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# Boyle's Law

Have you ever heard of "the bends"? Also called decompression sickness, it is a dangerous disorder that can harm divers. When divers go deep into the ocean, where the pressure is greater, their body adapts to this change. However, problems can arise when the diver starts to ascend. As the diver ascends, the pressure decreases, so the nitrogen gas in their blood expands. If the diver doesn't rise slowly enough for their body to release this gas, it can form bubbles in their blood and tissue, which causes "the bends".

So, why does the gas expand when the pressure decreases? Well, Boyle's law has the answer. Read on to find out more!

• First, we will review the components of Boyle's law: ideal gas, pressure, and volume.
• Next, we will define Boyle's law.
• Then, we will do an experiment to show how Boyle's law works.
• Subsequently, we will learn about the Boyle's law constant.
• Lastly, we will learn about an equation related to Boyle's law and use it in some examples.

## 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.

An ideal gas is a theoretical gas that follows these rules:

• They are constantly moving
• The particles have a negligible mass
• The particles have negligible volume
• They do not attract or repel other particles
• They have full elastic collisions (no kinetic energy is lost)

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.

Boyle's law states that for an ideal gas, the pressure of a gas is inversely proportional to its volume. For this relationship to be true, the amount of gas and temperature must be kept constant.

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

When we refer to a linear equation, we use the form y=mx+b, where b is the y-intercept. In our case, "x" (1/V) can never be 0 since we can't divide by 0. Therefore, there is no y-intercept.

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).

The ideal gas law: Boyle's law, when combined with other ideal gas laws (such as Charles's law and Gay-Lussac's law), forms the ideal gas law.

The formula is:

$$PV=nRT$$

Where P is pressure, V is volume, n is the number of moles, R is a constant, and T is temperature.

This law is used to describe the behavior of ideal gases, and therefore approximates the behavior of real gases. However, the ideal gas law becomes less accurate at low temperatures and high pressure.

## Boyle's Law Examples

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

A diver is deep underwater and is experiencing 12.3 atmospheres of pressure. In their blood, there is 86.2 mL of nitrogen. As they ascend, they are now experiencing 8.2 atmospheres of pressure. What is the new volume of nitrogen gas in their blood?

As long as we use the same units on both sides, we don't need to convert from milliliters (mL) to liters (L).

$$P_1V_1=P_2V_2$$

$$V_2=\frac{P_1V_1}{P_2}$$

$$V_2=\frac{12.3\,atm*86.2\,mL}{8.2\,atm}$$

$$V_2=129.3\,mL$$

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!

A container of neon gas has a pressure of 2.17 atm and a volume of 3.2 L. If the piston inside the container is pressed down, decreasing the volume to 1.8 L, what is the new pressure?

The first thing we need to do is solve for the constant using the initial pressure and volume

$$k=PV$$

$$k=(2.17\,atm)(3.2\,L)$$

$$k=6.944\,atm*L$$

Now that we have the constant, we can solve for the new pressure

$$k=PV$$

$$6.944\,atm*L=P*1.8\,L$$

$$P=3.86\,atm$$

## Boyle's Law - Key takeaways

• An ideal gas is a theoretical gas that follows these rules:
• They are constantly moving
• The gas particles have a negligible mass
• The gas particles have negligible volume
• They do not attract or repel other particles
• They have full elastic collisions (no kinetic energy is lost)
• Boyle's law states that for an ideal gas, the pressure of a gas is inversely proportional to its volume. For this relationship to be true, the amount of gas and temperature must be kept constant.
• We can use this equation $$P \propto \frac{1}{V}$$ to visualize Boyle's law mathematically. Where P is pressure, V is volume, and ∝ means "proportional to"
• We can use the following equations to solve for the change in pressure/volume due to a change in volume/pressure
• $$k=PV$$ (Where k is the proportionality constant)
• $$P_1V_1=P_2V_2$$

Boyle's law states that for an ideal gas, the pressure of a gas is inversely proportional to its volume. For this relationship to be true, the amount of gas and temperature must be kept stable.

When the top of a spray can is pressed down, it greatly increases the pressure inside the can. This increased pressure forces the paint outward.

To verify that Boyle's law is true, all we need to do is measure the pressure using a pressure gauge or other pressure reader. If the pressure of a gas increase when volume is reduced, Boyle's law is verified.

Both the amount of gas and the temperature of the gas are assumed to be constant.

No, since pressure increases with a volume decrease (i.e the relationship is indirect/inverse).

## Final Boyle's Law Quiz

Question

Which of the following is NOT a property of an ideal gas?

They have a fixed position

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Question

What is pressure?

Pressure is the force of the gas particles colliding with a wall, divided by the area of that wall.

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Question

True or False: When we refer to the volume of a gas, we usually mean the volume of the container

True

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Question

What is Boyle's law?

Boyle's law states that for an ideal gas, the pressure of a gas is inversely proportional to its volume.

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Question

What factors are assumed to be constant in Boyle's law?

Amount of gas

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Question

True or false: Volume and pressure have a direct relationship (when one increases, so does the other)

False

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Question

True or False: Pressure and inverse volume (1/V) have a direct relationship (when one increases, so does the other)

True

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Question

Why is it important that the temperature and amount of gas is stable when we look at Boyle's law?

The other variables can also affect pressure and volume, so it isn't clear what variable affects what. 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 is decreases (i.e more room for particles since there are less of them.)

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Question

Which of the following is the mathematical definition for Boyle's law?

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

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Question

What does Boyle's law constant (k) tell us?

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.

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Question

Which of the following shows the correct relationship for Boyle's law?

$$P_1V_1=P_2V_2$$

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Question

Boyle's law is combined with other laws to make what law?

The idea gas law

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Question

If a 2.3 L container of oxygen gas has a pressure of 1.23 atm. What is the new volume of the gas if the pressure is now 0.81 atm?

3.5 L

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Question

If the volume and pressure of a container of hydrogen gas have a Boyle's law constant (k) of 2.3 atm*L, what is the pressure of the system when the volume is 1.82 L?

1.26 atm

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Question

Can Boyle's law be applied to real gases?

Boyle's law is used for ideal gases. Ideal gases are used to approximate the behavior of real gases. This approximation becomes less accurate at high pressures.

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