Problem 64 - Chemistry The Central Science In Si Units

Question

The vapor pressure of pure water at \(70^{\circ} \mathrm{C}\) is \(31.2 \mathrm{kPa}\). The vapor pressure of water over a solution at \(70^{\circ} \mathrm{C}\) containing equal numbers of moles of water and glycerol \(\left(\mathrm{C}_{3} \mathrm{H}_{5}(\mathrm{OH})_{3}\right.\), a nonvolatile solute) is \(13.3 \mathrm{kPa}\). Is the solution ideal according to Raoult's law?

Step-by-Step Solution

Answer

No, the solution is not ideal according to Raoult's law.

Step 1: Understand Raoult's Law

Raoult's law states that the vapor pressure of a solvent in a solution is equal to the mole fraction of the solvent times the vapor pressure of the pure solvent. Mathematically, it can be expressed as \( P_{ ext{solution}} = X_{ ext{solvent}} \cdot P_{ ext{pure solvent}} \), where \( P_{ ext{solution}} \) is the vapor pressure of the solvent in the solution, \( X_{ ext{solvent}} \) is the mole fraction of the solvent, and \( P_{ ext{pure solvent}} \) is the vapor pressure of the pure solvent.

Step 2: Calculate the Mole Fraction of Water

Since the problem states that there are equal numbers of moles of water and glycerol, the mole fraction of water \( X_{ ext{water}} \) in the solution is given by \( X_{ ext{water}} = \frac{n_{ ext{water}}}{n_{ ext{water}} + n_{ ext{glycerol}}} = \frac{1}{1+1} = \frac{1}{2} \).

Step 3: Use Raoult's Law to Calculate Expected Vapor Pressure

According to Raoult's law, the expected vapor pressure of water in the solution should be \( P_{ ext{solution}} = X_{ ext{water}} \cdot P_{ ext{pure water}} = \frac{1}{2} \cdot 31.2 \text{kPa} = 15.6 \text{kPa} \).

Step 4: Compare Expected and Given Vapor Pressures

We expect the vapor pressure over the solution to be 15.6 kPa according to Raoult's law, but the given vapor pressure is 13.3 kPa. Since these values differ, the solution does not behave ideally according to Raoult's law.

Key Concepts

Vapor Pressure

Vapor pressure is a term used to describe the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. It's important in chemistry because it determines how volatile a liquid is. The higher the vapor pressure, the more likely the liquid is to evaporate.

• In pure substances, vapor pressure is unique at a specific temperature.
• For mixtures, vapor pressure will depend on the components and their interactions.

When a nonvolatile solute, such as glycerol, is added to water, fewer water molecules are present at the surface to escape into the vapor phase. This results in a lower vapor pressure compared to pure water. It's worth noting that vapor pressure decreases because the solute particles block the escape of solvent molecules into the gas phase.
This change in vapor pressure is crucial when examining solutions and their interactions.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is defined as the ratio of the moles of one component to the total moles of all components in the mixture. Mathematically, it can be written as:\[X_{\text{component}} = \frac{n_{\text{component}}}{n_{\text{total}}}\]Where:

• \(X_{\text{component}}\) is the mole fraction of the component.
• \(n_{\text{component}}\) is the number of moles of the component.
• \(n_{\text{total}}\) is the total number of moles in the solution.

In the original exercise, the mole fraction of water is calculated using the ratio of moles of water to the total moles in the solution. Since there are equal moles of water and glycerol, the mole fraction of water is \(0.5\). This simple ratio gives us insight into the proportion of the solvent in the solution.

Ideal Solution

An ideal solution is a mixture where the intermolecular forces between the different particles are similar to those in the pure components. In such solutions, Raoult’s Law perfectly predicts the behavior of the vapor pressure. This means that the mixture's vapor pressure can be calculated accurately using the mole fraction of the components.Raoult’s Law is stated as:\[ P_{\text{solution}} = X_{\text{solvent}} \cdot P_{\text{pure solvent}} \]For solutions that don't strictly follow Raoult’s Law, like the one in the exercise, the assumption of an ideal solution does not hold. This deviation usually occurs due to:

• Strong interactions between solute and solvent molecules that alter the expected behavior.
• Differences in molecular size and polarity.

In the given exercise, the observed lower vapor pressure compared to the expected value indicates the solution isn't ideal. This highlights the importance of considering molecular interactions and properties when predicting solution behavior.