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

01

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: \(P_{total} = \sum_{i=1}^n P_i\)

02

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: \(x_{acetone} = x_{CS_2} = 0.5\)

03

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: \(P_{acetone} = x_{acetone} P_{acetone}^* = 0.5 × 47.9\ kPa = 23.95\ kPa\) \(P_{CS_2} = x_{CS_2} P_{CS_2}^* = 0.5 × 66.7\ kPa = 33.35\ kPa\)

04

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: \(P_{total}^{ideal} = P_{acetone} + P_{CS_2} = 23.95\ kPa + 33.35\ kPa = 57.3\ kPa\)

05

Part (a): Vapor pressure of the ideal solution

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

06

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, \(\Delta H_{soln} < 0\).

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