Q.Calculate the number of moles of air in the inflated balloon at room temperature, as shown in the figure. The radius of the balloon is 10 cm, and the pressure inside the balloon is 180 kPa.

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).