Here is an explanation on how we do the iteration to calculate the pressure, the bubble point
pressure, and the mole fractions in the vapor phase that is in equilibrium with the mole
fraction that we know in the liquid phase and we know the starting system temperature.
This is of course, for the non-ideal case where we have to do iterations to solve.
So the first thing we do is guess the answer, we guess the pressure, we guess the mole fractions
in the vapor phase.
This allows us then, for example, to use the equations of state and we can calculate the
fugacity coefficients in the vapor phase ad the fugacity coefficients in the liquid phase.
So we could use, for example, use an equation of state to do this.
We are basically calculating fugacities, fugacity coefficient divided by pressure.
And then we are going to make use of the vapor-liquid equilibrium criteria.
Fugacity in the vapor phase of each component is the fugacity in the liquid phase of each
component.
And we can write the equation in terms of the mole fraction, the fugacity coefficient,
and the pressure in the vapor phase.
This is equal to the mole fraction in the liquid phase, the fugacity coefficient of
the liquid phase and the pressure which is negligible.
So yi over xi is equal to the fugacity coefficient of the liquid phase divided by the fugacity
coefficient of the vapor phase where we also use the symbol K to represent this.
So once we have calculated the fugacity coefficients, we can make use of equilibrium to say the
new yi value is Ki times xi which is just the modified Raoult's Law.
This is given, this we get because Ki is the ratio of fugacity coefficients, so the yi
we calculate is going to be different from the yi we guess unless some amazing miracle
happens.
So we could use the new yi values, go back to the equation of state, and therefore get
new fugacity coefficients.
However, before we do that, what we would do here is normalize.
So our new values are not going to add to 1, these are not the correct values.
So we are going to take, so I put a prime here to indicate the new values and we are
just going to normalize and force them to add to 1.
Once we have done that, we could use this new yi value and we could calculate the fugacity
coefficients in the vapor phase.
Plug in, we have corrected it, but they are not going to add to 1 so let's say sum of
the mole fractions is now less than 1.
What we want to do is therefore decrease the pressure.
That was our guess and just redo the calculation at the new pressure.
So we go through the iteration again and the objective is to get the sum of the yi's to
add to 1.
When this happens, then we have the correct pressure and we have the correct mole fractions
and of course in the spreadsheet, we could check, is the fugacity for the component in
the liquid phase is equal to the fugacity of the component in the vapor phase.
So we go through this iteration, and in the spreadsheet, we are adjusting the pressure.
So each time we choose a new pressure, it means we have new fugacity coefficient values
and therefore, we are going to get new yi values.