Have you ever wondered why the pressure in your car’s tires is higher after you have been driving a while? In this lesson, we are going to discuss the law that governs ideal gases and is used to predict the behavior of real gases: the ideal gas law.

## The Ideal Gas

Johnny is riding his bicycle and enjoying his vacation here on Ideal Island, where all gases behave ideally, meaning they move rapidly and randomly and the gas particles have no volume and no intermolecular forces. Keep in mind, real gases (like the ones you and I are familiar with) behave LIKE ideal gases at normal temperatures and normal pressures, so often when we are describing real gases, we will use the ideal gas characteristics to make approximations of our real gases.

## The Ideal Gas Equation

Now, back to Johnny on Ideal Island.

One thing Johnny really loves about this place is that all gases here are predictable and follow one law. Just like different states and countries have different laws, Ideal Island has its law of the land: the **ideal gas law**. All gas particles follow this law all of the time. The ideal gas law relates the temperature, pressure, number of moles, and volume of any gas.

So, let’s take the air in the tires in Johnny’s bicycle as an example. The air particles are all flying around inside of the tires, hitting the inner walls of the tires and applying *pressure* to the inside of the tire with each collision between the particle and the wall of the tire. These particles all have a certain velocity based upon the *temperature* of the gas (the hotter they are, the faster they will move). Also, these tires have a certain *number* of particles, and as a group they have a certain *volume*, meaning they take up a defined amount of space.These four variables are all related, meaning that if one increases or decreases, one or more of the others will change.

They can be related mathematically by the **ideal gas equation**, or *PV = nRT*. In this equation, *P* stands for the pressure inside the container (the bicycle tires), *V* stands for the volume of the container, *n* represents the quantity of particles in the container, *R* is the ideal gas constant equal to 0.0821 L atmospheres per mol K (which we will go into later) and *T* is the temperature of the gas in the container. Now, any number of units can be used for pressure, volume, number of particles, and temperature, but the most commonly used ones are atmospheres, liters, moles, and Kelvin.

And, we must use these units if we want to use 0.0821 as our *R* constant.

## What Does It Mean?

So, what does this equation mean? Well, let’s start by comparing the pressure and temperature. Remember that pressure is caused by the collision between a gas particle and the walls of its container. Say we have two containers; each is holding particles. If they are both at the same temperature, the pressure inside each container will be the same. Now, if we increase the temperature of one container, the velocity of the particles will increase and there will be MORE collisions with the inside walls of the container.

According to **Gay-Lussac’s Law**, this will result in a higher pressure. So, pressure and temperature are directly related to each other. We can represent this as *‘P* is proportional to *T*,’ and we can see this relationship in our car tires. As we drive, the temperature in our tires increases, which causes an increase in tire pressure.

We also know that if two containers have the same volume and the same temperature, the more particles that are in that container, the more collisions there will be with the inside walls of that container and the higher the pressure will be. This means that the pressure and the number of particles (or the number of moles of particles) is directly related. We see this every time we fill our tires with air. The more air we put in, the higher the tire pressure.

We can represent this as *‘P* is proportional to *n*‘ or combine the two relationships to make *‘P* is proportional to *n* times *T*.’Next, we will examine the relationship between the number of particles and the volume. According to **Avogadro’s Law**, if we increase the number of particles in a balloon (or a container that can expand), the volume of that container will also increase. So, the volume and the number of particles are also directly related, giving us *‘V* is proportional to *n*.’ Now, if we combine this with our previous statements, we get *‘PV* is proportional to *nT*.’ So, this *‘P* times *V* is proportional to *n* times *T*‘ also shows **Boyle’s Law**, which states that the pressure and the volume are inversely proportional to each other. And, it shows **Charles’ Law**, which states that volume and temperature are directly related to each other.

All of these separate combinations of gas laws fuse together to form the **ideal gas law**!

## R – The Ideal Gas Constant

So, where does this *R* come from? I’m going to start by taking the ideal gas law and rearranging it for *R*. So, if we do that, we need to divide both sides by *n* times *T*, giving us *R* = *P***V*/*n***T*. Now, we can find the *R* for any gas situation (each time getting the same number, of course). So, let’s solve it for a gas situation we have previously been introduced to: **Avogadro’s Law**, which states that 1 mole of an ideal gas at standard temperature and pressure (or 1 atmosphere and 273 K) takes up 22.4 liters of space.

So, if we fill in the values we know – *P* equals 1 atmosphere, *V* equals 22.4 L, *n* = 1 mol, and *T* = 273 K – we multiply across the top and bottom and then divide, rounding to three significant figures, and we will get 0.0821 L atm per mol K. So, you never really have to memorize *R* if you know Avogadro’s Law!

One thing that’s really tricky about this number is all the units attached to it.

When we solved for *R*, none of these units cancelled out, so whenever you use this *R*, you need to make sure you keep all four of those units on it. Also, if you do this when using the ideal gas law, you will always end up with the correct unit you are solving for. Another thing to mention about this constant is the number (0.0821) is different depending on which units of pressure and volume you use. So, if you always use liters, atmospheres, moles, and Kelvin, the 0.0821 will always work.

This means that there may be many situations when you will need to convert units like Celsius to Kelvin or torr to atmospheres and so on.

## Lesson Summary

If we combine EVERYTHING we already know about gases into one central governing principal, it would be the **ideal gas law**, which shows the relationship between pressure, volume, temperature and the number of moles of an ideal gas. It is represented using the **ideal gas equation** , or *PV = nRT*, where *P* is the pressure in atmospheres, *V* is the volume in liters, *n* represents the quantity of particles in the container, *T* represents the temperature in Kelvin, and *R* is the ideal gas constant equal to 0.0821 liters atmospheres per moles Kelvin. Johnny uses this equation in his gas experiments to calculate any one of these variables as long as the other three are known.

## Learning Outcome

At the end of this video, you should be able to explain the ideal gas law, provide an equation for it and solve a problem using that equation.