Introducing Chemical Kinetics - Rate Equations

By rate of reaction we mean the change in concentration of a reactant (or a product) in a given period of time. This might, for example, be over a short period of time anywhere during the reaction, or it might be at an instant of time (or corresponding concentration of a reactant).

Chemists need to understand chemical kinetics so that industrial reactions can be controlled and their mechanisms understood. It also helps them to make predictions about important reactions such as those that occur between gases in the atmosphere.

Progressive Curve Methods

Consider the reaction involving the decomposition of dinitrogen pentoxide dissolved in tetrachloromethane at 30 C. (Temperature must be kept constant since rate of reaction is also affected by temperature.)

N2O5(sol) ® 2NO2(sol) + O2(g)

As this reaction proceeds from start to finish, its rate will diminish as the N2O5 is used up and its concentration decreases. This changing rate could be observed by monitoring the concentration of N2O5(sol) (written as [N2O5(sol)]) during the reaction. For example, the increase in pressure of O2 is related to the decrease in concentration of N2O5.

For each mole of O2 formed, 1 mole of N2O5 is consumed.

The Concentration against Time graph for this reaction is shown below:

During the reaction, the rate at any instant of time can be determined by measuring the gradient of the tangent to the curve at that time. This also corresponds to the rate at an instant of concentration.

For the above reaction, if the rate is measured at a range of instants of concentration during the experiment, a Rate against Concentration graph can be plotted. The resulting graph is given below:

This graph shows that the rate of reaction is directly proportional to the concentration of [N2O5(sol)]:

rate a [N2O5(sol)]      or       rate = constant x [N2O5(sol)]

rate = k[N2O5(sol)]

This expression is called a rate equation, where k is called the rate constant. The concentration of N2O5(sol) is raised to the power 1. We say that the reaction is first order with respect to the [N2O5(sol)]. That is:

when the concentration of N2O5(sol) is doubled, the rate doubles.

The rate constant, k, can be calculated from the rate equation:

k = rate/[N2O5(sol)]

This shows that the value of k is equal to the gradient of the graph. In this case the units of k are s-1.

The units of k are dependent on the particular rate expression.

However, for the decomposition of [N2O5(sol)] a plot of rate against concentration is not needed to show that the reaction rate is first order with respect to [N2O5(sol)].

The 'concentration against time' curve shows repeating constant half-lives for a reaction
that is first-order with respect to a reactant.

Half-life is the time required for one-half of a given quantity of a reactant to react. It is commonly used as a measure of the rate of first-order reactions: the shorter the half-life, the faster the reaction. It indicates the kinetic stability of a reactant: the longer the half-life, the greater the stability. Read more about Half-life. Read more about Chemical Stability.

The rate equation has to be experimentally determined...

Consider the reaction between hydrogen peroxide and iodide ions in acidic solution.

H2O2(aq) + 2I-(aq) + 2H+(aq) ® 2H2O(l) + I2(aq)

The rate equation for this reaction can be written as:

rate = k[H2O2(aq)]a[I-(aq)]b[H+(aq)]c

The orders of reaction, a, b and c, have to be determined by experiment. This is why the rate equation is said to be experimentally determined.


A catalyst may appear in the experimentally determined rate equation...

but it is not usual to include it in the balanced chemical equation for the reaction. For example, the acid-catalysed hydrolysis of the ester methyl methanoate...

HCOOCH3 + H2O + H+ ® HCOOH + CH3OH + H+

rate = k[HCOOCH3][H+]

Here we are going to determine the order of reaction with respect to [H2O2(aq)]. For this to be done, only the [H2O2(aq)] can be allowed to change, so the concentrations of the other reactants have to be kept constant. Having them present in large excess so their concentrations change so little during the reaction that they remain effectively constant achieves this. Under these circumstances the rate equation becomes

rate = k'[H2O2(aq)]a

where k' is a modified rate constant which includes the constant concentrations of the other reactants. An instrumental method called colorimetry could be used to monitor the increasing concentration of I2(aq) during the experiment. This is related to the intensity of light reaching the photocell in the colorimeter. Knowing the starting concentration of H2O2(aq), its diminishing concentration can be determined at time intervals. Finding the order of reaction with respect to [H2O2(aq)] follows the method already described. It is first order with respect to [H2O2(aq)], that is, a = 1 in the equation above.

