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An in-depth analysis of the thermodynamic study of elementary reactions, focusing on the steady-state approximation and Transition State Theory (TST). the mathematical expressions for rate constants using TST, the Eyring equation, and the calculation of microscopic rate constants. It also discusses the differences between quasi-equilibrium and steady-state approximations for reversible elementary reactions.
Typology: Lecture notes
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Manuscript ID: ed-2016-00957b.R2 (Revised)
Departamento de Ciencia de Materiales y Quimica Fisica, Seccion de Quimica Fisica, Facultad de Quimica, Universidad de Barcelona, Marti i Franques 1, 08028 Barcelona, Spain
ABSTRACT: The elementary reaction sequence (^) A (^) ^ I Products is the simplest
mechanism for which the steady-state and quasi-equilibrium kinetic approximations can be applied. The exact integrated solutions for this chemical system allow inferring the conditions that must fulfil the rate constants for the different approximations to hold. A graphical approach showing the behavior of the exact and approximate intermediate concentrations might help to clarify the use of these methods in the teaching of chemical kinetics. Finally, the previously acquired ideas on the approximate kinetic methods lead to the proposal that activated complexes in steady state rather than in quasi-equilibrium with the reactants might be a closer to reality alternative in the mathematical development of Transition State Theory (TST), leading to an expression for the rate constant of an elementary irreversible reaction
given by conventional TST, and to an expression for the equilibrium constant of an elementary reversible reaction more compatible with that predicted by chemical thermodynamics.
KEYWORDS: Audience: Graduate Education/Research ; Domain: Physical Chemistry ; Pedagogy: Misconceptions ; Topics: Kinetics , Reactive Intermediates
The Exact Solutions
Let us start considering the following simple mechanism:
(^1 ) 1
Since the three reactions involved in that mechanism are unimolecular, the differential laws
corresponding to the reactant and intermediate concentrations are, respectively:
The advantage of illustrating the conditions required for the steady-state and quasi- equilibrium approximations to hold with a sequence of unimolecular (instead of bimolecular) reactions is that we know in this case the exact analytical solutions for the concentrations of the reactant and intermediate at different instants during the course of the reaction:
o 1 2 -^1 1 1 - 2 1 2
1 o -^2 - 1 1 2
where parameters 1 and 2 are algebraic combinations of the three rate constants involved:
2 12
2 12
The integrated laws (eqs 4 and 5) have been obtained by integration of the corresponding differential laws (eqs 2 and 3) using a matrix-based method.^10 The validity of these exact solutions can be verified by consulting different sources,^11 ,^12 and they have been experimentally confirmed for the oxidation of thiols by Cr(VI).^13 -^16 Otherwise, the integrated laws can be checked straightforwardly by differentiation, leading to the differential laws derived from the mechanism considered (eq 1 ). An interesting parameter to discuss later the application of the steady-state and quasi- equilibrium approximations is the time elapsed when the intermediate reaches its maximum concentration. By obtaining d[I]/d t from eq 5 and equating the result to zero, it can be inferred that:
max^1 1 2 2
First Approximation
Mathematical Approach to the Steady-State Method
Let us consider firstly the particular case corresponding to the mechanism for which the first, reversible step is slow in the forward direction and fast in the backward direction, whereas the second, irreversible step is also fast, that is k - 1 >> k 1 and k 2 >> k 1. Under these conditions, we
can approximate:
(^2 )
Replacing this result into eqs 6 and 7:
and the new results into eqs 9 and 10 (remember that k 1 is negligible against both k - 1 and k 2 ):
o^1
1 o 1 -1 2
From eqs 14 and 15:
1
Finally, from eq 16:
and from eq 3:
which is precisely the equation serving as definition for the steady-state approximation. We can thus conclude that application of this approximate method to a particular intermediate requires that it disappear in steps much faster than the one corresponding to its formation (for the particular case considered here, k - 1 >> k 1 and k 2 >> k 1 ). Thus, an intermediate in steady
state always is present in the reacting system in minute concentrations, since its tendency to disappear (reactivity) is much higher than its tendency to be formed. Small free radicals would be an excellent example of steady-state intermediates indeed.
