Considering a binary solution of components A and B obeys Raoult’s law, which of the following is true?

A plot of vapour pressures of both components gives a linear plot

The correct answer is option 3. A plot of vapour pressures of both components gives a linear plot.

When a binary solution of components A and B obeys Raoult’s law, the relationships involving vapor pressures and mole fractions are straightforward and linear. Here's a detailed explanation:

Raoult's Law states that the partial vapor pressure of each component in an ideal solution is directly proportional to its mole fraction. For a binary solution with components A and B:

\(P_A = x_A P_A^0 \)

\(P_B = x_B P_B^0 \)

where:

\( P_A \) and \( P_B \) are the partial vapor pressures of components A and B in the solution.

\( x_A \) and \( x_B \) are the mole fractions of A and B in the solution.

\( P_A^0 \) and \( P_B^0 \) are the vapor pressures of pure A and B.

The total vapor pressure of the solution, \( P_{total} \), is the sum of the partial pressures of the components:

\(P_{total} = P_A + P_B = x_A P_A^0 + x_B P_B^0 \)

Since \( x_A \) and \( x_B \) are related by \( x_A + x_B = 1 \) (because the sum of mole fractions in a binary solution must equal 1), we can express \( x_B \) as:

\(x_B = 1 - x_A \)

Substituting \( x_B \) in the total vapor pressure equation gives:

\(P_{total} = x_A P_A^0 + (1 - x_A) P_B^0 \)

\(P_{total} = x_A P_A^0 + P_B^0 - x_A P_B^0 \)

\(P_{total} = x_A (P_A^0 - P_B^0) + P_B^0 \)

This equation shows that \( P_{total} \) varies linearly with \( x_A \). The total vapor pressure is a linear function of the mole fraction of component A.

Analysis of Each Option

1. Total vapour pressure cannot be related to the mole fraction of only one component: This is incorrect because the total vapor pressure can indeed be expressed in terms of the mole fraction of one component due to the linear relationship.

2. Total vapour pressure of one component varies non-linearly with another component: This is incorrect because the partial pressures, and hence the total vapor pressure, vary linearly with the mole fractions in an ideal solution.

3. A plot of vapour pressures of both components gives a linear plot: This is correct because, according to Raoult's law, both \( P_A \) and \( P_B \) are linear functions of their respective mole fractions \( x_A \) and \( x_B \). Therefore, the total vapor pressure, which is the sum of these linear functions, will also be a linear function of the mole fraction.

4. Total vapour pressure of solution always decreases with an increase in mole fraction of a component: This is incorrect because the total vapor pressure can either increase or decrease depending on the vapor pressures of the pure components. It depends on whether \( P_A^0 \) is greater or less than \( P_B^0 \).

Summary: For a binary solution obeying Raoult’s law, the partial vapor pressures of both components and the total vapor pressure vary linearly with the mole fractions. Therefore, a plot of the vapor pressures of both components will give a linear plot. This linear relationship is key to understanding the behavior of ideal solutions as described by Raoult’s law.