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 at \(-0.56^{\circ} \mathrm{C}\) Calculate the number of moles 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.

Molality is a measure of solute concentration in a solution, expressed as the number of moles of solute per kilogram of solvent. Unlike molarity, which depends on the volume of the solution and can change with temperature, molality depends only on the mass of the solvent, making it temperature-independent.

Calculation of molality is straightforward: take the moles of solute and divide by the mass of the solvent in kilograms. In the formula form, it is represented as:
\[ m = \frac{\text{moles of solute}}{\text{kg of solvent}} \].

It is crucial in colligative properties chemistry because these properties are directly proportional to the solute concentration when expressed as molality. It ensures accurate predictions regardless of temperature changes.

Freezing point depression is one of the colligative properties that describes how the freezing point of a solvent is lowered when a solute is added. It is directly proportional to the molality of the solution and the nature of the solute.

The general formula for calculating the change in freezing point (\(\Delta T_f\)) is given by:
\[ \Delta T_f = K_f \times m \times i \],
where \(K_f\) is the freezing-point depression constant unique to each solvent, \(m\) is the molality of the solution, and \(i\) is the van't Hoff factor. This property is beneficial in practical applications such as de-icing roads in winter using salt or creating non-freezing solutions for various industrial processes.

The van't Hoff factor (\(i\)) is a dimensionless quantity used in the formulas for colligative properties to represent the number of particles a compound produces when it dissolves in a solvent. It accounts for compounds that may dissociate into multiple ions in a solution.

The ideal van't Hoff factor for a non-electrolyte is 1, as they don’t dissociate into ions. For electrolytes, the factor can either be an integer larger than 1 or occasionally a fraction when dealing with ion-pair formation or incomplete dissociation.

To determine the van't Hoff factor experimentally, one might measure a colligative property (such as the freezing point depression) and then use the known values of the other variables in the formula to solve for \(i\). It acts as an essential correction factor to obtain accurate measurements related to boiling point elevation, osmotic pressure, and freezing point depression.

A complex ion consists of a metal ion at its center with a number of other molecules or ions surrounding it. These surrounding particles, called ligands, are typically polar molecules or ions themselves and can donate electron pairs to the central atom, forming coordinate covalent bonds.

In the example of \(\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{3}\), the central Cobalt ion (Co) forms a complex with four Ammonia (NH3) ligands. The ligands create a specific geometry around the central ion - in this case, a tetrahedral structure. The complex ion would be written as \([\mathrm{Co}(\mathrm{NH}_{3})_{4}]^{2+}\) with chloride ions (Cl-) as counterions. Understanding the structure of complex ions is crucial for predicting properties such as magnetic behavior, reactivity, and color in coordination compounds.