Solid Lead nitrate is dissolved in $1 \, L$ of water. The solution was found to boil at $100.15^{\circ}C$. When $0.2 \, mol$ of $NaCl$ is added to the resulting solution,it was observed that the solution froze at $-0.8^{\circ}C$. The solubility product of $PbCl_2$ formed is $........... \times 10^{-6}$ at $298 \, K$. (Nearest integer) Given: $K_b = 0.5 \, K \, kg \, mol^{-1}$ and $K_f = 1.8 \, K \, kg \, mol^{-1}$. Assume molality to be equal to molarity in all cases.

Identify the correct statements :
$(A)$ The molality of $2.5 \text{ g}$ of ethanoic acid (Molar mass : $60 \text{ g mol}^{-1}$) in $75 \text{ g}$ of benzene solution is $0.556 \text{ m}$.
$(B)$ The molarity of a solution containing $5 \text{ g}$ of NaOH (molar mass : $40 \text{ g mol}^{-1}$) in $450 \text{ mL}$ of solution is $0.278 \text{ M}$ at $298 \text{ K}$.
$(C)$ Aquatic species are more comfortable in cold water.
$(D)$ The solubility of gas increases with decrease in pressure.
$(E)$ For a binary mixture of $A$ and $B$, the number of moles of $A$ and $B$ are $n_{A}$ and $n_{B}$ respectively. The mole fraction of $B$ will be $x_{B} = n_{A} / (n_{A} + n_{B})$.
Choose the correct answer from the options given below :

If a $1 \ m$ solution of benzoic acid in benzene has a freezing point depression of $2.56 \ ^{\circ}C$ $(K_f = 5.12 \ ^{\circ}C \ kg \ mol^{-1})$ and a boiling point elevation of $2.53 \ ^{\circ}C$ $(K_b = 2.53 \ ^{\circ}C \ kg \ mol^{-1})$,select the correct statement$(s)$:
Statement $I$: There is dimer formation when undergoing freezing.
Statement $II$: There is no change when undergoing boiling.
Statement $III$: Reverse of $I$ and $II$.
Statement $IV$: Dimer formation in freezing and boiling state.

The $pH$ of a $0.1 \ M$ monobasic acid is $2$. Its osmotic pressure at a given temperature $T \ (K)$ is: (Given that the effective concentration for osmotic pressure is $(1+\alpha) \times$ concentration of acid, where $\alpha$ is the degree of dissociation)

$1.22 \, g$ of an organic acid is separately dissolved in $100 \, g$ of benzene $(K_{b}=2.6 \, K \, kg \, mol^{-1})$ and $100 \, g$ of acetone $(K_{b}=1.7 \, K \, kg \, mol^{-1})$. The acid is known to dimerize in benzene but remain as a monomer in acetone. The boiling point of the solution in acetone increases by $0.17^{\circ} C$.
The increase in boiling point of solution in benzene in $^{\circ} C$ is $x \times 10^{-2}$. The value of $x$ is ..... .(Nearest integer)
$[$ Atomic mass : $C=12.0, H=1.0, O=16.0]$

The osmotic pressure of a solution of $NaCl$ is $0.10 \ atm$ and that of a glucose solution is $0.20 \ atm.$ The osmotic pressure of a solution formed by mixing $1 \ L$ of the sodium chloride solution with $2 \ L$ of the glucose solution is $x \times 10^{-3} \ atm.$ $x$ is.......... (nearest integer)