Boyle's Law, Charles' Law and Ideal Gas Law

For a fixed amount of gas at low density, two empirical laws combine into the ideal gas law. Boyle's law: at constant temperature, pressure and volume are inversely related, $P\propto 1/V$. Charles' law: at constant pressure, volume is directly proportional to absolute temperature, $V\propto T$. Together these give $PV=CT$; considering two identical gas containers merged into one (doubling both volume and particle count $N$ while $P,T$ stay fixed) shows $C$ must be proportional to $N$, i.e. $C=Nk$, where $k=1.381\times10^{-23}\ \text{J K}^{-1}$ is the universal Boltzmann constant -- giving $PV=NkT$. Writing $N=\mu N_A$ (with Avogadro's number $N_A=6.023\times10^{23}\ \text{mol}^{-1}$, the number of carbon atoms in 12 g of $^{12}\text{C}$) and $R=N_Ak=8.314\ \text{J mol}^{-1}\text{K}^{-1}$ gives the equation of state for $\mu$ moles, $PV=\mu RT$. One mole of an ideal gas occupies 22.4 …