Solutions · JEE Advanced previous year questions

At $300\text{ K}$, an ideal dilute solution of a macromolecule exerts osmotic pressure that is expressed in terms of the height ($\mathrm{h}$) of the solution ($\text{density}=1.00{\text{ g cm}}^{-3}$) where $\mathrm{h}$ is equal to $2.00\text{ cm}$. If the concentration of the dilute solution of the macromolecule is $2.00{\text{ g dm}}^{-3}$, the molar mass of the macromolecule is calculated to be $\mathbf{X}\times 10^4{\text{ g mol}}^{-1}$. The value of $\mathbf{X}$ is _________. $\text{Use:}$ Universal gas constant $(\mathrm{R})=8.3{\text{ J K}}^{-1}{\text{ mol}}^{-1}$ and acceleration due to gravity $(\mathrm{g})=10{\text{ m s}}^{-2}$

Vessel-1 contains ${\mathbf{w}}_2$ g of a non-volatile solute $\mathbf{X}$ dissolved in ${\mathbf{w}}_1$ g of water. Vessel-2 contains ${\mathbf{w}}_2$ g of another non-volatile solute $\mathbf{Y}$ dissolved in ${\mathbf{w}}_1$ g of water. Both the vessels are at the same temperature and pressure. The molar mass of $\mathbf{X}$ is 80% of that of $\mathbf{Y}$. The van’t Hoff factor for $\mathbf{X}$ is 1.2 times of that of $\mathbf{Y}$ for their respective concentrations. The elevation of boiling point for solution in Vessel-1 is ____ % of the solution in Vessel-2.

$50\text{ mL}$ of $0.2\text{ molal}$ urea solution ($\text{density}=1.012{\text{ g mL}}^{-1}$ at $300\text{ K}$) is mixed with $250\text{ mL}$ of a solution containing $0.06\text{ g}$ of urea. Both the solutions were prepared in the same solvent. The osmotic pressure (in $\text{Torr}$) of the resulting solution at $300\text{ K}$ is _____. [Use: Molar mass of urea $=60{\text{ g mol}}^{-1}$; gas constant, $\mathrm{R}=62{\text{ L Torr K}}^{-1}{\text{ mol}}^{-1}$ ; Assume, ${\Delta }_{\mathrm{mix}}\mathrm{H}=0$, ${\Delta }_{\mathrm{mix}}\mathrm{V}=0$]