Graphical Approach to the Steady-State Method
Now, we can represent the exact solution of [I] at different instants during the course of the reaction (eq 5) and compare it with the approximate solution provided by the steady-state condition (eq 16):
Figure 2. Comparison between the kinetic plots showing the [Intermediate]/[Reactant]o ratio
as obtained using either the steady-state approximation (green plots) or the exact rate law (purple plots) for the mechanistic scheme formed by two unimolecular consecutive reactions, the first reversible and the second irreversible, with k 1 = 1.00 10 -^4 s-^1 , and k - 1 = k 2 = 5.
10 -^5 s-^1 (A), 1.00 10 -^4 s-^1 (B), 2.00 10 -^4 s-^1 (C) and 4.00 10 -^4 s-^1 (D).
In order to reach conclusions about the relative values that must take the rate constants of the elementary reactions ( k 1 , k -1, and k 2 ) for the steady state approximation to hold, k 1 has
been kept constant, whereas the other two rate constants have been progressively increased keeping k - 1 = k 2. In Figure 2, the four bell-shaped curves showing a maximum (in purple)
A B o [I] / [A]
Time / min Time / min
o [I] / [A]
C
Time / min Time / min
[I] / [A] [I] / [A]
o o
D
0 600 1200 1800
0.0 0 600 1200 1800
0 600 1200 1800
0 600 1200 1800
correspond to the exact concentration ratio (eq 20), whereas the four continuously-decreasing curves (in green) correspond to the steady-state approximate concentration ratio (eqs 21 and 22). We can see that, for each set of rate constants, the exact and approximate curves cross exactly at the maximum of the intermediate concentration, given that eq 18 (the steady-state condition) is only exact at that maximum (the only point with a horizontal tangent). We can also see that each time rate constants k -1and k 2 are multiplied by a factor of 2 (keeping k 1
constant), the approximate curve gets closer and closer to the exact curve after its maximum. For the case k - 1 = k 2 = 4 k 1 (Figure 2, D) the steady-state approximate curve is almost
coincidental with the exact curve once the maximum is reached, meaning that the steady-state condition qualifies as an excellent approximation for this set of rate constants. Thus, we can consider that the time necessary for the steady-state condition to be fulfilled is coincident with
The ratio between the exact and approximate intermediate concentrations increases with time, reaching a plateau when t (Figure 3). The asymptotic value of the ratio is (from eqs 9 , 10 and 19 ):
-1 2 ss (^1 )
and, from eq 12, it can be inferred that:
-1 2
-1 2 ,^1
k k
Figure 4. Dependencies of the time elapsed when the intermediate reaches its maximum concentration (top) and the limit at infinite time of the ratio between the exact intermediate concentration ([I]) and that obtained using the steady-state approximation ([I]ss) (bottom) on the fast/slow ratio of rate constants (keeping k -1 = k 2 ) for the mechanistic scheme formed by
two unimolecular consecutive reactions, the first reversible and the second irreversible. The dashed line shows the limit corresponding to a perfect fulfilment of the steady-state approximation ([I] = [I]ss).
