Kinetics is the study of the rates at which chemical reactions take place, how fast reactants are used up and products are formed. It is also the study of how the conditions to which the reaction mixture is subjected, such as temperature or pressure, and addition of a catalyst affects this. In the transition between reactants and products, there may be a variety of intermediates formed through a number of individual sequential reactions. Studying the rates at which reactants are consumed and how varying the conditions affect this allows us to examine in detail these sequential steps and to deduce a mechanism for the reaction. This is of particular interest in the study of catalytic reactions where there may be direct applications in industry. A detailed understanding of the catalytic mechanism for a reaction is vital in any effort to improve it’s efficiency.

In order to begin understanding the reaction mechanism and we must first establish the ratios of the indiviual reactent and the ratios of reactant and product formed. In the reaction A + 2B → C, one molecule of A react with 2 molecules of B to form one molecule of C. This called the reaction stoichiometry. We must also determine whether there are any side reaction where the reactants can form more than one set of products. We then determine how the concentration of the reactants vary as a function time in the course of the reaction.

Expermimental Techniques

There a several ways to experimentally monitor the rates of chemical reactions. Some reactions may involve acids or bases and cause a change in the concentration of H⁺ ions. We may therefore be able to follow the rate of reaction by recording the change in pH of the reaction mixture over time. Since

pH = -log₁₀[H⁺]

where [H⁺] is the concentration of H⁺ (in chemistry the concentration of a compound X is denoted by surrounding X with square brackets ie [X]), we can then determine the change in concentration with time. If the reaction involves the production of gaseous products then we can also determine the rate of reaction by monitoring the change in pressure as pressure is a measure of concentration. Many chemical compounds also absorb in specific regions of the electromagnetic spectrum. Provided the reactant we wish to monitor has an easily observable absorption band in a particular part of the EM spectrum which is not coincident with absorption of other reactants or products, we can follow the reaction spectroscopically. For infrared, visible and ultraviolet techniques we can define the spectroscopic property absorbance, A. This is defined by the equation

A = -log(I/I_o) = εcl

where I_o is the intensity of light at a given wavelength applied to the mixture, I is the intensity after passing through the sample, l is the distance the light travels through the reaction mixture, ε is a constant of proportionality that describes how strongly it absorbs and c is the concentration of the compound responsible for the absorption band. We can see that absorbance is directly proportional to the concentration and therefore an equivalent measure.

Reaction Rates and Rate Laws

We now have a variety of experimental techniques at our disposal to monitor the rates of reactions and we choose the one that best suits the reaction under study depending on the nature of the reactant and product compounds. We define the rate of reaction as the change in the concentration of the reactants or products. However if we monitor the change in concentration of the reactants during a reaction, the slope of concentration with time on the decay becomes more shallow as they are consumed. Because the rate changes in the course of the reaction we need to consider the instantaneuos rate which is the slope of concentration versus time at a specific instant in time. If we measure the concentration in moles per litre, mol L^-1, then the rate is measured in moles per litre per second, mol L^-1 s^-1. We also have to consider the reaction stoichiometry, ie the proportions of the different reactants that participate in the reaction. In a reaction of the type

A + 2B → C

where two molecule of B react with one molecule of A, the rate of change in concentration of B will be twice that for A.

In general for most chemical reactions we find that the rate of reaction is proportional to the concentration of the reactants. So if we have a reaction

A + B → C

we can write an expression for the rate which is proportional to the concentration of A, [A], and the concentration of B, [B]

rate ∝ [A][B]

so

rate = k[A][B]

This equation is called a rate law and describes the rate of reaction in terms of the concentration of the species taking part. The coefficient k is called the rate constant and is a number specific to the reaction at the temperature at which the reaction is run and describes how fast the reaction will proceed. It’s unit depend on the reactants. If the rate is only dependent on the concentration of one species then the units are s^-1, if it dependent on the concentration of two reactant as in the above reaction then the units are L mol^-1 s^-1.

We also have to classify the reaction based on it’s order of the reaction. This defines rate of consumption of the individual species involved and how this affects the overall rate. If we have a reaction

A → products

We have rate law

rate = k[A]

and as one molecule of A is involved then the reaction is said to be first order in A an so first order overall. For a reaction of the type

A + B → products

rate = k[A][B]

the reaction is first order in A, first order in B and therefore second order overall. If we have a reaction

A + 2B → products

Where two molecules of B are involved the rate law is written as

rate = k[A][B][B] = k[A][B]²

the reaction if first order in A, second order in B and third order overall.

It must be noted that rate laws are experimentally determined and can’t be simply derived by looking at the balanced equation for the reaction. There may be equilibria between various reactant and intermediates and may also proceed to the products by different routes, the contribution from each affecting the overall rate. A classic example of this is the gas phase reaction of hydrogen with bromine (note that sometimes the rate law is also dependent on the concentration of products).

H_{2 (g)} + Br_{2 (g)}→ 2HBr _(g)

Rate of formation of HBr = (k[H₂][Br₂]^3/2) / ([Br₂] + k’[HBr])

It is also possible to have reactions that are zero order where the overall rate is independent of the reactant concentration.

Experimentally Determining the Rate Law

In order to determine the rate law for a reaction we need to be able to separate out the contribution from each of the reactants. We do this by the isolation method in which we concentrate on the rate of change of the concentration of one of the reactant while the concentration of the other reactants are in an appreciable excess. If we have a reaction

A + B → products

and monitor the change in concentration of A where B is in excess, [B] is so large in comparison to [A] that it hardly changes and can be treated as being constant throughout the reaction. At any time during the reaction we can say that [B] is equal to the initial concentration, [B]₀, and so the rate law can be written as

rate = k’[A]

where k’ = k[B]₀. The reaction may be second order overall but under these condition is said to be pseudo-first order. If the reaction is then carried out with a variety of values of [B], we should then get a straight line for rate versus [B] showing the first order dependence on the concentration of B.

So we can isolate the contributions to the overall rate and determine the form of the rate law and calculate the value of the rate constant and tell what the rate of reaction will be at a given point in time if we know the concentrations of the reactant at that time. However if we want to know what the concentration of a reactant is at any given time during the reaction, or if we want to know the concentrations at the start of the reaction we must form integrated rate laws.

Integrated Rate Laws

Integrated rate laws allow the concentrations of reactants to be calculated at any point in time during the reaction. By integrating a proposed rate law for a reaction, it can be used as a predictive test to verify the rate law comparing it to experimental data.

For a first order reaction where a reactant A forms products, the differential rate law is written as

-d[A]/dt = k[A]

At the start of the reaction when t = 0, the concentration of A is [A]₀. This equation is rearranged to give

d[A]/[A] = -kdt

If we integrate this equation from t = 0 where [A] = [A]₀ to the time of interest t and [A] = [A]_t, we get the equation

ln([A]₀/[A]_t) = kt

So the concentration at time t is given by

[A]_t = [A]₀e^-kt