IB Physics HLThe Behaviour of GasesPaper 1 & 2PV = nRT~9 min read
Ideal Gas Equation
The combined gas law told us that PV/T stays constant for a fixed amount of gas — but constant equal to what? The missing piece is simply how much gas you have. Feed that in and the mystery constant turns into a real number, giving one of the most useful equations in physics: PV = nRT. It even comes in a twin version that counts individual molecules.
📘 What you need to know
The ideal gas equation is PV = nRT (the “moles” form)
P in Pa, V in m3, n in mol, T in kelvin, and R = 8.31 J K−1 mol−1 (the ideal gas constant)
It comes straight from PV/T = constant, with that constant equal to nR
The molecular form is PV = NkBT, where N is the number of molecules and kB = 1.38 × 10−23 J K−1
The two forms are bridged by kB = R / NA and N = nNA
An ideal gas is one that obeys PV = nRT at all pressures, volumes and temperatures
From the gas laws to PV = nRT
Each empirical gas law fixes some quantities and lets the others vary — but notice they all quietly assume the amount of gasn stays fixed too.
Gas law
Relationship
Held constant
Boyle’s law
PV = constant
T, n
Charles’s law
V ∝ T
P, n
Pressure law
P ∝ T
V, n
Stitch all three together and you get PV/T = constant. That constant grows with how much gas is present, and it works out to be exactly n times the ideal gas constantR. Rearranging gives the star of the show:
The three gas laws fold into one. Since PV/T = constant and that constant is nR, we arrive at the ideal gas equation.
Ideal gas equationPV = nRT
where P is pressure (Pa), V is volume (m3), n is the number of moles, R = 8.31 J K−1 mol−1, and T is the temperature in kelvin. An ideal gas is simply defined as one that obeys this equation under all conditions.
WE 1
A gas at a pressure of 2.0 × 105 Pa fills a 0.025 m3 container at 300 K. How many moles of gas are there? (R = 8.31 J K−1 mol−1.)
Step 1 — rearrange PV = nRT for n
n = PV / (RT)
Step 2 — substitute (T already in kelvin)n = (2.0×10⁵ × 0.025) ÷ (8.31 × 300)n = 2.0 molCheck the units line up: Pa × m³ is a joule, and J ÷ (J K⁻¹ mol⁻¹ × K) leaves mol.
Two forms: moles or molecules
Sometimes you’d rather count individual molecules than moles. Swap n for the number of molecules N using N = nNA, and swap R for the Boltzmann constantkB — the “per-molecule” version of R:
Boltzmann constantkB = R / NA = 1.38 × 10−23 J K−1
Because nR = (nNA)(R/NA) = NkB, the equation becomes the molecular form:
Ideal gas equation (molecular form)PV = NkBT
The same law, two ways to count. Multiply moles by NA to get molecules; divide R by NA to get kB. R is the macroscopic (per-mole) constant; kB is the microscopic (per-molecule) one.
WE 2
A 0.010 m3 container holds gas at 1.0 × 105 Pa and 27 °C. How many molecules does it contain? (kB = 1.38 × 10−23 J K−1.)
Step 1 — convert temperature to kelvinT = 27 + 273 = 300 KStep 2 — use PV = NkBT, rearranged for N
N = PV / (kBT)
N = (1.0×10⁵ × 0.010) ÷ (1.38×10⁻²³ × 300)N ≈ 2.4 × 10²³ moleculesSame answer you’d get the long way: find n = 0.40 mol from PV = nRT, then × NA.
PV = nRT moles
n × NA = N R ÷ NA = kB
PV = NkBT molecules
🛠️ Using the ideal gas equation
List your quantities and convert: T to kelvin, pressures to Pa, volumes to m3.
Pick the form: moles given or wanted → PV = nRT; molecules → PV = NkBT.
Rearrange for the unknown, then substitute.
Need the other count? Hop across with N = nNA.
Sanity-check the size — a mole is ~6 × 1023 molecules.
💡 Top tips
Kelvin, always — add 273 to any temperature in °C first.
Match the constant to the count:R goes with moles n, kB goes with molecules N.
SI units throughout: Pa, m3, mol, K — watch for kPa, MPa, cm3 or litres.
R, kB and NA are in the data booklet — no need to memorise the values.
Don’t mix up the proportionality constant with kB: it’s nR or NkB, never a bare k.
⚠ Common mistakes
Temperature in °C instead of kelvin — the single most common error here
Pairing the wrong constant with the wrong count (R with N, or kB with n)
Forgetting to convert kPa/MPa to Pa, or cm3/litres to m3
Confusing the number of molesn with the number of moleculesN (they differ by NA)
Assuming a real gas is exactly ideal at high pressure or low temperature
Quick recap: The ideal gas equation is PV = nRT (moles, with R = 8.31) or equivalently PV = NkBT (molecules, with kB = 1.38 × 10−23). They’re linked by kB = R/NA and N = nNA. Always work in kelvin and SI units.
You’ve now got the equation that governs every ideal gas. But why should a gas obey it at all? That’s the beautiful part — it drops out of imagining the gas as tiny particles bouncing around. Next we build that picture in the Kinetic Theory of Gases, the model that explains where all this behaviour comes from.
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