term bYZ becomes negligible and v aY: the reaction is first order. Since v is always positive, parameter a will always be positive. At the start of the reaction, Y = 1 and v = a b, which makes ( a + b) positive as well, and so b > -a. The derivative of the rate as a function of Y is
-
+
+
dv/dY = a 2bY (2) This becomes zero for Y = -a/2b, and the rate then passes through a maximum. If -a < b < -a/2,this maximum occurs for l / 2 < Y < 1. If b > -a/2,the rate curves are monotonic and decaying. Expression 1 is integrated in the usual way, noting that l / ( a Y + b y 2 )= ( l / u ) ( l / Y - b / ( a
+ bY))
(3)
Thus l y d Y / Y - b l Y d Y / ( a + bY) = - a c d r
(4)
and hence In Y - ln(a
+ bY) + ln(a + b ) = -at
l - Y
Figure 1. Parabolicrelationshipbetween the. relative rateofthenormalid reaction -(dY/dt)/(a + b) and (1 - Y). The numbers arc the corresponding values of m = b / ( o + b).
(5)
which can be transformed into
+
Y = a / ( ( a b ) exp(at) - b ) Letting m = b / ( a b), expression 6 becomes
+
(6)
Y = (1 - m)/(exp(at) - m) (7) where the parameter m, which lies between 1 and --OD, characterizes the geometrical shape of the integrated curve (shape parameter). Relationship7 is representedin Figure 2 with the same parameters as described above. When m 1, Y- 1/( 1 + b f ) ,expression 7 behaves as a pure second-orderfunction.2 For 0 < m < 1, the kinetics are between second and first order. m = 0 correspondsto first order. For -1 < m < 0, the course is always monotonic, although the curve is stretched out. For m = -1, there is an inflection at the origin. If m < -1, the curve becomes sigmoid, whereas the induction period increases as m --OD.
-
-
Examples of Kinetic Schemes Giving Rise to a Single-Variable Quadratic Rate Equation We have collected together seven kinetic schemes whose rate equations can be represented by a single variable quadratic
0022-3654/93/2097-5321$04.00/00 1993 American Chemical Society
Lavabre et al.
5322 The Journal of Physical Chemistry, Vol. 97, No. 20, 1993
SCHEME 11: Parallel First and h e Second-Order Reactions u , = k,[A]
A-B A
time Figure 2. Graphical representation of eq 7 for different values of m. u = 1 for all the kinetics, time is in arbitrary units.
+A
-
C
u2 = k2[AI2
This process involves a reactant A of initial concentration [&I , which undergoes a monomolecular reaction (first order) as well as a dimerization (second order). For example, in radical reactions, A is a radical, which may be either trapped by the solvent in a pseudo-first-order process (kl)or dimerize (k2).3*4 The changes in concentrations of the species [A], [B], and [C] are given by -d[A]/dt = uI
equation of type 1 and which can be integrated to relationship 6. Each of these schemes is presented in its most general form from which particular cases such as first order or pure second order can be derived.
SCHEME I: General Second-Order Reaction A+B+C
+ 2u2
(16)
d[B]/dt = u,
(17)
This scheme leads to the normalized variable type 1 quadratic equation by putting Y = [A]/[&]:
u=k2[A][B]
-dY/dt = k, Y The corresponding kinetic equations are
+ 2k2[Ao]Y2
(19)
with
-d[A]/dt = -d[B]/dt = d[C]/dt = u
(8)
The conservation law is [&I - [AI = [Bo] - [Bl = [Cl - [Col where [&I, [Bo], and [Co]are theinitial concentrationsof species A, B, and C. If [Ao] > [BO],let Y = [Bl/[Bol = 1 + ([AI - [Aol)/[Bol = 1 + ([COI - [CI)/[Bol (9) Combining relationships 8 and 9, one obtainsa quadratic equation of type 1: -dY/dt = k2([Ao]- [Bo])Y+ k2[Bo]Y2
a = k,
(20)
In this case, the shape parameter is given by
m = 2k,[&I/(k, + Zk,[&l)
(0 < m
< 1)
(22)
It should be noted that neither [B] nor [C] are linear functions of y:
(10)
with [C] = k 2 [ B o ] 1 Y Zdt
Transformation to the observablevariableAbs is obtained from relationship 14. If EB and cc are f O , the observablekineticcurve is not a linear function of Y and eq 15 does not apply. On the other hand, if CB = ec = 0, the absorbance depends only on [A], and relationship 15 applies.
In this case, the value of the shape parameter is
m = [Bo]/[&]
(0 < m
< 1)
(13)
In practice, Yis not directly measurable, and so an observable
SCHEME III: Parallel First-Order and Autocatalytic Second-Order Reactions
signal is chosen that is a linear function of Y. For example, in UV/visiblespectrophotometry,the absorbance Abs of thereaction mixture is given by AbS
(€*[A] + €e[B] + +[C])I
(14) where care the molar extinction coefficients of the corresponding species and I the optical path length of the measurement cell (generally 1 cm). A linear relationship between Abs and Ycan be obtained from relationship 9 since [A], [B], and [C] are all linear functions of Y: 5
AbS = Y(Abs0 - Absi,,,)
+ Absi,,f
(15) where Abso = (cA[A0]+ c~[Bo]+ cc[Co])l at t = 0 and Absinf att--. Scheme I includes also pure second-order (if [Ao] [Bo], then m 1 and a 0) and pseudo-first-order reactions (if [ao] >> [BO]or [Bo] 0, then a k2[&], b 0, and m 0).
- -- -
-
(24)
-
u, = k,[A]
A-B A
+B
+
2B
u2 = k2[A] [B]
The first reaction is first order, while the second is catalyzed by product B and is second order. The set of reactions behaves as a spontaneousautocatalytic process, which starts without prior addition of the autocatalyst B. The kinetic equations for the species [A] and [B] are -d[A]/dt = d[B]/dt = u1
+ u2
(25) The normalized variable Y is obtained from the conservation equation [A] + [B] = [&I [Bo] by putting
+
y = [Al/[Aol = 1 - ([BI - [B,l)/[&b] (26) Combining relationships 25 and 26, the following type 1 quadratic
The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 5323
Particular Cases of a Single Kinetic Rate Law
order rate law as e ~ p e c t e d . ~If. ~k2 = kf2 (& = 1) and thus m = 0, the relaxation is still first order whatever the initialconditions. Transformation to the observable variable is carried out in the same way as for Scheme 111.
equation is obtained:
SCHEME V: Second- and First-Order Reversible Reactionip b = -k,[&I
(29)
In this case, the shape parameter is given by
m = -k,[&I/(k,
+ k,[BOI)
(--m
< m < 0) (30)
For values of m C -1, the curves are sigmoid. Transformation to the observable variable Abs is derived from relationship 14. Relationship 15 is valid as both [A] and [B] are linear functions of Y.
SCHEME N:Second-Order Reversible Reactions A +B
C+D
A+Bi=C
u=k,[A][B]-kl[C]
This scheme describes association equilibria (metal-ligand, drug-receptor interactions, charge-transfer complexes, etc.). The changes in concentrations of the species are governed by the following kinetic equations:
-d[A]/dt = d[B]/dr = d[C]/dt= u
(38)
Concentrations [A], [B], and [C] are related by the conservation equation
R = [&I - [AI = P o 1 - [BI = [CI - [Col
v = k,[A] [B] - k
