B.3 Gas lawsIB Physics HL: Revision notes

Pressure is the force acting perpendicular to a surface per unit area: P = F/A, measured in pascals (1 Pa = 1 N m⁻²).

The amount of substance n, in moles, is n = N/N_A, where N is the number of particles and N_A = 6.02 × 10²³ mol⁻¹ is the Avogadro constant. Keep N (a pure number of molecules) and n (moles) separate: many calculation errors come from swapping them.

For a fixed mass of gas, three laws were found by experiment:

• Boyle's law (constant T): PV = constant
• Charles's law (constant P): V/T = constant
• pressure law (constant V): P/T = constant

Combining them gives PV/T = constant. Including the amount of gas gives the ideal gas law: PV = nRT or PV = Nk_BT, where R = 8.31 J K⁻¹ mol⁻¹ and k_B = R/N_A = 1.38 × 10⁻²³ J K⁻¹. Temperatures must always be in kelvin.

An ideal gas is a model: an idealised system used to approximate real gases. Its assumptions are:

• a very large number of identical molecules in random motion with a range of speeds;
• molecular volume is negligible compared with the container;
• no intermolecular forces except during collisions;
• collisions are elastic and last a negligible time.

Every empirical gas law follows from these assumptions.

When a molecule rebounds from a wall, its momentum component perpendicular to the wall reverses, a change of 2mv. The wall exerts a force equal to the rate of change of momentum, and the molecule exerts an equal and opposite force on the wall. Averaging over the huge number of collisions gives

P = ⅓ρv²

where ρ is the gas density and v is the root mean square speed. Pressure rises if molecules move faster (more momentum per collision and more frequent collisions) or if there are more of them per unit volume.

In an ideal gas there are no intermolecular forces, so there is no intermolecular potential energy. The internal energy is the total random kinetic energy of the molecules. For a monatomic gas:

U = (3/2)Nk_BT = (3/2)nRT

The mean kinetic energy of one molecule is (3/2)k_BT, so absolute temperature is a measure of the average kinetic energy of the molecules. Because PV = nRT, you can also write U = (3/2)PV.

Real gases behave most like an ideal gas at low pressure, low density and high temperature (well above the temperature at which they liquefy). Then the molecules are far apart, so their volume and the forces between them are negligible.

At high pressure / high density or low temperature, molecules are close and move slowly: attractive forces reduce the pressure below the ideal value, and the molecules' own volume becomes significant. Near liquefaction the ideal model fails completely.