This example looks at vapor liquid equilibrium at constant pressure where we have piston
and cylinder arrangement. And we have liquid water and we have water vapor in equilibrium
at 1.2 bar. And what we are going to do is inject into the vapor phase, 5 cm^3 of air.
So we are going to keep the pressure constant because we have a constant weight on the piston
and we have this in a constant temperature bath. And the question is, when we get back
to equilibrium after injecting the air, what has happened? Has all the liquid vaporized?
Has all the water vapor condensed? Or is some of the liquid vaporized? Or some of the water
vapor condensed?
First we have to look at what assumptions we make for this system.
First, we assume the pressure is low enough that we can assume ideal gases. And then we
are going to assume the solubility of air in water is negligible. At 1.2 bar for liquid
water the temperatures above 100 degrees C and so the solubility is going to be pretty
low at this elevated temperature. And then we ask ourselves the question, what happens?
What about the air changes the system to determine the behavior and predict what the behavior
will be. Let's look at the liquid water. Temperature is constant, the air does not dissolve in
the water. So we haven't made any changes in the water so we can say the fugacity of
water in the liquid phases is constant for this system. It is not changing as we inject
the air. Water vapor, however, is different. The total pressure is 1.2 bar, when we start
we only have water vapor present. And so that means that the pressure of water in the vapor
phase, initially, is equal to the saturation pressure of water and the saturation pressure
of water is 1.2 bar. At a later time, the total pressure is constant and going to be
equal to the pressure of the water plus the pressure of the air. So the air must have
some partial pressure. And so once we've injected the air, then the pressure of the water is
less than 1.2 bar, which means the pressure of the water is less than the saturation pressure
of the water. Or in terms of fugacities, the fugacity of water in the vapor phase is less
than the fugacity of the water in the liquid phase because we did not change the fugacity
of water in the liquid phase. The fugacity of water in the liquid phase is going to be
about the same as its saturation pressure, 1.2 bar. And so we have a difference in fugacity.
And fugacity or chemical potential, is a driving force for mass transfer. So what this says
is we're going to have liquid water that is going to go into the vapor phase to try and
get back to equilibrium. It's a non-equilibrium situation when the fugacity of water in the
vapor phase is less than the fugacity of water in the liquid phase. We no longer have vapor
liquid equilibrium, water is going to evaporate. As water evaporates then the partial pressure
of water increases. But, and this is the important aspect, partial pressure of water is always
going to be less than 1.2 bar, which means that the fugacity of water in the vapor phase
is always going to be less than 1.2 bar. If the fugacity is less than 1.2 bar it says
we can't have liquid in equilibrium with it at this temperature, temperature is fixed.
So all the water will evaporate. This would have to be a very large piston and cylinder
because the water vapor occupies quite a bit more volume than the liquid water. so this
would have to extend up dramatically in order to not just have the piston come out of the
cylinder arrangement. And also we would have to add heat to the system to vaporize this
water. So we have it in a constant temperature bath. This means we get to equilibrium continuously
adding heat to maintain this constant temperature to supply the heat of vaporization. So therefore
the answer is that all the liquid evaporates to get back to equilibrium because of this
difference in fugacity. Important point here is that it's a constant pressure system.