The Rate Law of a Chemical Reaction

The Dependence of Rate on Concentration

One of the goals of chemical kinetics is to be able to predict the rates of reactions.  This is done by using mathematical equations known as rate laws.  A rate law is an equation that relates the rate of a reaction to the concentrations of reactants and catalyst raised to various powers.  For the general equation 

aA + bB + .. ==> gG + hH + ..

where A, B, ..., represent the reactants; G, H, ..., represent the products; and a,b,g,h,..., represent the coefficients in the balanced equation.  The rate law usually takes the form

The terms [A], [B],.., are molarities. The exponents m, n, ..., which must be determined experimentally, are usually small numbers, not necessarily related to the corresponding coefficients.  The proportionality constant, k, is called the rate constant.  The units of  k depend on the particular rate law.  Its numerical value depends on the particular reaction, the temperature, and the presence of a catalyst.  The larger the value of k the faster the reaction will go.

Reaction Order

We can see from the rate law above that rate would depend upon reactant concentration.  One way to predict how strongly the concentrations of the reactants affect the rate of a given reaction is to determine the order of a reaction.  The order of a reaction is determined by the values of the exponents in the rate equation.  In the rate law above, if m = 1, we say that the reaction is first order in A.  If n = 3, the reaction is third order in B, etc. The overall order of the reactions the sum of the exponents in the rate equation: m + n + ...

 Determining the Rate Law

The experimental determination of the rate law for a reaction requires that you find the order of the reaction with respect to each reactant and any catalyst.  There are several methods which can be used to establish the order of a reaction.  The one we will consider here and the one used in the companion drill problem, is the initial rate method.  This method consists of doing a series of experiments in which the initial concentrations are varied.   The initial rates are compared, from which the reaction orders can be deduced.  The method can best be explained by an example.

Consider the reaction A + 2 B + C ==> products
A series of four experiments was run at different initial concentrations of the reactants, and the initial reaction rate was determined (see table). From these data, determine the reaction orders with respect to each of the reactants. Assume that the rate law has the following form
Rate = k[A]m[B]n[C]p    
Initial Concentration (M) Init. Rate
[A] [B] [C] [M/s]
Exp. 1 0.0113 0.0113 0.113 1.12e-4
Exp. 2 0.0226 0.0113 0.113 2.24e-4
Exp. 3 0.0226 0.0226 0.113 8.96e-4
Exp. 4 0.0113 0.0113 0.226 4.48e-4
Solution

We will consider two experiments at a time. The experiments are chosen so that all reactant concentrations remain constant except one. Notice that in experiments 1 and 2 the concentration of only reactant A is changed, this will allow us solve for m, the order of the reaction with respect to A.
Write the rate law algebraically for experiment 1 and 2.
Rate1 = k[A]1m[B]1n[C]1p
Rate2 = k[A]2m[B]2n[C]2p
Next we divide the second equation by the first.
Rate2 k[A]2m[B]2n[C]2p
------- = -----------------------
Rate1 k[A]1m[B]1n[C]1p
Cancelling k and grouping terms we have
Rate2/Rate1 = ([A]2/[A]1)m([B]2/[B]1)n([C]2/[C]1)p
Substitute in values from experiment 1 and 2.
(2.22e-4/1.11e-4) = (0.0260/0.0130)m (0.0130/0.0130)n (0.130/0.130)p
This gives 2 = 2m from which we deduce that m = 1. We can obtain n by going through the same procedure using experiments 2 and 3, this gives us 4 = 2n hence n = 2. Using experiments 1 and 4, we find 4 = 2p and so p = 2.