VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
A. Given:
!!!
!!A
k1
C
k2
k-1
d[P]/dt = k2[C]
But what is [C]?
P
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
B. Approximations:
1. Equilibrium Approximation:
!!!
!!A
!!!
k1
C
k2
P
k-1
a. Assumption: k-1 >> k2
• The A-C equilibrium is maintained during the reaction.
• This is the essence of the equilibrium approximation;
C returns to A faster than it proceeds to product.
Keq = k1/k-1 = [C]/[A]
[C] = Keq[A]
Therefore...
d[P]/dt = k2[C] = k2Keq[A]
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
2. Steady State Approximation: Olivieri, A. C. J. Chem. Educ. 1987, 64, 1002.
a. Assumption: k-1 >> k1
[C] stays low irrespective of k2. Therefore...
!!!
!!A
!!!
k1
C
k2
P
k-1
d[C]/dt = k1[A] - k-1[C] - k2[C] ≅ 0
• While [C] can change, d[C]/dt stays small relative to other concentration
changes.
Solving for [C]...
[C] =
k1
[A ]
k"1 + k2
Consequently...
!
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
d[P]/dt = k2 [C] =
and...
!
k1k2
[A ]
k"1 + k2
!!!
!!A
!!!
k1
C
k2
P
k-1
d[P]/dt = -d[A]/dt
3. Summary:
• The steady state approximation requires low [C], but does not require an A/C
pre-equilibrium.
• The equilibrium approximation requires a fully established A/C preequilibrium, but does not require low [C].
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
C. Limiting Cases:
1. Case 1: k1 >> k-1, k2
!!!
!!A
!!!
k1
C
k2
P
k-1
d[P]/dt ≠ -d[A]/dt
• The steady state approximation does not apply; [C] builds
quickly on the time scales of product formation, and -d[A]/dt ≠
d[P]/dt. This produces two very separate steps...
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
a. Early reaction times:
!!!
!!A
!!!
k1
C
!!!
!!A
!!!
k1
C
k2
P
k-1
• -d[A]/dt = k1[A] = d[C]/dt (fast, 1st-order loss of A and formation of C)
b. Long reaction times:
k2
C
P
!!!
• Irreversible, 1st-order reaction that is relatively slow.
-d[C]/dt = d[P]/dt = k2[C]
• Slow, 1st-order loss of C and formation of P.
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
c. Caveat:
Keq = [C]/[A]
Therefore...
d[P]/dt = k2Keq[A]
• The rate is proportional to [A] even though A is not observable. In fact...
fast
A
C
P
observable
C
A
difficult
to
distinguish
P
observable
• It is difficult to know if C is on (or tangent to) the reaction coordinate.
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
!!!
!!A
!!!
2. Case 2: k-1 >> k1, k2
k1
C
k2
P
k-1
• A rapid, spectroscopically invisible pre-equilibrium is fully established.
" d[A]/dt =
k1k2
[A] #
k"1 + k2
123
steady state
approximation
!
!!!
!!A
kobsd
k1k2
k{
"1
equilibrium
appoximation.
C
[A ]
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
3. Case 3: k2 >> k1, k-1
-d[A]/dt =
!!!
!!A
!!!
k1k2
[A] = k1 [A]
k"1 + k2
k1
C
k2
P
k-1
• The steady state approximation applies since [C] stays low.
!
• Slow conversion of A to C is followed by fast reaction of C to P (fast
relative to return of C to A). Therefore, conversion of A to C is rate
limiting; the scheme reduces to...
!!!
!!A
!!!
k1
P
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
D. Non-Limiting Cases:
1. Case 1: k-1 << k1 ≅ k2 (within 10-fold)
!!!
!!A
!!!
k1
C
k2
k-1
!!!
k1
k2
!!A
C
P
!!!
• The pre-equilibrium is not established.
• C forms measurably, but not quantitatively.
a.
Rate Equation: Hughes, E. J. Chem. Educ. 1989, 66, 45.
i. Species A:
-d[A]/dt = k1[A]
[A] = [Ao ]e"k t
1
!
P
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
ii. Species C:
d[C]/dt = k1[A] - k2[C]
!!!
!!A
!!!
k1
C
k2
k-1
Substituting for [A] from eq (1)...
d[C]/dt = k1{[Ao ]e"k1 t } " k2 [C]
Integrate... (CRC 1,2,4,12 and the J. Chem. Educ. article)
!
!
[C] =
[Ao ]k1
k2
(e
"k
1
"k1 t
" e-k2 t )
P
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
iii. Species P:
d[P]/dt = k2[C]
(3)
!!!
!!A
!!!
k1
C
k2
k-1
Since...
[P] = [A0] - [A] - [C]
Substituting for [A] from eq (1) and for [C] from eq (2) and
rearranging (Appendix 2A)...
#
&
1
"k2 t
"k1 t
k1e " k2 e )'
[P] = [Ao ]$1+
(
% ( k2 " k1)
(
!
(4)
• We could have also reversed the order by substituting for [C] into
eq (3) and then integrating (Appendix 2B).
P
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
b. Graphics:
!!!
!!A
!!!
i. Low [C]: (k1 < k2)
[C] > [P]
C
k2
[P]
[A]
[X]'s
k1
•
•
[Cmax ] correlates with
the inflection point.
[C]
t (sec)
• [C] necessarily exceeds [P] at early reaction times since formation of
C is a prerequisite to formation of P.
• The rate of P formation, as indicated by the inflection point, will be
maximal as the [C] reaches a maximum.
P
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
ii. High [C]: (k1 > k2)
[A]
•
[P]
[X]'s
[C] > [P]
!!!
!!A
!!!
C
[Cmax] correlates with
the inflection point.
•
[C]
t (sec)
• C can be isolated in reasonable yield.
c. Quantitation: Frost & Pearson (There's a typo!)
• Discusses [Cmax], tmax, and some algebraic tricks.
• Also, see Appendix 2C.
k1
k2
P
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
2. Non-Limiting Case 2: k1 ≅ k-1 >> k2
!!!
!!A
!!!
k1
k2
C
P
k-1
• Both A and C are observable,
and the equilibrium is established completely before P forms.
-d[A]/dt ≠ d[P]/dt
d[P]/dt = k2[C]
i.e., the rate is proportional to [A] or [C].
P
[X]
A and C
t (sec)
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
Since Keq = k1/k-1= [C]/[A]
d[P]/dt = k2Keq[A]
i.e., the rate of P formation is proportional to [A] as well.
• The following scheme is indistinguishable under these assumptions:
!!!
!!C
A
P
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
!!!
!!A
!!!
k1
C
k2
k-1
3. Non-Limiting Case 3: k1 ≅ k-1 ≅ k2
• [C] builds up, reacting to give A and P competitively. This is
solvable numerically, but can be recognized graphically.
• Graphics look alot like A→ C→ P (Case 1). Other experiments are
necessary to distinguish this limiting behavior.
P
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
• Interestingly, the following scheme is now distinguishable...
C
P
A
[P]
[A]
[X]
[C]
t (sec)
VIII. 1st-Order Pre-equilibrium, 1st-Order Reaction:
• d[P]/dt is maximal at t = 0; there is no inflection point.
In addition, at early reaction times...
[P] > [C]
or...
[P] < [C]
This is not possible for...
!!!
!!A
C
P
in which [C] necessarily exceeds [P] at early reaction
times.