L. Volk, W. Richardson, K. H. Lau, M. Hall, and S. H. Linl Arizona State University Temp. 85281
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Steady State and Equilibrium Approximations in Reaction Kinetics
In the kinetics of complex reactions, the steady state and equilibrium approximationsare often used (I). However, little has been done to investigate the validity of these approximations on quantitative terms. I t is the purpose of this paper to study the limitation of the application of these approximations to the kinetics of complex reactions. For this purpose, we shall study the following two types of reactions
I:;.
3
.;:2.
kc
.,.
bz
A e B-C
*. ..-.
The reason for choosing these reactions is to investigate how the reversible reaction affects the applicability of the steady state approximation.
I 2
Reactlon (1)
To describe the approach we adopt in discussing the validity of the steady state approximation or equilibrium approximation, we shall go through reaction (1) rather in detail, although most of the results are well known. The rate equations for this case are given by
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1
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6
8
10
12
I4
16
ati Figurn 1. Plot of Dverrus At' far various k2'.
d A - -klA --
dt -d B - k l A - k2B dt dC - kzB -(3) dt and the exact solutions of these equations based on the use of the initial conditions A = Ao, B = C = 0 can he written
and B =k l A o ( e - b , t kz - k ,
- e-k2t)
(5)
+ +
Notice that A B C = Ao In order to test the validity of the steady state approximation as applied to B, we compare the exact expression for the rate equation for C with the approximate rate equation for C obtained from the use of the steady state approximation, i.e. D(ki,t) =-- - kn [I - e-,b2-k,)t] k,A k 2 - k ,
(6)
The factor D may be regarded as representing the extent of deviation for the steady state approximation. In other words, the factor D may be used to test the performance of the steady state approximation. However, it should be noted that this is not the only way to study the validity of the steady state approximation. It is useful to consider the time required for the steady state to he reached. If we let the time a t which the concentration of B is a maximum be denoted by t., then an explicit expression fort, can be obtained from setting dBldt = 0, i.e. t , = -log-kz k z - k ~ k~
(7)
At t = t,, D = 1. That is, a t this point the steady state approximation holds exactly. For the purpose of numerical comparison, it is convenient John Guggenheim Fellow.
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i 5
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10
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15
i 25
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30
Izf
Figure 2. Plot of ts'
versus ke'.
to use aset of dimensionlessvariables defined by kz* = k d k ~ t* = kit, and B* = B/Ao. In terms of these dimensionless variables. we have
and t.*
=L
-
l o g kz*
k2* - 1 where At* = t* - t,'. When t*
m,
eqn. (8) reduces to
* D(kz*, m ) = A kz* - 1 k
(9)
(10)
Both eqns. (8) and (10) indicate that the steady state approximation is valid when k2* >> 1or k z >> k , . In other words, for reaction (I), the steady state approximation holds under the condition in which the intermediate B disappears quickly once it is formed. Figure I is a plot of D versus At' for various kz*, where At* denotes the time after which 1,' has been reached. The upper Volume 54, Number 2, February 1977 1 95
limit for each curve is kz*lkz' - 1,while the lower limit is D = 1. The distance from the D = 1line to each D versus At* curve gives the deviation involved in the steady state approximation for a particular k2* value. For example, for the steady state approximation to hold within 5%,i t is required that kz* > 20 (cf. eqn. (10)). Figure 2 shows a plot oft.* versus kzl; i.e., it shows how rapidly the steady state is reached under various conditions. Sometimes it is important to estimate the relative magnitude of the intermediate B relative to A, i.e., to see if the reaction intermediate is detectable. For this purpose, it should he noted that BIA = Dlkz', which indicates that under the condition in which the steady state approximation is valid, the concentration of B is very small. Suppose we require the steady state approximation t o be valid within 1%(i.e., 0.99 5 D _< 1.01). In this case, from eqn. (10) a n d D = 1.01, we can determine kz* to be kzt = 101, and from eqn. (a), k2* = 101 andD = 0.99, we obtain the time t ~ * = log 101/2/100 after which the steady state approximation will he valid within 1%. tw* should be compared with t.' = log 1011100. It should be noted that the maximum concentration of the intermediate I3 can he calculated easily by setting dBldt = 0 to obtainlog B,/Ao = -kz*l(kz* - 1) log &*.In this case, , is equal we have B,I& = (101)-1.0'. Notice that a t t ~ *A* to exp (-(11100) log (10112)) or 0.962. In other words, in this case, the extent of reaction is quite small (-4%). Similarly, suppose if we require the steady state approximation to be , valid within 0.1%, then BJAo = (lOO1)-l.W1and a t t ~ *A* is equal to exp (-(111000) log (100112)). In this case, the extent of reaction is less than 1%. Readlon (2)
Reaction (2) corresponds to the Lindemann scheme and much bas been discussed about this mechanism ( 2 4 ) . This is a tint-order consecutive reaction with a reversible fmt step; the rate equations are dA- - -klA
dt
..-.
