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 in the sodium chloride solution at 25°C is 0.353. The vapor pressure of the solution at 45°C is 59.25 torr.

Step-by-step solution

Calculate mole fraction at 25°C

First, let's denote the mole fraction of water in the solution as \(x_{H_2O}\) and the mole fraction of sodium chloride ions (\(\text{Na}^+\) and \(\text{Cl}^-\)) as \(x_{NaCl}\). According to Raoult's law, the vapor pressure of the solution (\(P_{solution}\)) is given by: \[P_{solution} = x_{H_2O} \cdot P^{0}_{H_2O}\] Here, \(P^{0}_{H_2O}\) is the vapor pressure of pure water (23.8 torr at 25°C). We are given \(P_{solution}\) as 19.6 torr at 25°C, and we can use the above equation to calculate the mole fraction of water in the solution: \[x_{H_2O} = \frac{P_{solution}}{P^{0}_{H_2O}} = \frac{19.6}{23.8}\] Now, calculate the value of \(x_{H_2O}\): \(x_{H_2O} = 0.8235\) Since the mole fractions in the solution must sum up to 1, the mole fraction of sodium chloride ions in the solution is given by: \[x_{NaCl} = 1 - x_{H_2O} = 1 - 0.8235\] Calculate the value of \(x_{NaCl}\): \(x_{NaCl} = 0.1765\) Since sodium chloride exists as \(\text{Na}^{+}\) and \(\text{Cl}^{-}\) ions in the solution, the mole fraction of solute particles in the solution is twice the value of \(x_{NaCl}\): \(x_{solute} = 2 \cdot x_{NaCl} = 2 \cdot 0.1765\) Now, calculate the value of \(x_{solute}\): \(x_{solute} = 0.353\) The mole fraction of solute particles in the solution is 0.353.

Calculate vapor pressure at 45°C

Now, let's find the vapor pressure of the solution at 45°C. We are given the vapor pressure of pure water at 45°C, which is 71.9 torr. Using Raoult's law again, we can find the vapor pressure of the solution at 45°C: \[P_{solution_{45}} = x_{H_2O} \cdot P^{0}_{H_2O_{45}}\] Here, \(P^{0}_{H_2O_{45}}\) is the vapor pressure of pure water (71.9 torr at 45°C). Now, plug in the values of \(x_{H_2O}\) and \(P^{0}_{H_2O_{45}}\) in the above equation: \[P_{solution_{45}} = 0.8235 \cdot 71.9\] Calculate the value of \(P_{solution_{45}}\): \(P_{solution_{45}} = 59.25\) The vapor pressure of the sodium chloride solution at 45°C is 59.25 torr.

Key concepts

Vapor Pressure

Vapor pressure is an important concept in chemistry because it helps us understand how substances transition between phases. The vapor pressure of a liquid is the pressure exerted by its vapor when it is in equilibrium with its liquid phase at a given temperature. In simpler terms, it's the tendency of particles to escape from the liquid surface, entering the gas phase.
When we have a solution, such as sodium chloride in water, the vapor pressure is often lower than that of the pure solvent (pure water, in this case). This happens due to the presence of solute particles, which occupy space at the liquid's surface, reducing the number of solvent molecules that can escape into the vapor phase.
Raoult's Law describes this phenomenon quantitatively: \[ P_{solution} = x_{solvent} \cdot P_{solvent}^0 \] Here, \( P_{solution} \) is the vapor pressure of the solution, \( x_{solvent} \) is the mole fraction of the solvent, and \( P_{solvent}^0 \) is the vapor pressure of the pure solvent. Understanding vapor pressure variations is crucial for applications in distillation, colligative properties, and other phase transition studies.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It indicates the ratio of the number of moles of one component to the total moles of all components in the mixture. For a two-component system, such as a sodium chloride and water solution, it is defined for both solute and solvent.
The mole fraction of the solvent (water) is denoted as \( x_{H_2O} \), and the solute (sodium chloride) is \( x_{NaCl} \). Mole fractions sum up to 1 within the mixture:

• \( x_{H_2O} + x_{NaCl} = 1 \)

In the context of the sodium chloride solution, the presence of \( \text{Na}^+ \) and \( \text{Cl}^- \) ions means that the mole fraction of solute particles is twice that of \( x_{NaCl} \).
This is an essential consideration when utilizing Raoult's Law or calculating vapor pressure, as accurate mole fractions provide insights into the solution's behavior and interactions.

Sodium Chloride Solution

A sodium chloride solution is formed when table salt (\( \text{NaCl} \)) is dissolved in water. In the solution, \( \text{NaCl} \) dissociates into its constituent ions, \( \text{Na}^+ \) and \( \text{Cl}^- \).
This dissociation is significant because it determines how the solution behaves in terms of colligative properties like boiling point elevation, freezing point depression, and, relevant to our exercise, vapor pressure lowering.
Since individual water molecules can escape to the vapor phase, the presence of \( \text{Na}^+ \) and \( \text{Cl}^- \) ions in sodium chloride solution interferes with this process. Ionic dissociation results in a lower vapor pressure compared to pure water, as predicted by Raoult's Law.
The concept of vapor pressure reduction is pivotal in understanding why sodium chloride solutions behave differently from pure water. Through this, we recognize the impact of solute particles in practical applications, from cooking to chemical manufacturing. By understanding this basic chemistry, students can predict how other substances might alter a solvent's physical properties.