The approximate molality of ethylene glycol \(\left(\mathrm{C}_{2} \mathrm{H}_{6} \mathrm{O}_{2}\right)\) in an aqueous solution which freezes at a temperature no higher than \(-15^{\circ} \mathrm{C}\) is \(\left(k_{\mathrm{f}}\right.\) of water \(=1.86^{\circ} \mathrm{C}\) \(\mathrm{kg} \mathrm{mol}^{-1}\) ) (a) \(8.06 \mathrm{~m}\) (b) \(0.806 \mathrm{~m}\) (c) \(0.145 \mathrm{~m}\) (d) \(1.5 \mathrm{~m}\)

When exploring the effects of a solute on the freezing point of a solvent, the term molality becomes crucial. Molality (\(m\)) measures the concentration of a solution by defining the amount of solute in moles per kilogram of solvent, not the solution as a whole. This parameter is especially useful in freezing point depression because it does not change with temperature, unlike molarity, which is concentration expressed as moles of solute per liter of solution.

Molality is calculated by the formula: \(m = \frac{moles \, of \, solute}{kilograms \, of \, solvent}\). In the context of our exercise, to find the ethylene glycol's molality, we'd divide the number of moles of ethylene glycol by the mass (in kilograms) of water acting as the solvent.

The cryoscopic constant (\(k_f\)) is a proportionality constant that relates the molality of a solution to its freezing point depression—the drop in freezing temperature observed when a solute is dissolved in a solvent. This constant is unique for each solvent and is expressed in degrees Celsius kilogram per mole (\(^\text{o}C\text{ kg/mol}\)).

For water, the cryoscopic constant is typically \(1.86 ^\text{o}C\text{ kg/mol}\), which means that a 1 molal solution of a non-ionizing substance in water would lower the freezing point by 1.86°C. The calculation in our example uses this constant to determine the significant freezing point depression observed with the given concentration of ethylene glycol.

The van't Hoff factor (\(i\)) is a dimensionless quantity that represents the number of particles a compound splits into when it dissolves in a solvent. For non-electrolytes, which do not dissociate into ions, the van't Hoff factor is 1. For electrolytes, it can be greater than 1, reflecting the ionic character of the dissolved substance and how it separates into multiple particles.

For example, sodium chloride (\(NaCl\)) has a van't Hoff factor of about 2, because it dissociates into two ions in solution – one sodium ion (\(Na^+\)) and one chloride ion (\(Cl^-\)). In the instance of ethylene glycol (\(C_2H_6O_2\)), an organic compound that does not dissociate in solution, the van't Hoff factor remains 1. This factor is pivotal in the freezing point depression equation as it adjusts the effect observed based on the number of particles the solute becomes in the solution.