Question

At \(60^{\circ} \mathrm{C}\) the vapor pressure of pure water is \(149.44 \mathrm{mmHg}\) and that above an aqueous sucrose, \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{12},\) solution is \(119.55 \mathrm{mmHg} .\) Calculate the mole fraction of water and the mass in grams of sucrose in the solution if the mass of water is \(150 . \mathrm{g}\).

Short answer

The mole fraction of water is 0.80, and the mass of sucrose is approximately 713 g.

Step-by-step solution

Identify Given Information

The vapor pressure of pure water at 60°C is given as 149.44 mmHg. The vapor pressure of the solution is 119.55 mmHg. The mass of water is 150 g. We need to find the mole fraction of water and the mass of sucrose.

Use Raoult's Law to find Mole Fraction

Raoult's Law states that the vapor pressure of the solvent over a solution is the product of the mole fraction of the solvent in the solution and the vapor pressure of the pure solvent:\[ P_{ ext{solution}} = x_{ ext{water}} imes P_{ ext{pure water}} \]where \( P_{ ext{solution}} = 119.55 \text{ mmHg} \) and \( P_{ ext{pure water}} = 149.44 \text{ mmHg} \). Solve for the mole fraction of water \( x_{ ext{water}} \):\[ x_{ ext{water}} = \frac{P_{ ext{solution}}}{P_{ ext{pure water}}} = \frac{119.55}{149.44} \approx 0.80 \].

Calculate Moles of Water

Calculate the moles of water using its mass (150 g) and molar mass (18.015 g/mol):\[ n_{ ext{water}} = \frac{150 \text{ g}}{18.015 \text{ g/mol}} \approx 8.33 \text{ moles} \].

Find the Mole Fraction of Sucrose

The total mole fraction must equal 1; therefore, the mole fraction of sucrose is:\[ x_{ ext{sucrose}} = 1 - x_{ ext{water}} = 1 - 0.80 = 0.20 \].

Calculate Moles of Sucrose

Use the mole fraction of sucrose and the moles of water to find the moles of sucrose. Let \( n_{ ext{sucrose}} \) be the moles of sucrose:\[ \frac{n_{ ext{sucrose}}}{n_{ ext{success}} + 8.33} = 0.20 \]Solve for \( n_{ ext{sucrose}} \):\[ n_{ ext{sucrose}} = \frac{0.20 \times 8.33}{0.80} = 2.08 \text{ moles} \].

Calculate the Mass of Sucrose

Find the mass of sucrose using its moles and molar mass (342.30 g/mol):\[ \text{Mass of sucrose} = 2.08 \text{ moles} \times 342.30 \text{ g/mol} \approx 712.99 \text{ g} \].

Key concepts

Understanding Vapor Pressure

Vapor pressure is an essential concept in understanding the behavior of liquids and solutions. It is the pressure exerted by the vapor of a liquid when it is in equilibrium with its liquid phase. In simpler terms, it represents how much of the liquid has evaporated to form a gas.

• When a liquid is in an open container, it evaporates until the pressure of the vapor reaches a certain point, known as the vapor pressure.
• This equilibrium is key in measuring how volatile a liquid is. A higher vapor pressure indicates more evaporation.

In Raoult's Law, this concept is vital because the vapor pressure of a solvent above a solution depends on its mole fraction. Knowing the vapor pressure helps us determine the concentration of components in a solution.

Exploring Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of moles of one component to the total moles in the mixture.

• For example, in a solution of sucrose in water, if you want to find the mole fraction of water, you divide the moles of water by the total moles of water and sucrose combined.
• Raoult's Law utilizes this concept to connect the vapor pressure of a solution to the mole fraction of its components.

Mathematically, it is represented as:\[ x = \frac{n_{\text{component}}}{n_{\text{total}}} \]where \(n_{\text{component}}\) is the moles of the individual component and \(n_{\text{total}}\) is the total moles of all components. Understanding mole fraction helps in explaining how different substances affect the properties of the solution.

The Role of Sucrose Solution

Sucrose, a common sugar, when dissolved in water forms a solution. This solution behaves differently compared to pure water.

• Adding sucrose to water decreases the water's vapor pressure. This is because sucrose molecules occupy space at the surface, hindering some water molecules from evaporating.
• Raoult's Law quantifies this change in vapor pressure based on the mole fraction of water, which is reduced when sucrose is present.

In the provided exercise, understanding the behavior of the sucrose solution helps in calculating how much sucrose has been added. By observing the decrease in vapor pressure, we determine the concentration of water and, subsequently, the amount of sucrose needed to achieve that specific change.