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Mechanism of a Reaction - "an attempt at explaining"
Maple plot of a solution to a system of differential equations
describing the rates of change of concentrations of reactants
in a stoichiometric reaction.
As was mentioned earlier, the sequence of "molecular events" which rationalize the overall stoichiometric reaction that is investigated, is termed the mechanism of the reaction. So, a primary purpose of the mechanism is that it must be able to generate the experimentally determined rate law. The molecular events or reactions that comprise a mechanism are termed elementary reaction (E.R.) or elementary steps. Therefore, in order to see how a sequence of elementary reactions can generate the empirical rate law, we must talk about the rate of an elementary reaction and see how it relates to the observed rate (R) of the overall reaction.
We will denote the rate of an elementary reaction by r and the observed (initial) rate of the overall reaction by R. Since an elementary reaction (E.R.) denotes the molecular event, i.e. the molecular collisions taking place; the stoichiometric coefficients are telling more than just the relative proportions that must react to form products.
Consider an elementary reaction "i" of the form:
where X and Y are gaseous or aqueous species, x and y are the
stoichiometric coefficients and k_i is the elementary rate
constant for the elementary reaction of X and Y to form
products.
The above elementary reaction is telling us that x molecules
of X are colliding with y molecules of Y to give products - in
other words the E.R. is telling us about molecular
collisions, not just about stoichiometry. We define:
So, the overall order of the rate law for an elementary
reaction is termed the molecularity of the
elementary reaction. Thus, some examples of elementary
reactions are listed below (assume the elementary rate
constant for each of the E.R.'s below is "k" and that "P"
denotes "Products").
Consider the following sequence of reactions,
which are presumed to give the sequence of molecular
events (in temporal order) for the overall
(stoichiometric) reaction.
where we will call the rate constant for this overall
reaction, k_{exp}. Each reaction in this sequence of
molecular events is termed an elementary step or an
elementary reaction (E.R.). The collection of elementary
reactions is termed the mechanism of the overall reaction.
The mechanism is often a theoretical "rationalization" -
since evidence for each of the proposed elementary reactions
is often indirect. Also, sometimes additional experimental
data will disprove a previously proposed mechanism - then
we must often start over. Thus, a mechanism can never
be proven beyond reasonable doubt - it can only be disproven
(or sometimes, substantiated) by further experimental
evidence! Our proposed mechanism is listed below.
overall rxn. with rate constant k_{exp}.
Experimental Rate Law - determined by the isolation method:
R = Rate = k_{exp}[HBr][O_2]. This is first order
with respect to HBr(g), first order w/r to O_2(g) and
second order overall.
The rate constants, k_1, k_2, k_3, and k_4 above, are termed elementary rate constants, since they are rate constants for elementary reactions. Note that the sum of the E.R.'s must add up (in an algebraic manner) to the overall reaction. In the above mechanism, they do. Species that are generated during the course of the elementary reactions (not externally introduced), but do not appear in the overall reaction are termed intermediates. HOOBr(g) and HOBr(g) classify as intermediates in the above mechanism. What do the terms "Fast" and "Slow" appended after the elementary reactions mean? We'll see in a moment. But first, let's take a look at what the rate constants do to the formation of products for a relatively straightforward reaction. Observe the simulation provided at the University of California at Irvine of a simple reversible reaction.
The elementary rate constants are often indirectly determined - of course our hope is to be able to derive the value of these elementary rate constants from "first principles". This is often very difficult to do and can only be done so in an empirical or semi-empirical manner - even for the simplest of reactions! At least, we would like to be able to express the observed rate constant for the overall reaction, k_{exp}, as a function of the elementary rate constants (k_1, k_2, k_3, k_4) above. This leads us to the next requirement of a good or "plausible" mechanism. The mechanism must be able to generate the experimentally determined rate law. In order to see how it relates to the observed rate (R) of the overall reaction. We will denote the rate of an elementary reaction by r and the observed (initial) rate of the overall reaction by R. Remember, since an elementary reaction (E.R.) denotes the molecular event, i.e. the molecular collisions taking place; the stoichiometric coefficients are telling more than just the relative proportions that must react to form products.
Consider the elementary reaction (1) above:
Thus, the overall order of the rate law for an elementary reaction is termed the molecularity of the elementary reaction. The above elementary reaction is thus termed a bimolecular reaction - since two (2) molecules collide to form products.
So, writing down the (elementary) rate law for each of
the elementary reactions in our example, we have:
An n-molecular E.R. is always n-order (overall) - but not
all n-order (overall) reactions are n-molecular; because
not all reactions are elementary reactions!
