Finding Mole Fractions in a Two-Phase Equilibrium System

[FranciscoSili]

Hello everyone, I'm doing revision for a final exam in Thermodynamics and i found this exercise i can't solve:

A particular material has a latent heat of vaporization Δh, constant along the coexistence curve. One mole of this material exists in two.phase (liquid-vapor) equilibrium in a container of fixed volume V0, at an initial temperature T0 and a pressure P0. The system is heated at constant volume increasing its pressure to 2P0. The vapor phase can be treated as an monoatomic ideal gas, and the molar volume of the liquid can be neglected relativa to that of the gas. Find the initial and final mole fractions of the vapor phase. ( x≡Ng/(Ng+Nl) ).

I started using the Clapeyron equation to find the final temperature (Tf) of the gas (and so the complete system i think). I integrated from T0 to Tf in one side and from P0 to Pf in the other. The only unknown variable was Tf. Then using the ideal gas equation ( vg(molar volume of gas)=NRTf ) and dividing that by N = Ng + Nl I could leave the final mole fraction of the gas as a function of Pf, Tf and final vg, which i don't know.

So, how can i find the value of the final molar volume? This problem is in a book called Herbert Callen - "Thermodynamics and an Introduction to Thermostatistics" 2nd Edition. page 233 problem 9·3-7

Thank you and sorry i don't use formulas, i don't know how to write them here.

 

[John Park]

You're told the volume of the container, that its value is fixed, and that the volume of liquid is negligible. . . .

 

[Chestermiller]

You know the initial temperature, pressure, and volume. So how many moles of vapor do you have? You know the final temperature, pressure, and volume. So how many moles of vapor do you have?

 

[FranciscoSili]

You're told the volume of the container, that its value is fixed, and that the volume of liquid is negligible. . . .

Yes i know, but after the process both liquid and gas mole fractions changed. I need to know, that variation, specifically in the gas one.

 

[Chestermiller]

Yes i know, but after the process both liquid and gas mole fractions changed. I need to know, that variation, specifically in the gas one.

The vapor occupies virtually the entire volume. From the ideal gas law, how many moles of vapor do you have in the final state? You know the temperature, the pressure, and the volume.

 

[FranciscoSili]

The vapor occupies virtually the entire volume. From the ideal gas law, how many moles of vapor do you have in the final state? You know the temperature, the pressure, and the volume.

Oh, yes. Oh my god, it was very simple :) thank you very much! :D