In this explainer, we will learn how to use the formula (Boyleβs law) to calculate the pressure or volume of a gas that is allowed to expand or contract at a constant temperature.

First, let us understand what an βideal gasβ is. A gas is made up of very small particles that move around, occasionally colliding with each other. In an ideal gas, we assume that these particles are so small that they take up no individual volume and that there are no interactions between these particles.

### Definition: Ideal Gas

An ideal gas is made up of particles that occupy negligible space and do not interact with each other.

Recall that volume is a measure of how much space something occupies. When we deal with gasses, it can be hard to imagine the volume that lots of tiny particles take up, so it is often useful to think of the gas being placed inside a container. Recall also that a gas will expand to fill any container it is inside.

This gives us a measure of the volume of a gas; now we must understand pressure.

If we consider a small section of the wall of the container, we will see particles moving in random directions, and some of them collide with the wall.

Each collision exerts a small force on the wall. Over the whole surface of the container at any instant, there is a constant force pushing on the walls.

When we divide this force by the total area it is being acted on, we get the **pressure**.

Pressure has a unit of N/m^{2};
notice that this unit is **force divided by area**.

Now that we are familiar with volume, pressure, and some of the units of pressure, we can look at the relationship between volume and pressure.

If we expand our container, the volume the gas occupies increases. However, because we have not added or taken away any gas, the number of gas particles remains the same.

If we consider our small section of the wall again, we will see that the particles are much more spread out, and therefore there are less collisions with the wall of the container at each instant. The surface area of the walls of the container has also increased. Together, these mean that the number of collisions per small section of area has decreased; the pressure has decreased.

Similarly, if we shrink the container we will have the opposite effect. There will be more collisions per small section of area, meaning pressure will increase if volume decreases.

Let us look at an example question about this.

### Example 1: The Relationship between Pressure and Volume of an Ideal Gas at Constant Temperature

For a gas at a constant temperature, if the volume is , then the pressure .

- increased, stays the same
- decreased, stays the same
- increased, decreases
- increased, increases
- decreased, decreases

### Answer

To answer this question, we should imagine what happens to the particles colliding with the walls of a container filled with the gas.

Recall that there is a relationship between the volume and pressure of an ideal gas at constant temperature. If volume increases or decreases, volume will also change. This immediately rules out options A and B.

If the volume is decreased, as in options B and E, at any instant, more particles would be colliding with the walls of the containers.

As we have learnt, this means there is a higher amount of force being exerted on the walls per unit area, meaning pressure has increased.

This rules out option E and again rules out option B.

Now let us consider what happens when the volume of the container is increased, as in options A, C, and D. Particles are further apart, and the surface area of the container has increased, so at any moment there will be less collisions between the particles and the walls of the containers.

This means that the pressure of the gas decreases. This rules out options A and D and corresponds to option C.

The correct answer is therefore option C; for a gas at a constant temperature, if the
volume is *increased*, then the pressure *decreases*.

This relationship between pressure and volume was discovered in the 17th century, and the exact relationship is known as βBoyleβs law.β

Boyleβs law states that the pressure exerted by an ideal gas is inversely proportional to the volume it occupies if the temperature and amount of gas remain constant.

### Definition: Boyleβs Law

The absolute pressure exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies if the temperature and amount of gas remain unchanged within a closed system.

The term inversely proportional means that if pressure, , increases by some factor, volume, , decreases by the same factor. This can be written as

Another way to write this relationship is to include some constant, :

Multiplying both sides by , we get

So the pressure multiplied by the volume the gas is occupying is constant, provided we also keep the temperature and the amount of gas constant.

This has some interesting implications. If the volume the gas is occupying is increased massively, so there is a huge amount of distance between particles (like in space), then the pressure of the gas must be very very small (also known as a vacuum).

Or, going in the other direction, if we shrink the container down to a tiny volume, the pressure of the gas becomes very large.

A possible graph of this relationship is shown here.

