Introduction

The state of a fixed mass of gas is completely specified by a small set of macroscopic quantities -- its pressure $P$, volume $V$, temperature $T$, and internal energy $E$ -- and any equation relating these quantities for a given gas is called its equation of state.

Three separate experimental gas laws, each holding one variable fixed, describe how these quantities are linked for a fixed mass of gas:

• Boyle's law: $V \propto \dfrac{1}{P}$ at constant $T$
• Charles' law: $V \propto T$ at constant $P$
• Gay-Lussac's law: $P \propto T$ at constant $V$

Combining all three into a single relation for a fixed mass of gas gives $\dfrac{PV}{T} = \text{constant}$, so that for the same gas in two different states, $\dfrac{P_1V_1}{T_1} = \dfrac{P_2V_2}{T_2}$.

Expressing the fixed mass in terms of the number of moles $n = \dfrac{N}{N_A} = \dfrac{\text{mass of gas}(M)}{\text{molar mass}(M_0)}$, this generalises to the ideal gas equation

where $R$ is the universal gas constant, $8.314\ \text{J mol}^{-1}\text{K}^{-1}$, the same for every gas. Writing the mass in terms of the actual number of molecules $N$ instead of moles gives the equivalent microscopic form

where $k_B$ is the Boltzmann constant, related to $R$ by $R = N_Ak_B$ ($N_A$ being Avogadro's number, the number of molecules in one mole).

The gas laws of Boyle, Charles and Gay-Lussac are strictly valid only when the pressure is not too high and the temperature is not close to the gas's liquefaction temperature. A gas that obeys $PV = nRT$ at all pressures and temperatures, with no exception, is called an ideal gas -- a theoretical idealisation that real gases only approach under suitable conditions, explored further in the next section.