Question

Two liquids A and B have vapor pressures of 76 and \(132 \mathrm{mmHg},\) respectively, at \(25^{\circ} \mathrm{C}\). What is the total vapor pressure of the ideal solution made up of (a) \(1.00 \mathrm{~mol}\) of \(\mathrm{A}\) and \(1.00 \mathrm{~mol}\) of \(\mathrm{B}\) and (b) \(2.00 \mathrm{~mol}\) of \(\mathrm{A}\) and \(5.00 \mathrm{~mol}\) of \(\mathrm{B}\) ?

Short answer

(a) 104 mmHg, (b) 116 mmHg.

Step-by-step solution

Calculate Mole Fraction of Each Component for Part (a)

For an ideal solution of 1.00 mol of A and 1.00 mol of B:Calculate the mole fraction of A, \( X_A \), using the formula \[ X_A = \frac{n_A}{n_A + n_B} \] where \( n_A = 1.00 \text{ mol} \) and \( n_B = 1.00 \text{ mol} \).\[ X_A = \frac{1.00}{1.00 + 1.00} = 0.5 \]Similarly, calculate the mole fraction of B, \( X_B \):\[ X_B = \frac{1.00}{1.00 + 1.00} = 0.5 \]

Calculate Total Vapor Pressure for Part (a)

Use Raoult's Law to calculate the total vapor pressure:\[ P_{total} = P_A + P_B = X_A \cdot P^o_A + X_B \cdot P^o_B \]where \( P^o_A = 76 \text{ mmHg} \) and \( P^o_B = 132 \text{ mmHg} \).Substitute the given values:\[ P_{total} = (0.5 \cdot 76) + (0.5 \cdot 132) \]\[ P_{total} = 38 + 66 = 104 \text{ mmHg} \]

Calculate Mole Fraction of Each Component for Part (b)

For an ideal solution of 2.00 mol of A and 5.00 mol of B:Calculate the mole fraction of A, \( X_A \):\[ X_A = \frac{2.00}{2.00 + 5.00} = \frac{2}{7} \approx 0.286 \]Similarly, calculate the mole fraction of B, \( X_B \):\[ X_B = \frac{5.00}{2.00 + 5.00} = \frac{5}{7} \approx 0.714 \]

Calculate Total Vapor Pressure for Part (b)

Using Raoult's Law again:\[ P_{total} = X_A \cdot P^o_A + X_B \cdot P^o_B \]Substitute the calculated mole fractions and vapor pressures:\[ P_{total} = \left(\frac{2}{7} \cdot 76\right) + \left(\frac{5}{7} \cdot 132\right) \]\[ P_{total} = 21.71 + 94.29 = 116 \text{ mmHg} \]

Key concepts

Vapor Pressure

Vapor pressure is a critical concept when studying solutions and their behavior. It refers to the pressure exerted by the vapor when a liquid is in equilibrium with its vapor. This means that the rate of evaporation of the liquid is equal to the rate of condensation of the vapor.
Vapor pressure is influenced by temperature—higher temperatures usually increase it. Each substance has its own characteristic vapor pressure, and this property is crucial when discussing mixtures of liquids, such as in the exercise provided.
When two liquids like A and B are mixed to create an ideal solution, their individual vapor pressures combine, often leading to a different total vapor pressure for the solution. This total vapor pressure can be calculated using Raoult's Law, as seen in the original solution provided.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It represents the ratio of moles of one particular substance to the total number of moles in the solution.
To calculate the mole fraction, use the formula:

• For component A: \( X_A = \frac{n_A}{n_A + n_B} \)
• For component B: \( X_B = \frac{n_B}{n_A + n_B} \)

Here, \( n_A \) and \( n_B \) stand for the number of moles of components A and B, respectively.
The mole fractions help you understand the proportion of each liquid in the solution, which is essential for calculating properties like the total vapor pressure using Raoult's Law. In an ideal solution, the sum of the mole fractions always equals 1, which makes these calculations straightforward.

Ideal Solution

An ideal solution is a type of solution where the interactions between molecules of the different components are the same as those between molecules of the same component. This means that when dissolved, the solute and solvent interact in such a way that the solution behaves predictably.
In an ideal solution, each component contributes to the overall properties of the mixture according to its mole fraction. Raoult's Law, which states that the vapor pressure of a component in a solution is equal to its mole fraction multiplied by its pure component vapor pressure, is based on the concept of ideal solutions.
The calculations in the original exercise make use of the assumption that the system behaves as an ideal solution. This allows us to use Raoult’s Law to find the total vapor pressure by summing the contributions from each substance based on their respective mole fractions and vapor pressures. Understanding these properties and behaviors help in predicting how different compositions will behave in mixtures.