We're going to discuss in this screencast how the fugacity of each component in a binary
mixture changes if we start with a liquid and then raise the temperature so we have
vapor-liquid mixture and then we raise it further and have all vapor.
And what we're interested in is how the fugacity of each of the components change.
So let's first look at a diagram.
So on the left we have a T-x-y diagram, on the right we have fugacity as a function of
temperature.
So let's first look at the T-x-y diagram.
We're going to start with a mole fraction of 0.4, this point here, and the idea is we're
going to raise the temperature.
This T-x-y diagram is generated from Raoult's Law and there's a one line simulation that
will how you this diagram and some other related aspects of Raoult's Law.
What we're going to do is raise the temperature until we get to this point, the bubble point.
Then of course the vapor, this is Y of benzene, this corresponds to temperature to mole fraction
of the liquid phase of benzene.
As we continue to raise the temperature, if vapor is enriched in benzene, what's going
to happen is the mole fraction of benzene is decreasing so as we raise the temperature
the mole fraction of benzene in the two phase region goes down which means the mole fraction
of toluene is going up.
Likewise, the mole fraction in the vapor phase in equilibrium, these values here, are decreasing
for benzene and therefore they must be increasing for toluene.
And this happens until we get to the dew point.
Now the mole fraction is significantly lower than 0.4 for benzene and once we get in the
vapor phase of course now we have a mole fraction of 0.4 again for benzene and 0.6 for toluene.
So the question is, as we raise the temperature like this in a system that is at constant
pressure - this is an important aspect that we're doing this at constant pressure.
The plots on the left, this is the fugacity of benzene and this is the fugacity of toluene.
This region here, liquid phase, region here, vapor-liquid equilibrium, and then here is
vapor phase.
So as we raise the temperature, we're raising the saturation pressure exponentially and
the mole fraction Xb is constant for liquid, but the saturation pressure is increasing
exponentially, so the fugacity is increasing significantly.
Likewise, the mole fraction of toluene is constant but its saturation pressure is increasing
up to the bubble point, both of these are increasing with temperature.
We'll look at the two phase region last.
Let's look at the vapor phase region.
Well here is the fugacity of benzene, it's now equal to the mole fraction of benzene
in the vapor phase times the pressure.
So this is 0.4 times pressure.
And fugacity of toluene, again its mole fraction times the total pressure, and this is 0.6
so of course the fugacity is higher for toluene in the vapor phase.
The two phase region is a little more complicated so let's look at that on a fresh diagram to
make it a little easier to see.
So in the two phase region, remember as we raise the temperature, the mole fraction of
benzene is decreasing in the liquid phase and therefore the mole fraction of toluene
is increasing.
So benzene in the liquid phase and benzene in the vapor phase have the same fugacity,
so we can look at this in a couple ways.
But if we look at, let's refresh our memory here, this is fugacity of benzene, this is
fugacity of toluene.
In this region, as we continue to increase the temperature, we're increasing the saturation
pressure of toluene.
We're also increase the mole fraction of toluene.
So in the liquid phase, remember, we're moving this way, and the benzene mole fraction is
going down so the toluene mole fraction is going up which means the fugacity is increasing
for toluene for this region.
For benzene, if we look at it the same way with fugacity, well the mole fraction, so
we could say the fugacity here for benzene is the mole fraction of benzene times the
saturation pressure of benzene.
The saturation pressure goes up as we raise temperature but the mole fraction goes down
so it's not obvious from looking at the liquid phase, but remember the fugacity in the liquid
phase is identical to the gas phase for benzene when we are in vapor-liquid equilibrium.
And so the fugacity of benzene in the gas phase is the mole fraction times the total
pressure.
Total pressure is constant, the mole fraction of benzene, remember we're moving, here's
the mole fraction of benzene in the vapor phase at the bubble point, and it's continued
to decrease if you raise the temperature, and so therefore the fugacity of benzene is
decreasing as we raise the temperature.
And so I've included a figure at a different mole fraction.
The T-x-y diagram.
Let's suppose we start at 0.6.
Here's the corresponding plot of fugacity of each of the components versus temperature
and it would be good practice for you to on your own to make sure that you can one identify
which is benzene and which is toluene, and also explain the behavior for each of the
curves.