Q.Discuss the ideal gas laws.

Step 1. Boyle's law (constant temperature): the pressure of a fixed amount of gas is inversely proportional to its volume, $P\propto\dfrac{1}{V}$.

Step 2. Charles' law (constant pressure): the volume of a fixed amount of gas is directly proportional to its absolute temperature, $V\propto T$.

Step 3. Combining the two gives $PV=CT$ for some constant $C$. Considering two identical gas containers (same $P,V,T,N$) merged into a single combined system -- now with volume $2V$ and particle count $2N$, but unchanged $P,T$ -- shows that $C$ must scale with the particle number $N$: $C=Nk$, where $k=1.381\times10^{-23}\ \text{J K}^{-1}$ is the universal Boltzmann constant.

Step 4. This gives the ideal gas law $PV=NkT$. Writing $N=\mu N_A$ (Avogadro's number $N_A=6.023\times10^{23}\ \text{mol}^{-1}$) and $R=N_Ak=8.314\ \text{J mol}^{-1}\text{K}^{-1}$ (universal gas constant) turns this into the molar form $PV=\mu RT$ -- the standard equation of state for an ideal gas, valid at thermodynamic equilibrium. One mole occupies 22.4 L at STP and about 24.6 L at 300 K.

Combine the two proportionalities into $PV=CT$, then justify $C=Nk$ via the doubling-the-system argument.

• Stating the final law without deriving WHY $C=Nk$ -- the particle-scaling (doubling) argument is the key logical step connecting Boyle's/Charles' laws to the full ideal gas law.