In this explainer, we will learn how to use molar gas volume, under standard conditions, to calculate the volume and number of moles of a gas.

In 1811, Amedeo Avogadro hypothesized that gases of equal volume at equal temperatures and pressures contained the same number of gas particles. This hypothesis gave rise to what we now call Avogadroβs law. Avogadroβs law states that the volume of a sample of gas is directly proportional to the number of moles of gas at a constant temperature and pressure.

### Law: Avogadroβs Law

Avogadroβs law states that at a constant temperature and pressure, the volume and number of moles of a gas are directly proportional.

We can represent Avogadroβs law with the proportionality statement where is the volume of the gas and is the number of moles.

We must always recognize that this proportion is only true so long as the temperature and pressure remain constant. The proportionality statement indicates that as the number of moles of gas increases, so does the volume occupied by the gas, and vice versa.

### Example 1: Using Avogadroβs Law to Recognize the Relationship Between the Volume and Number of Moles of a Gas

According to Avogadroβs law, what happens to the volume a gas occupies if the number of moles increases?

- It stays the same.
- It increases.
- It decreases.

### Answer

Avogadroβs law states that the volume and number of moles of a gas are directly proportional at a constant temperature and pressure. We can express this as

When variables are directly proportional, increasing the quantity of one variable will increase the quantity of the other variable at a constant rate. Therefore, if the number of moles of the gas increases, the volume of the gas will also increase. The correct answer is choice B.

Let us consider the implication of Avogadroβs law on the following reaction under constant temperature and pressure:

According to the equation, two moles of hydrogen gas and one mole of oxygen gas combine to form two moles of water vapor. As the number of moles of gas is proportional to the volume of gas, the equation also indicates that two volumes of hydrogen gas (in units of litres or millilitres, for example) combine with one volume of oxygen gas to produce two volumes of water vapor.

The ratio of to to will be regardless of whether it is expressed in moles or units of volume.

In addition to the proportionality statement Avogadroβs law can be expressed as the proportional equation where is the molar volume, a proportionality constant. Molar volume is typically expressed with a unit of L/mol and indicates the volume occupied by one mole of gas at a specific temperature and pressure. The molar volume will change if the temperature and/or pressure of the gas is changed. If we measure the volume and number of moles of a gas at a specific temperature and pressure, we can determine the molar volume.

### Definition: Molar Volume (π_{π})

Molar volume is the volume occupied by one mole of gas at a specific temperature and pressure.

### Example 2: Calculating the Moles of a Gas in a Given Volume by Determining the Molar Gas Volume

A 12 L balloon contains 0.52 moles of helium gas. A second balloon at the same temperature and pressure has a volume of 18 L.

How many moles of helium gas does the second balloon contain? Give your answer to two decimal places.

### Answer

The volume and number of moles of a gas are related by the equation where is the volume, is the number of moles, and is the molar volume. The molar volume is a proportionality constant that indicates the volume of one mole of any gas at a particular temperature and pressure.

We can substitute the volume and number of moles of helium gas in the first balloon into the equation to give us

Then, we can determine the molar volume as follows:

The molar volume of the gas is 23.077 L/mol. As the second balloon is at the same temperature and pressure as the first balloon, the gas in the two balloons will have the same molar volume. This means that we can substitute the volume of the second balloon and the molar volume into the equation to give us

Then, we can determine the number of moles of helium in the second balloon as follows:

The number of moles of helium gas in the second balloon, rounded to two decimal places, is 0.78 moles.

As the volume and therefore the density of a gas is dependent on the temperature and pressure, it is useful to define a standard temperature and pressure that can be used as reference conditions when comparing different gases. Standard temperature is defined as and standard pressure is defined as 1 atm. Collectively, standard temperature and pressure are abbreviated as STP.

### Definition: Standard Temperature and Pressure (STP)

Standard temperature is and standard pressure is 1 atmosphere (atm).

As it turns out, one mole of any gas at STP will have a volume of 22.4 litres.

