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F. Gases and Atmospheric Chemistry · Gases and Atmospheric Chemistry

Avogadro's Hypothesis and Molar Volume

F3.6 F3.4 F2.3 F2.1About 70 minutes
Learning goals
  • I can state Avogadro's hypothesis and explain how it accounts for the law of combining volumes.
  • I can describe Avogadro's contribution to chemistry and how it was eventually accepted.
  • I can use Avogadro's law, V₁/n₁ = V₂/n₂.
  • I can use molar volume at STP and SATP to convert between volume, moles and mass.
  • I can calculate the density of a gas from its molar mass.

Words to know

law of combining volumes
At the same T and P, gases react in small whole-number ratios by volume (Gay-Lussac, 1808).
Avogadro's hypothesis
Equal volumes of gases at the same T and P contain equal numbers of particles.
Avogadro's law
At constant T and P, V is directly proportional to n: V₁/n₁ = V₂/n₂.
molar volume (Vₘ)
The volume of one mole of a gas: 22.4 L/mol at STP, 24.8 L/mol at SATP.
Avogadro's constant
6.02 × 10²³, the number of particles in one mole.
diatomic molecule
A molecule made of two atoms, such as H₂, O₂, N₂ or Cl₂.

A puzzle from 1808

The French chemist Joseph Louis Gay-Lussac measured the volumes of gases that react with each other. He found a surprising pattern: at the same temperature and pressure, gases always react in small whole-number ratios by volume. For example:

  • 2 L of hydrogen + 1 L of oxygen → 2 L of water vapour
  • 1 L of nitrogen + 3 L of hydrogen → 2 L of ammonia
  • 1 L of hydrogen + 1 L of chlorine → 2 L of hydrogen chloride

This is the law of combining volumes. But why should volumes come in neat ratios? And how can 1 L of hydrogen and 1 L of chlorine make 2 L of product? Many chemists of the time, including John Dalton, found these results hard to accept.

Avogadro's hypothesis (1811)

The Italian scientist Amedeo Avogadro proposed a bold explanation:

Equal volumes of all gases, at the same temperature and pressure, contain equal numbers of particles.

If that's true, then volume ratios are really particle ratios. And to explain how 1 volume of hydrogen + 1 volume of chlorine gives 2 volumes of product, Avogadro argued that hydrogen and chlorine must exist as diatomic molecules (H₂ and Cl₂) that split apart in the reaction: H₂ + Cl₂ → 2 HCl.

His idea was largely ignored for almost 50 years. In 1860, at a famous chemistry conference in Karlsruhe, Germany, Stanislao Cannizzaro showed how Avogadro's hypothesis allowed chemists to work out correct molecular formulas and a consistent set of atomic masses. That consistency helped make the periodic table possible. Today, Avogadro's constant (6.02 × 10²³ per mole) is named in his honour, although he never measured it himself.

+2 volumes of hydrogen, H₂1 volume of oxygen, O₂2 volumes of water vapourEach box has the same volume, temperature and pressure, so the same number of molecules (4 shown).
The law of combining volumes explained by Avogadro's hypothesis. Equal volumes contain equal numbers of molecules, so 2 volumes of H₂ and 1 volume of O₂ give 2 volumes of H₂O vapour: 2 H₂ + O₂ → 2 H₂O.

Avogadro's law

Avogadro's hypothesis leads to a fourth gas law. At constant temperature and pressure, the volume of a gas is directly proportional to the amount of gas (in moles):

V₁ ÷ n₁ = V₂ ÷ n₂ (constant T and P)

Blow more air into a balloon (more moles) and it gets bigger. KMT explanation: more particles means more collisions on the walls, so a flexible container expands until the pressure inside matches the outside again.

Molar volume

If equal volumes hold equal numbers of particles, then one mole of any gas must occupy the same volume at a given temperature and pressure. This is the molar volume, Vₘ:

  • at STP (0 °C, 101.325 kPa): Vₘ = 22.4 L/mol
  • at SATP (25 °C, 100 kPa): Vₘ = 24.8 L/mol

So for a gas at STP or SATP: n = V ÷ Vₘ and V = n × Vₘ.

