Thermodynamics of Separations

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Thermodynamics of Separations
This lecture primarily focuses on the
thermodynamics of separations. Well cover
Phase Stability and the Gibbs phase rule. A
simple separation based on a vapor-liquid phase
diagram. The lever rule. Equilibrium
ratios (K-values, distribution coefficients,
etc.). The activity, and activity
coefficients. Measures of separations.
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Conditions for Phase Stability
Integrate dU
From the definition of G
Substituting U into our expression for G
Gives the Gibbs Free Energy in terms of chemical
potentials and concentrations.
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The Gibbs-Duhem Equation
Consider a single phase in equilibrium
Starting with our expressions for the Gibbs free
energy We consider how it changes for an
infinitesimal process We see that G changes
because the amount of components changes
or because the chemical potential of the
components changes
G changes because T changes or P changes or
because the composition changes
Since these two expressions for dG are equivalent
we can equate them to find
The Gibbs-Duhem Equation which relates T, P and µ
at equilibrium in a single phase system
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Gibbs Phase Rule
Each phase of a system in internal equilibrium is
governed by its own Gibbs-Duhem equation
Each phase is described by C2 intensive
variables T, P and the C chemical
potentials. Since the Gibbs-Duhem expression
relates these C2 variables, only C1 of them are
independent. If ? phases are in equilibrium with
each other, then we have only one T and one P
for all the phases, so we still have C2
variables. However, we now have ? relationships
between the C2 variables since we have a
Gibbs-Duhem expression for each phase.
Note F is the number of degrees of freedom and P
is the number of equilibrium phases.
Gibbs Phase Rule
One can independently vary f intensive variables
for a system of C components and still keep ?
phases in equilibrium.
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Vapor-Liquid Equilibrium
Note that if we start with aliquid of
composition XB(L) andheat to the bubble point we
start to form a vapor with composition XB(V),
which is rich in the volatile component. This
vapor could be collected, and cooled to the
bubble point line to create a liquid
concentrated in the volatile species. Note
that this is not really a practicalprocess
because _______________?
Lever Rule
T
Dew point line
V
Bubble point line
L
XB(L)
XB(V)
XB
0
1
XB
T
Note that the liquid could be collected, and
heated to the dew point line to create a liquid
even more concentrated in the non-volatile
species.
If we start with a liquid of the
samecomposition, but heat above the bubble
point as shown we can performa separation
however the resultingproducts are less pure.
V
L
XB
0
1
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Equilibrium Ratios
This type of separation is possible becausethere
is an equilibrium ratio different from one. That
is,the K-value for the component of interest
issubstantially different from one. A K-value
is the ratio of the amount of component i in one
phase to another. For liquid-liquid equilibrium
it is oftencalled a distribution coefficient
T
V
L
XB
0
1
For liquid-vapor equilibrium it iscalled a
vapor-liquid equilibrium ratio
Note that the partial pressure of component i in
thevapor is used rather than the mole fraction,
which implies that we are assuming an ideal gas.
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Difficult Cases
T
Note that the vapor-liquid equilibrium shown on
the rightwill be a difficult system to separate
because the difference inconcentrations of the
vapor and liquid phases is small.
V
L
XB
0
1
T
V
Note that the azeotropic vapor-liquid equilibrium
shown on the rightwill be a difficult system to
separate because the difference inconcentrations
of the vapor and liquid phases is small.
L
Why are these phase diagrams like this?
XB
0
1
Azeotrope composition
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Ideal Mixtures
If the liquid phase is ideal, then the separation
would bedifficult because there would be no
tendency to vaporize onecomponent versus the
other. In other words, in order to usethe above
type of separation process the mixture should
notbe ideal.
We know that
And for mixing
By definition for an ideal mixture
Since there is no enthalpic effect, the Gibbs
free energy of mixingis only due to the
randomnesscreated by mixing. This is the same
for the any ideal mixture.
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Activity
The activity indicates how different the partial
pressures of a vapor in equilibrium with a
condensedphase are from the mole fractions of
the condensed phase.
By definition for an ideal mixture
Gas constant
Use fugacitiesif gas doesnt behave ideally.
For a non-ideal mixture
Activity
Activity coefficient
The activity coefficient is defined by
An activity coefficient equal to one indicates
that themixture is behaving ideally for that
component.
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Activity
Note that there is more of the component
indicated by the small green circles in the
vapor phase above its pure form than abovethe
mixture. Its activity is _____. The activity of
the other component is______? The enthalpic
contribution to non-ideality changes the amount
ofone species in the vapor phase relative to the
amount that you wouldexpect based on the
composition of the liquid phase in
equilibriumwith the vapor.
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Measures of Separation
The Split Fraction is the ratio of the amount of
component in a product stream to the feed stream.
Split Fraction
The Split Ratio is the ratio of the amount of
component in two product streams.
Split Ratio
Number of moles of i
The separation power is the a ratio of split
ratios. A Separation Power near 1 indicates a
poor separation.
Separation Power
Concentration
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Next Lecture
Next lecture will show how to apply some of our
thermodynamics to flowing systems, and show how
data of mixture compositions are
sometimes tabulated graphically. Specifically, we
cover  Energy and entropy balances in flowing
systems Availability and lost work Gibbs
Phase Rule for flowing systems and system
specification Graphical determination of
vapor-liquid equilibrium of hydrocarbon systems