Question

A solution of sodium chloride in water has a vapor pressure of 19.6 torr at \(25^{\circ} \mathrm{C} .\) What is the mole fraction of solute particles in this solution? What would be the vapor pressure of this solution at \(45^{\circ} \mathrm{C} ?\) The vapor pressure of pure water is 23.8 torr at \(25^{\circ} \mathrm{C}\) and 71.9 torr at \(45^{\circ} \mathrm{C},\) and assume sodium chloride exists as \(\mathrm{Na}^{+}\) and \(\mathrm{Cl}^{-}\) ions in solution.

Short answer

The mole fraction of solute particles (ions) in the sodium chloride solution is 0.353, and the vapor pressure of the solution at 45°C is 46.5 torr.

Step-by-step solution

Calculate the mole fraction of solute particles using Raoult's law

According to Raoult's law, we have: \(P_{solution} = X_{solvent} P^0_{solvent}\) Where \(P_{solution}\) is the vapor pressure of the solution, \(X_{solvent}\) is the mole fraction of the solvent, and \(P^0_{solvent}\) is the vapor pressure of the pure solvent. In our case, the solvent is water and the solute is sodium chloride (NaCl). For the given solution at 25°C, we have: - Vapor pressure of the solution, \(P_{solution} = 19.6 \,\text{torr}\) - Vapor pressure of pure water at 25°C, \(P^0_{water} = 23.8 \,\text{torr}\) We need to find the mole fraction of solute particles, so first, let's find the mole fraction of the solvent (water). We can rearrange the equation to find \(X_{solvent}\): \(X_{solvent} = \frac{P_{solution}}{P^0_{water}}\)

Solve for the mole fraction of solute particles

Now, we can solve for the mole fraction of solvent (water), \(X_{solvent}\): \(X_{solvent} = \frac{19.6}{23.8} = 0.8235\) Since the sum of mole fractions in a solution is always equal to 1, the mole fraction of solute particles, \(X_{solute}\), can be calculated as follows: \(X_{solute} = 1 - X_{solvent} = 1 - 0.8235 = 0.1765\) However, we should consider that sodium chloride dissociates completely into Na+ and Cl- ions in the solution. Therefore, the actual mole fraction of solute particles (ions) will be twice as much: \(X_{ions} = 2 \times X_{solute} = 2 \times 0.1765 = 0.353\)

Calculate the vapor pressure of the solution at 45°C

Now, we will use the same Raoult's law equation to find the vapor pressure of the solution at 45°C. We are given the vapor pressure of pure water at 45°C, so let's find the mole fraction of water at 45°C first: \[X_{water(45)} = 1 - X_{ions} = 1 - 0.353 = 0.647\] Now, we can use the Raoult's law equation to find the vapor pressure of the solution at 45°C: \(P_{solution(45)} = X_{water(45)} P^0_{water(45)}\) where \(P^0_{water(45)} = 71.9 \,\text{torr}\) is the vapor pressure of pure water at 45°C. Solve for the vapor pressure of the solution at 45°C: \(P_{solution(45)} = 0.647 \times 71.9 = 46.5 \,\text{torr}\)

Summary

In summary, the mole fraction of solute particles (ions) in the sodium chloride solution is 0.353, and the vapor pressure of the solution at 45°C is 46.5 torr.

Key concepts

Vapor Pressure

Vapor pressure is a critical concept in understanding the behavior of liquids and solutions. It refers to the pressure exerted by the gas phase of a substance when it is in equilibrium with its liquid phase at a given temperature. The higher the temperature, the higher the vapor pressure, as more molecules have the kinetic energy to escape from the liquid to the gas phase.

When a solute is dissolved in a solvent, the vapor pressure of the resulting solution generally decreases. This is because the solute molecules occupy space at the surface of the liquid, making it harder for solvent molecules to escape into the gas phase. Raoult's law helps us quantify this effect, stating that the vapor pressure of an ideal solution is directly proportional to the mole fraction of the solvent in the solution. This relationship is foundational in solution chemistry and is pivotal for exercises like understanding how the vapor pressure of water changes with the addition of sodium chloride.

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 the number of moles of a specific component to the total number of moles of all components in the mixture. The beautiful simplicity of the mole fraction is that it is a dimensionless quantity and adds up to 1 for all components in a solution.

To compute the mole fraction, one needs the number of moles of each substance involved. In solution chemistry, the mole fraction is particularly useful because it correlates directly to important properties like vapor pressure. According to Raoult's law, the vapor pressure of a solvent in a solution is equal to the vapor pressure of the pure solvent multiplied by its mole fraction. This means that as more solute is added, the mole fraction of the solvent decreases, as does the vapor pressure.

Dissociation of Ionic Compounds

When ionic compounds like sodium chloride are dissolved in water, they undergo dissociation. This means that the ionic compound separates into its constituent ions—for sodium chloride, this would be Na+ and Cl-. The process of dissociation effectively increases the number of particles in the solution, which is a crucial consideration when applying Raoult's law in solution chemistry.

In such a dissociation, for each formula unit of the ionic compound that dissolves, the concentration of solute particles doubles. Thus, it becomes necessary to adjust the mole fraction of the solute to account for this increase. When solving problems involving ionic compounds, as in the example provided, one must remember to multiply the mole fraction of the solute by the number of ions it produces upon dissociation. This adjustment is imperative for accurate calculations of vapor pressures and other colligative properties.

Solution Chemistry

Solution chemistry is a branch of chemistry focused on the study of solutions—homogeneous mixtures composed of two or more substances. In a typical solution, the substance present in the greatest amount is called the solvent, and the substance or substances present in smaller amounts are called solutes.

The interactions between solvent and solute can affect various properties of the solution, such as boiling point, freezing point, and vapor pressure. Understanding these interactions facilitates predictions about how a solution will behave under various conditions. Raoult's law is one of the fundamental principles governing solution chemistry, particularly in predicting how solutions will behave when it comes to vapor pressure. It applies to ideal solutions where the solute-solvent interaction is similar in strength to the solvent-solvent and solute-solute interactions. Non-ideal solutions may require additional considerations, but the basic concepts of Raoult's law give significant insight into the solution's behavior.