A solution made by dissolving \(0.875 \mathrm{~g}\) of \(\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{3}\) in \(25.0 \mathrm{~g}\) of water freezes \(0.56^{\circ} \mathrm{C}\) below the freezing point of pure water. Calculate the number of mole of ions produced when 1 mole of \(\mathrm{Co}\left(\mathrm{NH}_{3}\right){ }_{4} \mathrm{Cl}_{3}\) is dissolved in water, and suggest a structure for the complex ion present in this compound.

Colligative properties are a set of solution properties that depend entirely on the ratio of solute particles to solvent molecules in a solution, without regard to the identity of the solute particles. This means that these properties are not influenced by the chemical nature of the solute, but rather by how many particles it introduces into the solution.

Freezing point depression is one of the key colligative properties, in addition to boiling point elevation, vapor-pressure lowering, and osmotic pressure. When a solute is dissolved in a solvent, such as salt in water, it interferes with the formation of the solid phase, thereby lowering the temperature at which the solution will freeze. This is particularly important in phenomena like de-icing roads in winter or the creamy texture of ice cream, where control over freezing is essential.

The extent of the freezing point depression depends on the concentration of dissolved particles in the solution, which can be expressed as the molality of the solution. Intuitively, the more solute particles there are, the more significant the depression of the freezing point.

The Van't Hoff factor, denoted by 'i', is indispensable in quantifying the effect of a solute on the colligative properties of a solution. It represents the number of particles a compound dissociates into when it dissolves in a solvent.

For example, common table salt, NaCl, dissociates into two ions, Na+ and Cl-, in water. Therefore, the Van't Hoff factor for NaCl would be 2. If the solute does not dissociate or ionize, like sugar in water, then the Van't Hoff factor is 1. This factor is crucial in accurately calculating the change in colligative properties because it adjusts for the number of particles introduced by electrolytes that dissociate in solution.

Understanding the Van't Hoff factor helps scientists and engineers design antifreeze solutions, optimize salt quantities for de-icing, and it's fundamental to many industrial processes where the control of solution properties is necessary.

Molality is a measure of the concentration of a solution that expresses the moles of a solute per kilogram of solvent. This unit of concentration is especially useful when dealing with colligative properties because it doesn't change with temperature, as opposed to molarity (moles per liter) which can vary because the volume of liquids changes with temperature.

It's calculated by dividing the moles of solute by the mass of the solvent in kilograms. In contrast to molarity, which depends on the total volume of the solution, molality focuses solely on the amount of solvent, making it particularly advantageous in situations where temperature changes are involved or when solute-solvent interactions are strong.

In freezing point depression, knowing the molality of a solution allows for the precise prediction of how much the freezing point will be lowered. This characteristic makes it an invaluable concept in disciplines ranging from chemistry and biology to environmental sciences and engineering.