(In the unit Quantities in Chemical Reactions you learned that one mole contains 6.02 × 10²³ particles and that n = m ÷ M. Molar volume gives you a third way to count moles, by measuring volume.)

1 mol He22.4 L at STP6.02 × 10²³ particlesmass: 4.00 g1 mol N₂22.4 L at STP6.02 × 10²³ particlesmass: 28.02 g1 mol CO₂22.4 L at STP6.02 × 10²³ particlesmass: 44.01 gSame volume and same number of particles, but very different masses
One mole of helium, nitrogen and carbon dioxide each occupies 22.4 L at STP and contains the same number of particles, but their masses are very different. Avogadro's hypothesis is about particles, not mass.
0.00.51.01.52.001020304050Amount of gas, n (mol)Volume (L)SATPSTP24.8 L
At constant temperature and pressure, volume is directly proportional to the number of moles. The slope of each line is the molar volume: 24.8 L/mol at SATP and 22.4 L/mol at STP.
Worked example

Example 1: Combining volumes

Ammonia is made by the reaction N₂(g) + 3 H₂(g) → 2 NH₃(g). How many litres of nitrogen react with 30 L of hydrogen, and how many litres of ammonia form? All volumes are measured at the same temperature and pressure.

  1. By Avogadro's hypothesis, the volume ratio equals the mole ratio: 1 : 3 : 2.
  2. V(N₂) = 30 L H₂ × (1 N₂ ÷ 3 H₂) = 10 L
  3. V(NH₃) = 30 L H₂ × (2 NH₃ ÷ 3 H₂) = 20 L

10 L of N₂ reacts, producing 20 L of NH₃.

Worked example

Example 2: Volume to moles to mass

What is the mass of 5.60 L of oxygen gas at STP?

  1. n = V ÷ Vₘ = 5.60 L ÷ 22.4 L/mol = 0.250 mol
  2. M(O₂) = 2 × 16.00 = 32.00 g/mol
  3. m = n × M = 0.250 mol × 32.00 g/mol = 8.00 g

8.00 g of O₂

Worked example

Example 3: Mass to volume

What volume does 10.0 g of carbon dioxide occupy at SATP?

  1. M(CO₂) = 12.01 + 2(16.00) = 44.01 g/mol
  2. n = m ÷ M = 10.0 g ÷ 44.01 g/mol = 0.2272 mol
  3. V = n × Vₘ = 0.2272 mol × 24.8 L/mol = 5.64 L

5.64 L of CO₂ at SATP

Worked example

Example 4: Gas density

Find the density of carbon dioxide at STP, and explain why CO₂ from a fire extinguisher sinks.

  1. Take 1 mol: mass = 44.01 g, volume = 22.4 L.
  2. Density = m ÷ V = 44.01 g ÷ 22.4 L = 1.96 g/L
  3. Air is mostly N₂ and O₂, with an average molar mass of about 29 g/mol, so its density at STP is about 29 ÷ 22.4 = 1.29 g/L.

CO₂ is 1.96 g/L, about 1.5 times denser than air, so it sinks and can smother flames from below.

Hands-on activity

How many moles of air are in your room?

You need: a measuring tape (or a ruler and patience), a calculator, and some cardboard and tape (optional).

  1. Measure the length, width and height of your bedroom in metres.
  2. Volume (m³) = length × width × height. Convert to litres: 1 m³ = 1000 L.
  3. Assume your room is near SATP (about 25 °C and 100 kPa). Calculate n = V ÷ 24.8 L/mol.
  4. Air has an average molar mass of about 29 g/mol. Calculate the mass of air in your room in kilograms.
  5. Optional: a cube with sides of 29.2 cm has a volume of 24.8 L. Build one from cardboard: that's how much space one mole of gas takes up at SATP. (At STP, the side would be 28.2 cm.)

