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# Maxwell-Boltzmann Distribution

Have you ever considered how safety matches work? How does a small splint of wood suddenly burst into flame?

There are two parts to safety matches. The first is the match head, full of an oxidising agent such as potassium chlorate. The second is the rough surface on the side of the matchbox. This contains red phosphorous. When you strike the match head against this rough surface, you provide enough energy to turn some of the red phosphorous into white phosphorous vapour. The white phosphorous vapour spontaneously ignites in the air. This heat is enough to start decomposing the potassium chlorate inside of the match head, releasing oxygen which fuels the flame further. Your match is now lit.

But this reaction only happens because you provide energy - in this case, from the friction caused by rubbing the match head against the rough side of the matchbox. Providing more energy means that some of the particles meet the activation energy requirements of the reaction. This is where the Maxwell-Boltzmann distribution graph comes in.

The Maxwell-Boltzmann distribution is a probability function that shows the distribution of energy amongst the particles of an ideal gas.

It's a handy way of showing how particles within a substance vary in energy, including how many meet or exceed the activation energy of a reaction.

• We will start by explaining how you interpret a Maxwell-Boltzmann distribution graph. This will include looking at points like the average energy and the most probable energy.
• We'll then consider how the graph changes under certain conditions, such as when you increase the temperature, or add a catalyst.

## The Maxwell-Boltzmann distribution curve

As we mentioned above, the Maxwell-Boltzmann distribution is a probability function that shows the distribution of energy amongst the particles of an ideal gas. (See Chemical Kinetics for more on this subject.)

An ideal gas is a hypothetical gas made up of particles that don't interact. Although we rarely come across ideal gases, it's still useful to imagine their behaviour using a Maxwell-Boltzmann curve, because we can apply these conclusions to any gas or solution.

Put simply, a Maxwell-Boltzmann distribution graph shows how the energy of gas particles varies within a system. Don't worry - we'll go through this now.

Notice here that we are using the word 'particles'. This is because the Maxwell-Boltzmann distribution applies to all sorts of gaseous species, from atoms to ions to molecules.

### Axes

Here's a typical example of a Maxwell-Boltzmann distribution graph.

Fig. 1 - A Maxwell-Boltzmann distribution graph

On the x-axis, we have energy and on the y-axis, we have the number of particles.

You might find other values represented on the axes. For example, energy might be replaced with speed. However, speed is just a measure of kinetic energy. Here, they are similar enough that we can swap one for the other. Particles with a large amount of energy move at high speeds - it is as simple as that.

Likewise, on some graphs, the y-axis actually shows the probability of a gas particle having a particular energy. However, we can generalise this to the number of particles with each energy value. For example, if you had 100 gas particles and the probability of them having a certain energy was 0.05, then you could expect to find 5 particles with this energy.

#### Energy

Particles don't have a fixed amount of energy. Instead, their energy levels constantly change as they move around and collide with one another. A Maxwell-Boltzmann distribution simply shows us the different energies we could expect to see at any one moment in time.

#### Number of particles

The y-axis shows the number of particles with each particular amount of energy. A higher value means more particles have that energy. If you add up the number of particles with each energy value, you'll get the total number of particles. This is equal to the area under the graph.

By looking at the graph, we can see the following things:

• No particles have either negative energy or zero energy, shown by the left-hand limit of the curve, which goes through the origin.

• A few particles have a very large amount of energy, shown by the long right-hand tail of the curve. In fact, there is no upper limit to the energy a particle can have - the curve stretches on indefinitely.

• Most particles have an intermediate amount of energy, shown by the large peak in the middle of the curve.

Fig. 2 - The different energies of particles on a Maxwell-Boltzmann distribution graph

### Points on the graph

Let's look at our graph again. This time, we're going to mark certain points on it.

Fig. 3 - Points on a Maxwell-Boltzmann distribution graph

#### Most probable energy

The highest point of the graph's peak represents the most probable energy of the particles. Out of all the different energy values present, the greatest number of particles have this one particular energy.

#### Average energy

The line marked to the right of the most probable energy shows the average energy of the particles. To be more precise, this is the median energy value. Exactly half of the particles have more energy than this, while exactly half of the particles have less energy than this.

#### Activation energy

On the right-hand side of the graph lies the activation energy.

Activation energy is the minimum amount of energy needed to start a chemical reaction. It takes the symbol .

All the particles to the right of this point meet the activation energy requirements of this particular reaction. This means that they could potentially react. All the particles to the left of this point don't meet the activation energy. They don't have enough energy to react.

This is why matches are perfectly safe if left alone. The particles don't have enough energy to meet the activation energy requirements needed to start a reaction.

## Factors affecting Maxwell-Boltzmann distributions

Now we know what Maxwell-Boltzmann distribution graphs are, we can look at factors affecting them. These include:

• Temperature

• The presence of a catalyst

We can then apply this to rate of reaction.

### Temperature

First, let's look at the effect of heating a system.

#### Increasing the temperature

When we heat particles, we supply them with energy. This means two things.

• The particles have more energy overall, so a larger number of particles meet or exceed the activation energy.

• The particles also have more kinetic energy. On average, they move faster, and there are more collisions per second.

