Question

A mixture contains \(1 \mathrm{~mole}\) of volatile liquid \(\mathrm{A}\left(P_{\mathrm{A}}^{\circ}=100 \mathrm{~mm} \mathrm{Hg}\right)\) and \(3 \mathrm{moles}\) of volatile liquid \(\mathrm{B}\left(P_{\mathrm{B}}^{\circ}=80 \mathrm{~mm} \mathrm{Hg}\right)\). If the solution behaves ideally, the total vapour pressure of the distillate is (a) \(85 \mathrm{~mm} \mathrm{Hg}\) (b) \(85.88 \mathrm{~mm} \mathrm{Hg}\) (c) \(90 \mathrm{~mm} \mathrm{Hg}\) (d) \(92 \mathrm{~mm} \mathrm{Hg}\)

Short answer

The total vapour pressure is \(85.88~mm~Hg\).

Step-by-step solution

Calculate mole fraction of liquid A

Calculate the mole fraction of liquid A (\( X_A \) by dividing the number of moles of A by the total number of moles. Formula: \( X_A = \frac{moles~of~A}{total~moles} \).

Calculate mole fraction of liquid B

Calculate the mole fraction of liquid B (\( X_B \) by dividing the number of moles of B by the total number of moles. Formula: \( X_B = \frac{moles~of~B}{total~moles} \).

Apply Raoult's law to find partial pressure of A

Using Raoult's law, calculate the partial vapour pressure of A (\( P_A \) by multiplying the mole fraction of A (\( X_A \) by the pure vapour pressure of A (\( P_A^\circ \). Formula: \( P_A = X_A \times P_A^\circ \).

Apply Raoult's law to find partial pressure of B

Using Raoult's law, calculate the partial vapour pressure of B (\( P_B \) by multiplying the mole fraction of B (\( X_B \) by the pure vapour pressure of B (\( P_B^\circ \). Formula: \( P_B = X_B \times P_B^\circ \).

Calculate the total vapour pressure of the mixture

The total vapour pressure (\( P_{total} \) is the sum of the partial pressures of A and B. Formula: \( P_{total} = P_A + P_B \).

Key concepts

Mole Fraction Calculation

Understanding the mole fraction of a component in a mixture is a fundamental step in mastering solution chemistry. Mole fraction, denoted as \( X \), is a dimensionless quantity that represents the proportion of a substance in a mixture. It is calculated by dividing the number of moles of the substance by the total number of moles present in the mixture.

For a given mixture of two volatile liquids A and B, the mole fraction of liquid A (\( X_A \)) is determined by the formula:\[ X_A = \frac{\text{moles of A}}{\text{total moles}} \]Similarly, the mole fraction for liquid B (\( X_B \)) would be:\[ X_B = \frac{\text{moles of B}}{\text{total moles}} \]Since the two liquids are the only components in the mixture, their mole fractions must add up to 1 (\( X_A + X_B = 1 \)). This allows us to understand the behavior and composition of the mixture before further calculations.

Partial Vapour Pressure

The term 'partial vapour pressure' refers to the pressure exerted by a single component of a mixture of gases or vapors. According to Raoult's Law, the partial vapour pressure of a component in an ideal solution is directly proportional to its mole fraction. Mathematically speaking:\[ P_{\text{component}} = X_{\text{component}} \times P_{\text{component}}^\circ \]In this formula, \( P_{\text{component}} \) is the partial vapour pressure of the component, \( X_{\text{component}} \) is its mole fraction, and \( P_{\text{component}}^\circ \) signifies the pure component's vapour pressure at the same temperature.

Applying Raoult's Law

For our mixture containing liquids A and B, applying Raoult's law means we calculate the partial pressures for each, \( P_A \) and \( P_B \), using their respective mole fractions and pure vapour pressures given in the problem statement. This process aids in understanding how each component contributes to the total pressure.

Total Vapour Pressure

Total vapour pressure is a key indicator of a solution's volatility and is especially important in applications involving distillation or evaporation. In a binary mixture, Raoult's Law helps us conclude that the total vapour pressure is the sum of the partial pressures of the individual components. The equation is:\[ P_{\text{total}} = P_A + P_B \]This relationship implies a direct dependence of the total pressure on the composition of the mixture as long as the solution behaves ideally (follows Raoult's Law without deviations). For instance, in the given exercise, after calculating the individual partial pressures, you simply add them to compute the total vapour pressure above the mixture.Remember that the meticulous calculation of each partial pressure using their respective mole fractions is essential for an accurate determination of the total vapour pressure.

Ideal Solution Behavior

An ideal solution is a mixture where the intermolecular forces between unlike molecules are equal to those between like molecules. In other words, the enthalpy of mixing is zero, and the volume change upon mixing is negligible. This theoretical concept is crucial because it provides a baseline to understand real-world solutions, which often deviate from ideality to varying degrees.

Characteristics of Ideal Solutions

• They obey Raoult's Law across all concentrations.
• The chemical potential of each component in the solution is equivalent to that of the pure substance multiplied by its mole fraction.
• The total vapour pressure of the solution can be predicted accurately using the mole fraction and vapour pressures of the pure components.

These characteristics are assumed when solving Raoult's Law problems for simplicity and to provide a foundation upon which more complex behaviors can be studied as one advances in the study of chemistry.