Learn how pressure, volume, temperature, and the amount of a gas are related to each other.

## What is an ideal gas?

Gases are complicated. They're full of billions and billions of energetic gas molecules that can collide and possibly interact with each other. Since it's hard to exactly describe a real gas, people created the concept of an **Ideal gas** as an approximation that helps us model and predict the behavior of real gases. The term ideal gas refers to a hypothetical gas composed of molecules which follow a few rules:

**Ideal gas molecules do not attract or repel each other**. The only interaction between ideal gas molecules would be an elastic collision upon impact with each other or an elastic collision with the walls of the container.The phrase

**elastic collision**refers to a collision wherein no kinetic energy is converted to other forms of energy during the collision. In other words, kinetic energy can be exchanged between the colliding objects (e.g. molecules), but the**total kinetic energy before the collision is equal to the total kinetic energy after the collision**.A car crash where kinetic energy gets converted into heat energy and sound energy yielding two crumpled bumpers would not be elastic.

**Ideal gas molecules themselves take up no volume**. The gas takes up volume since the molecules expand into a large region of space, but the Ideal gas molecules are approximated as point particles that have no volume in and of themselves.

If this sounds too ideal to be true, you're right. There are no gases that are exactly *ideal*, but there are plenty of gases that are close enough that the concept of an ideal gas is an extremely useful approximation for many situations. In fact, for temperatures near room temperature and pressures near atmospheric pressure, many of the gases we care about are very nearly ideal.

If the pressure of the gas is too large (e.g. hundreds of times larger than atmospheric pressure), or the temperature is too low (e.g. *non-ideal gases* read this article.

## What is the molar form of the ideal gas law?

The pressure, **ideal gas law**. The simplicity of this relationship is a big reason why we typically treat gases as ideal, unless there is a good reason to do otherwise.

Where

For a typical room there is likely to be at least

This is far larger than the estimated number of stars in the Milky Way galaxy, and is even larger than most estimates of the number of stars in the observable universe.

Perhaps the most confusing thing about using the ideal gas law is making sure we use the right units when plugging in numbers. If you use the gas constant

If you use the gas constant

This information is summarized for convenience in the chart below.

Pressure in | Pressure in | |

Volume in | volume in | |

Temperature in | Temperature in |

Here is some useful information relating the different types of units.

To convert

Also, the term *STP* refers to "standard temperature and pressure" which are defined to be

## What is the molecular form of the ideal gas law?

If we want to use

Where

When using this form of the ideal gas law with Boltzmann's constant, we have to plug in pressure

Pressure in | |

Volume in | |

Temperature in |

## What is the proportional form of the ideal gas law?

There's another really useful way to write the ideal gas law. If the number of moles

This shows that, as long as the number of moles (i.e. molecules) of a gas remains the same, the quantity

This formula is particularly useful when describing an ideal gas that changes from one state to another. Since this formula does not use any gas constants, we can use whichever units we want, but we must be consistent between the two sides (e.g. if we use

## What do solved examples involving the ideal gas law look like?

### Example 1: How many moles in an NBA basketball?

The air in a regulation NBA basketball has a pressure of

a. **Determine the number of moles of air inside an NBA basketball.**

b. **Determine the number of molecules of air inside an NBA basketball.**

We'll solve by using the ideal gas law. To solve for the number of moles we'll use the molar form of the ideal gas law.

Given this choice of gas constant, we need to make sure we use the correct units for pressure (

Yes, we could have used the gas constant

We can convert the pressure as follows,

And we can use the formula for the volume of a sphere

The temperature

Now we can plug these variables into our solved version of the molar ideal gas law to get,

Now to determine the number of air molecules

Alternatively, we could have solved this problems by using the molecular version of the ideal gas law with Boltzmann's constant to find the number of molecules first, and then converted to find the number of moles.

### Example 2: Gas takes an ice bath

A gas in a sealed rigid canister starts at room temperature

**Determine the pressure of the gas after reaching a temperature of **

Since we know the temperature and pressure at one point, and are trying to relate it to the pressure at another point we'll use the proportional version of the ideal gas law. We can do this since the number of molecules in the sealed container is constant.

Notice that we plugged in the pressure in terms of