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.
It links them to how much gas there is, measured two ways: the amount in moles (n) or the raw number of molecules (N).
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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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.
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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