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Thermodynamics of VLE
Importance of VLE | Thermo.
of VLE | Lever Rule & VLE | Molecular Interactions | Molecular Simulation
Classical thermodynamics allows us
to obtain a relationship between the equilibrium mole fraction in the liquid
phase, x, and the vapor phase, y. Equilibrium between the
liquid and vapor phases requires that the temperatures, pressures, and
partial fugacities of each component be equal in the two coexisting
phases. By equating the partial fugacity of component 1 in the liquid
phase to its partial fugacity in the vapor phase, we can write
for the more volatile
component. (We use x for the liquid phase mole fraction of the more
volatile component and 1 - x for the mole fraction of the less volatile
component. We use a similar procedure for y, but we use subscript 1 for
the remaining properties of the more volatile component and subscript 2 for
those of the less volatile component.) This equation, and a similar
equation for the less volatile component is exact, but difficult to use
because it contains the partial fugacity coefficient at the system pressure,
the pure component fugacity coefficient at the corresponding pure-component
vapor pressure (sat), and the pointing correction factor F. For
most systems, these terms represent only small corrections, and for our
purposes the VLE equation can be simplified to
A similar equation can be written
for the less volatile component. Notice that there are mainly two
things that affect the relationship of y to x: the relative vapor pressures
of the two pure components, P1* and P2*, and
the nonidealities of the mixture as manifest in deviations of the activity
coefficient (gamma) from unity. This can perhaps better be seen by
adding the above equation to the similar equation for component 2 which gives
If the activity coefficients are
unity, then the system is ideal and is said to obey Raoult's Law.
Activity coefficients greater that unity give pressures that are larger than
the sum of the pure vapor pressures and are said to have positive deviations
from Raoult's Law. Notice that from the above equation that positive
deviations mean that the partial pressure in the vapor phase (left-hand side
of the equation) is larger than it would be for an ideal system; i.e., the
component "doesn't like" to be in the liquid phase. We shall
see that this corresponds to smaller attractions between molecules in the
mixture than in the pure component.
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Shown at the right are isotherms
of pressure versus composition for two different binary systems. The less
volatile component is the same in both cases, but the more volatile
component is different. Notice the interplay of the two factors that
affect the separability of the components or the ration of y to x.
In the case of the red line, the vapor pressures of the two components are
very close together and very small positive deviations from ideality can
cause there to be a maximum in the total pressure. This maximum is a
minimum-boiling azeotrope. (Recall that higher volatility corresponds
to a lower boiling temperature, so if a constant pressure plot were made
instead of a constant temperatre plot, the maximum in pressure would
correspond to a minimum in the boiling temperatures of the possible
mixtures). On the other hand, if the boiling points (or relative
volatilities) of the two pure components are quite different, then it takes
very large positive deviations from ideality to produce a minimum-boiling
azeotrope as shown by the blue line. The black lines show the ideal
pressures for the system and the difference between the colored lines and
the black lines are due to the nonidealities of the liquid mixtures as
expressed by the activity coefficients.
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A common way to show the
relationship of y to x is on a x-y plot. The
same systems shown above on a pressure plot are shown on an x-y plot at the
right by using the equation shown above. Here the magenta line
represents the pressure that would be observed for the system with
disparate vapor pressures if the activity coefficient were unity.
Note that even if the system is ideal that the large difference in vapor
pressures produces a large difference between y and x.
The system containing components of approximately the same volatility shows
very little difference between the x and y values.
Mixtures of these two components would be very difficult to separate by
distillation; the hypothetical magenta mixture could be separated by
distillation with only a few trays. However, the nonidealities also
have an affect upon the mixture separability, in this case producing an azeotrope
in both the red and blue system. The azeotrope point is the point
where x = y and the y-x plot crosses the 45-degree (black)
line. A liquid mixture of this composition when boiled gives off a
vapor of exactly the same composition as the liquid and no separation takes
place.
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Mixtures may also exhibit
negative deviations from Raoult's Law. We shall see that this is
produced when the attractions between the unlike molecules are greater than
those between identical molecules in the pure components. This added
attraction is described by classical thermodynamics with activity
coefficients that are less than unity. As can be seen from the
equations above, activity coefficients that are less than one lower the
partial pressure of the component below the ideal value for the given
composition; the molecules "prefer" to stay in the liquid phase.
A Pxy plot for a system exhibiting negative deviations is shown at
the right. This particular system does not have an azeotrope, but if
the negative deviations were large enough in comparison to the relative
volatilities of the two pure components, then a minimum pressure would
occur. This would correspond to a maximum-boiling azeotrope.
