In this screencast we're going to discuss how to calculate vapor-liquid equilibrium
for a multi-component system by using an equation of state, in this case the equation of state
will account for both the non-ideal vapor phase and the non-ideal liquid phase. The
particular example we'll look at is the bubble point pressure calculation, namely if we're
given the mole fractions in the liquid phase, xi and we're given the temperature of the
system we want to determine the mole fractions in the vapor phase yi and the pressure of
the system, and we'll do this with an iterative approach. So in this screencast we'll explain
the procedure and the actual calculations we'll use the spread sheet that has an equation
of state built in to it to demonstrate with numerical values in a separate screencast.
So to calculate this equilibrium what we're doing is setting the fugacity of each component
in the liquid phase equal to the fugacity of each component in the vapor phase and then
we can write this fugacity in the liquid phase as a mole fraction in the liquid phase the
fugacity coefficient in the mixture and that symbol indicates "in the mixture" for the
liquid times the pressure and then the fugacity in the vapor phase looks very similar but
it's the mole fraction now in the vapor phase the fugacity coefficient in the vapor phase
time the pressure, so the pressure drops out and we're going to use this equation where
we'll write yi over xi equals Ki the idea now we can calculate Ki from the fugacity
coefficients and Ki is the fugacity coefficient in the liquid phase over the fugacity coefficient
in the vapor phase, and this is for each component. So our approach, in order to calculate the
fugacity coefficients we need the pressure which we don't know, since the pressure effects
the fugacity coefficients and then to calculate the vapor phase values we also need the mole
fractions in the vapor phase which is also something we don't know and we're trying to
solve for. So what we're going to do is we're going to guess a pressure and guess each of
the mole fractions, we can certainly make intelligent guesses for the mole fractions
for example if we have information so we could use Raoult's law as a first approximation
and we could certainly do that but we're going to make that guess, we're then going to calculate
these fugacity coefficients so both the vapor phase and the liquid phase values, the next
step is to calculate each of the Ki values, once we have Ki we can calculate a new value
for the mole fraction in the vapor phase. And so what we're going to do is say yi, new
values for yi, Ki xi this is known, this we calculated from the fugacity coefficients
this now should be better than our original guess value, so a couple approaches we could
use, if the sum of yi equals one that means we actually guessed the completely correct
values that's of course not very likely and so most likely either less that one or greater
than one so let's say the sum is less than one, that means we want to make a new guess
of a lower value for the pressure, if it was greater than one we'd want to make a new guess
of a higher value for the pressure of course when we change the pressure we're going to
get new values for the fugacity coefficients, of course that means new values for the Ki
and therefore we'll get new values for yi, we can get better values for the fugacity
coefficients by normalizing the yi, namely take our yi values divide them by the sum
in order to normalize so that the sum is equal to one, use those yi values to calculate our
fugacity coefficients in the vapor phase, and now we have an iterative procedure, we
get new fugacity coefficients, if they don't sum to one when we calculate the new yi values
then we go back and adjust pressures and we'll continue that loop until the sum of the yi
equals one and then we would have determined the pressure and the yi values in vapor-liquid
equilibrium, of course with the spreadsheet this iterative approach is easier to do and
probably easier to visualize, so in a subsequent screencast we'll go through the actual numbers
using this procedure. Of course instead of bubble pressure, if we wanted bubble temperature
its essentially the same calculation but we're going to guess temperature because the pressure
is given and we're going to guess again the mole fractions in the vapor phase.