Problem 71 - A Complete Resource Book in Chemistry for JEE Main

Question

A solution has a 1: 4 mole ratio of pentane to hexane. The vapour pressures of the pure hydrocarbons at \(20^{\circ} \mathrm{C}\) are \(400 \mathrm{~mm} \mathrm{Hg}\) for pentane and \(120 \mathrm{~mm} \mathrm{Hg}\). for hexane. The mole fraction of pentane in the vapour phase would be (a) \(0.200\) (b) \(0.549\) (c) \(0.786\) (d) \(0.478\)

Step-by-Step Solution

Answer

The mole fraction of pentane in the vapour phase is approximately 0.455, so none of the options are a perfect match.

Step 1: Determine Moles in the Solution

The given mole ratio is 1:4 for pentane and hexane, meaning there is 1 mole of pentane for every 4 moles of hexane. Thus, if the total number of moles is 5, then there is 1 mole of pentane and 4 moles of hexane.

Step 2: Calculate Mole Fractions

The mole fraction of pentane \(X_{C_5H_{12}} = \frac{1}{1+4} = \frac{1}{5} = 0.2\)and the mole fraction of hexane \(X_{C_6H_{14}} = \frac{4}{1+4} = \frac{4}{5} = 0.8\).

Step 3: Calculate Partial Pressures Using Raoult's Law

Raoult's Law states: the partial pressure of each component in the mixture is equal to the vapor pressure of the pure component multiplied by its mole fraction.For pentane, \(P_{C_5H_{12}} = P_{pure} \times X_{C_5H_{12}} = 400 \, \text{mm Hg} \times 0.2 = 80 \, \text{mm Hg}\).For hexane, \(P_{C_6H_{14}} = P_{pure} \times X_{C_6H_{14}} = 120 \, \text{mm Hg} \times 0.8 = 96 \, \text{mm Hg}\).

Step 4: Calculate Total Vapor Pressure

The total vapor pressure of the solution is the sum of the partial pressures:\(P_{total} = P_{C_5H_{12}} + P_{C_6H_{14}} = 80 \, \text{mm Hg} + 96 \, \text{mm Hg} = 176 \, \text{mm Hg}\).

Step 5: Determine Mole Fraction of Pentane in Vapour Phase Using Dalton's Law

Using Dalton's Law of partial pressures, the mole fraction of pentane in the vapor phase can be determined by:\(Y_{C_5H_{12}} = \frac{P_{C_5H_{12}}}{P_{total}} = \frac{80}{176} \approx 0.455\). Note: This value approximates, with a small deviation from the exact answer.

Key Concepts

Mole Fraction

The mole fraction is a crucial concept in chemistry, especially when dealing with solutions. It represents the ratio of the number of moles of a given component to the total number of moles in a mixture. The mole fraction is a unitless number, and it helps to describe composition without involving masses or volumes.

In our context of pentane and hexane, the mole fraction gives us an idea of how much each component contributes to the entire mixture:

• For pentane, which has a 1:4 mole ratio with hexane, the mole fraction is calculated as: \( X_{\text{pentane}} = \frac{\text{moles of pentane}}{\text{total moles}} = \frac{1}{1+4} = 0.2 \).
• Meanwhile, hexane's mole fraction is \( X_{\text{hexane}} = \frac{4}{5} = 0.8 \).

This calculation shows that in the mixture, hexane is in a higher proportion compared to pentane, which influences how the solution behaves. Understanding mole fractions is essential for predicting other physical and chemical properties, such as vapor pressure.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid form. Each component in a mixture contributes to the total vapor pressure based on its inherent vapor pressure and its concentration in the mixture.

Raoult's Law describes how the vapor pressures of components in a solution are calculated:

• The partial vapor pressure for any component (like pentane or hexane) is the vapor pressure of the pure component multiplied by its mole fraction in the mixture.
• For pentane: \( P_{\text{pentane}} = P_{\text{pentane, pure}} \times X_{\text{pentane}} = 400 \, \text{mm Hg} \times 0.2 = 80 \, \text{mm Hg} \).
• For hexane: \( P_{\text{hexane}} = 120 \, \text{mm Hg} \times 0.8 = 96 \, \text{mm Hg} \).

These calculations reveal the individual contributions of pentane and hexane to the total vapor pressure of the solution, which is the sum of their partial pressures. The total vapor pressure becomes \(176 \, \text{mm Hg}\) as these individual pressures combine in the mixed vapor.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is critical for understanding how gas mixtures behave. This law states that the total pressure of a gaseous mixture is the sum of the partial pressures of each individual gas component. This principle allows us to predict how the components of our solution will vaporize.

In the exercise, after determining the individual vapor pressures using Raoult's Law, Dalton's Law helps us find the mole fraction of pentane in the vapor phase:

• The mole fraction in the vapor phase is calculated by dividing the partial pressure of pentane by the total vapor pressure: \( Y_{\text{pentane}} = \frac{P_{\text{pentane}}}{P_\text{total}} = \frac{80}{176} \approx 0.455 \).

This result indicates the proportion of pentane in the vapor compared to hexane. Dalton’s Law gives a clear picture of how gases are distributed above mixtures, which can differ from their distribution in the liquid state due to differences in volatility.