Application of Kinetic Approximations to the A B ^ C Reaction System -
Gregory
I.
Gellene
Texas Tech University, Lubbock, TX 79409-1061
Two of the most often discussed procedures for simplifying the kinetic analysis of chemical reaction systems are the steady-state (ss) and the preequilibrium (pe) assumptions (1). The domains of applicability of these kinetic assumptions as applied to the title reaction system are determined by a comparison of the exact solution of the rate equations to the corresponding approximate solutions.
The Simple Steady State The steady-state approximation is often introduced (1, 2) by considering the consecutive first-order reaction system,
Using eq 8 to eliminate [A]ss from eq rangement [B]ss
kx + k2
(EA]0 +
[B|0 + [C]0
IBl!S
=
[CP
=
+
+
[Bl0
=
d[Cl
kx\A]-
(3)
,
(4) can
be
analytically solved by elementary techniques to
(5)
IBI
[C]
=
=
I
Bitj
e** +
[Cfe +
f k2-kx
lI'A|° (e*'1
-e**)
(6)
[B^(l-e^) + [A]0 f,1 + k^-k^S) k2~ki I
t
{
.
)
(7)
The steady-state assumption for the reaction intermediate (i.e., d[B]ss/dt 0) is usually applied to the special case of [B]0 Of however, this condition is not required. Because
rear-
(11)
(12)
+ [BJ0 + |C]0 [CP)e*J r7r,[A,« kx + k2 -
[C]0 + [Bio + [A]0(l
-
e"H
-
([Bio + tC]0
(13) -
[Cg5)
e*^ (14)
where kss =kxk2l(kx + k2). These approximate results can be compared to the exact solutions (eqs 5-7) to identify the domain of applicability of the steady-state assumption as k2» kx (i.e., k2±kx =k2 and kss kx) and t » l/k2 (i.e., e~k2< 0 and [CJ5E = [B]0 + [C]0). Under these conditions the familiar steady-state solutions for this reaction system are recovered. Although this approach cannot be generalized to explore the conditions under which the steady-state approximation for an arbitrary kinetic system is valid (3), the analysis does reveal the two critical minimal requirements: an intermediate that is destroyed faster than it is produced and sufficient time for the steady state to be established. If the reaction scheme is extended to include the reverse of the first step of reaction 1, that is, -
yield the exact solutions.
[CD
-
[C]0-[C]gs) e*’*
(2)
^
gives upon
Substitution of eq 11 into eq 9 allows [Clss to be determined by direct integration. This result allows [B]ES to be determined from eq 11, which in turn allows [A]ss to be determined from eq 8. The final results are
IAP= —([A]0 k\+k2 for which the corresponding coupled differential equations
=
10
---
=
=
the general approach for applying kinetic approximations is not widely treated and will be used throughout, it will be outlined here. If the steady-state equations (8)
IS!! =^|BI di
(9)
the validity of the preequilibrium assumption can be similarly explored by this pedagogically satisfying approach of comparing exact and approximate solutions of the coupled rate equations. Although a numerical analysis of this kinetic system is available (4), an analytical approach appears not to have been explicitly treated previously (5) and is thus considered here. Although the solution to the more general case, which includes the reverse of the second step, that is,
(where the superscript denotes the kinetic approximation invoked) are combined with the mass-balance relationship |A|5S + [B]ss + |C]SS
=
[A]0
+
[B]0
+
[C]0
(10)
(which must hold for any valid kinetic approximation), a steady-state solution can be obtained for arbitrary initial concentrations. 196
Journal of Chemical Education
is available (6), the rate equations for k.% = 0 case are easier to solve and better suited for treating the preequilibrium assumption. With apologies to those readers familiar with the techniques of solving elementary differential equations, some intermediate steps will be explicitly indicated
clarify the presentation for those with cated mathematical background. to
less sophisti-
a
Approximate Solutions Preequilibrium
The Exact Solution The coupled rate equations for the title reaction system
The equations describing the application of the preequilibrium approximation
MAP MBF-
are
(29)
=
dlA]
A ;|
—
df
d|Bl
A]
A j-LAJ
df
did
+
+ A2)l
(k_t
-
BI
(16)
A3[B1
df
d|Cp
(15)
(17)
EAp
_J_d[BJ + k_i+k2 (18)
EBP
which upon substitution into eq 15 followed by differentiation and rearrangement yields
[Cp
dal B
]
+
At2
{k.I
+
k
df
+
k1+k2)^
k1k.J[B\-
(19)
This homogeneous second-order differential equation with constant coefficients is readily solved by substitution of the assumed solution, [B] ert, into eq 19, which generates the characteristic equation,
(30)
combined with the mass-balance relationship (eq 10 with the superscript “ss” replaced by “pe”), can be simultaneously solved by the largely algebraic approach that led to eqs 12-14. When this is done the results are
which can be solved exactly by the following approach (7). Solving eq 16 for IAJ gives
kj
fc2[BP
df
=
T~~r «-1 + «1
k]
= ,
.
