Question

A solution is made by mixing \(50.0 \mathrm{g}\) acetone \(\left(\mathrm{CH}_{3} \mathrm{COCH}_{3}\right)\) and 50.0 g methanol (CH_OH). What is the vapor pressure of this solution at \(25^{\circ} \mathrm{C} ?\) What is the composition of the vapor expressed as a mole fraction? Assume ideal solution and gas behavior. (At \(25^{\circ} \mathrm{C}\) the vapor pressures of pure acetone and pure methanol are 271 and 143 torr, respectively.) The actual vapor pressure of this solution is 161 torr. Explain any discrepancies.

Short answer

The vapor pressure of the solution at \(25^{\circ} \mathrm{C}\) is approximately \(181.9 \, \mathrm{torr}\). The composition of the vapor is expressed as a mole fraction of approximately \(0.5330\) for acetone and \(0.4670\) for methanol. The discrepancy between the calculated vapor pressure and the actual vapor pressure (161 torr) suggests that the ideal behavior assumption may not be accurate for this solution, as real interactions between molecules can lead to deviations from theoretical predictions.

Step-by-step solution

Calculate the moles of acetone and methanol in the solution

First, we need to calculate the moles of acetone and methanol in the solution. We can do this by using the molar masses of each compound. Molar mass of acetone (CH3COCH3) = \(12 + (3\times1) + 12 + 16 + 12 + (3\times1)\) = 58 g/mol Molar mass of methanol (CH3OH) = \(12 + (3\times1) + 16 + 1\) = 32 g/mol Next, we can calculate the moles of each compound in the solution. Moles of acetone = (50.0 g) / (58 g/mol) = 0.8621 mol Moles of methanol = (50.0 g) / (32 g/mol) = 1.5625 mol

Calculate the mole fractions of acetone and methanol

Now, we can calculate the mole fractions of each compound in the solution. Mole fraction is the number of moles of a particular component divided by the total number of moles in the mixture. Mole fraction of acetone (X_A) = \( \frac{0.8621 \mathrm{mol}}{(0.8621 \mathrm{mol} + 1.5625 \mathrm{mol})} =\) 0.3556 Mole fraction of methanol (X_B) = \( \frac{1.5625 \mathrm{mol}}{(0.8621 \mathrm{mol} + 1.5625 \mathrm{mol})} =\) 0.6444

Calculate the vapor pressure of the solution using Raoult's law

Using Raoult's law, we can calculate the vapor pressure of the solution by adding the product of the mole fraction and vapor pressure of each compound. The given vapor pressures of pure acetone and methanol are 271 torr and 143 torr, respectively. Vapor pressure of the solution (P_solution) = (X_A × P_A) + (X_B × P_B) = (0.3556 × 271 torr) + (0.6444 × 143 torr) P_solution ≈ 181.9 torr

Calculate the mole fractions in the vapor phase

To find the composition of the vapor, we'll calculate the mole fractions of acetone and methanol in the vapor phase. We can use the expression: Mole fraction in vapor = (mole fraction in solution × vapor pressure of pure component) / vapor pressure of solution Mole fraction of acetone in vapor (Y_A) = (0.3556 × 271 torr) / 181.9 torr ≈ 0.5330 Mole fraction of methanol in vapor (Y_B) = (0.6444 × 143 torr) / 181.9 torr ≈ 0.4670

Compare the calculated vapor pressure to the actual value

Finally, we can compare the calculated vapor pressure of the solution (181.9 torr) to the actual vapor pressure (161 torr). Since the values don't match, it suggests that the assumption of ideal behavior may not be accurate for this solution. In real situations, interactions between molecules and deviations from ideal behavior can lead to differences between theoretical and actual vapor pressures.

Key concepts

Raoult's Law

Raoult's Law is a principle that provides a way to calculate the vapor pressure of a solution. It states that for a solution with multiple components, the vapor pressure exerted by a particular solvent is equal to the vapor pressure of the pure solvent multiplied by its mole fraction in the solution.

Raoult's Law is represented by the formula: \( P = X_{solvent} \cdot P^{\circ}_{solvent} \) where \(P\) is the partial vapor pressure of the solvent in solution, \( X_{solvent} \) is the mole fraction of the solvent, and \( P^{\circ}_{solvent} \) is the vapor pressure of the pure solvent. For solutions containing more than one volatile component, the total vapor pressure is the sum of the partial pressures of each component:

\( P_{total} = \sum (X_i \cdot P^{\circ}_i) \) where \(X_i\) is the mole fraction and \(P^{\circ}_i\) is the vapor pressure of the \(i\)-th component.

Mole Fraction

The mole fraction represents the proportion of a component in a mixture and is defined as the number of moles of that component divided by the total number of moles in the mixture. It is a dimensionless quantity and is useful because it doesn't change with temperature or pressure, unlike concentration terms that depend on volume.

The mole fraction (\(X_i\)) of component \(i\) in a solution is given by the formula: \( X_i = \frac{n_i}{n_{total}} \) where \(n_i\) is the number of moles of the \(i\)-th component and \(n_{total}\) is the total number of moles in the solution. Mole fractions are a central part of calculating vapor pressures using Raoult's Law and are also essential for determining the composition of the vapor phase.

Ideal Solution Behavior

An ideal solution is one where the intermolecular forces between unlike molecules are equal to those between like molecules. This means the molecules mix evenly without any change in enthalpy or volume. In an ideal solution, Raoult’s Law can be applied perfectly due to the assumption that there are no deviations caused by molecular interactions.

In reality, few solutions exhibit truly ideal behavior. Deviations occur due to differences in molecule size, polarity, or shape, leading to changes in the solution properties. For the solution we analyzed, the discrepancy between the calculated and actual vapor pressure suggests a deviation from ideal behavior, indicating that intermolecular forces in the mixture were not the same as in the pure substances.

Colligative Properties

Colligative properties are those that depend on the number of solute particles, rather than their identity. Vapor pressure lowering is one such property, along with boiling point elevation, freezing point depression, and osmotic pressure. The addition of a non-volatile solute to a solvent will lower the solvent's vapor pressure in a predictable manner.

For solutions that do not show ideal behavior, colligative properties calculations require correction factors. These are derived from measuring the extent of the deviation and accounting for the intermolecular forces. Colligative properties provide insight into the solution's behavior and can help to explain why the observed vapor pressure of a solution may differ from the value predicted by Raoult's Law for an ideal solution.