Chemical Kinetics

Chemical kinetics is the study of the speed at which chemical and physical processes take place. In a chemical reaction it is the amount of product that forms in a given interval of time or it can be defined as the amount of reactant that disappears in a given interval of time. Scientists that study rates at which processes occur are called kinetists.

Reactions occur at many different rates.  Some reactions in geological processes occur at imperceptably slow speeds where product change takes place in intervals of years or even decades. Other reactions occur so rapidly that they are measured in nanoseconds. Most reactions fall somewhere in the middle. For example the combustion of a hydrocarbon occurs fairly rapidly
whereas the hydrolysis of the bismuth(III) ion in a chloride containing solution does not produce the white BiOCl for several minutes in some cases depending upon a number of factors.

Reaction Rates

Instantaneous Rates are the change in product formation at a moment in time. If we plot the concentration of a product forming against time we will get a curve. The tangental slope at any place on that curve would be the instantaneous rate at that moment in time. The tangental slope can be determined by drawing a straight line that just touches the concentration vs time curve at the moment in time of which you wish to determine the instantaneous rate.  It we draw such a line then we can determine the slope of the tangental line.

Rate = -[Reactant] /t or [Product] /t

To relate the rate with respect to different reactants or products, stoichiometry is used.  For example, for reaction:

2 A  3 B

2[B] /t = -3[A] /t

This slope value is the instantaneous rate at the time indicated at the point which the tangental line touches the concentration curve. The best instantaneous rate is the rate at t = 0, or the initial rate.

Reaction rates can be affected in a number of ways. The four factors are:

  1. Concentration
  2. Phase of the reactants
  3. Temperature
  4. The presence of a catalyst

Concentration

The concentration dependence of the kinetics of a chemical reaction is described by its rate law.  The rate law for a reaction in the form of:

aA  +  bB  +  cC...   dD  +  eE  +  fF....

is usually:

rate = k[A]x [B]y [C]z ....

where k is the rate constant, [ ] is the molarity of the reactant, and x, y, and z are the reaction orders with respect to A, B and C, respectively.  The overall order of the reaction is x+y+zThe rate law is always determined experimentally.  There are several ways to determine the rate law for a particular reaction.

Method of Initial Rates

For a reaction:

BrO3-  +  5Br-  +  6H   3 Br2  +  3H2O

a set of experiments are performed, with different initial conditions.  The experiments are designed to investigate the concentration effect of one of the reactants, while keeping the others constant.
 

Experiment [BrO3-] (M) [Br-] (M) [H+] (M) Initial Rate (M/s)
1 0.10 0.10 0.10 8.0 x 10 -4
2 0.20 0.10 0.10 1.6 x 10 -3
3 0.20 0.20 0.10 3.2 x 10 -3
4 0.10 0.10 0.20 3.2 x 10 -3

In experiment 1 and experiment 2, Br- has a a concentration of 0.10 M and H+ has a concentration of 0.10 M. Now look at the concentrations of BrO3- and decide how it was changed. The BrO3- concentration is doubled. Then you have to compare it to the quotient of the initial rates. So:

1.6 x 10 -3 / 8.0 x 10 -4 = 2 times faster
[2BrO3-]x = 2 times rate, so x = 1; the order for BrO3- is first.

In both experiment 2 and experiment 3, BrO3- has a concentration of 0.20 M and H+ has a concentration of 0.10 M. Now look at the concentration of Br- and decide how it was changed and what effect it had on the reaction rate. The concentration1 was doubled, and the rate doubled. This is found by taking the quotient of the initial rates:

3.2 x 10 -3 / 1.6 x 10 -3 = 2 times faster
[2Br-]y = 2 times rate, so y = 1; the order for Br- is first.

Finally, In both experiment 1 and experiment 4, BrO3- has a concentration of 0.10 M and Br- has a concentration of 0.10 M. Now look at the concentrations of H+ and decide how it was changed and what effect it had on the reaction rate. The concentration1 was doubled, and the rate goes up by four times.  This is found by taking the quotient of the initial rates:

3.2 x 10-3 / 8.0 x 10-4 = 4 times faster
[2H+]z = 4 times rate, so z = 2; the order for H+ is second.



