Ideal gas law

Remarks

The law combines three older laws. Boyle's law says that $pV$ is constant at fixed $T$ and $n$. Charles's law says that $V$ is proportional to $T$ at fixed $p$ and $n$, and Avogadro's law that $V$ is proportional to $n$ at fixed $p$ and $T$. In kinetic theory it is derived for non-interacting point particles. The pressure is the momentum given to the walls per unit area and time, which gives $pV = \tfrac{2}{3} N \langle E_{\mathrm{kin}} \rangle$. The average kinetic energy is $\tfrac{3}{2} kT$, so $pV = NkT$ with $R = N_A k$. Example: one mole at $T = 273.15\,\mathrm{K}$ and $p = 101\,325\,\mathrm{Pa}$ has $V = RT/p \approx 0.0224\,\mathrm{m^3}$. The low-density hypothesis is needed: at high pressure, or near condensation, the attraction and the size of molecules give large deviations, measured by the compressibility factor $Z = pV/(nRT) \ne 1$.

Going deeper

For a student

The ideal gas law is the simplest equation of state. Most of the elementary thermodynamics of gases, such as work in isothermal processes and Mayer's relation, is computed with it. The fact that $pV/n$ tends to a common limit for all gases is also the basis of the gas thermometer, which realises the thermodynamic temperature scale. Corrections for real gases come from the virial expansion $pV_m/(RT) = 1 + B(T)/V_m + \cdots$ or from model equations such as the van der Waals equation.