In this video we will try to describe the idea of fugacity, why it's important, where
it comes from, why we are interested, and using it.
And we'll start out by looking at two phases, an alpha phase and a beta phase.
And we won't specify any more about the phases right now.
But what we are showing is the concentration of component a in each of the phases and initially
these phases are not in equilibrium.
And the question is, what direction does component a move.
What's the direction of mass transfer to get to equilibrium.
We are always looking at how systems approach equilibrium and in this case we might look
and say well, very high concentration of a in this phase, we might expect mass transfer
to go to the lower concentration.
However, for this particular system, if this was, this phase was water vapor, and this
phase was liquid water, and we were at 45 degrees C, then mass transfer would be in
this direction, because this concentration, if we look at steam tables, is actually higher
than saturation pressure, so equilibrium would be a decrease to saturation pressure.
And so this means we can't use concentration as our determination of direction of mass
transfer.
Certainly concentration matters and that affects fugacity but we are going to use fugacity
or chemical potential as our determining factor.
If the chemical potential is higher in the alpha phase than the beta phase, so chemical
potential is higher in the alpha than the beta and therefor mass transfer goes from
alpha to beta.
And another way of looking at this means that fugacity of water is higher in the alpha phase
than the beta phase.
So the chemical potential is related to the Gibbs free energy and we use Gibbs free energy,
we'll look at this a couple ways, but the G in the vapor phase would be enthalpy per
mole or per kg, vapor phase, minus the temperature times the entropy in the vapor phase.
And at equilibrium that H liquid phase minus TS.
So G in the vapor phase, G in the liquid phase, those are equal at equilibrium, that turns
out to be our criteria for equilibrium.
And if we rearrange this equation we notice H vapor minus H liquid over T is S vapor minus
S liquid which means delta H vaporization over T is equal to delta S vaporization.
Heat of vaporization over temperature is the entropy change of vaporization.
This is how we originally derived entropy where delta S we defined as dQ reversible
over T. And for phase change, T is constant, Q is delta H when we carry out vaporization
at constant pressure.
And so this equation follows from our definition of entropy and it shows the G is the same
of both phases.
So let's look at how we get fugacity from G. And we'll go back to the first law, closed
reversible system, change in internal energy, differential change is the heat added, the
work added.
If it's reversible then we can write dQ in terms of entropy change and again if it's
reversible we can write work as -PdV.
Now delta H is defined as delta U plus delta PdV or dH is dU plus dPV.
So we can make the substitution form dU, these two terms cancel and dH is TdS plus VdP.
And now one more step, dG, dH.
So we are making the same type of substitution, dH, TdS plus VdP minus TdS minus SdT.
So the change, differential change in G, and again you will notice this term cancels.
VdP minus SdT.
And so what we are going to look at for a lot of our calculations is what happens if
we look at changes at constant temperature.
Then dG is VdP.
So we can calculate the change in G from volume and pressure changes at constant T. And when
we look at, for example, an ideal gas, what the change is going to look at, dG for an
ideal gas is RT over P dP.
Remember the volume we are using here is the specific volume, the volume per mole.
So RTdP/P. So this means change in G for an ideal gas is RTdlogP.
And so what has been done then for real gas is to say, we like this form so we are going
to introduce a new variable, the fugacity.
It's the value we put in here to make the G change correct for a real gas.
So if we were to subtract these two equations, the real and the ideal, and just divide by
T to make dimensionless, then on the right we would have dlog fugacity/P and this is
the fugacity coefficient (phi) a dimensionless variable.
So if we were to go back and remember that our constant T dG is VdP, we could subtract
that equation for real gas, so that's saying all this is real ideal gas.
And we could write this in terms of a compressibility factor, z and compressibility factor is 1
for an ideal gas.
So Z minus 1, RTdP/P tells us this change in difference between ideal gas and real gas,
and note that as pressure approaches 0 then G approaches the ideal gas value, so the ideal
gas value and real gas value for G are the same at low P where gases become ideal gases.
So this means an ideal gas, the fugacity is just the pressure.
For a real gas we have to use information like the equation of state, integrate this
to determine the fugacity because we have change in internal energy in terms of z.
We also have it in terms of fugacity and P. And so if we combine these equations, I won't
go through the details, then fugacity at a given pressure for an ideal gas is equal to
this integral, 0 to the pressure we are interested in, (z-1)dP/P and so we use an equation of
state we can do this integration and determine fugacity for a real gas at any pressure.
Ideal gas we know the fugacity so the question is, for a liquid what is the fugacity.
So the thing that is critical is to keep in mind what we've showed, way back here, the
G of the vapor and the liquid are equal at equilibrium, so we are looking at a one component
system.
If the G in a liquid equals the G in a vapor, that means, from our definitions the fugacity
in the liquid is the fugacity in the vapor.
So at vapor liquid equilibrium we have both phases present, the f vapor is going to be
the saturation pressure if it's an ideal gas.
That means the f liq is the saturation pressure if indeed that saturation pressure is low
enough that we can assume ideal gas.
So now we have f in the vapor for an ideal gas.
We have a way of calculating f for a gas that is not ideal, if we have more information
about how non-ideal it is.
And then for liquid at saturation conditions the f is the pressure.
For a liquid at higher pressure, so for a liquid that the pressure is greater than the
saturation pressure.
The first thing to recognize is not much changes.
So the properties don't change much, H, U, S, don't change, and since G is H - TS, if
H and S don't change much it means G doesn't change very much.
And so there is indeed a small change, and this small change in the interest of time
here, the f would be f at saturation pressure, which is going to be very close the saturation
pressure and then just from that definition we have this exponential.
This term is close to 1 so it takes high pressures to change the fugacities very much for the
liquid phase.