Lesson 1 · 25 min

Gas laws: where pressure comes from

Pressure is the drumming of countless molecules on the walls. Change volume, temperature, and particle number in a molecular-dynamics box, plot the measured pressure, and the ideal gas law emerges on its own.

In this lesson

Explain gas pressure as molecules striking the walls
Obtain Boyle’s, Gay-Lussac’s, and Avogadro’s laws from measured data
Use $PV = nRT$ to find pressure, volume, temperature, or amount
Understand why gas calculations must use absolute temperature (K)

Put a half-empty balloon in the fridge and it sags further; set it in the sun and it swells. Press a suction cup onto glass and it can hold a bag of rice. Gases are invisible and intangible, yet they push pistons, hold up tyres, and keep aircraft aloft. The force behind all of it comes from about $10^{23}$ molecular impacts on every square centimetre, every second.

Predict first

At constant temperature, squeeze a gas into half its volume. What happens to the pressure?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

Unchanged Doubles Quadruples Halves

Pressure is made by collisions

Pressure is force per unit area, $P = F/A$ , measured in pascals (Pa = N/m²). Gas molecules never stop moving; each one that bounces off a wall hands it a momentum of $2mv_x$ . The momentum delivered per unit time is the force. So pressure depends on two things: how often molecules hit (proportional to the number density $N/V$ ) and how hard each hit is (set by the molecules’ kinetic energy, i.e. the temperature).

Everyday units: 1 standard atmosphere (atm) = 101.325 kPa = 760 mmHg; 1 bar = 100 kPa.

Three empirical laws, one equation

Between the 17th and 19th centuries chemists found, one by one:

Boyle’s law (1662): at fixed $T$ and $n$ , $P \propto 1/V$ ;
Charles’s and Gay-Lussac’s laws (1787–1802): at fixed $P$ , $V \propto T$ ; at fixed $V$ , $P \propto T$ ;
Avogadro’s law (1811): at the same $T$ and $P$ , $V \propto n$ — equal volumes of gas hold equal numbers of molecules.

Combine the three and you get the ideal gas equation:

PV = nRT

R = 8.314\ \mathrm{J\,mol^{-1}\,K^{-1}} = 0.08206\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}

$P$: pressure
$V$: volume
$n$: amount of gas (mol)
$R$: molar gas constant
$T$: absolute temperature (K)

Real 2D molecular dynamics · pressure measured from wall hits

Gas molecular-dynamics box

Temperature · 300 K Intermolecular attraction

Pressure (measured)…

T0 K

PA / NkT…

Equilibrating…

Recorded measurements

Ideal gas PA = NkT
Recorded measurements

Hold the volume fixed and raise the temperature, recording a point each time. Extend the line: where does it meet the temperature axis?

Worked example

The air in a tyre

A 2.0 L bicycle tyre is pumped at 20 °C to a gauge pressure of 400 kPa (about 501 kPa absolute). How many moles of air does it hold? In 40 °C sunshine (constant volume), what is the absolute pressure?

Concept check

What is the volume of 1 mol of ideal gas at 0 °C and 1 atm?

1.0 L 8.3 L 22.4 L 24.8 L

Try the calculation

A 5.00 L steel cylinder holds 0.500 mol of nitrogen at 27 °C. What is the pressure in atm?

Key takeaways

Gas pressure comes from wall collisions: it depends on how often molecules hit (number density) and how hard (temperature).
Boyle’s, Charles’s/Gay-Lussac’s, and Avogadro’s laws combine into $PV = nRT$ .
Gas calculations need kelvin; the $P$ – $T$ line extrapolates to 0 K.
At 0 °C and 1 atm, 1 mol of ideal gas occupies 22.4 L.

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Data and references

E. Tiesinga et al., “CODATA recommended values of the fundamental physical constants: 2018”, Rev. Mod. Phys. 93, 025010 (2021) — R, k_B.
P. Atkins, J. de Paula, J. Keeler, Atkins’ Physical Chemistry, 11th ed. (2018), Focus 1 (the properties of gases).
D. Frenkel, B. Smit, Understanding Molecular Simulation, 2nd ed., Academic Press (2002) — velocity Verlet and pressure from wall collisions.