So, the experiments referred to above have been used to find the order of reaction with respect to a reactant. The experimental method used is a progressive curve method.

Let's summarise rate equations¼

Consider the reaction:

aA + bB ® products

Experiments show that the reaction rate can be related to the concentration of individual reactants by a rate equation of the form

rate = k[A]m[B]n

where m and n are constants whose values are usually 0, 1 or 2, and k is the rate constant, with units depending upon the particular rate equation. m is the order of reaction with respect to reactant A, and n with respect to reactant B. The overall order of reaction is the sum of the powers of the concentrations of the individual reactants in the rate equation, that is (m + n).

Reactions may commonly be zero order, first order or second order with respect to a particular reactant. A reaction is zero order with respect to a reactant when changing the concentration of that reactant has no effect on the reaction rate. A reaction that is first order with respect to a particular reactant has its rate doubled when the concentration of that reactant is doubled. Where doubling the concentration of a reactant results in a quadrupling (x4) of the rate, the reaction is second order with respect to that reactant.

Order of ReactionConcentration doubled. The rate:Concentration tripled. The rate:
ZeroDoes not changeDoes not change
1stDoubles (x2)Triples (x3)
2ndQuadruples (x4)x9

You can establish the order of reaction with respect to a reactant from a concentration against time graph. However, it can sometimes be difficult to decide if a reaction is first-order or second-order from the concentration-time graph. A rate-concentration graph quickly reveals the order with respect to a reactant.

Compare 'concentration-time' and 'rate-concentration' graphs...

Concentration of a Reactant against Time
 Zero Order
Rate of Reaction against Concentration of a Reactant
 Zero Order

Concentration-time and Rate-concentration graphs.

Remember, rate equations are experimentally determined. The following reaction occurs at 800 °C:

2H2(g) + 2NO(g) ® 2H2O(g) + N2(g)

rate = k[NO(g)]2 [H2(g)]

Notice that the orders do not correspond to the numbers in the balanced chemical equation. The reaction is third order overall.

Initial Rate Methods

Another method of determining the order of reaction with respect to a particular reactant is the initial rate method (as opposed to the continuous method). This involves repeating the same experiment several times, but using a different initial concentration of the reactant in question, and keeping all other reactant concentrations (and temperature) the same in each. Consider the decomposition of H2O2(aq):

H2O2(aq) ® H2O(l) + O2(g)

For each initial concentration of H2O2(aq), a concentration against time curve is obtained. The [H2O2(aq)] may be plotted against time, or alternatively the volume of oxygen collected, for example, can be used instead. A tangent is drawn to each curve at time t = 0 (where the curve is virtually a straight line) to find the initial rate corresponding to each concentration of H2O2(aq). A rate against concentration graph can now be plotted to determine the order of reaction with respect to the H2O2(aq).

Clock reactions...

The initial rate method is conveniently employed for clock reactions. For each experiment (corresponding to each initial concentration of reactant) the time taken for a definite, small amount of a product to be formed at the start of the reaction (when its rate is most rapid) is measured. This gives a measure of the initial rate of the reaction because the shorter the time taken for it to form, the faster the rate. Rate and time are inversely related, or rate a 1/time. A rate against concentration curve is obtained by plotting 1/t against concentration, revealing the order of reaction with respect to the reactant concerned.

