Thermal physics

Ideal gas equation of state

The ideal-gas state links pressure, volume, absolute temperature, and amount of substance through pV=nRT.

For a fixed amount of gas behaving ideally, pressure p, volume V, and absolute temperature T are not independent. The equation of state determines an unknown when the others are known.

One state, many paths

An equilibrium state is specified by p, V, T, and n. For fixed n, two states of the same gas satisfy $p_1V_1/T_1=p_2V_2/T_2$. The path between them depends on how the gas is heated, compressed, or expanded.

Quick check

For a fixed amount of gas, the equation of state lets you check a state when three variables are known. If temperature rises while volume is fixed, pressure rises in proportion to absolute temperature; at fixed pressure, warming the gas increases its volume. These variables are not independent: their values are constrained by the amount of gas and the gas constant. Use kelvins, cubic metres, and pascals with $R=8.314$ J mol⁻¹ K⁻¹ for consistent units. The equation can be written $pV=Nk_BT$ when counting molecules $N$, or $pV=nRT$ when using amount $n$ in moles. The forms are connected by $R=N_Ak_B$. Real gases depart from this model when molecular volume or intermolecular forces become important, especially near condensation. Thus the ideal-gas law is a useful approximation with a physical range of validity, not a universal identity for every gas state.

For example, one mole of ideal gas at 300 K in a 0.024 m³ vessel has pressure $p=nRT/Vapprox104$ kPa. This is close to atmospheric pressure. If using $R$ in joules per mole-kelvin, convert litres to cubic metres before substituting.