Avogadro's law: moles and volume of gases

Blowing up balloons: volume follows amount

Blowing up a balloon is Avogadro's law in action. Each breath adds a fixed crowd of air molecules, and the balloon grows with every one. The temperature and the outside pressure barely change, yet the volume climbs steadily, because volume tracks the amount of gas:

$$\frac{V_1}{n_1} = \frac{V_2}{n_2}$$

Double the moles and the volume doubles. The law completes the gas-law family: Boyle linked pressure and volume, Charles linked volume and temperature, Gay-Lussac linked pressure and temperature, and Avogadro links volume to amount.

A worked example: a balloon holds 2.00 L of helium containing 0.0800 mol at room temperature and pressure. A second, identical balloon is inflated with 0.120 mol. Its volume is $V_2 = V_1n_2/n_1 = 2.00 \times 0.120/0.0800 = 3.00$ L. Fifty percent more moles gives fifty percent more volume, and the ratio $V/n = 25.0$ L/mol is the same for both balloons, which is the law's real content.

Equal numbers, equal volumes: the particle view

Why should a mole of wispy hydrogen occupy the same volume as a mole of heavy xenon? Because at fixed temperature and pressure, what matters is the number of particles, not their identity. Each particle, on average, needs the same elbow room to keep the collision rate, and therefore the pressure, unchanged.

Add particles to a flexible container and the collision rate with the walls rises. The walls move outward until the rate per unit area falls back to balance the outside pressure. Equilibrium is restored when each mole once again has its characteristic volume.

A worked example with particles: one mole contains Avogadro's number, $6.022 \times 10^{23}$ particles. At STP (0 °C, 1 atm) that crowd fills 22.4 L, so each particle commands on average $22.4 \div (6.022 \times 10^{23}) \approx 3.72 \times 10^{-23}$ L, or about 37 cubic nanometers. The particles themselves occupy a tiny fraction of that; the rest is the empty space kinetic theory insists on.

Solving V1/n1 = V2/n2 problems

Avogadro's-law problems are proportions, so the routine is short:

1. Confirm temperature and pressure are constant.
2. Match units: moles with moles, liters with liters.
3. Solve: $V_2 = V_1n_2/n_1$ or $n_2 = n_1V_2/V_1$.
4. Check direction: more gas means more volume.

A worked example: a 5.00 L flask holds 0.200 mol of argon at constant temperature and pressure. Gas is added until the flask, fitted with a movable piston, expands to 7.50 L. The new amount is $n_2 = n_1V_2/V_1 = 0.200 \times 7.50/5.00 = 0.300$ mol, so 0.100 mol was added. The volume grew by half, the moles grew by half, and the piston moved just enough to keep every particle's elbow room unchanged.

Molar volume: the 22.4-liter shortcut

The law's most famous child is molar volume: at standard temperature and pressure, 0 °C and 1 atm, one mole of any ideal gas occupies 22.4 L. Chemists use it as a conversion factor between gas volume and moles, no $R$ required.

A worked example: a reaction produces 11.2 L of hydrogen gas collected at STP. Moles of hydrogen: $11.2 \div 22.4 = 0.500$ mol. Particles: $0.500 \times 6.022 \times 10^{23} = 3.01 \times 10^{23}$ molecules. Mass: $0.500 \times 2.016 = 1.01$ g. One measured volume unfolds into moles, particles, and mass in three multiplications.

But STP is a specific address, not a universal truth. At room conditions, about $25^\circ\text{C}$ and 1 atm, the molar volume is roughly 24.5 L. A second worked check: using $PV = nRT$ with $R = 0.08206$, $V = (1 \times 0.08206 \times 298)/1 = 24.5$ L. Always verify the conditions before reaching for 22.4 L.

Limits: rigid tanks and real gases

The law assumes ideal behavior: particles with negligible volume and no attractions, at constant temperature and pressure. It also assumes the container can actually change size; in a rigid tank, adding gas raises pressure instead, which is the domain of the ideal gas law.

Real gases stray at high pressure, where the particles' own volume steals space and a mole occupies slightly less than predicted, and at low temperature, where attractions pull particles together.

A worked example of the rigid-container trap: 0.100 mol of gas sits in a sealed 2.00 L bottle. Another 0.100 mol is pumped in at the same temperature. A student applying $V_2 = V_1n_2/n_1$ predicts 4.00 L, but the bottle cannot grow. The correct prediction is doubled pressure, from the ideal gas law: $P_2 = n_2RT/V$ is twice $P_1$. Avogadro's law answers 'how big?' only when the container is free to answer; otherwise it answers 'how pressurized?' through $PV = nRT$.

Common mistakes

Real-world applications

• Sizing helium tanks for party balloons

• Dosing reactant gases in industrial synthesis

• Calibrating flow meters that measure gas by volume

• Estimating lung capacity from exhaled gas volumes