What is the composition of the benzene vapour which is at equilibrium at 30 degree celcius with a benzene-toluene solution with a mole fraction of benzene of 0.400?

To determine the composition of benzene vapor at equilibrium with a benzene-toluene solution, we can apply Raoult's Law. This law states that the partial vapor pressure of each component in a solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution. Let's break this down step by step.

Understanding the Components

In this scenario, we have a solution containing benzene (C6H6) and toluene (C7H8). The mole fraction of benzene in the solution is given as 0.400, which means that the mole fraction of toluene is 0.600 (since the total mole fraction must equal 1).

Gathering Vapor Pressure Data

Next, we need the vapor pressures of pure benzene and pure toluene at 30 degrees Celsius. These values are typically found in tables:

• Vapor pressure of pure benzene at 30°C: approximately 12.3 kPa
• Vapor pressure of pure toluene at 30°C: approximately 3.8 kPa

Calculating Partial Pressures

Using Raoult's Law, we can calculate the partial pressures of benzene and toluene in the vapor phase:

• Partial pressure of benzene (P_benzene):

  P_benzene = (Mole fraction of benzene) × (Vapor pressure of pure benzene)

  P_benzene = 0.400 × 12.3 kPa = 4.92 kPa

• Partial pressure of toluene (P_toluene):

  P_toluene = (Mole fraction of toluene) × (Vapor pressure of pure toluene)

  P_toluene = 0.600 × 3.8 kPa = 2.28 kPa

Finding Total Vapor Pressure

Now, we can find the total vapor pressure (P_total) of the system:

P_total = P_benzene + P_toluene = 4.92 kPa + 2.28 kPa = 7.20 kPa

Determining Mole Fractions in the Vapor Phase

To find the mole fractions of benzene and toluene in the vapor phase, we use the partial pressures we just calculated:

• Mole fraction of benzene in vapor (y_benzene):

  y_benzene = P_benzene / P_total = 4.92 kPa / 7.20 kPa ≈ 0.683

• Mole fraction of toluene in vapor (y_toluene):

  y_toluene = P_toluene / P_total = 2.28 kPa / 7.20 kPa ≈ 0.317

Final Composition of the Vapor

At equilibrium, the composition of the benzene vapor above the solution at 30 degrees Celsius will be approximately:

• Benzene: 68.3%
• Toluene: 31.7%

This analysis shows how the mole fraction of components in a liquid solution can influence the composition of the vapor phase above it, demonstrating the principles of vapor-liquid equilibrium. If you have any further questions or need clarification on any part of this process, feel free to ask!