Chemical Kinetics
Gary L. Bertrand
Professor Emeritus of Chemistry
Missouri University of Science and Technology
(formerly University of Missouri-Rolla)


Chemical Kinetics deals with the rates of chemical processes - how concentrations change with time during a reaction. The concentrations of some substances decrease, others increase, and others are not changed. These changes are all interconnected by the chemical reactions that are occurring and their stoichiometry, but some of these relationships may be very complex. In studying these processes, we try to choose conditions which will simplify the relationships as much as possible. Some of these conditions might include:

a) The initial concentrations of components may be set in their stoichiometric ratio, so that this ratio of concentrations is maintained throughout the reaction.
b) The concentrations of all but one reactant may be set in large excess over that component. Under these conditions, the concentrations of the other components are essentially constant while that of the component of interest goes from a finite concentration to almost zero.
c) The temperature may be chosen so as to minimize the effects of other, competing reactions.

These conditions allow the study to be conducted by studying the changes in concentration of a single component with time. This allows a simpler mathematical description of the changes that are occurring. For a generic chemical reaction:

νA A + νB B  →  νC C + νD D
the rate of reaction is defined as:

rate of reaction = - (1/νA)d[A]/dt = - (1/νB)d[B]/dt = (1/νC)d[C]/dt = (1/νD)d[D]/dt

for reactants and products that are not involved in other reactions.

The rate of reaction is known to depend on the concentrations of some or all or the reactants, in some cases on the concentrations of some of the products, and possibly on the concentrations of other materials such as catalysts. Temperature usually has an effect (rates normally increase with temperature). The medium (gas phase, solvents, etc.) and even the shape of the container may have an effect. For simple reactions, the rate may often be described mathematically as a rate expression:

rate of reaction = k[A]a[B]b ,

With k (the reaction rate constant) depending on temperature, the medium, concentrations of catalysts or inhibitors, and other factors. The constants a and b are expected to have integral or half-integral values between 0 and 3. Occasionally a negative value is encountered, indicating that a reactant may also be acting as an inhibitor. The constants a and b are referred to as "order", a is the order for component A and b is the order for component B . The overall order for the reaction is the sum (a + b) of the orders for the individual components.

In order to simplify the mathematical relationships, experiments are often conducted under special conditions. One such situation is to have the intial concentrations of the reactants in their stoichiometric ratio:
[A]t=0A = [B]t=0B ,

so that the concentrations of A and B remain in that ratio as the reaction progresses:
[A]tA = [B]tB .

The rate expression may then be simplified (?) to:
rate of reaction = k(νBA)b[A](a + b) .

and may be treated as if there is only one reactant (A) with a pseudo-order of a + b ,
and an apparent rate constant k', with
k' = k(νBA)b .

Another simplifying technique is to have one reactant in large excess over the other, so that the concentration of the excess component does not change appreciably during the reaction:

[B]t=0B  ≥  10 x [A]t=0A .
The rate expression then simplifies to:
rate of reaction = k[B]t=0b [A]a ,

and may also be treated as if there is a single reactant (A) with a pseudo-order of a and an apparent rate constant (k')
k' = k[B]t=0b .

These simplified forms all have the basic relationship:

- d[A]/dt = νAk'[A]n .

This mathematically gives two integrated forms:

For n = 1 (first-order or pseudo-first-order):

ln ([A]t) = ln ([A]t=0) - νAk't

and for any other value of n:

1/[A]tn-1 - 1/[A]t=0n-1 = (n - 1) νA k't .

The initial concentration of the component is not normally known with great accuracy because of uncertainties associated with starting the reaction. These equations are then considered to have two unknowns, the initial concentration ([A]t=0) and the apparent rate constant (k').

If the order or apparent order of the reaction is known, the concentration of component A may be measured as the reaction occurs, preferably covering the times in which the concentration changes from 2/3 of the initial value to 1/3 of the initial value. The apparent rate constant may then be determined graphically or by regression of the appropriate function of concentration
(ln[A]t or 1/[A]tn-1) vs time (t).

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