The constant in our equation, , depends on many other factors, such as the gas we are considering and the temperature.

Let us take a look at a series of volume changes at constant temperature.

As we have learnt, the pressure multiplied by the volume at each of these points is constant. That means that

Plotting these points on a graph of pressure and volume, we can see that all of these points lie on the same line: .

Using this relationship, we can calculate the pressure of a gas after a volume change at constant temperature.

If we know the pressure, , and volume, , of a gas before a volume change and the volume afterward, , then we can calculate the pressure afterward, .

Starting with we can divide both sides by to give an expression for the pressure after the volume change:

Let us work through an example question looking at the pressure change when a gas is compressed at constant temperature.

### Example 2: Using Boyleβs Law to Find the Pressure of a Gas

A gas with a volume of 2 m^{3}
is at a pressure of 500 Pa.
The gas is compressed at a constant temperature to a volume of
0.5 m^{3}.
What is the pressure of the gas after it is compressed?

### Answer

Boyleβs law states that the absolute pressure, , exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies, , if the temperature and amount of gas remain unchanged within a closed system.

This can be written as

In this question, we are asked to consider two points in time: before and after a compression.

At point 1, we are given and . At point 2, we are given , and we are asked to calculate .

Because is constant, we can write

If we divide both sides by we are left with an expression for :

We can now substitute the values given to us into the equation:

Similar to calculating the pressure of a gas after a volume change at constant temperature, Boyleβs law can also be used to calculate the volume of a gas after a pressure change at constant temperature.

If we know the pressure, , and volume, , of a gas before a pressure change and the pressure afterward, , then we can calculate the volume afterward, .

Starting with we can divide both sides by to give an expression for the volume after the pressure change:

Let us work through an example question looking at the volume change when a gas is compressed at constant temperature.

### Example 3: Using Boyleβs Law to Find the Volume of a Gas

A gas with a volume of 3 m^{3}
is at a pressure of 500 Pa.
The gas is compressed at a constant temperature until it is at a pressure
of 1βββ500 Pa. What is the volume
of the gas after it is compressed?

### Answer

Boyleβs law states that the absolute pressure, , exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies, , if the temperature and amount of gas remain unchanged within a closed system.

This can be written as

In this question, we are asked to consider two points in time: before and after a compression.

At point 1, we are given and . At point 2, we are given , and we are asked to calculate .

Because is constant, we can write

If we divide both sides by , we are left with an expression for :

We can now substitute the values given to us into the equation:

We can use Boyleβs law to calculate pressure and volume changes after multiple compressions or expansions at constant temperature. This is because the pressure multiplied by the volume of the gas remains constant throughout.

For changes, the following can be written:

Let us work through an example question where multiple compressions or expansions take place.

### Example 4: Using Boyleβs Law to Calculate Volume Changes over Multiple Compressions and Expansions

A gas initially has a pressure of 800 Pa
and a volume of 2 m^{3}.
It is compressed at a constant temperature until its volume is half its initial value.
At this point it has a pressure . It is then allowed to expand again until the
pressure is . What is the final volume of the gas?

### Answer

Starting with Boyleβs law,

In this question, we have three points, which we will call points 0, 1, and 2:

The question tells us that , , is unknown, , , and is unknown.

This is a lot of information, but there is a trick here that will simplify the problem for us a lot.

Let us consider the expansion from point 1 to point 2:

Although we do not know , is given to us in terms of :

Substituting this into our equation from point 1 to point 2, we get

Dividing both sides by , this becomes

Dividing both sides by 0.25, we get our expression for :

The question tells us that and that , so

### Key Points

- Boyleβs law relates the pressure and volume of a constant amount of an ideal gas at constant temperature.
- Boyleβs law states that, for an ideal gas at constant temperature in a closed system, the pressure of the gas is inversely proportional to the volume of the gas: Another way to write this is to include a constant, :
- This can be used to relate the pressure and volume at different stages in the compression and expansion of a gas: