Problem 28 - Chemistry and Chemical Reactivity

Question

Pure iodine \((105 \mathrm{g})\) is dissolved in \(325 \mathrm{g}\) of \(\mathrm{CCl}_{4}\) at \(65^{\circ} \mathrm{C}\) Given that the vapor pressure of \(\mathrm{CCl}_{4}\) at this temperature is \(531 \mathrm{mm}\) Hg, what is the vapor pressure of the \(\mathrm{CCl}_{4}-\mathrm{I}_{2}\) solution at \(65^{\circ} \mathrm{C} ?\) (Assume that \(\mathrm{I}_{2}\) does not contribute to the vapor pressure.)

Step-by-Step Solution

Answer

The vapor pressure of the solution is approximately 445 mm Hg.

Step 1: Determine the Moles of Iodine and CCl4

Firstly, calculate the number of moles of iodine (\( \mathrm{I}_2 \)) and carbon tetrachloride (\( \mathrm{CCl}_4 \)) present. The molar mass of iodine (\( \mathrm{I}_2 \)) is approximately 253.8 g/mol, and the molar mass of carbon tetrachloride is 153.8 g/mol. For iodine:\[ n_{\mathrm{I}_2} = \frac{105 \, \text{g}}{253.8 \, \text{g/mol}} \approx 0.414 \, \text{mol} \]For carbon tetrachloride:\[ n_{\mathrm{CCl}_4} = \frac{325 \, \text{g}}{153.8 \, \text{g/mol}} \approx 2.113 \, \text{mol} \]

Step 2: Calculate the Mole Fraction of CCl4

The mole fraction of a component in a solution is the ratio of the moles of that component to the total number of moles of all components. The mole fraction of \( \mathrm{CCl}_4 \) is calculated as follows:\[ x_{\mathrm{CCl}_4} = \frac{n_{\mathrm{CCl}_4}}{n_{\mathrm{CCl}_4} + n_{\mathrm{I}_2}} = \frac{2.113}{2.113 + 0.414} \approx 0.836 \]

Step 3: Apply Raoult's Law to Find Solution's Vapor Pressure

Raoult's Law states that the vapor pressure of a solution is equal to the vapor pressure of the pure solvent multiplied by its mole fraction in the solution. In this case:\[ P_{\text{solution}} = x_{\mathrm{CCl}_4} \times P^{\circ}_{\mathrm{CCl}_4} \]Given that the vapor pressure of pure \( \mathrm{CCl}_4 \) is 531 mm Hg, the vapor pressure of the solution is:\[ P_{\text{solution}} = 0.836 \times 531 \, \text{mm Hg} \approx 444.876 \, \text{mm Hg} \]

Key Concepts

Vapor Pressure Lowering

Vapor pressure lowering is a key concept in understanding how solutions behave differently from pure substances. When a non-volatile solute like iodine is added to a solvent such as carbon tetrachloride ( CCl_4), the vapor pressure of the resulting solution is reduced. This is because fewer solvent molecules are present at the surface to evaporate, due to the presence of the solute molecules. According to Raoult's Law, vapor pressure lowering can be quantitatively described by lowering the vapor pressure of the solvent based on the mole fraction of the solvent in the solution. This effect is why the vapor pressure of the solution of CCl_4 and iodine is less than that of pure CCl_4, as calculated in the exercise.

Solution Composition

The composition of a solution refers to the proportions of the solute and solvent, which together affect various properties of the solution, including vapor pressure. In this example, iodine serves as the solute and carbon tetrachloride as the solvent. Solution composition is expressed in terms of the amount of each component, typically in grams or moles. A precise knowledge of solution composition is necessary for calculations such as determining vapor pressure or performing stoichiometric calculations in reactions involving solutions. Understanding the balance between solute and solvent is crucial for predicting the solution's behavior.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a solution. It is the ratio of the moles of one component to the total moles of all components in the solution. In our problem, the mole fraction is used to determine how much of the vapor pressure of the pure solvent contributes to the total vapor pressure of the solution. To find the mole fraction of carbon tetrachloride, CCl_4, the calculation uses the moles of CCl_4 divided by the total moles of both CCl_4 and iodine. The mole fraction ( x_{CCl_4}) becomes a key factor in calculating the vapor pressure using Raoult's Law. This calculated mole fraction reflects how much of the solvent is present relative to the total solution, directly affecting the extent of vapor pressure lowering.

Carbon Tetrachloride Solvent

Carbon tetrachloride ( CCl_4) is the solvent in this exercise, often used in organic chemistry due to its non-polar properties and ability to dissolve a wide variety of organic compounds. In the context of solutions and Raoult's Law, the main interest in CCl_4 is its original vapor pressure and how it changes when a solute is added. Carbon tetrachloride's vapor pressure at a specific temperature, in this case 531 mm Hg at 65°C, is a crucial parameter used in the calculations. By understanding CCl_4's behavior as a solvent, students can better comprehend how solutions form and affect vapors over a range of conditions.