Now we are ready to see how to "derive" the experimental rate
law from the elementary reactions and elementary rate constants
of the proposed mechanism. This can be easily done if the rate
of one and only one E.R. is much slower than all of the other
E.R.'s. If this is the case, then we assume:
This is a very important relationship for aiding us in deriving
an experimental rate law from its proposed mechanism. We note
that there is not always a step that is very much slower than
the others. In these cases, it may be much more difficult -
or impossible - to derive an experimental rate law from its
proposed mechanism. We will consider some situations where
there is not a single "slow" step later. Now we know the
purpose for the "slow" and "fast" designations. Step (1)
is denoted as the "slow" step (E.R.); all others are
considered as "much faster".
This matches the experimentally determined rate
law above. Hence, by correspondence,
Also, we have all bimolecular steps and the mechanism adds up to the correct overall reaction. This does NOT prove that the mechanism is correct - only that the mechanism is plausible. Note: Steps (3) and (4) are listed separately - because we mean to communicate that we have two bimolecular steps, not one tetramolecular step! Tetramolecular steps, i.e. four (4) molecules simultaneously colliding with enough energy and the proper orientation - in the gas phase - are quite implausible!
What about reactions that are "reversible", i.e.
reactions that can proceed from products back to reactants?
It turns out that most chemical reactions - once we get
past the "initial" period - can proceed backwards, i.e.
are reversible. Consider the elementary reaction "1":
Of course, the rate of this E.R. is: r_1 = k_1[A]² [B] . Consider now, the reverse of E.R. 1, which we will call E.R. "-1":
Of course, the rate of this E.R. is:
In general, if there is not a single slowest (rate-determining) step controlling the time evolution of the system (and hence the experimental rate law), we must solve the system of differential equations (simultaneously) that arise from writing down the differential rate law for each elementary reaction. This is often a tedious task and - more often than not - leads to cumbersome solutions that cannot be reduced to a tractable form or cannot be solved analytically! Before we discuss what to do in those cases, let's write down an example of a simple mechanism that can be solved exactly (and very easily with Maple). Here we will write down the system of differential equations and the solutions. In a separate handout, the Maple procedure for solving this system of differential equations will be carefully laid out. Please refer to it.
Example #1: A mechanism of two (2) consecutive first-order (unimolecular) steps.
Consider the following two-step mechanism in which X produces the intermediate Y in the first step and then Y subsequently forms Z in the second step. (Note that neither step is presumed reversible).
Prove for yourself that the overall (stoichiometric)
reaction is:
We want to determine the concentration of X, of Y, and of
Z as a function of time (t). Therefore, we want to
determine [X(t)], [Y(t)], and [Z(t)].
Solution:
Because these steps are mechanism steps, the stoichiometry
of each step determines the differential rate law - as
discussed above. Since we have three (3) unknown functions,
we need a system of three (3) differential equations which
can easily be solved simultaneously (hopefully). After
looking at the E.R.'s, it is easy to see that step 1
directly controls the [X], and that step 2 directly
controls the [Z]. However, steps 1 and 2 directly control
the [Y] - the intermediate. From what was discussed
earlier, it is easy to see that:
Therefore, we can immediately write:
What about the differential rate law for [Y]? E.R. 1
tells us that its production rate
As discussed in your coverage of differential equations, in order to obtain specific solutions, we must specify the initial conditions for [X], [Y], and [Z], i.e. [X]_0, [Y]_0, and [Z]_0. The accompanying Maple handout mentioned above considers the case in which we start off with some X, i.e. [X]_0 = "A", but no Y nor Z, i.e., [Y]_0 = 0 and [Z]_0 = 0. The solutions are (refer to the accompanying Maple handout for the details) shown below. (Note: exp(x) = ex).
To see a plot of these solutions, refer to the accompanying Maple handout.
In order to allow you to further explore this material and
to help you to develop your facility with Maple, some
homework problems have been provided. Even though Maple
allows you to solve some fairly sophisticated systems,
consider the following:
What if:
- there is not an "obvious" rate-determining
("slow") step,
OR
- the system of differential equations gives rise to
very cumbersome solutions,
OR
- the system of differential equations cannot be solved
analytically by conventional mathematical methods?
As is usual in science, approximations and/or simplifications
are made to the mathematical equations that describe our
model - our mechanism in this case - which allow us to either
write down analytical solutions and/or allow us to gain some
physical intuition about the system. In the case of reaction
mechanisms, one useful approach is termed the steady-state
approximation (SSA). We discuss this technique below.
Do problems 3-15
through 3-18.