Question

In a solution with carbon tetrachloride as the solvent, the compound \(\mathrm{VCl}_{4}\) undergoes dimerization: $$ 2 \mathrm{VCl}_{4} \rightleftharpoons \mathrm{V}_{2} \mathrm{Cl}_{\mathrm{s}} $$ When \(6.6834 \mathrm{~g} \mathrm{VCl}_{4}\) is dissolved in \(100.0 \mathrm{~g}\) carbon tetrachloride, the freezing point is lowered by \(5.97^{\circ} \mathrm{C}\). Calculate the value of the equilibrium constant for the dimerization of \(\mathrm{VCl}_{4}\) at this temperature. (The density of the equilibrium mixture is \(1.696 \mathrm{~g} / \mathrm{cm}^{3}\), and \(K_{f}=29.8^{\circ} \mathrm{C} \mathrm{kg} / \mathrm{mol}\) for \(\mathrm{CCl}_{4} \cdot\) )

Short answer

The equilibrium constant for the dimerization of \(\mathrm{VCl}_{4}\) at this temperature is \(9.54\).

Step-by-step solution

Calculate the molality of the \(\mathrm{VCl}_{4}\) solution

First, we will determine the molar mass of \(\mathrm{VCl}_{4}\): \(\mathrm{VCl}_{4} = 1 \times \mathrm{V} + 4 \times \mathrm{Cl} = 1 \times 50.94 + 4 \times 35.45 = 193.74 \mathrm{g/mol}\) Now, we will calculate the moles of \(\mathrm{VCl}_{4}\): Moles of \(\mathrm{VCl}_{4}\) = \(\frac{6.6834 \mathrm{~g}}{193.74 \mathrm{g/mol}} = 0.0345 \mathrm{mol}\) Then, we will calculate the molality of the solution: Molality = \(\frac{0.0345 \mathrm{mol}}{100.0 \mathrm{~g} \times \frac{1 \mathrm{kg}}{1000\,\mathrm{g}}} = 0.345 \frac{\mathrm{mol}}{\mathrm{kg}}\)

Calculate the molality of dissociated solute (dimerized)

We will use the freezing point depression equation: \(\Delta T_{f} = i K_{f} \cdot m\) , where: - \(\Delta T_{f} = 5.97^{\circ} \mathrm{C}\) (given) - \(i\) = van't Hoff factor - \(K_{f} = 29.8^{\circ} \mathrm{C} \mathrm{kg} / \mathrm{mol}\) (given) - \(m\) = molality of dissociated solute For the dimerization reaction, the van't Hoff factor \(i = 1\) (half the number of original particles). Therefore, we have: \(5.97^{\circ} \mathrm{C} = 1 \times 29.8^{\circ} \mathrm{C} \mathrm{kg} / \mathrm{mol} \cdot m\) Solving for the molality: \(m = \frac{5.97^{\circ} \mathrm{C}}{29.8^{\circ} \mathrm{C} \mathrm{kg} / \mathrm{mol}} = 0.200 \frac{\mathrm{mol}}{\mathrm{kg}}\)

Calculate the equilibrium constant of the dimerization reaction

Let \(x\) be the molality of dissociated \(\mathrm{VCl}_{4}\) and \(y\) be the molality of the dimer \(\mathrm{V}_{2}\mathrm{Cl}_{8}\). We have: \(x = 0.345 - 0.200 = 0.145 \frac{\mathrm{mol}}{\mathrm{kg}}\) \(y = 0.200 \frac{\mathrm{mol}}{\mathrm{kg}}\) The equilibrium constant for the dimerization reaction is given by: \(K = \frac{[\mathrm{V}_{2}\mathrm{Cl}_{8}]}{[\mathrm{VCl}_{4}]^{2}} = \frac{y}{x^{2}}\) Calculating the equilibrium constant: \(K = \frac{0.200}{(0.145)^{2}} = 9.54\) Therefore, the equilibrium constant for the dimerization of \(\mathrm{VCl}_{4}\) at this temperature is \(9.54\).

Key concepts

Dimerization

Dimerization is a process where two identical molecules, known as monomers, combine to form a single molecule called a dimer. This chemical reaction is significant in various fields such as organic chemistry and biochemistry. For the compound \( \mathrm{VCl}_{4} \), dimerization can be observed when two \( \mathrm{VCl}_{4} \) molecules join to create \( \mathrm{V}_{2}\mathrm{Cl}_{8} \).

This reaction is reversible, meaning it can proceed in both directions, from monomers to dimers and vice versa. The equilibrium constant describes the ratio of the concentration of the dimer to the square of the concentration of the monomer in this state. In practical terms, understanding dimerization helps in explaining the behaviors of different substances when they interact with each other.

• Reversible reaction: Monomers can revert back from dimers.
• Impacts substance properties: Dimer formation can alter physical and chemical properties.
• Applications in industry: Dimerization is important in polymer production and pharmaceuticals representation.

Freezing Point Depression

Freezing point depression is an important colligative property of solutions. It refers to the lowering of a solvent's freezing point due to the addition of a solute. In the case of the \( \mathrm{VCl}_{4} \) dissolved in carbon tetrachloride, the freezing point of the solvent decreases by \( 5.97^{\circ} \mathrm{C}\).

The degree of freezing point depression is determined by the concentration of solute particles, not their identity. This property allows chemists to calculate molar masses and determine solute concentration in solutions.

• Depends on the quantity, not type: It's the number of particles that matter.
• Utilizes temperature difference: The change in freezing point aids calculations.
• Applied in antifreeze solutions: Helps prevent liquid freezing under low temperatures.

Molality

Molality is a concentration unit used in chemistry, defined as the number of moles of solute per kilogram of solvent. It differs from molarity as it is not affected by temperature or pressure changes, making it a more stable unit of measure in diverse conditions. In the case of \( \mathrm{VCl}_{4} \), we calculated its molality to be \( 0.345 \frac{\mathrm{mol}}{\mathrm{kg}} \).

Molality is particularly essential in colligative property calculations, like freezing point depression. This is because it provides a consistent basis for understanding how solute concentration impacts solutions, irrespective of external environmental conditions.

• Independent of temperature: Ideal for calculations involving temperature changes.
• Crucial for colligative properties: Often used in studies involving boiling and freezing points.
• Standard unit in experiments: Offers consistency in experimental results across variable environments.

Van't Hoff Factor

The van't Hoff factor \( i \) is a dimensionless number used in colligative properties to represent how many particles a solute forms in a solution. For instance, it quantifies the effect of solute particles on boiling point elevation or freezing point depression. In the context of \( \mathrm{VCl}_{4} \), which dimerizes, the van't Hoff factor is considered 1 because the dimerization reduces the number of particles to half the original count.

Understanding the van't Hoff factor is critical in precisely predicting the behavior of solutions and their colligative properties. It helps in adjusting calculations to reflect real-world scenarios more accurately when solutes dissociate or associate.

• Reflects dissociation or association: Adjusts for the number of resultant particles.
• Key in solution chemistry: Vital for calculating boiling and freezing point changes.
• Adjusts theoretical predictions: Ensures calculations match experimental realities.