The big idea: Blow more air into a balloon and it swells; leave it in the sun and it swells further. A gas's pressure P, volume V and temperature T all move together, tied by one equation: the ideal gas law.
Two ways to measure the gas: It links them to how much gas there is, measured two ways: the amount in moles (n) or the raw number of molecules (N).
What a mole is: A mole is just a fixed-size 'pack' of particles — one mole = 6.02 × 10²³ of them (the Avogadro constant).
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Spot it: Same P, V and T → same number of molecules N — no matter what the gas is.
So two equal boxes at the same conditions hold the same N, even if one gas is heavier; the heavier gas just has the bigger mass and density.
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The ideal gas law is given in the data booklet in two equal forms — use moles n with R, or molecules N with the Boltzmann constant kB. They describe the same gas:
- pressure (Pa)
- volume (m³)
- amount of gas (mol)
- gas constant = 8.31 J K⁻¹ mol⁻¹ (given)
- absolute temperature (K — always kelvin)
- number of molecules (no unit — just a count)
- Boltzmann constant = 1.38 × 10⁻²³ J K⁻¹ (given)
Two things to get right: 1. Temperature T is always in kelvin — add 273 to a Celsius value first.
2. Use n with R (8.31), or N with kB (1.38 × 10⁻²³) — never mix them.
To swap between the amount in moles n and the raw number of molecules N, use the Avogadro constant NA. It's a simple quotient, so a formula triangle helps:
- amount of gas (mol)
- number of molecules (just a count)
- Avogadro constant = 6.02 × 10²³ mol⁻¹ (given)
N = number of molecules, n = moles, NA = Avogadro constant. Cover the one you want: two side by side → multiply; one above the other → divide. So N = n × NA, and n = N ÷ NA.
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A sealed flask holds gas at a pressure of 1.2 × 10⁵ Pa in a volume of 8.0 × 10⁻⁴ m³ at 300 K. Find the number of gas molecules N. (kB = 1.38 × 10⁻²³ J K⁻¹.)
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One atom: its mass and its volume: A mole has a mass equal to the molar mass M (in kg), and holds NA particles, so one atom has mass M ÷ NA. Copper: 0.0635 ÷ 6.02 × 10²³ = 1.05 × 10⁻²⁵ kg.
In a solid of density ρ, one mole fills M ÷ ρ, so one atom takes up M ÷ (ρNA). Aluminium (M = 0.027 kg mol⁻¹, ρ = 2700 kg m⁻³): 1.66 × 10⁻²⁹ m³.
Reading a gas-law graph: Put the gas law in the form y = (gradient) × x. At constant volume p = (nR/V) T, so the gradient of p against T (in kelvin) is nR/V: with n known, V = nR ÷ gradient.
For a monatomic gas U = 3/2 nRT, so the gradient of U against T is 3/2 nR.
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How this is tested — the ideal gas law is one of the most-tested ideas in Theme B:
Paper 1A
- Quick MCQs comparing two samples — equal N → compare density/pressure.
- Or which sample of equal P, V, T has the smallest mass.
Paper 1B / Paper 2
- Calculate the number of molecules or moles in a sample.
- Or a ratio of amounts between two containers.
The classic trap: Leaving the temperature in degrees Celsius — it must be in kelvin (add 273). Another: mixing n with kB or N with R.
Comparing two samples: Write PV = NkT for each sample.
Whatever is the same (P, V or T) cancels when you divide one equation by the other, leaving a simple ratio of the rest. This is how every 'compare the two containers' question is solved.
The words to know
| Word | What it means |
|---|---|
| Amount of substance n | Number of particles ÷ NA, in moles |
| Avogadro constant NA | 6.02 × 10²³ particles per mole |
| Molar mass M | Mass of one mole (kg mol⁻¹) |
| Ideal gas law | pV = nRT = NkBT, with T in kelvin |
Container X holds gas at pressure P, volume V and temperature 300 K. Container Y holds the same gas at the same pressure P, but volume 2V and temperature 400 K. Find the ratio of the amount of gas in Y to that in X (nY ÷ nX).
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Same p, V and T: same number of molecules: N = pV ÷ kBT does not depend on what the gas is. So equal volumes at the same pressure and temperature hold equal numbers of molecules.
Their masses then follow the molar mass: the same volume of CO₂ (44 g mol⁻¹) is heavier and denser than one of N₂ (28 g mol⁻¹).