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When plotted for constant
pressure conditions, the VLE data better illustrate the reason for the
terminology "maximum- and minimum-boiling" azeotropes. Thus,
a minimum on the Txy plot, as in the figure at the right,
corresponds to a minimum-boiling azeotrope, or the minimum possible boiling
temperature over the whole range of compositions for this mixture. This
minimum temperature would of course be a maximum in system pressure at
constant temperature on a Pxy plot.
The plots generated in this
module's simulations are all at constant pressure like the one on the upper
right. The corresponding yx diagram is also generated by the
simulator. The mixture at the right is ethanol-toluene at 756 mm
Hg. The corresponding yx plot is also shown in the lower
right. At this point you should be able to say whether this system
exhibits positive or negative deviations from Raoult's Law. Can you also
say from a molecular view point why it exhibits the deviations that it
does?
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In addition to Txy
and yx plots, the module simulator also generates activity
coefficient and Gibbs energy plots. Several mixtures will be seen to
have large enough nonidealities relative to the separation in
pure-component vapor pressures that they exhibit azeotropy. Of
particular importance in viewing the simulations is for you to identify the
types of molecular interactions (e.g., strong cross interactions).
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The Gibbs excess energy is related
to the individual activity coefficients by
The second equation shows
that the change in Gibbs energy upon mixing two components consists of the
ideal mixing term and the excess term. Even if all of the interactions
between molecules are identical, there is an increase in entropy,
consequently a decrease in Gibbs energy, just due to the ability to randomize
the molecules in the mixture relative to two separate regions of all one kind
of molecule. This is the ideal change in Gibbs energy represented by
the first term. The excess term then results from non-ideal behavior;
i.e., differences in intermolecular interactions due to the unique electron
density structure of the individual molecules and correlational effects
between the those electron populations. These molecular effects result
in nonunity activity coeffients and either positive or negative excess Gibbs
functions.
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A plot of the excess Gibbs energy
and the change in Gibbs energy upon mixing, similar to those generated by
the module simulator, is shown at the right for the case of ethanol +
toluene (positive deviations from Raoult's Law). Remember that
positive deviations occur because the molecules don't like each other in
the mixture as much as within the pure components. This produces
activity coefficients greater than one and gE >
0. What does this mean about the liquid mixture. You might
recall from your thermodynamics class that favorable processes at constant T
and P proceed with a decrease in the Gibbs energy. Note from
the above equations that the ideal mixing term is always negative since
mole fractions are always less than one. Thus two components that
form an ideal mixture are always completely soluble because the free energy
goes down with the mixing process. However, gE can be positive and
large enough that the mixing Gibbs energy actually bows up. In such a
case, the Gibbs energy of the mixture can be minimized by splitting into two
phases rather than following the upward trend of the single-phase Gibbs
energy. The plot at the right is approaching the limit of
liquid-liquid partial miscibility, but the curve does not yet have a region
that bows up. Negative deviations, gE < 0, can never lead to
liquid-liquid insolubility because the Gibbs energy of mixing is even more
negative than the ideal case. This is because the molecules
"like each other" in the mixture.
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A sample of an activity
coefficient plot similar to those generated by the module's simulator is
shown at the right. Often activity coefficients are compared on
semilog plots as they are here. Activity coefficients must go to
unity as the pure component limit is approached for each of the two
components. This is because they physically indicate deviations from ideal
behavior for the component relative to the pure component standard
state. That is, just adding a molecule or two of a second component
doesn't change the behavior of the liquid much from the pure-component
case, so the activity coefficient will be very close to one. But,
when you have mostly component 2, then the activity coefficient of
component 1 may be quite different than pure component 1 because molecule 1
is almost completely surrounded by molecules of component 2 and the 1-2
interactions will dominate the behavior of the activity coefficient of
component 1. Here note that the infinite dilution values of the
activity coefficients are close are both close to 1.6.
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The module simulator
also generates plots of the relative volatility (usually given the symbol
alpha). The distribution coefficient, K = y/x, shows the
volatility of an individual component; i.e., the composition in the vapor
phase relative to the composition in the liquid phase. The relative
volatility, on the other hand, compares the volatility of one component to
the other. It is defined by
for a two-component system
(where as usual, we use x and y for the more volatile
component mole fractions). A relative volatility of 1 indicates that
both components are equally volatile and no separation takes place.
The plot at the right shows a y-x plot for different values of the
relative volatility. As can be seen, the higher the relative
volatility, the more separable are the two components. This means
fewer stages in a distillation column in order to effect the same
separation between the overhead and bottoms streams.
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More Background: Importance
of VLE | Thermo. of VLE | Lever Rule & VLE | Molecular
Interactions | Molecular Simulation
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