The Ideal Gas Equation

The three gas laws are three views of one relationship. Combining them gives the equation of state for an ideal gas.

pV = nRT

n is the number of moles, and R is the molar gas constant, 8.31 J mol⁻¹ K⁻¹. T must be in kelvin.

The mole — the amount of substance containing as many particles as there are in 12 g of carbon-12. That number is the Avogadro constant, N_A = 6.02 × 10²³ mol⁻¹.

N = n N_A

The same equation, counted in particles — dividing R by the Avogadro constant gives the energy per particle per kelvin rather than per mole. That is the Boltzmann constant, k = R / N_A = 1.38 × 10⁻²³ J K⁻¹, and the equation becomes:

pV = NkT

Use pV = nRT when the question gives moles or a molar mass; use pV = NkT when it gives a number of particles. They are the same statement.

Example: 2.0 mol of an ideal gas at 300 K in a container of volume 0.050 m³. p = nRT / V = (2.0 × 8.31 × 300) ÷ 0.050 = 9.97 × 10⁴ Pa, a little under atmospheric pressure.

What "ideal" assumes — the particles have negligible volume compared with the container, there are no forces between them except during collisions, all collisions are perfectly elastic, and the motion is random. Real gases follow the equation closely at low pressure and high temperature, where the particles are far apart. They depart from it near liquefaction, where the particles are close enough for the attractive forces to matter and their own volume is no longer negligible.