Fugacity. It's just one of those thermodynamic properties that sometimes cause headaches.
So hopefully this screen-cast will provide you background on where it came from and why
we care about it. So to accomplish this we are going to start with ideas of chemical
potential and reference states, go back a little bit. First, let's all agree, that property
relations for mixtures of ideal gases or solutions are important as references in the analysis
of real mixtures. And this will form the basis for introducing fugacity. The second thing
I want to touch on is that we have known that for two phases to be in equilibrium, between
a liquid and a vapor for instance, the pressure and the temperature must be equal. Hopefully
by now you have also heard of chemical potential and known that it must also be equal in both
phases for each species to be in equilibrium. So we go back in time to an ideal gas mixture.
We've seen the PV=nRT, I'm going to rewrite this just in terms of pressure. I'm going
to write this V as Vtotal. The number of moles of species "i" designated at Ni would take
up the same total volume alone at the same temperature. Now this will yield the pressure
of that species. I will label this equation 1 and this equation 2. If we divide 2 by 1
then we are going to get the pressure species "i" over the total pressure of the system.
Since RT and V will all cancel out. Now you have hopefully seen this before. We designate
this Pi as the partial pressure. This is the gas mole fraction times the total pressure.
So that partial pressure is the pressure a species would take up if it was alone and
didn't have any interactions, behaved ideally. And we know that the sum of all species' partial
pressures is the total pressure of the system. To advance we need to discus the partial molar
properties, such as the partial molar volume of species i. And we will go into details
here but it is defined as such. This is going to be a constant pressure, temperature, and
keeping other species' moles constant. So this represents the change in molar volume
of a species in a mixture to the addition of that species keeping everything else constant.
For example, you could try and picture this, this of course is of course an abstract kind
of equation, right. But if you think about adding ethanol to equal molar water/ ethanol
solution. You might assume that the change in volume of the total mixture, would just
be the volume of the ethanol that's added. This isn't true, because in a real solution,
as you see in your lab doing this, interactions between ethanol and water molecules changes
the partial molar volume of each species, thus affecting the total volume. Alright back
on track here, we can rewrite this here as RT over P times change in moles with respect
to change in species i keeping other moles constant. And in an ideal case we can just
rewrite this as RT/P. So that the partial molar volume in an ideal gas mixture is equal
to the pure species molar volume at the same temperature and pressure. We could do this
because we are assuming two things. For an ideal gas they are comprised of imaginary
molecules having a volume of zero. And two, there are no interactions. Now of course in
most systems neither of these are true. Pretty boring really, but again this provides a great
reference. And it is something to keep in mind. So the summary so far: a partial molar
quantity (other than the volume) of a species, in an ideal gas mixture, is EQUAL to the molar
properties of the species as a pure ideal gas at the same mixture temperature but at
a P that is equal to the partial pressure that is in the mixture. This is pretty much
summarizing what we have seen so far. This works with other partial molar mass properties
as long as it is evaluated at the proper pressure, that being the partial pressure. So what does
this mean? Well let's take the case of enthalpy. The enthalpy of a species i in an ideal gas
mixture has some temperature and pressure. It is just going to be the enthalpy of the
pure species at a temperature T and pressure P i. For entropy it's a little different.
Same pressure, temperature but this is going to equal the entropy of the ideal gas mixture
at a temperature and pressure P, but we are going to subtract out this difference R time
the natural log of yi. And this comes into play because entropy is not independent of
pressure. And I won't get into where this comes from you can look it up in your thermodynamics
textbook. So here is where all the mathematical fun happens. So I am going to try and skip
through each part kinda quickly just to save some time. So the Gibbs energy of an ideal
gas mixture is defined as the following. Again I designated all of the ideal gas values with
an ig property. So the partial Gibbs is going to be very similar. At this point we could
replace everything from these two equations into those equations. So hopefully you see
how we got that equation. At this point you should recognize this as the ideal gas Gibbs
energy. So we could say that the partial Gibbs energy is equal to the ideal gas Gibbs energy
plus RT natural log of yi. Now this is going to be an important equation to keep in mind
as we progress. Now when you go back and look at your notes on partial Gibbs energies in
a mixture, we know that this is also considered to be the chemical potential. And so we will
set that equal to the following. And that is an equation that we are going to use heavily
as we progress into fugacity- and no we didn't forget about fugacity. So this is the first
equation we are going to use, the second, if you recall, changing Gibbs is equal to
the volume times the change in pressure, minus the entropy times the change in temperature.
If we assume constant T. And we can rewrite this for our mixture, and using a substitution
for the molar volume, we can plug that back in. And then we can take this equation and
integrate it with no boundaries, just using an integration constant we get the following.
And this is that integration factor at constant T. And every book might use a different symbol,
go ahead and use the capital gamma. We can go back to that equation I starred before
for the chemical potential. You see that when we plug everything in we get the following.
We simplify it further and say that the chemical potential is equal to some integration factor
plus RTln(yiP) which we know is the partial pressure. And the sum of each of the species
is also going to be the Gibbs energy of the mixture. And it looks like the following.
And so we expect this to describe ideal gas behavior tremendously well, which it does.
So can we start talking about fugacity? I believe you may be ready. So we are going
to start with these two equations since I said before that they were pretty important.
