Question

An ideal solution was obtained by mixing methanol and ethanol. If the partial vapour pressures of methanol and ethanol are \(2.8\) and \(4.2 \mathrm{kPa}\) respectively, the mole fraction of methanol in the vapour at equilibrium is (a) \(0.67\) (b) \(0.4\) (c) \(0.6\) (d) \(0.33\)

Short answer

The mole fraction of methanol in the vapour at equilibrium is 0.4.

Step-by-step solution

Understand Dalton's Law of Partial Pressures

Dalton's Law of partial pressures states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. The partial pressure of each gas is directly proportional to its mole fraction in the mixture.

Calculate the total vapour pressure

Add the partial vapour pressures of methanol and ethanol to find the total vapour pressure. Total vapour pressure, P_total = P_methanol + P_ethanol.

Calculate the mole fraction of methanol

Using Dalton's Law, the mole fraction of methanol in the vapour, X_methanol, can be found by dividing the partial pressure of methanol by the total vapour pressure: X_methanol = P_methanol / P_total.

Insert the given values and compute

Insert the given partial pressures into the formula from Step 3 to calculate the mole fraction of methanol in the vapour. Insert P_methanol = 2.8 kPa and P_ethanol = 4.2 kPa into the formula and calculate X_methanol.

Key concepts

Understanding an Ideal Solution

An ideal solution is a mixture where the forces between the molecules are similar, hence when two liquids are mixed, there's no change in enthalpy or volume. This concept is crucial in creating mixtures in pharmaceuticals, beverages, and other industries requiring precise formulations.

For an ideal solution, the components obey Raoult's law, where each component's vapour pressure is directly proportional to its mole fraction. This means if you have a 50-50 mixture, each component contributes to the total vapour pressure based on its pure vapour pressure and its proportion in the mixture.

Vapour Pressure Explained

The vapour pressure of a substance is the pressure exerted by its vapour when it is in equilibrium with its liquid or solid form. It's important for understanding how substances will behave when they are heated or mixed with others.

For instance, when you're cooking and want to boil water, the vapour pressure of the water increases with temperature until it matches atmospheric pressure, leading to boiling. When dealing with ideal solutions, knowing the vapour pressures helps predict how the mixture will behave under different conditions, which is crucial for designing processes in chemical engineering.

Mole Fraction and Its Significance

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the number of moles of a component divided by the total number of moles of all components in the mixture.

Mole fractions are used in calculating partial pressures, which can then be used to refine processes in laboratories and industries. In the context of Dalton's Law, mole fraction directly relates to how the pressure of each gas contributes to the total pressure of a gas mixture. Understanding the mole fraction aids in determining the composition of phases in equilibrium and is invaluable in fields like material science and meteorology.

Physical Chemistry Competitive Examinations

Students preparing for physical chemistry competitive examinations need to have a deep understanding of theoretical concepts and practical applications. These exams often include problems related to ideal solutions, vapour pressures, and mole fractions, which require a clear comprehension of the underlying principles.

Topics such as Dalton's Law of Partial Pressures are commonly tested, as they have real-world applications in industries related to gas production, chemical manufacturing, and even environmental science. A strong grasp of these concepts can make a significant difference in a student's performance on these competitive examinations.