Problem 113 - Chemistry: The Central Science in SI Units, Global Edition

Question

At \(35^{\circ} \mathrm{C}\) the vapor pressure of acetone, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{CO},\) is 47.9 \(\mathrm{kPa}\), and that of carbon disulfide, \(\mathrm{CS}_{2}\), is \(66.7 \mathrm{kPa}\). A solution composed of an equal number of moles of acetone and carbon disulfide has a vapor pressure of \(86.7 \mathrm{kPa}\) at \(35^{\circ} \mathrm{C} .(\mathbf{a})\) What would be the vapor pressure of the solution if it exhibited ideal behavior? (b) Based on the behavior of the solution, predict whether the mixing of acetone and carbon disulfide is an exothermic \(\left(\Delta H_{\text {soln }}<0\right)\) or endothermic \(\left(\Delta H_{\text {soln }}>0\right)\) process.

Step-by-Step Solution

Answer

The vapor pressure of the solution if it exhibited ideal behavior would be \(P_{total}^{ideal} = 57.3\ kPa\). Since the actual vapor pressure of the solution is greater than the ideal vapor pressure (\(86.7\ kPa > 57.3\ kPa\)), the mixing of acetone and carbon disulfide results in a decrease in the attractive intermolecular forces between the particles compared to their pure states, releasing energy and leading to an exothermic mixing process. Therefore, the mixing of acetone and carbon disulfide is an exothermic process with \(\Delta H_{soln} < 0\).

Step 1: Understand Raoult's Law and its application

Raoult's Law states that for an ideal solution, the partial vapor pressure of each component is proportional to its mole fraction times its vapor pressure when pure. Mathematically, it can be represented as: \(P_i = x_i P_i^*\) where \(P_i\) is the partial vapor pressure of component i, \(x_i\) is the mole fraction of component i in the solution, and \(P_i^*\) is the vapor pressure of the pure component i. The total vapor pressure of an ideal solution is given by the sum of the partial vapor pressures of its components: \(P_{total} = \sum_{i=1}^n P_i\)

Step 2: Calculate the mole fractions for the ideal solution

Given that the solution is composed of an equal number of moles of acetone and carbon disulfide, their mole fractions in the ideal solution are: \(x_{acetone} = x_{CS_2} = 0.5\)

Step 3: Calculate the partial vapor pressures of acetone and carbon disulfide in the ideal solution

Using Raoult's Law, we can calculate the partial vapor pressures of acetone and carbon disulfide in the ideal solution as follows: \(P_{acetone} = x_{acetone} P_{acetone}^* = 0.5 × 47.9\ kPa = 23.95\ kPa\) \(P_{CS_2} = x_{CS_2} P_{CS_2}^* = 0.5 × 66.7\ kPa = 33.35\ kPa\)

Step 4: Calculate the vapor pressure of the ideal solution

The total vapor pressure of the ideal solution can be calculated as the sum of the partial vapor pressures of acetone and carbon disulfide: \(P_{total}^{ideal} = P_{acetone} + P_{CS_2} = 23.95\ kPa + 33.35\ kPa = 57.3\ kPa\)

Step 5: Part (a): Vapor pressure of the ideal solution

The vapor pressure of the solution if it exhibited ideal behavior would be: \(P_{total}^{ideal} = 57.3\ kPa\)

Step 6: Part (b): Determining exothermic or endothermic behavior based on the vapor pressure

To determine whether the mixing process is exothermic or endothermic, we compare the actual vapor pressure of the solution with the vapor pressure of the ideal solution. \(P_{actual} = 86.7\ kPa\) Since \(P_{actual} > P_{total}^{ideal}\), it implies that the actual solution has a greater tendency to vaporize than the ideal solution. This suggests that the mixing of acetone and carbon disulfide results in a decrease in the attractive intermolecular forces between the particles compared to their pure states, which releases energy and results in an exothermic mixing process. Therefore, the mixing of acetone and carbon disulfide is an exothermic process, \(\Delta H_{soln} < 0\).

Key Concepts

Vapor Pressure and Raoult's Law

Vapor pressure is a measure of a liquid's tendency to evaporate into a gas. It's the pressure exerted by the vapor in equilibrium with its liquid phase at a given temperature. In simpler terms, it's how much a liquid wants to "escape" into the air as a gas.

Raoult's Law helps us understand vapor pressure in mixtures. It states that the vapor pressure of each component in an ideal solution is directly proportional to its molar fraction and the vapor pressure of the pure component. Mathematically, it's expressed as: \[ P_i = x_i \times P_i^* \] where \( P_i \) is the partial vapor pressure, \( x_i \) is the mole fraction, and \( P_i^* \) is the vapor pressure of the pure component.

In an ideal solution, the total vapor pressure is simply the sum of all partial vapor pressures:\[ P_{\text{total}} = \sum (x_i \times P_i^*) \]

This concept helps us compare the calculated ideal vapor pressure with the actual observed pressure to infer interactions in the solution, crucial for determining exothermic or endothermic processes.

Understanding Ideal Solutions

An ideal solution is a hypothetical mix where interactions between dissimilar molecules (those of different substances, like acetone and carbon disulfide) are the same as interactions between similar molecules (those of the same substance).

This means the solution behaves predictably according to Raoult's Law.

• The components obey Raoult's Law across the entire concentration range.
• There are no volume changes upon mixing and no heat effects.

Ideal solutions are rare in real life but serve as a valuable reference. They help us understand how the actual solution deviates and gives insights into molecular interactions.

Exothermic Processes in Mixing

When mixing two substances, the process can either absorb or release energy. An exothermic process releases energy, resulting from the formation of new intermolecular forces that are stronger than those in the pure components.

In our case of mixing acetone and carbon disulfide, the actual vapor pressure of 86.7 kPa is higher than what Raoult's Law predicts for an ideal solution (57.3 kPa). This indicates that the mixture has a higher tendency to vaporize, suggesting weaker intermolecular forces between different molecules compared to those in the pure components.

• This release of energy makes the process exothermic, with a negative enthalpy change \( (\Delta H_{\text{soln}} < 0) \).
• Exothermic mixing implies that the new interactions formed are less stable energetically but release energy due to the disruption of the original stronger interactions.

Understanding this concept is essential in predicting energy changes during chemical solution processes.