Question

The compound \(\mathrm{CoCl}_{2} \cdot 2 \mathrm{H}_{2} \mathrm{O} \cdot 4 \mathrm{NH}_{3}\) may be one of the hydrate isomers \(\left[\mathrm{CoCl}\left(\mathrm{H}_{2} \mathrm{O}\right)\left(\mathrm{NH}_{3}\right)_{4}\right] \mathrm{Cl} \cdot \mathrm{H}_{2} \mathrm{O}\) or \(\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{2}\left(\mathrm{NH}_{3}\right)_{4}\right] \mathrm{Cl}_{2} .\) A \(0.10 \mathrm{M}\) aqueous solution of the compound is found to have a freezing point of \(-0.56^{\circ} \mathrm{C} .\) Determine the correct formula of the compound. The freezing-point depression constant for water is \(1.86 \mathrm{mol}\) \(\mathrm{kg}^{-1}\) \(^{\circ} \mathrm{C}\), and for aqueous solutions, molarity and molality can be taken as approximately equal.

Short answer

The correct formula for the compound is \([\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{2}\left(\mathrm{NH}_{3}\right)_{4}\right] \mathrm{Cl}_{2}\)

Step-by-step solution

Calculate the depression in freezing point

The depression in freezing point (\(\Delta T_f\)) is the difference in temperature of the pure solvent and the solution. But the freezing point of water is \(0^{\circ}C\). So, \(\Delta T_f = 0^{\circ}C -(-0.56^{\circ}C) = 0.56^{\circ}C\)

Determine the molality of the solution

We can calculate molality by using the formula \(\Delta T_f = K_f \cdot molality\), rearranging the formula gives us molality} = \(\Delta T_f / K_f\). Substituting \(\Delta T_f = 0.56^{\circ}C\) and \(K_f = 1.86 kgmol^{-1}C^{-1}\), we get molality = \(0.56/1.86 = 0.30 mol/kg\)

Calculate the van 't Hoff factor

The van 't Hoff factor (i) is the ratio of the actual concentration of particles produced when the substance is dissolved, to the concentration of a substance as calculated from its mass. It can be calculated by dividing the molality of the solution by the molarity. So \(i = 0.30/0.10 = 3\)

Identify the correct formula

By looking at the formulas of the two possible isomers, the formula \([\mathrm{Co}\left(\mathrm{H}_{2}\mathrm{O}\right)_{2}\left(\mathrm{NH}_{3}\right)_{4}\right] \mathrm{Cl}_{2}\) implies 3 discrete particles (\(Co(H_2O)_2(NH_3)_4^{2+}\) and \(2 Cl^-\)) when dissolved in water, which corresponds to the calculated van't Hoff factor. So, \([\mathrm{Co}\left(\mathrm{H}_{2}\mathrm{O}\right)_{2}\left(\mathrm{NH}_{3}\right)_{4}\right] \mathrm{Cl}_{2}\) is the correct formula of the compound

Key concepts

Freezing Point Depression

Understanding the phenomenon of freezing point depression is crucial for chemists and anyone involved in solutions chemistry. It is observed that the freezing point of a solvent decreases when a solute is dissolved in it. This occurs because the solute particles disrupt the formation of a solid structure, making it more difficult for the solvent to solidify, hence lowering the freezing temperature.

In a practical sense, when you add antifreeze to your car's engine coolant, what you're doing is manipulating the freezing point depression to prevent the engine's fluids from freezing. In laboratory settings, the freezing point depression can help in identifying the molar mass of solutes and understanding the properties of a solution.

Van't Hoff Factor

The van't Hoff factor (\( i \)) is essential in quantifying the effect of solutes on the colligative properties of solutions, which include boiling point elevation, freezing point depression, and osmotic pressure. It provides insight into the number of particles a compound creates in solution. For example, a salt like NaCl splits into two particles (Na\textsuperscript{+} and Cl\textsuperscript{-}), having a theoretical van't Hoff factor of 2. However, in reality, the factor may vary due to ion interactions in the solution.

Knowing the van't Hoff factor aids in the accurate prediction of solution properties based on the quantity of solute present, thus simplifying tasks such as determining the respective substances' concentrations in various scientific and industrial processes.

Molality Calculation

Molality is a measure of concentration that indicates the moles of solute per kilogram of solvent (\( mol/kg \)). This unit is especially useful because, unlike molarity, molality is not temperature-dependent, as mass doesn't change with temperature.

To calculate molality, you divide the moles of the solute by the mass (in kilograms) of the solvent. In our case, you can deduce the number of particles in solution using molality and correlate it with the van't Hoff factor to further understand the behavior of the solute in the solution. This is particularly important in tasks like adjusting the concentrations in medicine and research.

Coordination Compounds

Coordination compounds are a class of compounds where transition metal ions are surrounded by an array of molecules or anions, termed ligands. These complexes exhibit intriguing chemistry that is pivotal in areas ranging from catalysis to biological processes.

Understanding coordination compounds involves recognizing how the metal ions bind with ligands to form specific geometries and how these influence the physical and chemical properties of the complex. For example, the color changes associated with different metal-ligand complexes are exploited in both analytical applications and creating pigments in art. Additionally, the knowledge of these compounds' formulae can allow scientists to determine their behavior, including their solubility and the role they play in reactions.