B.3.5—Ideal gas equations

Molar form

For an ideal gas,

$$PV=nRT$$

Use n in mol, T in kelvin and $$R=8.31\,\mathrm{J\,K^{-1}\,mol^{-1}}$$.

Particle form

Using the number of particles N, the same law is

$$PV=Nk_BT$$

where $$k_B$$ is the Boltzmann constant. The forms are equivalent because $$R=N_Ak_B$$ and $$N=nN_A$$.

Choose the form from the data

Use the molar form when amount of substance is given. Use the particle form when a question asks for the number of molecules or gives microscopic quantities. Rearrange before substituting, for example $$N=PV/(k_BT)$$.

Worked example from local Question Bank row 22708

$N=2.7\times10^{15}$ helium atoms occupy $V=1.3\times10^{-5}\,\mathrm{m^3}$ at $18\,^{\circ}\mathrm{C}=291\,\mathrm{K}$. The particle form matches the data:

$$P=\frac{Nk_BT}{V}=\frac{(2.7\times10^{15})(1.38\times10^{-23})(291)}{1.3\times10^{-5}}=0.83\,\mathrm{Pa}$$

The very small pressure is consistent with the tiny number of atoms compared with a macroscopic mole.

Common trap

Do not use Celsius in either equation, and do not mix n with N. Pressure must be in pascals and volume in cubic metres for SI results.