A Test of the Validity of Steady State and Equilibrium Approximations in Chemical Kinetics Vincente Viossat and Roger I. Ben-Aim Universite P. et M. Curie - 75005 Paris (France) Kinetic studies focus on various interests, including physical parameters (temperature, initial concentrations, etc.) and kinetic data (orders, activation energies, etc.). Such studies in closed and homogeneous systems a t coustant volume and temperature generally lead to nonlinear ordinary differential equations: the chemical rate equations. These equations have the following form.
This mechanism is often encountered in chemical kinetics. Some examples are well-known:
a acid-base catalysis ( I ) enzyme catalysis (2) transition-state theory (3) quasi-unimolecular reaction theory by Lindemann (41 nucleophilic and electrophilic substitutions (5) Two bfferent cases corresponding to the general method of approximation described above will be considered: Quasi-Stationary State Approximation (QSSA) r Quasi-EquilibriumApproximation (QEA)
where the concentrations of the species are denoted as XI, ..., x"; and the corresponding time derivatives are denoted by x,,...,i,,. In general these differential systems are difficult to solve analytically, so approximate or numerical solutions are often used. A general method for obtaining such appmximate solutions is to suppose that some of the functions f,, ..., fi are equal to zem. The system thus becomes f1 = 0
The exact solution for this mechanism is obtained when the reactions are first-order. Afterwards, the analytical expressions of the solutions calculated using the above ap~mximationsare derived. Finallv the solutions are compared either analytically or by"numerica1 simulations. Thus, the conditions of validity of the a ~ ~ r o x i m a t i overv ns commonly used in chemistry are p i t ' i n evidence an; tested. Kinetic Equations and Analytical Solutions Exact Theory The rates of reactions 1, 2, and 3 are denoted by u,, u,, and US. The concentrations of A, B, and C are denoted by the same symbol as the corresponding chemical species. The elementary reaction rates are assumed to be firstorder. The overall rate is calculated either for the reactant de-
let ion "=-A="
This system of j algebraic equations and n -j differential equations clearly has a simpler solution than the initial system. This method is similar to the theom of dvnamical svstems where information about the system is obtained from stationarv solutions of all the variables. These stationarv solutionsdefine the "attractor" that the system approaches when the time tends to i n f i t y . Evidently, relations such as fi = 0 and fi = 0 allow expression o f j concentrations as functions of the other n -j concentrations that are time-dependent. This is why the X I , ..., x, concentrations, which are now implicit functions of time, are called "quasi-stationary". Solving the remaining n - j differential equations leads to a solution that is an appmximation of the initial system. In this paper we investigate the following very simple mechanism, from this point of view. 1 A ~
3 B -
where A is the reactant; B is the intermediate compound; and the C i s product.
732
or for the accumulation of final U ' = C =3 ~ Generally the rates u and u' are different due to the presence of the intermediate compound B. The kinetic equations are A = -vl+ uz =-k,A+ k,B
Journal of Chemical Education
(11
B = V ~ - U ~ - U ~ = ~ , A - ~ , B - ~ ~system B 1 (2) C=u3=kSB
(3)
System 1 clearly corresponds to a particular case of the general mechanism reported in the introduction. I t is customary to consider the following initial conditions. A(O) = %
B(0)= C(0)= 0
c
2
1-02
The solution of system 1has been given by Lowry and John (63,Johnson (71,Rodiguin and Rodiguina (8). A = (b- A,)-'(& - kl) et'~t) - (XI - kl) et"lt) -
4
with With conditions ii, the solution is
Quasi-Stationary State Approximation
In the QSSA(9) one assumes ul-"~-U3=0
Equation 2' replaces eq 2 in the differential system 1. It follows that the overall reaction rates u and u' calculated for the reactant or the product are the same. The differential system is A=-k,~+k@
"
(2')
C = (kz + k,) k l + k , + k 3 [I -
AE])
(9')
We may notice that eqs 7',8',and 9' are deduced from eqs 7,8,and 9 by changing & into
(1) system 2 (2')
The solution of system 2 depends on integration constants, which in turn are specified by the initial time. However, the exact initial conditions, A(0) = %
As noted previously, it is evident that eq 8 does not satisfy the condition B = 0 at t = 0.For this reason QSSA is applied only during a certain time interval &er the reaction begins, as usually specified in textbooks. Quasi-EquilibriumApproximation
In the QEA (10)one assumes v,-v,=o
do not hold for eq 2'. It is absolutely necessary to adopt initial wnditions consistent with eq 2'. which wnstitutes the basis of the approximation. Two simple different conditions come to mind: -i: A(0) = % =$
and C(0) = 0
(At t = 0 the concentrations of A and C are the exact ones, whereas the concentrationof B is modified. As a result, the mass balance of the whole system is not obeyed.)
(1')
Equation 1' replaces eq 1 in the differential system 1, which becomes 0=-klA+k2B
B = klA - (k,
( 1')
+ k,)B
system 3 (2)
The overall reaction rate u for reactant depletion is always null, whereas u', the reaction rate for product accumulation, is nontrivial and always positive (except at the beginning and the end of the reaction). This surprising result wmes from the hypothesis that, at each time, the equilibrium between A and B is reached while B reads to form C. Solution of system 3 evidently depends on integration constants. At the initial time, the exact initial conditions 4 0 ) =%
do not hold for eq 1'.We decide to maintain eq 1' and one of the following conditions: (Mass is conserved but the initial conditions are modified for Aand B.) Acwrding to the wnditions adopted, it is easy to obtain the integrated form of system 2. With conditions i, the solution is
and C(0) = 0
(Mass is not conserved at t = 0, and the initial condition for B is modified.) Volume 70 Number 9 September 1993
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8 8
the differential system 1 (A+B + c = 0 ) the integrated expressions of A, B, and C
8
simple chemical considerations
or
The parameter r = A + B + C is introduced for approximate solutions, and the ratio rl& is compared to 1.
and
Quasi-Stationary State Approximation
(Mass is conserved, but the initial conditions are modified for A and B.) According to the condition adopted, it is easy to obtain the integrated form of system 3. With conditions i, the solution is
With QSSA and initial conditions i,
The limited expansion for a short time interval is
The condillon that makes this expression close to 1 for the initial time and fir the shon time intervals is k l