The rate law for the reaction, therefore is:

Rate = k[BrO3-]1 [Br-]1 [H+]2  or  rate = k[BrO3-] [Br-] [H+]2

Integrated Form of the Rate Law

If the rate law for a reaction is known to be of the form

rate = k[A]n

where n is either zero, one or two, and the reaction depends (or can be made to depend) on one species and if the reaction is well behaved, the order of the recation can be determined graphically.
 
Order Rate Law Integrated Rate Equation Linear Plot Slope Units for k
0 rate = k [A]0 - [A]t = k [A] vs time -k M / s
1 rate = k [A] ln ([A]t/[A]0) = - kt ln [A] vs time -k s-1
2 rate = k [A]2 (1/[A]t) - (1/[A]0) = k (1/[A]) vs time k M-1 s-1

The following data were collected for a hypothetical reaction whose rate is known to only depend on the concentration of a single reactant, A. The order of the rate law with respect to A is unknown.
 
Time (s)  [A] (M)
0 1.00
5 0.82
10 0.67
15 0.55
20 0.45
40 0.20
60 0.09

This data can be used to determine the order of the reaction. We assume that the reactions is either zeroth, first or second order.

Is the reaction 0th order?

If the reaction is 0th order, a plot of concentration versus time will result in a straight line. Since this plot is clearly non-linear, the reaction is not 0th order.

Is the reaction 1st order?

If the reaction is 1st order, a plot of the natural log (ln) of concentration versus time will result in a straight line. Since this plot is clearly linear, the reaction is 1st order.

Is the reaction 2nd order?

If the reaction is 2nd order, a plot of concentration versus time will result in a straight line. Since this plot is also clearly non-linear, the reaction is not 2nd order.

Phase of The Reactants

Reactions produce products by having the reacting molecules come into contact with one another.  The more often they collide, the more likely the chance that product will form. If the reacting molecules moving more rapidly and in the gaseous state then product will have a more likely chance to form. This is part of an over riding theory that forms the foundation of all kinetics work. This theory is called the Collisional Theory of Reaction Rates. Reactions usually occur more rapidly when the reactants are in the gaseous state. The reacting molecules dispersed in a solution is the next most favorable way for product to form at a reasonable speed. Reactions do occur in pure liquids or in solid form but the rates tend to be rather slow because the reacting molecules are very restricted in their movement among one another, and therefore, do not come into contact as often. The relative rates are roughly in this manner:

gases > solutions > pure liquids > solids

Temperature

The activation energy of a reaction is the amount of energy needed to start the reaction. It represents the minimum energy
needed to form an activated complex during a collision between reactants. In slow reactions the fraction of molecules in the
system moving fast enough to form an activated complex when a collision occurs is low so that most collisions do not produce a
reaction. However, in a fast reaction the fraction is high so that most collisions produce a reaction. For a given reaction the rate
constant, k, is related to the temperature of the system by what is known as the Arrhenius equation:

k = Ae-Ea/RT

where R is the ideal gas constant (8.314 J/mole-K), T is the temperature in Kelvin, Ea is the activation energy in
joules/mole, and A is a constant called the frequency factor; which is related to the fraction of collisions between reactants1
having the proper orientation to form an activated complex.  The exponential equation can be converted to the linear form by taking the logarithm of both sides:1

lnk = -Ea /RT + lnA

Thus, a plot of lnk vs. 1/T is a straight line which can be used to determine the activation energy of the reaction from the slope.  The rate constant at different temperatures can then be evaluated.

For example, in the hydrolysis of 2-chloro-2-methylpropane:

the following data was obtained:
 

T (K) 303 298 293 288 283
k (x102) 9.41 3.59 3.08 2.44 1.82

A plot gives the following results: