The molal boiling point elevation constant of water is \(0.513^{\circ} \mathrm{C} \mathrm{kg} \mathrm{mol}^{-1}\). When \(0.1\) mole of sugar is dissolved \(200 \mathrm{~g}\) of water, the solution boils under a pressure of 1 atm at (a) \(100.513^{\circ} \mathrm{C}\) (b) \(102.565^{\circ} \mathrm{C}\) (c) \(100.256^{\circ} \mathrm{C}\) (d) \(101.025^{\circ} \mathrm{C}\)

Colligative properties are those properties of solutions that depend on the number of solute particles in a given amount of solvent and not on the nature of the chemical species present. These properties arise because the solute particles disrupt the solvent's molecular interactions, which affects various physical aspects of the solution. Boiling point elevation is one such colligative property, alongside others like freezing point depression, osmotic pressure, and vapor pressure lowering.

Understanding how the addition of solute to a solvent can change the boiling point is critical in many industrial and laboratory processes, such as antifreeze in car radiators and the cooking of food at high elevations. Boiling point elevation occurs because the solute particles interfere with the escape tendency of solvent molecules from the liquid to the gas phase, thus requiring a higher temperature to boil.

Molality is a measure of the concentration of a solute in a solution. It is expressed as the number of moles of solute per kilogram of solvent, not the volume of the solution, making it temperature-independent. This is an important factor to consider, as the volume of liquids can change with temperature.

To calculate the molality, you divide the moles of solute by the mass of the solvent in kilograms. For example, dissolving 0.1 moles of sugar in 200 grams (0.2 kilograms) of water results in a molality of 0.5 moles per kilogram. In the context of boiling point elevation, molality provides a direct measure that, when multiplied by the molal boiling point elevation constant, gives us the increase in boiling point due to the presence of the solute.

The van't Hoff factor, symbolized by 'i', represents the number of particles into which a compound dissociates in solution. For example, common table salt (NaCl) dissociates into two particles (Na+ and Cl-), giving it a van't Hoff factor of 2. On the other hand, a molecule like sugar, which does not dissociate into ions when dissolved, has a van't Hoff factor of 1.

The van't Hoff factor is crucial in calculating the extent of boiling point elevation or freezing point depression in solutions. It's the multiplying factor in the equation that connects the number of particles resulting from a solute to the magnitude of the colligative effect observed. However, it is also essential to note that real solutions can have van't Hoff factors that deviate from their ideal values due to solute-solute and solute-solvent interactions.