Coupling Raoult's & Antoine's Equations for Mole Fraction & Temp

[Scrim]

Homework Statement

Given the equation for Roaults law and Antoine equation for 2 species I need to recast these into a pair of coupled non linear equations to compute the mole fraction of species one and the temperature for given values of vapor fraction and the total pressure, Antoines constants are also given.

These paired equations are eventually used in a 2x2 Jacobian so I am assuming that will have the mole fraction in terms of vapor fraction and total pressure, as well as the Temperature in terms of vapor fraction and total pressure.

Homework Equations

Raoults Law: x(1)*P(1_sat)=y(n)*P(total)
[1-x(1)]*P(2_sat)=[1-y(1)]*P(total)
Antoine Equation: log(base10)(Psat of species i) = Ai - (Bi/(T + Ci) where Ai,Bi,Ci are constants

The Attempt at a Solution

I have no clue how to separate these

Thank you so much for your help!

 

[KataKoniK]

This is an interesting problem that requires some careful consideration. Let's start by defining our variables:

x(1) = mole fraction of species 1
y(1) = mole fraction of species 1 in the vapor phase
P(1_sat) = saturation pressure of species 1
P(2_sat) = saturation pressure of species 2
P(total) = total pressure
T = temperature

Now, let's start by rearranging the Raoults Law equations:

x(1) = y(1)*P(total)/P(1_sat)
1-x(1) = [1-y(1)]*P(total)/P(2_sat)

Next, we can substitute the Antoine equation into the saturation pressure terms:

P(1_sat) = 10^(Ai - Bi/(T+Ci))
P(2_sat) = 10^(Aj - Bj/(T+Cj))

Where Ai, Bi, Ci are the Antoine constants for species 1 and Aj, Bj, Cj are the Antoine constants for species 2.

Substituting these equations into the Raoults Law equations, we get:

x(1) = y(1)*P(total)/10^(Ai - Bi/(T+Ci))
1-x(1) = [1-y(1)]*P(total)/10^(Aj - Bj/(T+Cj))

Now, we have two equations with two unknowns (x(1) and T). We can solve these equations using a numerical solver or by using the Jacobian method mentioned in the forum post. The final equations will look something like this:

f(x(1),T) = x(1) - y(1)*P(total)/10^(Ai - Bi/(T+Ci))
g(x(1),T) = 1-x(1) - [1-y(1)]*P(total)/10^(Aj - Bj/(T+Cj))

The Jacobian matrix for these equations will be:

J(x(1),T) = [1, -P(total)*ln(10)*[Bi/(T+Ci)^2], y(1)*P(total)*ln(10)*[Bi/(T+Ci)^2]
[-1, P(total)*ln(10)*[Bj/(T+Cj)^2], -[1-y(1)]*P(total)*ln(10)*[Bj/(T+Cj)^2]]

Solving these