Gas | |||
---|---|---|---|

Temperature | |||

Pressure | 1 atm | 1 atm | 1 atm |

Amount | 1 mole | 1 mole | 1 mole |

Mass | 32 g | 40 g | 16 g |

Number of Particles | molecules | atoms | molecules |

Volume | 22.4 L | 22.4 L | 22.4 L |

### Example 3: Determining Which Quantity of Gas will Occupy the Largest Volume at STP

Under standard temperature and pressure (STP), which of the following quantities of gas will occupy the largest volume?

- 1 mole of
- 5 moles of
- 0.5 moles of
- 2 moles of
- 3 moles of

### Answer

At standard temperature and pressure ( and 1 atm), one mole of gas will occupy 22.4 litres. This is true regardless of the gas being used. As volume and the amount of moles of a gas are directly proportional, two moles of any gas at STP should have double the volume as one mole:

At STP, the following graph of volume and number of moles can be constructed.

Using the graph, we can determine the volume occupied by each of the choices as follows:

- 1 mole of has a volume of 22.4 L.
- 5 moles of have a volume of 112 L.
- 0.5 moles of have a volume of 11.2 L.
- 2 moles of have a volume of 44.8 L.
- 3 moles of have a volume of 67.2 L.

The quantity of gas that will occupy the largest volume at STP is 5 moles of , which is choice B.

For any gas at standard temperature and pressure, we can substitute the volume of 22.4 litres and one mole into the proportional equation and determine the molar volume of any gas at STP as follows:

The molar volume of any gas at STP is 22.4 L/mol. This is the standard molar volume of a gas. Thus, at STP,

This equation can be used to determine the volume or number of moles of a gas at standard temperature and pressure. It is important to recognize that the standard molar volume can only be used when the gas is held at a constant temperature of and a constant pressure of 1 atm.

### Example 4: Calculating the Number of Moles of Gas Molecules at STP Given a Volume

Under standard temperature and pressure (STP), a gas occupies a volume of 2 L. How many moles of gas molecules are there? Give your answer to 2 decimal places.

### Answer

The volume and number of moles of a gas are related by the equation where is the volume, is the number of moles, and is the molar volume. The molar volume is a proportionality constant that indicates the volume of one mole of any gas at a particular temperature and pressure.

In this problem, the gas is under standard temperature and pressure (STP). Standard temperature and pressure are and 1 atm respectively. Any gas at STP will have a standard molar volume of 22.4 L/mol.

We can substitute the volume of the gas given in the question and the standard molar volume into the equation to give us

Then, we can solve for the amount in moles as follows:

Two litres of a gas at STP will contain 0.08928 moles of gas molecules. Rounding to two decimal places, we find that our final answer is 0.09 moles of gas molecules.

### Example 5: Calculating the Volume of Ammonia Gas at STP Given the Mass

What volume would 8.5 g of gas occupy at standard temperature and pressure (STP) taking the molar gas volume to be 22.4 L/mol? Give your answer in litres. [ = 14 g/mol, = 1 g/mol]

### Answer

The volume and number of moles of a gas are related by the equation where is the volume, is the number of moles, and is the molar volume. We need to solve for the volume of ammonia (). We are given the molar gas volume at STP, but we do not know the number of moles of ammonia. However, we can convert a mass of ammonia into moles of ammonia by using the equation where represents the number of moles, is the mass in grams, and is the molar mass in g/mol.

In order to solve for the moles of ammonia, we need to determine its molar mass. The molar mass of can be calculated by summing the average molar masses of the constituent atoms:

The mass given in the question and the molar mass can be substituted into the equation to get which gives us that the amount of in moles is

Now, we can substitute the number of moles and the molar volume into the equation to get which gives us that the volume is

The volume occupied by 8.5 g of at standard temperature and pressure is 11.2 L.

### Key Points

- Avogadroβs law states that the number of moles of a gas and the volume of the gas are directly proportional.
- Avogadroβs law can be represented by the equation where is the volume of the gas, is the number of moles, and is the molar volume.
- Standard temperature and pressure (STP) are and 1 atm respectively.
- At STP, all gases have a molar volume of 22.4 L/mol.