In this problem we're dealing with a liquid phase mixture of acetic acid and acetone that
is in equilibrium with its vapor and we're going to do a few calculations having to do
with vapor liquid equilibrium. So first it's useful to actually draw out the situation
onto a phase diagram. In this problem we have a constant temperature of 100 degrees C and
we therefore might want a plot a phase diagram at variable pressure and composition, and
we'll do this for component 1 because this problem asks about the acetic acid. This diagram
has been sketched a certain way so that you notice as the composition of acetic acid increases,
the pressure decreases. So hence the pressure for equilibrium decreases. So this indicates
that acetic acid is the less volatile component and so this, we expect to correspond to the
saturation pressure of acetic acid and conversely this is equal to the saturation pressure of
acetone which is the more volatile component. And so at the pure component, so at X equals
1 and X equals 0, we expect that the pressure of the system will be equal to the saturation
pressure at vapor-liquid equilibrium. So we draw the diagram this way, the higher pressure,
the higher pressure states or the higher pressure side of the equilibrium region is the liquid
region. The lower pressure or less compressed side is vapor and in between we have vapor-liquid
equilibrium. Furthermore, we'll stat plotting some points here and we'll just do this in
a qualitative way to just see where we are. We know intentionally our liquid composition
is given as 0.40 so we know that our liquid, which is described by the top line here on
our two phase region when it's in equilibrium with its vapor. We can describe its composition
and its pressure by a point right here. Okay, so now the first question is to determine
the fugacity of the acetic acid. We're told we can assume ideal gas and ideal solution
behavior. And so therefore we can say that the fugacity of component 1, if it's ideal
solution, then we can assume Raoult's Law for the liquid phase, and so that would be
fugacity for the liquid phase is equal to X1 or the mole fraction in the liquid phase
of component 1 times the saturation pressure of component 1. Now if we want to know the
saturation pressure, we know this is a function of temperature only and we already know the
temperature which is 100 degrees C, then we can use an Antoine relationship so we'll raise
to the power of 10 the quantity A minus B over T plus C. And we have values for these
As, Bs, and Cs that you can find in the back of most textbooks and so for component 1,
for acetic acid, we have A is equal to 8.021. B I find to be equal to 1936.01. And then
C we find to be 258.451. We can plus in these values at the known temperature of 100 degrees
C and if we plug those in, we can find when you look up your relations for Antoine coefficients,
they actually use temperature in degrees Celsius for the numbers that I have here. The P1-sat
is equal to 0.55 atm. I'm keeping track of things in atmospheres. And what the direct
output of this calculation is is in terms or Torr or mmHg so it's equal to 417 Torr.
So now to find the fugacity we simply have to multiply by the mole fractions. So we take
P1-sat, multiply by 0.40 and we find that the fugacity of component 1 in the liquid
phase is equal to 0.22 atm. And I say that's the fugacity of the liquid phase but of course
at vapor-liquid equilibrium we have the fugacities of each component of each phase should be
equal. Alright, so that's what we're dealing with here. So this is just the fugacity of
component 1. We'll see that it will be useful for a subsequent step of the problem to determine
the fugacity of component 2 or at least the saturation pressure of component 2. So P2-sat
is again equal to this same relationship with different values for the constants. And I'll
go ahead and write those down quickly here. Okay so those are the values of A, B and C
and then P2-sat if we just plug in those numbers along with T equals 100 degrees C, then we
calculate the saturation pressure in Torr and then just do a straight unit conversion
to atm which turns out to 3.59 atmospheres. So component 2 is quite a bit more volatile
than component 1. So now to determine the fugacity of component 1. It's not something
we would often plot on a phase diagram. Fugacity isn't of course a truly measurable concept,
it's just a useful concept for computing phase equilibrium as we will see. So what we're
going to determine in part B is the pressure at which the liquid will be in equilibrium
with its vapor at 100 degrees C if it has the composition of 40% acetic acid with a