k-i+fh =
-1cin
(Mo + IB|0 +
[C]0
([A]0 + [B]0 +
tCIo-1Clg6)eV
-e“V)
[C]0 + ([AJ0 + [B]0) (1
(31)
([C]0
-
-
(32)
|C]g^)
e“V
(33)
where kve= k\k2/(k-X + kx) and the familiar preequilibrium results are recovered for IC Iff = [Clo. Anticipating that the domain of applicability of eqs 3133 will require the equilibrium reactions to be fast (in a sense to be determined) relative to the rate of C formation, we rewrite eq 21 as
=
r2
+
(k_x + kx +
+
k2)r
kxk2
(20)
0
=
(k_x + kx +k2) ± ((A
=
+
1
l
2k2
.
j
kx+k2f -4kxk2)h-
+
£-1 + ^1
IB1 A
A!
L|
+
c+(r+
|
A1 now
c+erJ + c_er
=
k_x +k2)er'1
I
A
j0
=
(21)
B](j
=
be given as
can
Ail B lo AjlAlg
(r_ + A1
—
)|
A |0
+ ^_1 +
k2)erJ
(23)
(^4- +
[B | =-e1*+-e
r+
2
1 +
h_x +
k2
hx
kx k_x + kx
Neglecting terms beyond first-order in k2/(k.x the approximate results =
—kxk2/(k_x
+
kp
=
+
+
k_x
-f-
A_j
r_
(25)
k2)
A2)l B ^
kx) gives
k2/(k_x
+
kxY)
~
k\)
(36)
+ A1)
(371
+
hx
k-i
Substitution of eqs 36 and 37 into eqs 26-28 yields
,
(26) —
—k_x( 1 +
—
(k2 «
IA|
A_i([A]0
+
lB](j)
AjLAJq
,
=
AiLA]a
Ai(lAj0 |B|0) k_x+kx
-
—
A_1[B(t)
ill, *h,)l
kx+kx
k_x + k +
r+- r.
-r.
+
—k
(k2«k-i
—
(r_ + k_x + A2)lB]q
+ kx)
(24)
c+ + c_
cjr_
ex-
(35)
r+
(r++ &i)[Alo A_jl BJj r+-r_.
r+~r-
j
be assigned
can
+-e
*
(A„i
(22)
+c_(r_
c+(r+ + k_x + k2) +
k2) ±
+ kx +
2r± = -(&_]
When this is done, the exact solutions in their most symmetric form are given by [A | =—--e
AS
1
'
where the arbitrary constants, c+ and c_, by simultaneously solving the equations
A [!
(k_x+kx)2
If k2 « (k-\ + kx), the square root in eq 34 can be panded in powers of k2/(k_ + kx) with the result —
The general solutions for | B] and
=
4A |)
^2(^2
(34)
having the solutions 2r±
2r±=-(k_x+kx + k2)±(k_x + kx)\
^
^
A_|1
Blt| A
„
1
(38)
-
+ a-
A|)f
(39)
A1
(27)
(r 1C]
=
+
MBIo
k_x +
|C|0
r+(r. AxIAIq
+-k2
—
-
—
A j
IA
[C]*[C]0 + ([A]0 + [B]0)(l-Kk__\ + k \ + k2). With the expectation that the validity of the steady-state approximation will require k\ « k-i + k2 (for which kss = k\k2l(k-i + k2)), eq 21 is rewritten as 2r±
I-
k
A comparison of eqs 49-51 with eqs 43-45 indicates that
are
t
[B]
+
(51)
k\«(k_\ I.Af8
ki [Al0 k-i
(50)
(42)
&2)[B]“
„-k,j
k-l + kx (k^ + kxf
to the approximate solutions obtained under the assumptions of preequilibrium and steady-state indicate that the preequilibrium assumption is valid for k2 « (k-\ + k{! and t » IKk-i + k\), whereas the steady-state assumption is valid for k\ « (k-\ + k2) and t » \l(k-\ + k2).
These conditions are not mutually exclusive. For example, both kinetic
assumptions
are
Table 2. Relative Magnitudes of k-1, kz Required for the Validity of the and Steady-State (ss) Kinetic Approximations Pre-equilibrium (pe)
simultaneously
valid under the conditions k\ « k2 Relative constants « k_i and t » 1 l(k. + k\) as can be 42 reseen readily by noting that eq k-i « k\ « k2 duces to eq 29 under these condik-i « k2 « k tions. Less obvious, perhaps, is that k\ « k2 « k-1 the two kinetic assumptions are also simultaneously valid for the condiki « k-1 « k2 tions k2 « kt