Consider again the following reaction:

H2O2(aq) + 2I-(aq) + 2H+(aq) ® 2H2O(l) + I2(aq)

The same amount of iodine is known to be formed in each experiment by including the same volume of sodium thiosulphate solution of known concentration in each experiment. A little starch solution is also included in the reaction mixture. To begin with, the iodine formed immediately reacts with thiosulphate ions.

2S2O22-(aq) + I2(aq) ® 2I-(aq) + S4O62-(aq)

This happens until that instant when all the thiosulphate has reacted, at which point the I2 formed reacts immediately with the starch forming a deep blue-black colour. The time taken at this instant is noted. For each experiment the total volume has to be made the same by adding the required volume of water. This ensures that the [H2O2(aq)] is proportional to the volume of its solution used. A rate against concentration curve can be obtained by plotting 1/t againts volume. Now see if you can follow this actual kinetics experiment which uses an initial rates method.

Have a go at this question¼

The reaction between NO(g) and Cl2(g) has been studied at 50 C, recording the initial rate of formation of NOCl(g) for the initial concentrations of reactants as shown in the table.

NO(g) + Cl2(g) ® NOCl(g)

[NO(g)] (mol dm-3) [Cl2(g)] (mol dm-3)Initial Rate (mol dm-3 s-1) 
0.250 0.250 1.43 x 10-6
0.250 0.500 2.86 x 10-6
0.500 0.500 11.4 x 10-6

The rate equation takes the form:

rate = [NO(g)]m [Cl2(g)]n

The order of reaction with respect to NO(g) is:
    
The order of reaction with respect to Cl2(g) is:
    
The overall order of reaction is:
    
The value of the rate constant, k, at 50 C is:
mol-2 dm6 s-1
  
The rate of formation of NOCl when [NO(g)] and [Cl2(g)] are both equal to
0.110 mol dm-3 is:
mol dm-3 s-1
  
The rate at which NO is reacting, at the instant when Cl2 is reacting at the rate of
2.21 x 10-7 mol dm-3 s-1 is:
mol dm-3 s-1 
  
The rate at which NOCl is forming, at the instant when Cl2 is reacting at the rate of
2.21 x 10-7 mol dm-3 s-1 is:
mol dm-3 s-1 
  

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Reaction Mechanisms

Consider the hydrolysis of a tertiary halogenoalkane, for example, (CH3)3C-Br.

R-Br + OH-    ®    R-OH + Br-

Changing the concentration of the OH- ions is found to have no effect on the rate of the reaction. The experimentally determined rate equation is:

Rate = k[RBr]   (or   Rate = k[RBr][OH-]0   since [OH-]0 = 1)

So why is it that in some reactions changing the concentration of a particular reactant has no affect on the rate?

Where this is the case, there must be at least two steps involved in the reaction. Here we are talking about reaction mechanisms. The mechanism of a reaction is a proposal about the series of steps involved in going from reactants to products. It is a suggestion based on the scientific evidence available for the reaction. Part of the experimental evidence is the rate equation. Where there are two or more steps, it is the slowest step that determines the overall rate. This is called the rate-determining step (RDS). The numbers before the formulae in the balanced chemical equation for the rate-determining step respectively correspond to the powers to which the reactant concentrations are raised in the rate equation. A zero order reactant does not appear in the chemical equation for the rate-determining step. The term molecularity refers to the number of chemical species in the rate-determining step. Read more about Mechanisms.

For the hydrolysis of a tertiary halogenoalkane the following two-step mecahanism is proposed:

Step 1 R-Br    ®    R+ + Br-slow  RDS 
Step 2 R+ + OH-    ®    R-OHfast 

Step 1 is the 'difficult' one in which the halogenoalkane ionises. Once the carbocation (R+) is formed an OH- ion nucleophile quickly adds to it forming the alcohol product. Overall, the mechanism for the reaction is nucleophilic substitution, but it is also known as an SN1 mechanism (substitution nucleophilic unimolecular) because the molecularity of the rate determining step is 1.