If you heat a gaseous system, there are more collisions per second, and on average, more of the colliding particles meet the activation energy. This means that the rate of reaction increases.

Let's look at the effect of this on a Maxwell-Boltzmann graph.

Fig. 4 - The effect of increasing the temperature on a Maxwell-Boltzmann distribution graph

Notice that the tip of the peak, which shows the most probable energy, flattens and moves to the right. The most probable energy of the particles has increased. You can also see that more particles now meet the activation energy. This contributes to an increased rate of reaction, as we explored above.

This is how we light matches. Rubbing them against the rough side of the matchbox causes friction, which provides the particles within the match head with energy. Now, a significant number of them have enough energy to react - they meet the activation energy requirements of the reaction.

#### Decreasing the temperature

If we decrease the temperature, the reverse happens - the peak of the graph shifts up and left, and fewer particles now meet the activation energy. This decreases the rate of reaction.

Fig. 5 - The effect of decreasing the temperature on a Maxwell-Boltzmann distribution graph

### The presence of a catalyst

Adding a catalyst doesn't change the energy of any of the particles. Instead, it reduces the activation energy of the reaction. This means that a greater number of particles now meet or exceed the activation energy. This increases the rate of reaction.

Fig. 6 - The effect of adding a catalyst on a Maxwell-Boltzmann distribution graph

How does concentration affect a Maxwell-Boltzmann distribution graph?

Well, concentration is simply a measure of the number of particles in a given volume. To increase concentration, we can decrease the volume of a system. The total number of particles stays the same - they are just crammed into a smaller space.

You'll notice that we haven't done anything to change the energy of the particles. This means that the Maxwell-Boltzmann distribution stays exactly the same. Concentration has no effect on a Maxwell-Boltzmann distribution.

What if we increase the concentration by keeping the volume the same but increasing the number of particles in the system? This is easier to understand if we swap the number of particles on the y-axis for probability. Because we haven't done anything to change their energy, the probability of each particle having a specific energy stays the same, so the graph remains unchanged.

But you might also know that increasing the concentration of a species increases the rate of reaction. How does this work, if concentration doesn't affect a Maxwell-Boltzmann distribution?

Remember, concentration is a measure of the number of particles in a given volume. A more concentrated system has the same number of particles but in a smaller container. On average, the particles are closer together, and on average, they travel less between collisions. This means that there are more collisions per second, and thus a faster rate of reaction.

## Maxwell-Boltzmann Distribution - Key takeaways

• The Maxwell-Boltzmann distribution is a probability function that shows the distribution of energy amongst the particles of an ideal gas. You'll find the number of particles on the y-axis, and energy on the x-axis.
• Maxwell-Boltzmann distribution graphs often show the most probable energy, average energy, and activation energy.
• No particles have zero energy. A few particles have a very large amount of energy. However, most particles have an intermediate amount of energy.
• Increasing the temperature shifts the peak of the Maxwell-Boltzmann distribution to the right and increases the number of particles that meet or exceed the activation energy. This increases the rate of reaction.
• Adding a catalyst reduces the activation energy and so increases the number of particles that meet or exceed the activation energy. This increases the rate of reaction.
• Different particles within a gas all have differing levels of energy. Some have a lot of energy, whilst some only have a little. Most fall somewhere in between. On a Maxwell-Boltzmann graph, energy increases from left to right, so the particles on the right-hand side have more energy than those on the left.

The Maxwell-Boltzmann distribution is a probability function that shows the distribution of energy amongst the particles of an ideal gas. It predicts how many particles within a system have a particular energy.

The Maxwell-Boltzmann distribution is a probability function that shows the distribution of energy amongst the particles of an ideal gas. Put simply, a Maxwell-Boltzmann distribution graph shows how the energy of gas particles varies within a system.

Concentration doesn't affect a Maxwell-Boltzmann distribution. Increasing the concentration doesn't change the energy of the particles in a system, so the distribution stays the same.

The Maxwell-Boltzmann distribution is a probability function that shows the distribution of energy amongst the particles of an ideal gas. Put simply, a Maxwell-Boltzmann distribution graph shows how the energy of gas particles varies within a system. Kinetic energy is directly related to speed, so this is just a measure of the speeds of particles.

Yes, the presence of a catalyst increases the rate of reaction.

## Final Maxwell-Boltzmann Distribution Quiz

Question

What is a Maxwell-Boltzmann distribution?

A probability function that shows the distribution of energy amongst the particles of an ideal gas.

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Question

What is on the y-axis on a Maxwell-Boltzmann distribution graph?

Number of particles

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Question

What is on the x-axis on a Maxwell-Boltzmann distribution graph?

Energy

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Question

What does the area under the curve in a Maxwell-Boltzmann distribution graph represent?

The total number of molecules.

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Question

Increasing the temperature of a system ______.

Increases the rate of reaction

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Question

Decreasing the temperature of a system ______.

Decreases the rate of reaction

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Question

Decreases the activation energy

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Question

Explain why a small increase in temperature leads to a large increase in rate of reaction.

Increasing the temperature increases the energy of the particles. This means that a greater number meet or exceed the activation energy so there is an increased chance of a successful collision when they collide. It also means that the particles move faster and so have more collisions per second.

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