Question

If the freezing point of a \(0.01\) molal aqueous solution of a cobalt(III) chloride-ammonia complex (which behaves as a strong electrolyte) is \(-0.0558^{\circ} \mathrm{C}\), the number of chloride(s) in the coordination sphere of the complex is \(\left[\mathrm{K}_{\mathrm{f}}\right.\) of water \(\left.=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right]\)

Short answer

The number of chloride ions in the coordination sphere is found to be the integer part of the calculated van't Hoff factor minus 1.

Step-by-step solution

Understand the Concept of Freezing Point Depression

The freezing point depression (\(\text{Δ}T_f\)) is a colligative property which is dependent on the molality (m) of the solution and the van't Hoff factor (i), which indicates the number of particles the solute dissociates into in solution. The change in freezing point is given by the formula \(\text{Δ}T_f = i \times K_f \times m\), where \(K_f\) is the cryoscopic constant of the solvent.

Identify Given Values and the Unknown

We have been given the molality \(m=0.01\) mol/kg, the freezing point depression \(\text{Δ}T_f = 0.0558^\circ C\), and the cryoscopic constant of water \(K_f = 1.86\) K kg/mol. We need to find the van't Hoff factor (i), which will then allow us to determine the number of chloride ions in the coordination sphere.

Rearrange the Freezing Point Depression Equation to Solve for i

Use the equation \(\text{Δ}T_f = i \times K_f \times m\) and rearrange it to solve for the van't Hoff factor (i): \(i = \frac{\text{Δ}T_f}{K_f \times m}\).

Plug in the Known Values and Calculate i

Now substitute the known values into the rearranged equation: \(i = \frac{0.0558}{1.86 \times 0.01}\).

Calculate the van't Hoff Factor

Perform the calculation to find the value of i. This will give you the total number of particles the complex dissociates into when dissolved.

Determine the Number of Chlorides in the Complex

Knowing that the complex behaves as a strong electrolyte and dissociates completely, the number of chloride ions will be the whole number part of the van't Hoff factor minus 1 for the cobalt(III) ion itself.

Key concepts

Colligative Properties

Colligative properties are characteristics of solutions that depend solely on the number of dissolved particles in the solution, rather than on the identity of those particles. These properties include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure.

For instance, when a non-volatile solute is dissolved in a solvent, the solution's freezing point decreases; this phenomenon is called freezing point depression. It is important for students to understand that this effect is a result of the solute particles disrupting the orderly structure of the solvent molecules, which makes it more difficult for the solvent to form a solid. Therefore, the presence of solute particles requires a lower temperature to achieve the structured arrangement necessary for the solid phase.

In practical scenarios, colligative properties are used for purposes like anti-freezing in engines or making homemade ice cream. In the context of the exercise, understanding how the freezing point depression works is crucial to solving the problem related to the freezing point of the cobalt(III) chloride-ammonia complex solution.

Van't Hoff Factor

The van't Hoff factor, denoted as 'i', is a dimensionless quantity that is integral to the study of colligative properties. It represents the number of particles a solute yields when it dissolves in a solvent. For non-electrolytes, which do not dissociate into ions, the van't Hoff factor is typically 1, indicating that one particle of solute remains as one particle in solution.

In contrast, strong electrolytes dissociate completely into ions, so for example, table salt (NaCl) dissociates into two particles, Na+ and Cl-, yielding a van't Hoff factor of 2. However, the factor can be more complex for coordination compounds or solutes that associate or dissociate in solution.

In our exercise, determining the correct van't Hoff factor is key to identifying the number of chloride ions in the coordination sphere of the cobalt(III) chloride-ammonia complex. A common misconception is that strong electrolytes always have a van't Hoff factor of 2; this is not always true, as in this complex, where the actual value of 'i' will indicate the total number of particles in solution, including both chloride ions and the cobalt-ammonia complex.

Cryoscopic Constant

The cryoscopic constant, symbolized by 'K_f', is a property specific to each solvent and represents the change in freezing point per molal concentration of solute particles. It is measured in degrees Celsius per molal or kelvins per molal. The cryoscopic constant is a fundamental parameter when calculating the freezing point depression of a solution.

For water, the cryoscopic constant is relatively high, 1.86 K kg/mol, which implies that a small amount of solute can significantly decrease the freezing point. The knowledge of the cryoscopic constant enables us to predict how a solute will affect the freezing point of the solvent.

In the exercise, the cryoscopic constant is used as part of the equation to calculate the van't Hoff factor, which in turn helps us determine the number of chloride ions in the coordination sphere. As students tackle such problems, they must pay careful attention to the units of each quantity and ensure consistency throughout their calculations.