What to look for: a room 4.0 m × 3.0 m × 2.5 m holds 30 m³ = 30 000 L of air, which is about 1200 mol, with a mass of about 35 kg. The air in a bedroom has about the mass of two full 20 L water-cooler jugs, even though you can't feel it.

Just for fun

Chemists celebrate Mole Day every October 23, from 6:02 a.m. to 6:02 p.m., in honour of 6.02 × 10²³. Why do chemists bring just one stuffed mole to the party? Because a whole mole of them, 6.02 × 10²³, each the size of a 1 L milk carton, would cover the whole Earth more than 1000 km deep.

Your turn

1

State Avogadro's hypothesis.

2

What volume do 2.50 mol of nitrogen occupy at SATP?

3

How many moles are in 12.0 L of oxygen at STP?

4

For 2 H₂(g) + O₂(g) → 2 H₂O(g), what volume of hydrogen reacts with 6.0 L of oxygen, and what volume of water vapour forms (same T and P)?

5

A balloon contains 0.080 mol of helium and has a volume of 2.0 L. If 0.040 mol more helium is added at the same T and P, what is the new volume?

6

Calculate the density of oxygen gas at SATP.

More practice: warm-up

1

What is the molar volume of a gas at STP?

2

What is the molar volume of a gas at SATP?

3

How many molecules are in 22.4 L of any gas at STP?

4

If you double the moles of gas at constant T and P, what happens to the volume?

5

Who first showed chemists how to use Avogadro's hypothesis to find correct atomic masses, in 1860?

More practice: on your own

1

What is the mass of 5.00 L of sulfur dioxide (SO₂) at SATP?

2

A gas has a density of 1.96 g/L at STP. Find its molar mass and decide whether it is CH₄, O₂ or CO₂.

3

For C₃H₈(g) + 5 O₂(g) → 3 CO₂(g) + 4 H₂O(g), what volume of oxygen is needed to burn 2.0 L of propane, and what volume of CO₂ forms (same T and P)?

4

Explain why 1 mol of helium and 1 mol of carbon dioxide have the same volume at STP but different masses.

5

A party balloon holds 4.5 L of helium at SATP. How many helium atoms does it contain?

Challenge

1

A 0.500 L sample of an unknown gas at SATP has a mass of 0.645 g. Find its molar mass and suggest what the gas is.

2

Gay-Lussac found that 1 volume of nitrogen reacts with 1 volume of oxygen to give 2 volumes of nitrogen monoxide. Explain why this result means nitrogen and oxygen cannot be single atoms.

3

Why can't you use 22.4 L/mol for liquid water at 0 °C? Support your answer with a calculation.

Check your understanding

Quiz

1.

Avogadro's hypothesis states that equal volumes of gases at the same T and P contain equal:

2.

What volume does 0.500 mol of any gas occupy at STP?

3.

How many moles are in 49.6 L of gas at SATP?

4.

For N₂ + 3 H₂ → 2 NH₃ (same T and P), 4.0 L of N₂ produces how many litres of NH₃?

5.

At STP, which has the greatest mass?

6.

Avogadro's hypothesis was finally widely accepted after:

For parents and teachers

Big idea (F3.6, F3.4, F2.3): Avogadro's hypothesis explained the law of combining volumes, revealed diatomic molecules and led to correct atomic masses. It gives the molar volume (22.4 L/mol at STP, 24.8 L/mol at SATP), which links gas volumes to moles.

How to help: the "air in your room" activity makes the numbers real. Ask your teen to explain why 1 mol of every gas takes up the same space but has a different mass.

Common difficulties:

  • Using molar volume at other conditions. 22.4 and 24.8 L/mol are only for STP and SATP. For other conditions, use the ideal gas law (next lesson).
  • Using molar volume for liquids or solids. It only applies to gases.
  • Mixing up "equal numbers of particles" with "equal masses".

This lesson relies on the mole and molar mass from the unit Quantities in Chemical Reactions; if those feel shaky, review that unit's lessons on the mole first.