\(1.00 \mathrm{~g}\) of a non-electrolyte solute (molar mass \(\left.250 \mathrm{~g} \mathrm{~mol}^{-1}\right)\) was dissolved in \(51.2 \mathrm{~g}\) of benzene. If the freezing point depression constant, \(\mathrm{K}_{\mathrm{f}}\) of benzene is \(5.12 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\), the freezing point of benzene will be lowered by:

Molality is a measure of the concentration of a solute in a solvent. Unlike molarity, which depends on the volume of the solution, molality is defined as the number of moles of a solute per kilogram of the solvent. The formula to calculate molality (m) is given by
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \]
This means that if we dissolve the solute in a fixed amount of solvent, the molality remains constant, regardless of changes in temperature or pressure. In our exercise, the mass of the solute, temperature, and the molar mass are given, allowing us to calculate the molality of the solution, hence helping us evaluate the freezing point depression.

The van't Hoff factor (i) represents the number of particles into which a compound dissociates in solution. For non-electrolytes, which do not dissociate into ions, the van't Hoff factor is 1. For electrolytes, this factor can be greater than 1, as they break into multiple ions. For instance, NaCl would have a van't Hoff factor of 2, because it dissociates into Na+ and Cl- ions.
When calculating freezing point depression, the van't Hoff factor is crucial, as it directly modifies the extent of the freezing point change. The freezing point depression equation, incorporating the van't Hoff factor, is: \[ \Delta T = i \cdot K_f \cdot m \]
Here, \( \Delta T \) is the freezing point depression, \( K_f \) is the freezing point depression constant of the solvent, and m is the molality of the solution. In our case, since the solute is a non-electrolyte, the value of i is 1, thereby not altering the freezing point depression calculated through this formula.

Colligative properties are physical properties of solutions that depend on the ratio of the number of solute particles to the number of solvent molecules in a solution and do not depend on the nature of the chemical species present. They include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure.
Freezing point depression is a colligative property that particularly refers to the phenomenon where the addition of a solute to a solvent decreases the temperature at which the solvent freezes. The more solute particles present, the lower the freezing point will be. This property is independent of the chemical identity of the solute and is directly proportional to the molality (molal concentration) of the solute particles.

A non-electrolyte solute, such as the one described in our exercise, does not dissociate into ions when dissolved in a solvent. Substances like sugar or certain organic compounds are often non-electrolytes. Their van't Hoff factor (i) equals 1 because these substances remain intact as molecules in the solution.
This is important for calculations involving colligative properties, since the presence of ions (from electrolytes) would affect the number of particles in solution and thus the magnitude of properties like freezing point depression. However, when dealing with a non-electrolyte solute, one can simply calculate the effect on the freezing point by assuming that each molecule of solute will result in one particle in the solution.