Ideal Gas Law

Rearrange pV = nRT for the volume: V = nRT/p. Insert the amount of substance n in mol, the universal gas constant R = 8.314 J/(mol·K), the absolute temperature T in kelvin and the pressure p in pascals, then the volume comes out in cubic metres. Example: 5 mol of nitrogen at 300 K and 1 atm = 101 325 Pa give V = 5·8.314·300/101 325 ≈ 0.123 m³, that is 123 litres. Make sure to give the pressure in pascals and the temperature in kelvin; atm, bar or Celsius must be converted first. One cubic metre equals 1000 litres.

Because in the ideal gas law volume and pressure are directly proportional to the absolute temperature. This proportionality holds only from absolute zero, which the Kelvin scale sets at 0 K. Doubling the absolute temperature at constant pressure doubles the volume. On the Celsius scale this would be nonsensical, because a change from 10 °C to 20 °C is by no means a doubling of the particle energy. If you accidentally insert Celsius, you get grossly wrong and, for negative values, even unphysical results. Therefore always convert first: T in kelvin equals T in Celsius plus 273.15.

At constant volume and constant amount of substance, pV = nRT implies that the pressure is directly proportional to the absolute temperature: p ∝ T. Doubling the temperature in kelvin doubles the pressure. This is the law of Amontons or Gay-Lussac. Intuitively, at higher temperature the gas particles collide with the walls faster and more often, which raises the pressure. That is why the pressure in a closed container, such as a spray can, rises when heated and can become dangerous. For the calculation p₁/T₁ = p₂/T₂ holds, each with temperatures in kelvin.

The ideal gas law is an approximation that makes two assumptions: the gas particles have no volume of their own and exert no attractive forces on each other. These assumptions break down at high pressure and low temperature. At high pressure the particles own volume becomes relevant, at low temperature the attractive forces become noticeable, until the gas finally condenses. Near the boiling point or for very dense gases the law shows clear deviations. For real gases one then uses more accurate models such as the van der Waals equation. For dilute gases at room conditions, however, the ideal approximation is very good.

The amount of substance n is given in moles and counts portions of particles. One mole contains exactly the Avogadro number of particles, that is about 6.022×10²³. The number of particles N is the actual count of individual atoms or molecules and is linked to the amount of substance through N = n·N_A. In the ideal gas law pV = nRT you work with the amount of substance and the gas constant R. Alternatively there is the form pV = N·k_B·T with the number of particles and the Boltzmann constant k_B = R/N_A. Both forms are equivalent; you choose, depending on the task, whether moles or the particle number is given.