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# How Far? How Fast? - Kinetics

## Kinetics - Introduction

Kinetics is the study of how fast (the rate) a reaction progresses. The rate of a reaction is a physical property of a reaction and is measured by the change in some reaction quantity (e.g. volume, mass, concentration) per unit time.

The most common method used to calculate the rate of reaction is to measure the change in concentration of the reactant(s) per second. The rate becomes,

rate = - ( change in concentration of reactant in mol dm-3 ) / time in s

{or, rate = change in concentration of product / time}

This gives the unit of mol dm-3 s-1 for the rate of a reaction.

The rate of a reaction may be represented by a mathematical equation related to the chemical equation for a reaction.

e.g. for the simple hydrolysis reactions of haloalkanes,

SN1 : the most important reaction, or the rate determining step, is the breaking of the C-halogen bond -

The rate equation for this reaction is written as,

where, k = the rate constant for the reaction

and [(CH3)3CX] = the concentration, in mol dm-3, of the haloalkane.

SN2 : the rate determining step here is the displacement of the halogen atom with a hydroxyl group,

The rate equation here is,

In general for the mechanism equation

where, [ ] = concentrations of the various reactants

and m,n = the numbers of molecules of each reactant involved in the rate determining step also known as the orders of each reactant.

Units : rA = mol dm-3 s-1, [ ] = mol dm-3 and k = variable units depending on m and n.

In the examples you will meet in exam questions m and n will be 0, 1 or 2 with m+n = 2.

When, rA = k[A] the units of k are s-1.

When, rA = k[A]2 the units of k are dm3 mol-1 s-1.

N.B.: k is a constant only for a particular reaction and a particular set of reaction conditions.

## Kinetics - Graphs and Calculations

### (1) 0 order reaction graph :

If a reactant is said to be of 0 order in a reaction it doesn't effect the rate of the reaction. So graphical plots of concentration vs. time and rate vs. concentration would look like this,

N.B.: The rate of reaction at any particular time is found as the gradient of the concentration vs. time graph at the particular time. The gradient of rate of reaction vs. concentration is zero as the reactant has no effect on the overall reaction until after the rate determining step.

### (2) 1st order reaction graph :

If a reactant is said to be of 1st order, it has a uniform effect on the rate of reaction. So graphical plots of concentration vs. time and rate vs. concentration would look like this,

N.B.: The gradient of the concentration vs. time graph is not constant and gradually decreases as the concentration decreases. This produces a rate vs. concentration graph that is a straight line. The gradient of this line will be rate/concentration or k, the rate constant for the reaction.

### (3) Calculating initial rate using concentrations :

For the following set of data about the change in concentration of a reactant with time,

 Time (s) Concentration (mol dm-3) 0 0.50 5 0.43 10 0.37 20 0.27 30 0.20 40 0.15

a graph can be plotted,

The rate of reaction at various times can be found by taking tangents to the curve above and calculating their gradients.

(2) Calculating order of reaction from initial rates :

For a reaction involving three reactants, A, B and C, the following experimental data was found,

 Experiment Concentration of A (mol dm-3) Concentration of B (mol dm-3) Concentration of C (mol dm-3) Initial rate (mol dm-3 s-1) 1 0.01 0.01 0.01 0.05 2 0.01 0.02 0.01 0.10 3 0.02 0.01 0.01 0.10 4 0.01 0.01 0.02 0.05

A comparison of successive experimental runs shows that if the concentration of A is doubled (and all other concentrations are kept constant) then the rate doubles. This would follow if rate = k[A], therefore A must be a 1st order reactant.

If the concentration of B is doubled, with all the other concentrations remaining constant, then the rate is also doubled. This shows that B must be a 1st order reactant as well.

However, when C is looked at in a similar manner, a change in concentration has no effect on the rate. C is therefore of 0 order and takes no part in the rate determining step or the rate equation.

The rate equation for this reaction is, rA = k[A][B].

The reaction is 1st order with respect to A, 1st order with respect to B and 2nd order overall.

### (3) Calculating a rate constant from initial rates :

From the previous set of experimental data, rA = k[A][B]. Taking the information from experimental run 1, [A] = 0.01 mol dm-3, [B] = 0.01 mol dm-3 and rA = 0.05 mol dm-3 s-1,

therefore, k = 500 dm3 mol-1 s-1.

The same value of k is obtained no matter which set of experimental data are used.

