Thermodynamics problem regarding fugacity and fugacity coefficient.

Question

 

 

Hey, I'm at my wit's end when it comes to thermodynamics. Apparently, I was given this little assignment from my lecturer to solve it at home. I need help in solving this, really x_x

A binary liquid mixture of species 1 and 2 is in equilibrium with its vapor at 100C. The overall mole fraction of species 1 in the system is z1 = 0.58 and its liquid-phase mole fraction is x1 = 0.45 . Given :

ln γ1 = 0.71 (x2)^2
ln γ2 = 0.71 (x1)^2

At 100C ;

P1,sat = 65.24 kPa
P2,sat = 19.08 kPa


Conversion Unit ; 1 bar = 101.325 kPa
Use ideal gas constant R = 8314 (cm3 kPa mol-1 K-1)

For species 1,
ω = 0.191
Tc = 420K
Pc = 40.43bar
Zc = 0.277
Vc = 239.3 (cm3 mol-1)
Normal boiling point = 275K

i) Determine the molar fraction of vapor V of the sysmtem
ii) Determine the fugacity and fugacity coefficient of pure liquid species 1 at its normal boiling point.

 

Thank you in advance :)

Answer 1

Well, you have a lot going on here.

First, you must recognize the type of problem you are working with when it comes to equilibrium thermodynamics. Here, you have two different TYPES of problems. The first is a VLE problem, with the second as a vapor fugacity problem. Luckily, they are unrelated, so we can tackle each separately.

After you identify a problem as VLE, the next step is assumptions. What can we assume? This is going to determine if we have Raoult's Law for PhD's or Raoult's Law for Kids or something in between.

Here are the assumptions to go through. So, yes or no:

Are we at ideal gas conditions (high temperature and low pressure)?

Is this an ideal solution?

Is the system pressure ~ as the saturation pressure?

Is the liquid incompressible?

We are at IG conditions (or we can first assume and if the pressure comes out too high we can always go back. However, unless an azeotrope forms, we know the pressure has to be between the saturation pressures of the two components. I would say 373.15 K (none of that Celsius nonsense) and 0.5 atm (ish) is high temperature and low pressure.

What does that mean. Well, most of the terms simplify the problem to RLA (Raoult's Law for Adults):

yi*P=xi*gamma_i*Psat_i.

Now we check, what do we know, what do we not know, and what do we have a value for:

Know: T, xi, gamma_i, Psat_i

Don't know: P and yi

As such, you have three unknowns and two equations. Unknowns: P, y1 and y2. Equations: you have RLA for both species. However, you, in a sense have a third equation. What is it and write it down.

Yes, the summation of all mole fractions equals 1. As such y1+y2 = 1. You know have three equations and three unknowns, so it is an algebra problem not a thermo problem. From here you can get y1, y2, and P.

But the question asks for the vapor fraction. This comes from the lever rule. The formula for lever rule is:

z-x

y-x = Vf

At this point, you have z, x, and y, so you can solve.

Unfortunately, I cannot help with part two, because you have to apply an equation of state to solve for those.