.
The expression for ts* can he obtained from applying the condition dBldt = 0 to eqn. (13) t.' = log (A**/A,*)/(Xz*- A,*)
(20)
Assuming equilibrium between the reactant A and the intermediate B, i.e., the equilibrium approximation, we have B.,lA = kllk-, dC* = X A . dt* s. k-l*
(-1 and
De,(ki*t*) = k-l*B*IA*
(23)
From eqns. (23) and (la), D., is related to D,, by D,,(ki', t*) =
kL,* k..,* k2* Ddki*, t*)
+
(24)
The limiting form of D., is given hy D.,(k,*,
m)
= 1 - X2*
(25)
Plots of D, versus t* and Dq versus t * are shown in Figure 3 and 4, respectively. Both plots are based on values of the ratio k ~ * l k - ~larger ' than and smaller than one. From Figure 3, we see that for k ~ * l k - ~ 1, the larger value of k ~ * l k - ~gives proximation. Figure 4 shows that kz*lk-I* < 1the equilibrium approximation improves with decreasing kzxlk-I* and for k2*lk-'* > 1,the equilibrium approximation does not apply (see also eqn. (24)). The plot of B*IAx versus t* is shown in
+ kLIB
dB - - k,A - k-,B dt -- - kzB
;~.. :
- k2B (11)
Either the l.apla(wtransform or the matrix method ran be aonlied to solve ean. (11) exactlv: the resulting dimensionless
where and
A,* = %(l
Figure 3. Plat at D,, vsrsus t'
+ k-I' + kz* + d ( 1 + k-,* + k2*)2 - 4k2*)
with A* = AIAo, B* = B/Ao, and t* = klt. Here i t is assumed that at t = 0, B = C = 0. For reaction (2),both steady state and equilibrium approximations can be applied. Applying steady state approximation to B (i.e., dB/dt = O), eqn. (11) is reduced to B..* =
A* k_,* + k2*
(16)
and dCldt becomes
dC* -kz* A' dt' kL,* + k?*
(17)
Following the discussion for reaction (1) the deviation expression D,, can he obtained from eqns. (16) and (17) as D,(kis, t*) = (k-,*
+ k2*)B*IA*
(18)
where B* and A* are given in eqns. (12) and (13), respectively. When t* approaches infinity, eqn. (18) reduces to 96 I Journal of Chemical Education
ov 0
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05
1.9
Figure 4. Plot at D, versus t'.
2.0 tX(k,xt)
1.5
25
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3.0
1
35
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4.0
KI:
c Figure 6. Plot of',1
versus k2'.
Figure 5. In a previous paper ( 6 )we have applied the singular perturbation method7 to solve the Lindemann scheme and investigate the validity of the steady state and equilibrium approximations analytically. Plot oft.* versus k9* with k-T* varied from 0.1-35 is shown in Figure 6: From thisplot, one can see that a t small values of k-
