3.2.2 (f,g) The Boltzmann Distribution

Syllabus

(f) qualitative explanation of the Boltzmann distribution and its relationship with activation energy (see also 3.2.1 c)

(g) explanation, using Boltzmann distributions, of the qualitative effect on the proportion of molecules exceeding the activation energy and hence the reaction rate, for:

(i) temperature changes

(ii) catalytic behaviour (see also 3.2.2 c).

{Use of Boltzmann distribution model to explain effect on reaction rates.}

What does this mean?

Drawing and labelling

A Boltzmann distribution looks like this →

If you understand nothing about its meaning you should at least be able to draw it and label the axes - examiners like to ask that.

You may well see some with PARTICLE SPEED on the x-axis but Kinetic Energy and speed are proportional to each other.

It's tempting to think that at any temperature all the particles in a gas would have exactly the same amount of energy.

But this wouldn't last because the particles are colliding multiple times every second, exchanging energy.

So no particles have 0 energy - which is why the curve must start from the origin.

But there is some chance of a particle having any other energy we can think of - which is why the curve must never touch the x-axis again.

Vp - The most common energy (mode) is at the top of the curve

The shape is important too.

It shouldn't be a symmetrical, bell-shaped curve.

So the mean energy is to the right of the peak.

Changing Temperature

The standard question gives you a Boltzmann Distribution and asks you to draw one at a higher or lower temperature.

You should remember that the area under the graph represents the total number of particles.

This won't change with temperature - so the area under your new curve should be clearly similar to the one you were given.

If we increase the temperature there will certainly be more particles with higher energies.

So there must be fewer particles with lower energies.

The consequence is that the peak moves to the right but down.

The curve is below the lower temp at low energy and above it at high energy.

The two lines shouldn't meet.

Conversely, to draw the distribution for a lower temperature move the peak left and up.

Boltzmann and Activation Energy

The activation energy is the minimum energy needed to react.

If it is high there will be few particles with enough energy and there will be a slow reaction.

Catalysts lower activation energies.

This means that there will be more particles with enough energy to react.

And so the reaction will proceed faster.

An exam question will often ask you to use a Boltzmann Distribution to show the effect of a catalyst on the rate of reaction.

You should draw a distribution and mark an E_Act somewhere towards the right.

Then mark a lower E_Act somewhere to its left and make sure you label this one as "with catalyst".

Shade in the area to the right of the first E_Act and explain that this represents the number of particles with sufficient energy.

Then shade the area between the two E_Acts and explain that it represents the number of extra particles with sufficient energy once the catalyst has been added.

The other standard question asks you to explain the effect of temperature on the rate of reaction using a Boltzmann Distribution.

You should draw two distributions for different temperatures on the same axes.

You should then mark an E_Act somewhere on the right.

Shade in the area beneath the curve representing low temp. and to the right of E_Act and explain that this represents the number of particles with sufficient energy to react at that temp.

Then shade in the area beneath the two curves and to the right of E_Act and explain that this represents the extra number of particles with sufficient energy to react at that higher temp.

Video

Exam-style Questions

1. The diagram below shows the Maxwell–Boltzmann distribution of molecular energies in a sample of a gas.

(a) (i) State which one of X, Y or Z best represents the mean energy of the molecules.

............................................................................................................................................................................

(ii) Explain the process that causes some molecules in this sample to have very low energies.

............................................................................................................................................................................

.........................................................................................................................................................................(3)

(b) On the diagram above, sketch a curve to show the distribution of molecular energies in the same sample of gas at a higher temperature. (2)

(c) (i) Explain why, even in a fast reaction, a very small percentage of collisions leads to a reaction.

............................................................................................................................................................................

(ii) Other than by changing the temperature, state how the proportion of successful collisions between molecules can be increased. Explain why this method causes an increase in the proportion of successful collisions.

Method for increasing the proportion of successful collisions ........................................................................................

............................................................................................................................................................................

Explanation ...........................................................................................................................................................

.........................................................................................................................................................................(4)

(Total 9 marks)

2. The curve below represents the distribution of molecular energies at a temperature T₁ for a mixture of gases which react with each other. A is the activation energy for the reaction.

(a) (i) Label the vertical axis.

(ii) Explain the meaning of the term activation energy.

.............................................................................................................................................................................

(iii) Draw a second curve on the same axes, and label it T₂, for the same mixture at a lower temperature.

(iv) By reference to the curves, state and explain in molecular terms the effect of reducing the temperature on the rate of reaction.

Effect .....................................................................................................................................................................

Explanation ............................................................................................................................................................

.............................................................................................................................................................................

.........................................................................................................................................................................(8)

(b) The reaction is repeated in the presence of a catalyst.

Mark on the energy axis a possible activation energy B for the catalysed reaction. (1)

(c) Give an example of a catalyst and a reaction which it catalyses.

Catalyst .................................................................................................................................................................

Reaction using this catalyst ......................................................................................................................................

.........................................................................................................................................................................(2)

(d) Several small pieces of calcium carbonate (an excess) were allowed to react with 100 cm3 of 0.1 M Hydrochloric acid. The volume of Carbon Dioxide formed during the experiment is shown on the graph below.

Using the same axes, sketch curves labelled A, B and C for the following experiments which were all carried out at the same temperature as the original experiment:

A 50 cm3 0.2 M Hydrochloric acid with an excess of small pieces of Calcium Carbonate;

B 50 cm3 0.1 M Hydrochloric acid with an excess of small pieces of Calcium Carbonate;

C 50 cm3 0.1 M Hydrochloric acid with an excess of powdered Calcium Carbonate. (6)

(Total 17 marks)

Answers

1. (a) (i) Z 1

(ii) Collisions 1

Cause some molecules to slow down or lose energy 1

(b) Curve starts at origin and is displaced to the right 1

Curve lower and does not touch energy axis 1

(c) (i) Only a small percentage/very few collisions have E > Ea 1

(ii) Add a catalyst 1

Lowers Ea 1

More collisions/molecules have energy > Ea 1

[9]

2. (a) (i), (iii) and (b)

(ii) Minimum (1) energy needed to cause reaction (1)

(iv) Effect reduces rate (1)

Explanation fewer molecules (1)

have sufficient energy (1) 8

(b) Mark on the energy axis a possible activation energy B for the catalysed

reaction. 1

(c) Catalyst e.g. MnO2 (1)

Reaction using this catalyst decomposition of H2O2 (1) 2

(d) Shape (1) final vol (1) for each curve

[17]

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