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 \( \mathrm{CCl}_4-\mathrm{I}_2 \) solution is approximately 444 mm Hg.

Step 1: Calculate the moles of Iodine

To find the moles of iodine, use the formula for moles: \( n = \frac{m}{M} \), where \( m \) is the mass in grams, and \( M \) is the molar mass. The molar mass of iodine \( \mathrm{I}_2 \) is approximately \( 254 \text{ g/mol} \). Hence, the moles of iodine are \( n = \frac{105}{254} \approx 0.413 \text{ moles} \).

Step 2: Calculate the moles of \( \mathrm{CCl}_4 \)

The moles of \( \mathrm{CCl}_4 \) can be found similarly. The molar mass of \( \mathrm{CCl}_4 \) is approximately \( 153.82 \text{ g/mol} \). Therefore, the moles of \( \mathrm{CCl}_4 \) are \( n = \frac{325}{153.82} \approx 2.113 \text{ moles} \).

Step 3: Calculate the mole fraction of \( \mathrm{CCl}_4 \)

The mole fraction of \( \mathrm{CCl}_4 \) \( (X_{\mathrm{CCl}_4}) \) is calculated using the formula: \( X_{\mathrm{CCl}_4} = \frac{\text{moles of } \mathrm{CCl}_4}{\text{total moles}} \). The total moles are approximately \( 0.413 + 2.113 = 2.526 \text{ moles} \). Thus, \( X_{\mathrm{CCl}_4} = \frac{2.113}{2.526} \approx 0.836 \).

Step 4: Apply Raoult's Law

Raoult's Law states that the vapor pressure of a solution is \( P_{\text{solution}} = X_{\mathrm{CCl}_4} \times P^\circ_{\mathrm{CCl}_4} \), where \( P^\circ_{\mathrm{CCl}_4} \) is the vapor pressure of pure \( \mathrm{CCl}_4 \) which is given as \( 531 \text{ mm Hg} \). Thus, the vapor pressure of the solution is \( P_{\text{solution}} = 0.836 \times 531 \approx 444 \text{ mm Hg} \).

Key Concepts

Mole Fraction

Understanding the mole fraction is vital in solution chemistry as it tells us how many moles of a component exist compared to the total number of moles in the solution. To calculate the mole fraction of a component, you use the formula:

• For component A, the mole fraction is determined by dividing the moles of A by the total moles of all substances in the solution.
• It's expressed as a number less than one, where the sum of the mole fractions of all components equals one.

This concept is crucial when using Raoult's Law, as the vapor pressure of a solution is directly linked to the mole fraction of the solvent. In the original exercise, the mole fraction of carbon tetrachloride (\( X_{ ext{CCl}_4} = \frac{2.113}{2.526} \approx 0.836\)) gives us insight into the predominant component in the solution.

Vapor Pressure

Vapor pressure refers to the pressure exerted by the vapor of a liquid in equilibrium with its liquid phase at a given temperature. When discussing solutions, it's important to know that different substances have their unique vapor pressures, which depend on their molecular characteristics and temperature. Raoult's Law uses mole fraction to predict the lowering of a solvent's vapor pressure when a non-volatile solute is added. In the given problem, iodine acts as the non-volatile solute, meaning it doesn't significantly contribute to the solution's vapor pressure. Hence, the vapor pressure of the solution can be calculated using Raoult’s Law, emphasizing the solvent's vapor pressure reduction of pure \( CCl_4 \) from \(531 ext{ mm Hg}\) to \(444 ext{ mm Hg}\). This reduction is attributed to the presence of iodine, as depicted by the mole fraction applied in the solution.

Solution Chemistry

Solution chemistry explores how different substances interact when they dissolve. In particular, it looks into how solutes and solvents affect properties such as boiling and melting points, density, and vapor pressure. In the context of the exercise, iodine and \( CCl_4 \) form a solution resulting in a lower vapor pressure. Iodine dissolves in carbon tetrachloride, demonstrating how solute-solvent interactions can change the physical properties of a solution. Although iodine does not contribute to vapor pressure since it is non-volatile, its presence still impacts the solution by altering the solvent’s ability to escape as vapor, which can be helpful in various practical applications.

Molar Mass Calculation

Molar mass is a fundamental concept in chemistry that signifies the mass of one mole of a substance, usually expressed in g/mol. To solve the problem of vapor pressure in the exercise, accurate molar mass determinations of iodine \( (254 ext{ g/mol})\) and \( CCl_4 \, (153.82 ext{ g/mol})\) are used. This information is integral to calculating the moles of each substance using the formula \( n = \frac{m}{M} \). For instance, determining both the moles of iodine and \( CCl_4 \) separately is crucial for subsequent steps, like finding mole fractions and applying Raoult’s Law. Such calculations ensure precise measurements and applications in laboratory conditions and theoretical explorations.