Which one of the following solution has least vapour pressure? (a) \(0.01 \mathrm{M} \mathrm{CaCl}_{2}\) (b) \(0.01 \mathrm{M}\) glucose (c) \(0.01 \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}\) (d) \(0.01 \mathrm{M} \mathrm{Na}_{3} \mathrm{PO}_{4}\)

Understanding Raoult's Law is crucial when studying solutions' vapor pressure. It asserts that the vapor pressure of the solvent in an ideal solution is directly proportional to the mole fraction of the solvent. In mathematical terms, the law is expressed as:
\( P = P^\ast \times X \)
where \( P \) is the vapor pressure of the solution, \( P^\ast \) is the vapor pressure of the pure solvent, and \( X \) is the mole fraction of the solvent. In simple terms, as the amount of solute in a solution increases, the mole fraction of the solvent decreases, leading to a decrease in the solution's vapor pressure. This is because the solute molecules occupy space at the surface of the solution, preventing as many solvent molecules from evaporating.

This intimate connection between vapor pressure and mole fraction illustrates the importance of Raoult's Law in predicting how solutions will behave under different conditions, such as during the process of distillation or in chemical equilibrium scenarios.

The van't Hoff factor, denoted as \( i \), is a dimensionless quantity used in chemistry to account for the effect of solute particles on various colligative properties, such as boiling point elevation, freezing point depression, osmotic pressure, and, pertinent to this exercise, vapor pressure. The van't Hoff factor essentially tells us how many moles of particles (ions or molecules) a substance will produce in solution.

For example, a non-electrolyte like glucose doesn't dissociate in solution, so its \( i \) is 1. On the other hand, electrolytes like \( CaCl_2 \) dissociate into three ions (\( Ca^{2+} \) and two \( Cl^- \) ions), so its \( i \) is 3. Consequently, the presence of more particles from dissociation leads to a greater reduction in vapor pressure. Understanding the van't Hoff factor is vital for predicting a solution’s behavior, particularly in scenarios where the dissociation of compounds significantly influences the solution’s properties.

The mole fraction is a way of expressing the concentration of a component in a mixture. Defined as the ratio of the moles of a specific component to the total moles of all components in the mixture, the mole fraction is a unitless quantity that provides insight into the composition of solutions. Given the symbol \( X \), it is calculated for a substance A in a solution as:
\( X_A = \frac{n_A}{n_{total}} \)
where \( n_A \) is the number of moles of the substance A, and \( n_{total} \) is the sum of the moles of all the substances in the mixture. The mole fraction does not change with temperature or pressure, distinguishing it from other concentration measures like molarity.

In the context of vapor pressure, the mole fraction effectively determines the extent to which the solvent’s vapor pressure will be lowered upon the addition of a solute. As the solute is added, the mole fraction of the solvent decreases, which, as Raoult’s Law describes, results in a lower vapor pressure for the solution. This is a key concept to grasp when identifying the volatility of components in a solution and predicting how mixtures will behave in different thermodynamic processes.