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The Ideal Gas Law
Thermodynamics
The ideal gas law links the four quantities that describe a gas: pressure, volume, temperature and amount. It is one of the most useful equations in thermodynamics.
What an ideal gas is
An ideal gas is a model in which the particles are point-like, move randomly, collide elastically, and exert no forces on each other except during collisions. No real gas is perfectly ideal, but most gases at ordinary temperatures and low pressures behave very closely to this model. The ideal gas law, PV = nRT, combines three earlier laws: Boyle's law, that pressure and volume are inversely related at constant temperature; Charles's law, that volume is proportional to absolute temperature at constant pressure; and the rule that volume is proportional to the amount of gas. The constant R is the universal gas constant, 8.31 J/mol K.
Using the law
The law can be used to find any one of pressure, volume, temperature or amount when the other three are known, provided the temperature is in kelvin and the units are consistent. If the amount of gas is fixed, the combined gas law relates two states directly, which is convenient for problems about a gas being compressed or heated. At constant temperature, doubling the pressure halves the volume; at constant pressure, doubling the absolute temperature doubles the volume. These proportionalities are a quick check on any calculation: if the temperature rises, the pressure or the volume must rise with it, not fall.
Where you see it
Car tyres, scuba tanks, spray cans and weather balloons are all governed by the gas law. A tyre's pressure rises on a hot day because the trapped air is heated at nearly constant volume. A balloon rises because the gas inside is less dense than the surrounding air, and it expands as the outside pressure falls with altitude. The law also explains why a sealed container must not be heated: the pressure climbs until the container fails. In the laboratory it lets chemists calculate how much gas a reaction will produce or consume from a measured volume change.
Worked example: pressure of an ideal gas
A container holds n = 2.0 mol of gas at T = 300 K in a volume V = 0.050 m^3. Use P V = n R T, with R = 8.31 J/mol K. Rearrange: P = n R T / V. Substitute: P = 2.0 x 8.31 x 300 / 0.050. Numerator: 2.0 x 8.31 x 300 = 4986. P = 4986 / 0.050 = 99,720 Pa. The pressure is about 1.0 x 10^5 Pa, roughly one atmosphere.
Common mistakes
- Using Celsius instead of kelvin: the gas law needs absolute temperature, so add 273 to a Celsius value.
- Mixing up pressure units: use pascals in SI calculations, converting from atmospheres or bar first.
- Forgetting that n is moles, not mass or number of grams: divide the mass by the molar mass to get n.
Practice questions
- What volume does 1.0 mol of gas occupy at 300 K and 1.0 x 10^5 Pa?
Show answer
V = nRT/P = 1.0 x 8.31 x 300 / (1.0 x 10^5) = 2493 / 100000 = 0.0249 m^3, about 25 litres.
- A sealed container of gas is heated from 300 K to 600 K. What happens to its pressure?
Show answer
Pressure doubles, because P is proportional to T at fixed volume and amount.
- When does a real gas stop behaving like an ideal gas?
Show answer
At high pressure or low temperature, when molecules are close together and their own volume and attractions matter.
Frequently asked questions
Must temperature be in kelvin in the ideal gas law?
Use kelvin. A temperature in Celsius must have 273 added, because the law uses absolute temperature.
What volume does one mole occupy at standard conditions?
One mole of an ideal gas occupies about 22.4 litres at 0 degrees C and 1 atmosphere, or about 24 litres at 20 degrees C.