Q.Obtain an ideal gas law from Boyle's and Charles' law.

For a given amount of gas at low density, two empirical laws describe its behaviour. Boyle's law (Robert Boyle, 1662): at constant temperature, the pressure of a gas is inversely proportional to its volume, $P \propto \dfrac{1}{V}$. Charles' law (Jacques Charles): at constant pressure, the volume of a gas is directly proportional to its absolute temperature, $V \propto T$. Combining the two gives $PV = CT$, where the constant $C$ turns out to be proportional to the number of gas particles $N$ (considering two identical gas containers merged into one doubles both $V$ and $N$ while $P,T$ stay fixed): $C = Nk$, where $k$ is the universal Boltzmann constant, $k=1.381\times10^{-23}\ \text{J K}^{-1}$. This gives the ideal gas law, $PV = NkT$. Writing $N = \mu N_A$ (with Avogadro's number $N_A = 6.023\times10^{23}\ \text{mol}^{-1}$, the number of carbon atoms in exactly 12 g of $^{12}\text{C}$) and defining the universal gas constant $R = N_A k = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$ turns this into $PV = \mu RT$ for $\mu$ moles of gas -- the equation of state for an ideal gas, valid only at thermodynamic equilibrium. One mole of any ideal gas occupies 22.4 L at STP (273 K, 1.013×10⁵ Pa) and about 24.6 L at room temperature (300 K).