PV = nRT — the ideal gas law, explained variable by variable
The ideal gas law, PV = nRT, links pressure, volume, amount of gas and temperature into a single equation that describes how gases behave. This guide breaks down every variable, shows when the law applies, and walks through real worked examples — from bike tyres to weather balloons.
What is the ideal gas law?
The ideal gas law states that PV = nRT — pressure (P) multiplied by volume (V) equals the number of moles of gas (n) multiplied by the gas constant (R) multiplied by absolute temperature (T). It’s the single most useful equation in introductory thermodynamics because it combines three earlier gas laws — Boyle’s, Charles’s, and Gay-Lussac’s — into one formula.
It describes an ‘ideal’ gas: one where gas particles are treated as point-like with no volume of their own and no attractive forces between them. Real gases like air, oxygen, and nitrogen follow this law closely enough at everyday temperatures and pressures to make it extremely useful.
The law breaks down at very high pressure or very low temperature, where gas molecules start taking up meaningful space and attracting each other — conditions where a gas approaches becoming a liquid.
Breaking down every variable in PV = nRT
Each letter carries specific units — mixing them up is the single most common source of wrong answers.
| Concept | Formula | What it means | Real-world example |
|---|---|---|---|
| Pressure (P) | Pa (pascals) | Force per unit area exerted by gas molecules colliding with container walls | Air pressure inside a bicycle tyre |
| Volume (V) | m³ (cubic metres) | Space the gas occupies | The internal volume of a scuba tank |
| Moles (n) | mol | Amount of gas, measured in moles (6.022×10²³ particles per mole) | The amount of helium in a party balloon |
| Gas constant (R) | 8.314 J/(mol·K) | A fixed constant connecting the other variables | Same value in every ideal gas law problem |
| Temperature (T) | K (kelvin, always!) | Absolute temperature — never Celsius or Fahrenheit in this formula | Room temperature ≈ 293 K (20°C) |
The three laws hiding inside PV = nRT
The ideal gas law isn’t arbitrary — it’s built from three simpler relationships discovered over 150 years.
01 Boyle’s Law: P and V
At constant temperature, pressure and volume are inversely related — squeeze a gas into a smaller space and its pressure rises. This is why a syringe gets harder to push as the plunger nears the end.
02 Charles’s Law: V and T
At constant pressure, volume increases with temperature — a hot air balloon expands as the air inside it heats up. Cool the same gas down and it contracts.
03 Gay-Lussac’s Law: P and T
At constant volume, pressure rises with temperature — this is why an aerosol can warns against being left in a hot car; the sealed volume can’t expand, so pressure climbs instead.
Where PV = nRT shows up in real life
- Checking tyre pressure: Tyre pressure rises on a hot day and drops in cold weather because temperature directly affects pressure at roughly constant volume.
- Scuba diving: Divers calculate how much air remains in a tank using the ideal gas law, since pressure changes dramatically with depth.
- Weather balloons: As a balloon rises into thinner air, external pressure drops and the balloon expands — engineers use PV = nRT to predict burst altitude.
- Aerosol cans: The ‘do not incinerate’ warning exists because heating a sealed can raises internal pressure to dangerous levels.
- Cooking with a pressure cooker: Sealing the pot lets pressure build with temperature, allowing water to boil above 100°C and cook food faster.
- Car airbags: A rapid chemical reaction generates a large number of gas moles almost instantly, inflating the airbag in under 50 milliseconds.
Common mistakes when using the ideal gas law
- Using Celsius instead of Kelvin: PV = nRT only works with absolute temperature — always convert: K = °C + 273.15.
- Mismatched units for R: If you use R = 8.314 J/(mol·K), pressure must be in pascals and volume in cubic metres, not litres or atmospheres, unless you switch to a different value of R.
- Confusing moles with mass: n is the number of moles, not grams — convert mass to moles using the substance’s molar mass first.
- Forgetting the law assumes an ‘ideal’ gas: Real gases deviate from PV = nRT at very high pressure or very low temperature, where molecular volume and intermolecular forces start to matter.
- Mixing up which variable is held constant: Before applying Boyle’s, Charles’s, or Gay-Lussac’s law individually, double check which variable the problem is holding fixed.
Key takeaways
- PV = nRT links pressure, volume, moles of gas, and absolute temperature in a single equation.
- The gas constant R = 8.314 J/(mol·K) stays fixed across every ideal gas law problem.
- It combines Boyle’s Law (P vs V), Charles’s Law (V vs T), and Gay-Lussac’s Law (P vs T).
- Temperature must always be in kelvin, never Celsius or Fahrenheit.
- The law assumes an ‘ideal’ gas with no molecular volume or intermolecular forces — real gases deviate at extreme conditions.
- Everyday uses range from tyre pressure to scuba tanks, weather balloons, and pressure cookers.
Frequently asked questions about the ideal gas law
What does PV = nRT stand for?
P is pressure, V is volume, n is the number of moles of gas, R is the universal gas constant (8.314 J/(mol·K)), and T is absolute temperature in kelvin.
What is the value of R in the ideal gas law?
R = 8.314 J/(mol·K) when using SI units (pascals, cubic metres). A common alternative value is R = 0.0821 L·atm/(mol·K) when working in litres and atmospheres.
Why does temperature have to be in Kelvin?
The ideal gas law requires absolute temperature because it’s derived from the physical relationship between molecular kinetic energy and temperature, which is only proportional when measured from absolute zero.
What is an ‘ideal’ gas?
An ideal gas is a theoretical gas whose particles have no volume and experience no intermolecular forces. Real gases like air behave very close to ideal at everyday temperatures and pressures.
When does the ideal gas law break down?
It becomes inaccurate at very high pressure or very low temperature, where gas molecules take up meaningful space and start attracting each other — conditions approaching the gas turning into a liquid.
Ready to run the numbers?
Reading about PV = nRT is the easy part — plugging in real values is where it clicks. Try the interactive calculator, or continue to the kinetic theory of gases to see what’s happening at the molecular level.