Ideal Gas Law — PV = nRT Explained with Examples

High School⏱ 10 min readThermodynamics

PV = nRT unites pressure, volume, temperature and amount of gas in one equation. Learn each variable, when the law holds, and how to solve gas problems.

Quick answer

The ideal gas law, PV = nRT, links a gas's pressure (P), volume (V), amount in moles (n), and absolute temperature (T) through the gas constant R = 8.314 J/(mol·K). It predicts how gases respond to squeezing, heating, and expanding.

$PV = nRT$

Unit: P in Pa, V in m³, T in K, n in mol

Example: A 2.0 mol sample at 300 K in a 0.050 m³ tank has pressure P = nRT/V = (2.0 × 8.314 × 300)/0.050 ≈ 99,800 Pa — about one atmosphere.

What you will learn

State PV = nRT and define each symbol with SI units
Convert temperatures to kelvin before any gas calculation
Solve for an unknown variable in the ideal gas law
Explain Boyle's, Charles's, and Gay-Lussac's laws as special cases
Describe the ideal-gas assumptions and their limits

One equation for every gas

Squeeze a balloon and the pressure inside rises. Warm it and it swells. Let some air out and it shrinks. Three separate observations — one equation ties them together.

The ideal gas law says that for a fixed sample of gas, pressure times volume is proportional to absolute temperature:

PV = nRT

P = pressure (Pa), V = volume (m³), n = moles of gas, R = 8.314 J/(mol·K), T = absolute temperature (K)

Reading the variables

Each symbol has a physical story:

P — pressure: the battering of countless molecules against the walls, in pascals.
V — volume: the space the gas occupies, in cubic meters.
n — amount: moles of gas particles. More particles, more collisions, more pressure.
T — temperature: must be absolute (kelvin). It measures average molecular kinetic energy.
R — the gas constant: 8.314 J/(mol·K), the bridge between the particle world and the macroscopic world.

Pressure is trillions of molecular collisions per second. Heating the gas speeds the molecules up, raising the pressure unless the piston moves.

Three classic laws in one

The famous historical gas laws fall out immediately:

Boyle's law (n, T fixed): $P_1 V_1 = P_2 V_2$ — squeeze the volume, pressure rises.
Charles's law (n, P fixed): $V_1/T_1 = V_2/T_2$ — heat the gas, it expands.
Gay-Lussac's law (n, V fixed): $P_1/T_1 = P_2/T_2$ — heat a sealed tank, pressure climbs.

They are not rivals of PV = nRT; they are its children. If a problem holds two variables constant, reach for the shortcut — otherwise use the full law.

Solving it step by step

Worked example. A sealed 0.020 m³ tank holds 1.5 mol of air at 25 °C. What is the pressure?

First convert: $T = 25 + 273 = 298\ \mathrm{K}$ .

$P = \frac{nRT}{V} = \frac{1.5 \times 8.314 \times 298}{0.020}$

$P \approx 185{,}800\ \mathrm{Pa} \approx 1.83\ \mathrm{atm}$

Nearly twice atmospheric pressure — which is why you never heat a sealed container casually.

Unit trap

If you use R = 8.314, V must be in m³ and P comes out in Pa. If you prefer liters and atmospheres, switch to R = 0.08206 L·atm/(mol·K). Pick one system and commit.

Where the model breaks

The law assumes gas particles are points with no volume and no attraction — the ideal gas model. Real gases stray from it at high pressure (particles crowd each other) and near condensation (attractions matter). But for air at everyday conditions, the error is typically under a percent.

Kinetic theory goes deeper, deriving pressure from actual molecular collisions — the natural next stop after this law.

Variables and units

Symbol	Name	Unit	Meaning
$P$	Pressure	Pa	Force per unit area from molecular collisions
$V$	Volume	m³	Space occupied by the gas
$n$	Moles	mol	Amount of gas
$R$	Gas constant	J/(mol·K)	Universal gas constant
$T$	Temperature	K	Absolute temperature — always kelvin here

Common mistakes

✗ The gas constant R is always 8.314.

8.314 J/(mol·K) is one form. R = 0.08206 L·atm/(mol·K) is equally valid — you must match R to your units. Mixing R = 8.314 with liters and atmospheres is the classic exam error.

✗ Boyle's, Charles's, and Gay-Lussac's laws are separate from PV = nRT.

They are special cases of it. Hold n and T constant and PV = nRT becomes Boyle's law (P₁V₁ = P₂V₂). Hold n and P constant and you get Charles's law (V₁/T₁ = V₂/T₂).

✗ Doubling the temperature in °C doubles the pressure.

Gas laws need absolute temperature. Doubling from 20 °C to 40 °C is 293 K → 313 K, only a 7% rise. Always convert to kelvin first.

Real-world applications

Weather balloons: meteorologists compute how much a balloon will swell as it rises into thinner, colder air.
Scuba diving: tanks filled in a cool shop read higher pressure on a hot boat deck — PV = nRT predicts the rise.
Car tires: tire pressure climbs on hot highways; the law says exactly how much.
Chemistry labs: gas syringes, reaction vessels, and vacuum chambers are all sized using the ideal gas law.

Frequently asked questions

What is an ideal gas?

An ideal gas is a model: point-like particles with no volume that never attract or repel each other, colliding elastically. Real gases behave nearly ideally at low pressure and high temperature — ordinary air at room conditions is close enough for most problems.

Must temperature always be in kelvin?

Yes — but only for the temperature. Pressure, volume, and moles can use any consistent units as long as R matches them. With R = 8.314 J/(mol·K), use pascals, cubic meters, and moles. With R = 0.08206 L·atm/(mol·K), use atmospheres and liters.

When does the ideal gas law fail?

At very high pressure the particles' own volume matters, and at very low temperature attractive forces matter. Then the van der Waals equation (or real-gas data) replaces PV = nRT. For classroom problems at near-atmospheric conditions, the ideal law is fine.