Boyle's Law

Boyle's Law Overview

Before we talk about Boyle's law, let's talk about the components involved: ideal gases, pressure, and volume.

First up, let's talk about ideal gases.

When looking at this law and other related gas laws, we are typically applying them to ideal gases.

Ideal gases are a way to approximate gas behavior since "real" gases can be a bit tricky. However, the ideal gas model is less accurate than the behavior of a Real Gas at low temperatures and high pressure.

Next up, let's talk pressure. Since (ideal) gases are constantly in motion, they often collide with each other and the walls of their container. Pressure is the force of the gas particles colliding with a wall, divided by the area of that wall.

Lastly, let's discuss volume. Volume is the space a substance takes up. Ideal gas particles are approximated to have negligible volume.

Boyle's Law Definition

The definition of Boyle's law is shown below.

In other words, if volume decreases, pressure increases and vice-versa (assuming gas amount and temperature have not changed).

Boyle's Law Experiment

To get a better understanding of this law, let's do an experiment.

We have a 5L container of 1.0 mol of hydrogen gas. We use a manometer (pressure reading instrument), and see that the pressure inside the container is 1.21 atm. In a 3 L container, we pump in the same amount of gas at the same temperature. Using the manometer, we find that the pressure in the container is 2.02 atm.

Below is a diagram to illustrate this:

Fig.1-Diagram of Boyle's law

As the volume decreases, the gas has less room to move. Because of this, the gas particles are more likely to collide with other particles or the container.

This relationship only applies when the amount and temperature of the gas are stable. For example, If the amount decreased, then the pressure might not change or even decrease since the ratio of moles of gas-particle to volume decreases (i.e there is more room for particles since there are fewer of them).

Boyle's Law Constant

One way to visualize Boyle's law mathematically is this:

$$P \propto \frac{1}{V}$$

Where,

• P is pressure

• V is volume

• ∝ means "proportional to"

What this means is that for every change in pressure, the inverse volume (1/V) will change by the same amount.

Here's what that means in graph form:

Fig.2-Boyle's law graph

The graph above is linear, so the equation is \(y=mx\). If we put this equation in Boyle's law terms, it would be \(P=k\frac{1}{V}\).

So, what's the point of this? Well, let's rearrange our formula:

$$P=k\frac{1}{V}$$

$$k=PV$$

The constant (k) is a proportionality constant, which we call Boyle's law constant. This constant tells us how the pressure value will change when the volume does and vice-versa.

For example, let's say we know that k is 2 (atm*L). This means we can calculate the pressure or volume of an ideal gas when given the other variable:

Given a gas with a volume of, 1.5 L, then:

$$k=PV$$

$$2(atm*L)=P(1.5\,L)$$

$$P=1.33\,atm$$

On the other hand, if we are given a gas with a pressure of, 1.03 atm, then:

$$k=PV$$

$$2(atm*L)=1.03\,atm*V$$

$$V=1.94\,L$$

Boyle's Law Relationship

There is another mathematical form of Boyle's law, which is more common. Let's derive it!

$$k=P_1V_1$$

$$k=P_2V_2$$

$$P_1V_1=P_2V_2$$

We can use this relationship to calculate the resulting pressure when the volume changes or vice-versa.

It's important to remember that this is an inverse relationship. When variables are on the same side of an equation, that means there is an inverse relationship (here P₁ and V₁ have an inverse relationship, and so do P₂ and V₂).

Boyle's Law Examples

Now that we know this mathematical relationship, we can work on some examples

We can also solve this problem (and others like it) using the Boyle's law constant equation we used earlier. Let's try it out!