Maxwell-Boltzmann Distribution

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.)

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.

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.

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.

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