In the first video entitle "What is Fugacity?" single component fugacity is discussed and
in this video we will consider multi-component by looking at a two component system. So consider
two phases - an alpha phase and a beta phase - and the mole fractions of component one
and two, so it's a binary mixture, component one and two in the alpha phase and beta phase
are given, and the mole fractions are different. And so the question is, since this system
is assumed to not be at equilibrium, what is the direction of mass transfer? Does component one
move in this direction or does it move in this direction? And how do we determine that?
So that fact that the mole fractions are not equal is not enough to tell us whether or
not it's in equilibrium. The mole fractions are not going to be the same in equilibrium.
What determines the direction of mass transfer is the chemical potential of component one
in the alpha phase and the beta phase. If the chemical potential in the alpha phase
is great than the beta phase, then component one is going to move from the alpha phase
to the beta phase. Another way of stating that is that the fugacity in the alpha phase
is greater than the fugacity in the beta phase, then component one moves from the alpha phase
to the beta phase. And the fugacity notation that is used in a number of textbooks for
mixtures puts a small hat over the fugacity term. So the question then becomes how do
we determine the fugacity? So if we look at the case of an ideal gas, one of the simpler
cases, let's look at this system and let's assume that this is actually a gas phase and
this is a liquid phase and we want to determine equilibrium between gas and the liquid which
is a common situation when we're looking at separations. So we'll assume the gas phase
is an ideal gas. So it's an ideal gas mixture. We want the fugacity then of component one
in the alpha phase, the gas phase. Well it's going to be equal to the mole fraction of
component one in the alpha phase times the total pressure. Or the more normal notation
we might use is the mole fraction for a gas is Y times the total pressure. So this is
the partial pressure. So fugacity of an ideal gas, fugacity of a given component, so we
can do the same in the alpha phase component two is Y2 times the pressure, partial pressure
component two. So for this system, of course, then the fugacity of component one in the
alpha phase - in the vapor phase - would be equal to 0.7 times the total pressure. Notice
that the units of fugacity are the units of pressure. We're assuming then that the beta
phase is a liquid phase and we'll look at the simplest case for liquid phase of an ideal
solution. And for an ideal solution, we calculate the fugacity relative to the saturation pressure
for the pure component. So component one in the beta phase is the liquid phase in this
case. So the mole fraction of component one in the beta phase, the liquid phase, times
the saturation pressure of component one. So this would be in this case 0.4 times the
saturation pressure. If they were at equilibrium, then we would expect the fugacity in the alpha
phase for component one would be equal to the fugacity in the beta phase for component
one. And so that means that the mole fraction in the gas phase for component one, that's
the alpha phase, times the total pressure is equal to the mole fraction in the liquid
phase - beta phase - times the saturation pressure of component one. And this is Raoult's
Law, which is ideal solution and ideal gas. And so in our system, mass transfer is going
to occur either component one going from the beta phase to the alpha phase or component
one going from the alpha to the beta, until we get to equilibrium as defined by this criteria
here, so both X1 and Y1 could change if we're doing this at fixed pressure and fixed temperature,
which means Psat is also constant. If we don't have an ideal liquid solution, in other words
the interactions between the components are different than just some average interaction
and certainly that's a reasonably common occurrence, then component one fugacity in the beta phase
- and let me just make it clear, that's the liquid phase - would be X1 times gamma one
times P1-sat. So this gamma is basically a correction. It's referred to as the activity
coefficient and it's correcting for the fact that we don't have an ideal solution, we need
some information about the interactions in order to be able to calculate this. So if
we just look now in general of how changing variables affects fugacity in a mixture. If
we raise the pressure, well for the gas phase the relation is clear that fugacity of component
one in a vapor phase is the mole fraction times the pressure. Again, this is ideal gas.
And so as we raise the pressure we're going to raise fugacity. For liquid phase, as we
raise the pressure it's not so clear and it's going to be the same type of behavior that
we saw fugacity for single component. It's fugacity at saturation for the liquid times
the exponent of the pressure above the saturation pressure times RT and times the molar volume,
fugacity of component one at saturation conditions, that's the pure component, and then for the
ideal solution that we're looking at, that's saturation pressure. And this is a very small
correction, so it's going to be very close to our original value of approximately X1
P1-sat, but this is a small correction. It will only become important at high pressures,
for example, the high pressures that are used in reverse osmosis to purify salt water. If
we look at a mixture, what happens as we raise the temperature? Well for the liquid phase,
saturation pressure goes up as the temperature goes up, and therefore the fugacity will increase
as we raise the temperature, so we raise the mole mole fraction. So X1 increases and the
fugacity in general is also going to increase. So as we increase the concentration, we tend
to increase the fugacity. Keep in mind, this is increasing the mole fraction. Increasing
the amount of the phase does not increase fugacity. Fugacity, like many of the variables
we look at are intensive variables. So if we have two systems, liquid and vapor, where
we have very different amounts of liquid and vapor, and we are at equilibrium, X1 is 0.4,
here X1 is 0.4, the vapor let's say equilibrium is 0.65. Well then at equilibrium this would
be 0.65. The amounts of the phases are not important. The concentrations, which are of
course related to mole fractions, are important.