And here's the twist: since Gibbs energy and chemical potential were both defined in relation
to the internal energy and entropy of a species, both which are permanent values and absolute
quantities, can't be measured and are unknown this means that absolute values of chemical
potential do not exist. More so, let's look at this first equation. What happens to mixtures
when one species composition approaches zero, or the pressure of the system approaches zero,
such in cases of very high vacuum. Taking the limit the chemical potential would approach
negative infinity. So what does that even mean? I mean, negative infinity. Now these
aren't reasons to scrap chemical potential but it is a reason to add more quantities
to thermodynamics. So we could describe phases in chemical equilibrium much easier under
all conditions for both pure species and mixtures if we create a parameter called fugacity.
So why fugacity you ask? Well a little history of fugacity stems from the Latin root meaning
to flee or escape, like fugitive. So think of the escaping tendencies of a species. You
could probably think of a lot of jokes about a student's fugacity in class and probably
should mention it next opportunity you get. But for now we will continue to define it
in cases of equilibrium. So we are going to label fugacity as F. And let's substitute
fugacity into that second equation for pressure. So to do that for a real gas the G of each
species in the mixture is going to be equal to the following where I've designated f as
the fugacityof each species. Now looking at that previous equation you could probably
guess what the units of f are going to be. That's right they are going to be units of
P. Now let's compare the real to the ideal. Since an ideal case, again, is a nice reference
state that we can typically calculate and measure and a real case makes it a little
bit harder. So if we do that, and compare this real situation to an ideal situation
we get the following. Now the left side of this equation should probably look pretty
familiar. This is the residual G and we have relationships for that that are covered earlier
in textbooks. Now looking at the right side of the equation we are going to take that
term in parenthesis and give it a new symbol. So this symbol phi is going to the be f coefficient.
So the residual G equals RTlnphi and the f of an ideal gas is just going to be the P.
So we know the residual G in an ideal gas situation is going to be zero and phi is going
to be one. And so I am going to rewrite this here and kind of rearrange so you can see
that the ln of phi is just the residual G over RT. And this right side hopefully looks
fairly familiar. If we go back to how we defined the residual G it was related to the compressibility
factor of a species i as such. Hopefully that looks kind of familiar. And we can plug that
right side into the equation so that we can get a relationship for our f coefficient in
terms of the compressibility factor z. So what does this mean already? Well we could
go to PVT data or volume explicit equation of state like some virial equation of state.
I will chose this one where we have z related to a second virial coefficient Bii so we can
plug that in to our equation and have an integral that looks like the following. So hopefully
now you can see kind of how, where f stemmed from and how we can start using it. Because
now we have an equation that relates a f coefficient to a virial, second virial coefficient and
we could perform lab experiments to solve for this to and get a relationship for that
under certain T and P. So why we care about this in terms of equilibrium is something
I will touch on kind of quickly. So when we talk about equilibrium, let's start by writing
the G of the vapor as such. Now this relates the G of the vapor species i to the f of the
species in the vapor phase. We could do the same thing for the liquid so that we use the
f of the species i in the liquid phase. Now if we do the same thing that we did before
relating these two G and subtracting them like such we end up with the following. So
we know that if we have vapor liquid equilibrium, this quantity on the left is going to be equal
to zero. And if that is equal to zero as some T and P, and it is going to be the Saturation
P, we know that the f of the species in the vapor phase must be the f of the species in
the liquid phase. So we will call this f saturated for species i. So now we kind of have a new
rule to go on, such that, for pure species, existing in liquid and vapor, if they are
to be in equilibrium, then the T, P, and f in both phases must be the same. And we could
also say that f coefficient, saturation, is just going to be related to f saturation over
P saturation and this is going to be equal to the liquid phase and the vapor phase. And
again this is all at vapor liquid equilibrium. So we care because we wish to know if the
system is at equilibrium and this is one of the ways to determine that among other things.
F kind of acts as a correction factor for P in real systems. And we've seen that relationship
compared to the ideal cases. So let's look at two quick estimates of f. First we could
look at how to calculate the f of a compressed liquid. Now I will go back to this relationship
between phi and z that I showed you before. We are going to evaluate this at a P at saturation
so Psat. And so we know that the f coefficient in the vapor phase is going to be the f coefficient
sat. And then we could also use the relationship that f sat for species i is going to be the
phi sat times the Psat. So this reference would be for the saturated liquid. We calculate
the f change from the P increase since we are compressing the liquid. So we are going
from P sat to P or some pressure P, as we go from a saturated liquid to a compressed
liquid. So this is relating that change from the saturation conditions to our new values.
So we look at that left side and that left side is that residual that we have seen before
and we know that that is equal to RTln f/fsat. And if we substitute that in we have the following
equality. Since the molar volume in the liquid phase, the Vi over here, is a very weak function
of P as long as the T is under the critical T. We can assume that the molar volume of
the liquid phase is just the molar V of the pure species and that is a value we could
look up. So now this becomes the following and all that I have done is integrate that
right side from Psat to P and if we use this relationship that I showed you before for
the saturation f and substitute in the following we will get a relationship to solve for f.
And all I've done is rearrange a little bit and have f equal the f coefficient saturation
time P sat times this exponential on the right side and this exponential is known as the
Poynting exponential. And pretty much everything there you could look up or know from what
you are trying to figure out. F coefficient at saturation you could look up at tables
and so forth for P sat. So hopefully this gives you an idea of how to calculate the
f in the liquid phase for a compressed liquid. The gas states just depend on the equation
of state you decide to use. So if you model your equation with peng robinson or others
you just substitute that back in at the beginning and go through the calculations. So at least
you hopefully know where to answer those and I will leave those for your own exercise.
And remember, again, we care about this because at equilibrium the f of the liquid phase is
going to equal the f in the vapor phase. So we could determine specific parameters of
our real system if we know how our other phases are behaving.