JEE Advanced 2021 Paper 1 Chemistry: Paragraph 2

If no dissociation occurred, the pressure in the rigid vessel would be $P_0=\dfrac{nRT}{V}=\dfrac{1.00\times0.08314\times300}{10.0}=2.494\ \mathrm{bar}$. For degree of dissociation $\alpha$ the amounts are $1-\alpha$ of $\mathrm{N_2O_4}$ and $2\alpha$ of $\mathrm{NO_2}$, and because volume and temperature are fixed, the partial pressures are $p_{\mathrm{N_2O_4}}=P_0(1-\alpha)$ and $p_{\mathrm{NO_2}}=2\alpha P_0$. Then $K_p=\dfrac{(2\alpha P_0)^2}{P_0(1-\alpha)}=\dfrac{4\alpha^2P_0}{1-\alpha}=0.17$. Solving $9.977\alpha^2+0.17\alpha-0.17=0$ gives $\alpha=0.122\approx0.12$. Using the constant-pressure formula $\alpha=\sqrt{K_p/(K_p+4P)}$ would be wrong here, because it is the volume and not the total pressure that is fixed. The equilibrium total pressure is therefore $P_0(1+\alpha)=2.80\ \mathrm{bar}$, higher than the initial pressure, as expected when the number of gas moles increases.

The van't Hoff equation gives $\ln\dfrac{K_2}{K_1}=\dfrac{\Delta H^\circ}{R}\left(\dfrac1{T_1}-\dfrac1{T_2}\right)$. Here $\ln(0.71/0.17)=1.4296$ and $\dfrac1{300}-\dfrac1{320}=2.0833\times10^{-4}\ \mathrm{K^{-1}}$, so $\Delta H^\circ=\dfrac{8.314\times1.4296}{2.0833\times10^{-4}}=5.705\times10^4\ \mathrm{J\,mol^{-1}}$. At $300\ \mathrm{K}$, $\Delta G^\circ=-RT\ln K_p=-8.314\times300\times\ln0.17=+4420\ \mathrm{J\,mol^{-1}}$. From $\Delta G^\circ=\Delta H^\circ-T\Delta S^\circ$, $\Delta S^\circ=\dfrac{57046-4420}{300}=175.42\ \mathrm{J\,mol^{-1}K^{-1}}$. The positive value is expected because one mole of gas gives two moles of gas, so the disorder increases, and the endothermic dissociation is therefore entropy driven. Note that $\Delta H^\circ$ must first be obtained from the two equilibrium constants, and that $K_p$ is dimensionless when expressed relative to the standard pressure of $1\ \mathrm{bar}$, so the logarithm is valid. If the sign of $\Delta G^\circ$ were mistakenly reversed, the entropy would come out near $204.9\ \mathrm{J\,mol^{-1}K^{-1}}$, which is a typical slip in this type of calculation.