Raoult's Law Problem: Estimate Mole Fractions at 65K

[Samuel1321]

Homework Statement

Consider air to be made up of only N2 (78 mole %), O2 (21%) and Ar (1%)
Estimate the mole fractions of each in both the liquid and gas phases if 1 mole of air at 300 K and 4 atm. total pressure were cooled to 65 K in the same volume

Homework Equations

Raoult's law Pi = Pi* xi

The Attempt at a Solution

So I know that the atmospheric pressure is 760 torr. So before they are cooled

vapor pressure of N2 would be (0.78)(760) torr

Vapor Pressure of O2 would be (0.21)(760) torr

Vapor Pressure of Ar would be (0.01)(760) Torr

I'm not sure how should I proceed. Maybe I should use the Clausius Clapeyron equation to find the final pressure after cooling it down? But what delta H value should I use?

 

[mjc123]

Look at the question again; what is the initial total pressure?
You need to know the boiling points and enthalpies of vaporisation of the gases, but these should be easy to look up.

 

[Chestermiller]

What are the Antoine equations for these three substances?

 

[Samuel1321]

Look at the question again; what is the initial total pressure?
You need to know the boiling points and enthalpies of vaporisation of the gases, but these should be easy to look up.

The initial total pressure is 4 atm.

Should I just find the final total pressure first using the clasius clapeyron equation? The problem is if I use the CC equation, which enthalpy of vaporization should I use?

 

[Samuel1321]

What are the Antoine equations for these three substances?

I have not learned that equation yet, is there another way to solve this problem?

 

[Chestermiller]

I have not learned that equation yet, is there another way to solve this problem?

The Antoine equation is an 3-parameter very accurate fit to the integrated Clausius-Clapeyron equation. The parameters are available for common substances like oxygen, nitrogen, and argon, so you don't have to integrate the CC equation to describe the VLE behavior of the pure species. Google it, and get the Antoine equation parameters for your 3 species. This way, you don't need to know the boiling point, and the heat capacities of the vapor and liquid as functions of temperature, and you don't need to integrate the CC equation.

Of course, to solve your problem, the VLE behavior of the pure species is only the first step. Given the overall mole fractions of the 3 species, you need to determine the split of liquid and vapor, and the mole fractions of the 3 species in the liquid and in the vapor.

 

[Chestermiller]

@Samuel1321 Do you have any idea how to go about setting up the equations for this problem? It is pretty challenging. There will be two main unknowns: the fractional split of liquid and vapor and the final pressure. I can lead you through how to set up the two key equations, if you are willing to work with me. Any interest?