The first step in delineating the mechanism of a reaction is to determine what reactant species must come together to produce the activated complex of the rate-determining step.

**REACTION KINETICS**

The
first step in delineating the mechanism of a reaction is to determine what
reactant species must come together to produce the activated complex of the
rate-determining step. This can be done by determining the order of the
reaction with respect to each of the reactants in the process. If the rate of a
reaction is dependent on the concentration of a particular reactant, then that
reactant is involved in the transition state of the rate-determining step of that
reaction. This provides important structural information about the activated
complex since it reveals which chemical species are present in the activated
complex.

The
order of a reaction is found by determining the relationship between rate and
concentration for each reactant. Thus for the elementary process A → B, the rate of reaction *ν* can be expressed as the decrease in
the concentration of reactant A with time or the increase in the concentration
of product B with time:

Further,
the rate of the reaction can be expressed as a function of the concen-tration
of A, where *k* is the rate constant
for the process:

This
differential rate expression shows that the reaction rate is directly dependent
on the concentration of A — the greater is [A], the faster is A converted to B.
The reaction is said to be first order with respect to A because the exponent
of [A] is 1.

The
above expression is the first-order differential rate law for the conver-sion
of A to B. The change in concentration of A over the complete course of the
reaction is given by the integrated rate law, which is found by solving the
differential rate law:

The
integrated rate law shows that the natural logarithm of the concentra-tion of
the starting material A decreases linearly with time. By determining the
concentration of A at various times [A* _{t}*
] and plotting ln[A]

Second-order
reactions occur when two reagents must collide in solution to produce the
activated complex. Thus for the reactions

2A
→ B

A+Β
→ C

each
reaction is second order because the sum of the exponents of species in the
rate law is 2:

This
means that in the first instance two molecules of A must collide to produce the
activated complex. In the second case a molecule of A and B collide to produce
the activated complex. In each case the second-order dependence requires that *both* of the colliding molecules are a
part of the activated complex of the rate-determining step.

Integration
of these differential rate laws gives

Again
plotting concentration versus time using these integrated second-order rate
laws gives linear plots *only if the
reaction is a second-order process*. The rate constants can be determined
from the slopes. If the concentration – time plots are not linear, then the
second-order rate equations do not correctly describe the kinetic behavior.
There are integrated rate laws for many different reaction orders.

It
is often possible to simplify the rate law of a second-order process by
employing pseudo-first-order conditions. For a second-order reaction where

the
integrated rate law for a second-order reaction can be used to plot the kinetic
data. Alternatively, if the concentration of one of the reactants, for example
A, is much larger than the other (10-fold excess minimum, 20-fold better), then
its concentration does not change significantly over the course of the
reaction; thus [A]* _{t}* ≈ [A]

*d*P/*d*t = *k*[A][B] = *k*[A_{0}][B] = *k’* [B]

where
*k* = *k*[A]_{0}.
This new rate law is a first-order equation which is much easier to deal with
and plot. A plot of ln[B] versus *t*
will give a straight line of slope *k*’.
From *k’* and [A]_{0} (which is
known), the rate constant can easily be determined. This is a much simpler
method for determining the rate constant than the normal second-order
treatment. The order of the reaction with respect to A can be checked by using
several different initial concentrations of [A]_{0} and the
relationship

* k*’= *k*[A]_{0}

Therefore

log
*k’* = log[A]_{0} + log *k*

Plots
of log *k’* versus log[A]_{0}
will have unit slope if the reaction is first order in A. Alternatively,
doubling [A]_{0} should double the rate if the reaction is first order
in A.

In
a multistep process involving a reactive intermediate, the rate law for the
overall reaction cannot be written down a priori because the step in which the
reactants disappear is different than the step in which the products are formed
(Figure 5.15). In a large number of cases, the intermediate is of high energy
and reacts very rapidly — either returning to reactant or going on to product.

In
such cases the steady-state approximation can be used to derive a rate
expression that can be tested. Thus for a reaction process involving an
intermediate [I]

The
concentration of the intermediate [I] which gives product is given by the
difference between the rate of its formation from A and the rate of its
conversion back to reactant A or forward to product P

*d*I/*d*t = *k*_{1}[A] − *k*_{−}_{1}[I] − *k*_{2}[I] = 0

The
steady-state approximation assumes that since I is very reactive, its
concen-tration will be very low at any time during the reaction and it will not
change appreciably. Therefore *d*I*/d t* = 0. Solving the above expression for
the concen-tration of I and substitution into the rate law for the formation
product gives

where
the observed rate constant *k*_{obs}
= *k*_{2}*k*_{1}*/ k*_{−}_{1} + *k*_{2}.
This is a first-order rate expression in reactant A and can be integrated and
plotted normally.

Several
limiting cases can be envisioned for such a multistep process. If *k*_{2 }*>> k*_{−}_{1}* *(the intermediate
goes on to product more rapidly than it returns to reactant),_{}

This
represents the case where the first step is rate determining. (If this
situation is known beforehand, one need not work through the steady-state
approximation but merely write down *d*P*/d t* = *k*_{1}A
since the first step is rate limiting and irreversible.)

This
represents the case where there is a fast preequilibrium preceding the
rate-determining step. It is again not necessary to work through the
steady-state approximation if this situation is known to exist. If the intermediate
I is in equilibrium with the reactant A, then

substitution
for [I] in the rate expression gives

This
is actually the correct way to think about this case since the steady-state
approximation requires that the concentration of I is low, which it may not be
if there is a fast preequilibrium.

If
*k*_{2} ≈ *k*_{−}_{1}, then the full
steady-state rate expression is needed to describe the rate of reaction:

Consider
once again the solvolysis of a tertiary bromide in methanol.

This
reaction can be reduced to the kinetic scheme.

The
rate of product formation is given by the pseudo first-order expression in
which *k*_{2} = *k*[CH_{3}OH]
(normally *k*_{2} is taken as
the rate constant because the concentration of methanol is constant),

*d*P/*d*t = *k*_{2}[I]

and
the steady-state approximation is written as

*d*P/*dt* = O = *k*_{1}[A] − *k*_{−}_{1}[I][Br^{−}] − *k*_{2}[I] *d t*

Since
the return of intermediate I to reactants is a second-order reaction, solving
for [I] and substitution into the rate law give

If
k_{2} >> k_{−1}[Br^{−}], then ν = k_{1}[A]
and simple first-order behavior is found. If k_{−1}[Br] >> k_{2},
then there is a rapid ionization preequilibrium and

The
rate of product formation will be directly proportional to [A] but inversely
proportional to [Br^{−}].
By using an excess of Br^{−}
so that its initial concentration [Br^{−}]_{0} does not change
appreciably over the course of the reaction, pseudo-first-order behavior can be
achieved with *k*_{obs} = *k*_{2}*K*_{eq}*/*[Br^{−}]_{0}.

If
*k*_{−}_{1}[Br^{−}] ≈ *k*_{2},
then the full rate expression will be needed to describe the kinetic behavior.
The rate will be first order in [A] and the rate will slow down in the presence
of added bromide but not in a simple inverse relationship.

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