The Ideal Gas Law: PV = nRT From Molecules to Balloons

Three laws named after three scientists, one equation that contains them all: PV = nRT. Pressure times volume equals the amount of gas times a constant times absolute temperature. It looks like a formula to memorise, but it is better understood as a single sentence about what a gas is — a swarm of independently flying molecules whose collective behaviour is reckoned by four dials, any three of which determine the fourth.

What each letter actually measures

P · V = n · R · T
P = pressure        (Pa)
V = volume          (m³)
n = amount          (mol)
R = 8.314 J/(mol·K) — the gas constant
T = temperature     (K, always absolute)

The constant R is the conversion factor that makes the units agree; 8.314 J/(mol·K) pairs with pascals and cubic metres, while chemists' tables often quote 0.082057 L·atm/(mol·K) — the same constant dressed in laboratory units. Both give identical answers as long as you keep each system's units together — a good solver converts everything to SI first, then converts back.

Three famous laws hiding inside one

Every named gas law is PV = nRT with two of the four variables frozen:

• Boyle's law — fix n and T: pressure is inversely proportional to volume. Squeeze a gas into half the volume and it pushes twice as hard, because the same number of molecules now hit the walls twice as often.
• Charles's law — fix n and P: volume is proportional to absolute temperature. Heat a balloon and the molecules move faster, demand more room, and the balloon swells.
• Avogadro's law — fix P and T: volume is proportional to the amount of gas. Equal volumes of gas at the same conditions hold equal numbers of molecules.

The ideal gas law is not a fourth law — it is the general case, and the historical trio are its special cases. That is also why "solve for any missing variable" is the natural interface: holding two things constant and varying a third is just the full equation read from a particular angle.

Why kelvin is non-negotiable

The temperature in PV = nRT is measured from absolute zero, the point where molecular motion stops. Celsius zero and Fahrenheit zero are convenient sea-level conventions, not physical zeros, and using them directly gives answers that are not merely wrong but nonsensical. The conversion is simple:

K = °C + 273.15
K = (°F + 459.67) × 5/9

So 0 °C is 273.15 K, and "doubling the temperature" means doubling the kelvin figure — 0 °C to 0 °C is nothing, while 300 K to 600 K really does double the pressure of a sealed container. This single habit prevents the most common gas-law exam error.

A worked example: the molar volume at STP

Set P = 1 atm (101325 Pa), n = 1 mol and T = 273.15 K, and solve for V = nRT/P:

V = (1 × 8.314 × 273.15) / 101325
  = 2270.97 / 101325
  ≈ 0.02241 m³  =  22.41 litres

That is the famous 22.4 L/mol — one mole of any (ideal) gas at standard temperature and pressure. It is the bridge between the gas phase and everything else: 22.4 L of oxygen is one mole, which by the molar-mass math is 32 grams. Reverse the question and it becomes an assay: 100 kPa of gas in a 1 L vessel at 273.15 K holds n = PV/RT = 0.044 mol, about 1.4 grams of air.

Where "ideal" stops being true

The model assumes molecules are points with no mutual attraction. At room temperature and atmospheric pressure, permanent gases — nitrogen, oxygen, helium, argon — obey it to within a percent or so, which is why it rules homework and most everyday engineering. Push toward condensation and it fails: at high pressure the molecules' own volume matters, at low temperature their attractions matter, and near the boiling point the whole picture collapses into liquefaction. The van der Waals equation and its cousins patch this with substance-specific corrections. The law also needs a gas that neither reacts nor condenses — water vapour, for instance, drops out at the dew point rather than obeying the ideal ratio.

Try it yourself

The Ideal Gas Law Calculator solves PV = nRT in any direction with per-variable unit menus — pressure in kPa, atm, psi or mmHg, volume in litres or millilitres, temperature in K, °C or °F — and shows every value in both your units and SI. To weigh out the moles you just computed, the Molar Mass Calculator turns a formula into grams per mole, and the Dilution Calculator handles the same "any three, solve the fourth" pattern for solutions instead of gases.