0.9 13
105
140
210
70
0
175
26 39 52 65
35
t^
/ min max
k (^) -1/ k (^) 1 = k (^) 2 / k 1
lim ([I] / [I]
)ss 8 t
Mathematical Approach to the Quasi-Equilibrium Method
Let us consider as a second particular case the one corresponding to the mechanism (eq 1) for which the first, reversible step is fast in both directions whereas the second, irreversible step is slow, that is (^) k 1 >> k 2 and k - 1 >> k 2. Under these conditions, we can approximate:
(^2 ) ( k (^) 1 + k -1 (^) + k (^) 2 ) 4 k k 1 2 (^) k (^) 1 + k -1 (25)
and replacing this result into eqs 6 and 7:
1 k (^) 1 + k -1 (26)
and the new results into eqs 9 and 10 (remember that this time k 2 is negligible against both k 1
and k - 1 ):
-1 o^2 1 -
k^ k^ t
1 o^2 1 -
k^ k^ t
From eqs 28 and 29:
Graphical Approach to the Quasi-Equilibrium Method
We can represent the exact solution for [I] at different instants during the course of the reaction (eq 5) and compare it with the approximate solution provided by the quasi- equilibrium condition (eq 30):
qe^1
Figure 5. Comparison between the kinetic plots showing the [Intermediate]/[Reactant]o ratio
as obtained using either the quasi-equilibrium approximation (green plots) or the exact rate
A B o [I] / [A]
Time / min Time / min
o [I] / [A]
C
Time / min Time / min
[I] / [A] [I] / [A]
o o
D
0 600 1200 1800 0.
0 600 1200 1800
0 600 1200 1800 0.
0 600 1200 1800
law (purple plots) for the mechanistic scheme formed by two unimolecular consecutive reactions, the first reversible and the second irreversible, with k 1 = k -1 = 2.00 10 -^4 s-^1 (A),
4.00 10 -^4 s-^1 (B), 8.00 10 -^4 s-^1 (C) and 1.60 10 -^3 s-^1 (D) , and k 2 = 1.00 10 -^4 s-^1.
directly proportional to [A]o, it will be enough to compare the exact ratio (eq 20) with the
quasi-equilibrium approximate ratio (from eqs 4 and 33):
qe (^1) o (^) -1 1 2 ( )
where f ( t ) has the same meaning than in eq 22. In order to reach conclusions about the relative values that must have the rate constants of the elementary reactions ( k 1 , k -1and k 2 ) for the quasi-equilibrium approximation to hold, k 2
has been kept constant, whereas the other two rate constants have been progressively increased keeping k 1 = k - 1. In Figure 5 , the four bell-shaped curves showing a maximum (in
purple) correspond to the exact concentration ratio (eq 20 ), whereas the four continuously- decreasing curves (in green) correspond to the quasi-equilibrium approximate concentration
ratio (eq 34 ). We can see that each time rate constants k 1 and k -1are multiplied by a factor of
2 (keeping k 2 constant), the approximate curve gets closer and closer to the exact curve after
its maximum. For the case k 1 = k - 1 = 16 k 2 (Figure 5 , D) the quasi-equilibrium approximate
curve is almost coincidental with the exact curve once the maximum is reached, meaning that the quasi-equilibrium condition qualifies as an excellent approximation for this set of rate constants. Thus, we can consider again (as happened with the steady-state condition) that the
qe (^1 )
and, from eq 26, it can be inferred that:
In other words, when k 2 is kept constant and both k 1 and k - 1 are gradually increased, the limit
of the asymptotic value of the ratio [I]/[I]qe increases approaching unity (Figure 7 , bottom), corresponding to a perfect fulfilment of the quasi-equilibrium approximation. Simultaneously, the time elapsed when the intermediate reaches its maximum concentration (coincident with the time interval required for the quasi-equilibrium approximation to hold) decreases approaching zero (Figure 7 , top). We can, therefore, define an intermediate in quasi- equilibrium with the reactants as an intermediary chemical species formed in a fast, reversible step and that has already reached its maximum concentration.
Figure 7. Dependencies of the time elapsed when the intermediate reaches its maximum concentration (top) and the limit at infinite time of the ratio between the exact intermediate concentration ([I]) and that obtained using the quasi-equilibrium approximation ([I]qe) (bottom) on the fast/slow ratio of rate constants (keeping k 1 = k -1) for the mechanistic scheme
formed by two unimolecular consecutive reactions, the first reversible and the second irreversible. The dashed line shows the limit corresponding to a perfect fulfilment of the quasi-equilibrium approximation ([I] = [I]qe).
0.75 65
60
100
40
0
80
130 195 260
20
t^
/ min max
k (^) 1 / k (^) 2 = k (^) -1/ k 2
lim ([I] / [I]
)qe 8 t