## Kinetics - Half-lifes

The half-life of, t½, for a reaction is the time taken for the concentration of a reactant (or reactants) to halve exactly.

For a first-order reactant the half-life is both (i) a constant for the reaction and (ii) independent of the initial concntration of the reactant.

Therefore, knowing the half-life for a first-order reaction enables a graph of concentration vs. time to be plotted,

SN1 reactions and radioactive decay are typical examples of first-order reactions.

For a second-order reactant the half-life is both (i) not a constant for the reaction and (ii) dependent on the initial concntration of the reactant.

SN2 reactions are typical examples of second-order reactions.

## Kinetics - Reactions Mechanisms

Most reactions do not follow simple, single-step pathways. They involve two, three or more individual steps. However, one of these steps is always slower than the other and that individual stage is called the rate determining step - it is a limiting factor for the reaction rate. Only reactants appearing in the rate determining step, or in fast steps before the rate determining step, appear in the rate equation.

So, for a reaction involving A, B and C following the mechanism below,

The rate equation overall is,

rA = k[A2][B]

i.e. 1 moles of A2 and 1 mole of B are involved in the critical rate determining step and previous fast steps.

## Kinetics - Collision Theory

The simplest way to think of a reaction occurring is with the collision between two atoms or molecules.

The easier or faster these collisions occur, the faster the rate of reaction will be. As an example of this, an increase in the concentration of the reactants increases the frequency of collisions between species (there are more molecules present in a certain volume) and therefore increases the reaction rate.

The rate constant, k, for a rate equation brings in the idea of collisions between species into the rate equation itself. It gives some measure of how easy/effective the collisions are in forming the desired product.

#### Boltzmann distribution of energies :

At a given temperature a collection of molecules do not all have the same energy. Their energies follow the Boltzmann distribution,

A lot of molecules have a similar energy; however, there are some with less and some with more than this average value.

As well as depending on the simple collision of two molecules, the rate of reaction depends on the molecules having a certain amount of energy for a successful reaction to occur. This energy value is called the activation energy for a reaction - EA,

All the molecules in the shaded area have more than enough energy for the a successful reaction to occur between them when they collide.

As the temperature is increased the average energy possessed by the molecules increases,

Now more molecules have energy greater than or equal to EA so collisions are more effective. The molecules are moving faster, so the frequency of collisions is also greater. Both these factors lead to an increase in the rate constant, k, and thereby to an increase in the rate of reaction.

In fact, the rate constant and the activation energy are linked by this equation,

where, R = the gas constant, 8.314 J K-1 mol-1.

## Kinetics - Catalysis

### (1) General :

A catalyst is a compound that increases the rate of a reaction by providing an alternative reaction mechanism, for a chemical process. This alternative mechanism has a lower activation energy,

Now more molecules have energy greater then or equal to EA and so any collisions will be more effective, so the rate constant is higher and the rate of reaction is increased.

### (2) Homogeneous catalysts :

These are catalysts which are in the same phase as the reactants e.g. catalysts that are soluble in the same solvents as the reactants. With this sort of catalysis there is direct action between the catalyst and reactant.

e.g. Crown ethers are an example of this type of catalyst. They form complexes with ionic compounds, enabling the ionic compounds to be dissolved in organic solvents so they can carry out reactions there e.g. oxidation using potassium manganate(VII).

### (3) Heterogeneous catalysts :

This process uses catalysts which are in a different phase to the reactants e.g. a solid catalyst in a reaction between gases or a solid catalyst in a reaction occurring in solution.

With this type of catalysis the reactants have to be absorbed either into the solid or just onto the surface of the solid. The reaction proper can then occur on/in the solid catalyst.

e.g. In the hydrogenation of alkenes a nickel catalyst is used. The hydrogen gas is absorbed onto the surface of the nickel and the H-H bonds break to form single hydrogen atoms bound to the nickel's surface. These atoms are picked up by a passing alkene molecule, turning it into an alkane.

### (4) Enzymes :

These are biological catalysts which are often reaction specific. They have large, complex 3-D structures which provide sites for only certain molecules to interact with them.

They may also require a certain level of pH, moisture or other external factors.

### (5) Uses of catalysts :

Nitrogen fixation

Haber process, fertiliser production

Petroleum processing, cracking/reforming

Immobilised enzymes in industry

Catalytic converters in cars, lead as a poison