remainder acetone. So we're going ot find the value P that is right here. For part B,
we need to equate the fugacity of the vapor phase for component 1 to the fugacity of the
liquid phase for component 1. These kinds of fugacity relationships are really what
allow us to calculate vapor-liquid equilibrium. And if the fugacity for an ideal gas is equal
to Y1 time P, so in other words the partial pressure of component 1, and the fugacity
in the liquid for an ideal solution, as we've already said, is equal to X1 times P1-sat
at our temperature, which in this case is 100 degrees C. We can write a similar relationship
for component 2 and then we can then sum those up and we can realize now that Y1 and Y2,
if we have a two component system these are just the gas phase mole fractions of component
1 and 2, that they have to sum up to equal one. So we have that our pressure is equal
to the liquid phase fugacities for the two components. I have already seen that this
is equal to 0.22 atm from part A. And then here we never calculated it directly but we
have 60% acetone in our liquid or 60% of component 2 in our liquid, which is 3.59 atm. From that
sum, we get a total pressure of 2.37 atm. So we can come back up and label our plot
which is again quantitative. So this, we now know that pressure is 2.37 atm. We saw earlier
that this pressure is 0.55 and this pressure 3.59. Drawing is not to scale but it makes
sense that we find the pressure for this intermediate ideal solution and ideal gas behavior between
our two saturation pressures, as we'd expect based on the overall shape of this curve.
Next we want to move to part C. So we found the pressure by writing fugacity relationships
and we just use those again in a different way to solve for C, so we need the vapor phase
mole fraction of acetic acid. And we see that now that we know the pressure, that we know
P, X1 is 0.4 and we know P1-sat is 0.55 and so we can just take this with those known
values and solve for Y1 which would be equal to 0.092. So a little bit less than 10% on
the mole basis, acetic acid in the vapor phase. And so this number we find to be 0.4, this
number we find to be about 0.09, if we let down and plotted it on this axis. And again,
this drawing is not to scale. Okay, so we just found that composition in the vapor phase.
And so then the final question that we're asked is if we have an entire system and we
measure the mole fraction of the acetic acid to be 30%, how much of that total system is
vapor compared to liquid? So in other words, do we have half of our system that's in the
vapor phase on the mole basis, a quarter of the system, three quarters of the system?
So basically the idea here is that we're told that the total system composition is 30% acetic
acid. So that means we have vapor-liquid equilibrium at this temperature and pressure. We want
to know how much liquid and vapor add up to give that 30%. And so here's where we can
use the Lever Rule. Basically the Lever Rule is a form of a mole balance. We'll do this
graphically, you can also solve this completely algebraically. So what we'll realize according
to the Lever Rule is that the fraction that is equal to the vapor is equal to the length
of the opposite side, so the side that is adjacent to the liquid, that's what the Lever
Rule says, divided by the total length of this line segment. Okay, and so we'll go ahead
and apply this and come back and look at whether this makes sense. The total amount, the total
number of moles of vapor divided by the total number of moles in the entire system is equal
to the length of that line segment that was on the right which that line segment ends
at an X value of 0.40 and it begins at the mole composition value of 0.30. We're talking
about this segment right here. That line segment has a length of 0.1. And then we divide that
by the total line length which extends from 0.092 to 0.40 and we find the fraction of
total moles that's in the vapor phase is equal to 0.33. So in other words, if we had a mole
of this 30% acetic acid mixture in the total system, then one third of it at this temperature
and pressure would be in the vapor phase, or one third of the mole would be in the vapor
phase and two thirds would be in the liquid phase. So if we come up here we see that that
makes sense, according to our plot, that we're at a 30% composition of acetic acid, we're
inside our phase envelope, we're a lot closer to the liquid side of the phase envelope than
we are to the vapor side of the phase envelope. And that means that we'd expect to have more
liquid than vapor and we see that we do by a roughly 2:1 amount.