Q.An ideal gas passes from one equilibrium state $(P_1, V_1, T_1, N)$ to another equilibrium state $(2P_1, 3V_1, T_2, N)$. Then

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 ex …