Question

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.

Short answer

The complex dissociates into 3 ions: [Co(NH3)4Cl]^{2+} and 2 Cl^{-}.

Step-by-step solution

Calculate Molality of Solution

First, we need the freezing point depression formula: \[ \Delta T_f = i \cdot K_f \cdot m \]where \( \Delta T_f \) is the change in freezing point, \( i \) is the van't Hoff factor, \( K_f \) is the freezing point depression constant of water \((1.86\, \text{°C/molal})\), and \( m \) is the molality.Given \( \Delta T_f = 0.56\, \text{°C} \), calculate molality:\[ m = \frac{\Delta T_f}{K_f} = \frac{0.56}{1.86} = 0.301 \text{ mol/kg} \]

Calculate Moles of Solute

Use the molality to calculate the moles of solute. Molality is defined as:\[ m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \]Using \( m = 0.301 \text{ mol/kg} \) and mass of water \( = 25.0 \text{ g} = 0.0250 \text{ kg} \):\[ \text{moles of solute} = m \times \text{kilograms of solvent} = 0.301 \times 0.0250 = 0.007525 \text{ moles} \]

Calculate Molar Mass of Co(NH3)4Cl3

Find the molar mass of \( \text{Co(NH}_3\text{)}_4 \text{Cl}_3 \):- Co: \( 58.93\, \text{g/mol} \)- N: \( 14.01\, \text{g/mol} \), H: \( 1.01\, \text{g/mol} \) (each \( \text{NH}_3\) contributes 3 hydrogens)- Cl: \( 35.45\, \text{g/mol} \)Calculate total:\[ \text{Molar mass} = 58.93 + 4(14.01 + 3.03) + 3 \times 35.45 = 262.87\, \text{g/mol} \]

Determine Van't Hoff Factor

The van't Hoff factor \( i \) is calculated using:\[ i = \frac{\text{Experimental moles of ions}}{\text{Theoretical moles of solute}} = \frac{\text{0.007525} \; \text{(ions from molality)}}{\frac{0.875}{262.87}} \]Find theoretical moles of solute:\[ \frac{0.875}{262.87} = 0.003328 \text{ moles} \]Calculate \( i \):\[ i = \frac{0.007525}{0.003328} = 2.26 \]

Interpret Van't Hoff Factor

The van't Hoff factor (\( i = 2.26 \)) suggests the complex dissociates to produce 2 ions. However, expected values for full dissociation are integers; thus, suggest partial dissociation into a major complex ion \([\text{Co(NH}_3\text{)}_4\text{Cl}]^{2+}\) and two chloride ions \(\text{Cl}^-\).

Key concepts

Van't Hoff Factor

The van't Hoff factor, symbolized as \( i \), is a crucial concept in understanding how solutes affect colligative properties like freezing point depression. It represents the number of particles a compound dissociates into when dissolved. In simple terms, if a compound fully dissociates into its constituent ions in solution, the van't Hoff factor will equal the number of resulting particles.
However, sometimes the van't Hoff factor is not an integer. This is often the case with complex ions or incomplete dissociation. In the example of the \(\text{Co(NH}_3\text{)}_4\text{Cl}_3\), the factor \( i = 2.26 \) indicates that the compound partially dissociates, producing around 2.26 particles on average per formula unit.

• If full dissociation were to occur, we might expect three particles: one complex ion and two chloride ions.
• However, given the non-integer factor, some of the compound remains as non-dissociated molecules.

Understanding the van't Hoff factor helps predict and explain the extent of dissociation of solutes, which is vital for calculating changes in freezing point, boiling point, and osmotic pressure.

Molality

Molality (\( m \)) is a common concentration unit in chemistry, particularly useful in solving colligative property problems. It is defined as the number of moles of solute divided by the mass of the solvent in kilograms. This unit is handy because, unlike molarity, it does not change with temperature.
In the given exercise, we found that the molality of the solution is\( 0.301 \, \text{mol/kg} \). This value was crucial for determining the extent of the freezing point depression, helping us calculate the van't Hoff factor.

• Molality does not depend on the volume of the solution, which can change with temperature.
• Using molality simplifies colligative property calculations, ensuring accurate measurements by relying on mass rather than volume, which can be distorted by thermal expansion.

To determine molality, always remember to convert the mass of the solvent to kilograms and use the defined formula for precise calculations.

Complex Ion

A complex ion forms when a central metal atom or ion binds with one or more molecules or ions, typically ligands. These ligands donate electron pairs to the metal, resulting in a coordination compound.
The compound \(\text{Co(NH}_3\text{)}_4\text{Cl}_3\) features cobalt as the central metal ion bound to four ammonia ligands, creating a complex. The different ions or molecules attached to the metal center can significantly affect the chemical properties, such as solubility and reactivity.

• A typical complex ion has a positive or negative charge, depending on the constituent ions' charges.
• In our exercise, the suggested structure of the complex ion is \([\text{Co(NH}_3\text{)}_4\text{Cl}]^{2+}\).
• The coordinate bond formation involves electron donation from \(\text{NH}_3\)to \(\text{Co}^{3+}\).

Complex ions are often part of transition metal chemistry and affect how compounds like \(\text{Co(NH}_3\text{)}_4\text{Cl}_3\)dissolve in solvents.

Molecular Dissociation

Molecular dissociation refers to the process by which a compound separates into its constituent ions or molecules when dissolved in a solvent. This phenomenon is critical in understanding the behavior of electrolytes, which break apart in solutions to conduct electricity.
In the problem of \(\text{Co(NH}_3\text{)}_4\text{Cl}_3\), molecular dissociation might not be complete, given the fractional van't Hoff factor of \(2.26\).

• Dissociation illustrates how many particles a solute contributes to the solution.
• Incomplete dissociation can lead to fewer ions in solution than expected.

In this case, it appears the compound does not dissociate entirely into \([\text{Co(NH}_3\text{)}_4\text{Cl}]^{2+}\)and \(2\, \text{Cl}^-\).
A deeper understanding of dissociation helps chemists predict the physical properties of a solution and informs the design of chemical processes and applications.