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9. LLE Calculations
For two liquid phases at equilibrium
the fugacity of each component in
the phases must be equal.
For the binary case shown:
x1 1  x1 1
(1  x1 )  2  (1  x1 )  2
are the two relationships that
govern the partitioning of species
1 and 2 between the two phases.
CHEE 311
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Binary LLE Separations
The equivalent of a VLE flash calculation can be carried out on
liquid-liquid systems.
Feed
@ T, P 
L, x1, x2
z1 , z2

L, x1, x2
Given:
T, P and the overall composition of the system
F, z1, z2
Find:
L, x1, x2
L, x1, x2
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Binary LLE Separations - Governing Eqn
Solving these problems requires a series of material balances:
Using a unit feed as our basis, an overall material balance yields:
(A)
F  1  L   L
A material balance on component 1 give us:
z1(1) 
x1 L 

(B)
x1 L
Substituting for L from A into equation B:
(C)
z1  x1 L  x1 (1  L )
An analogous material balance on component 2, yields:
z2 
x 2 L

(D)
x2 (1  L )
We have two equations (C,D) and three unknowns (L, x1 and x1).
 We need an equilibrium relationship between xi and xi
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Binary LLE Separations - Governing Eqn
Our LLE expression is:
xi  i  xi  i
or
x1 
x1
1
1
(14.10)
 
x

2 2
x2  
2
and
(E)
The governing equation we require to solve the problem is
generated from a final material balance on one of the liquid phases:
(F)
x  x  1
1
2
Substituting equations C, D, E into the material balance F gives us
the final equation:
z1
z2

1


 1

 2

1  L     1 1  L     1
 1

 2

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Solving Binary LLE Separation Problems
Given:
T, P,F, z1, z2
Find:
L, x1, x2
L, x1, x2
The solution procedure follows that of binary VLE flash calculations
very closely.
 You can immediately solve for x1 and x1 using the LLE
relationships
Or
 You can solve the governing equation by iteration, starting
with estimates of x1 and x1 to determine activity coefficients,
and refining these estimates and L by successive
substitution.
z1
z2

1






1  L   1  1 1  L   2  1
 1

 2

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Vapour-Liquid-Liquid Equilibrium (VLLE)
In some cases we observe
VLLE, in which three
phases exist at
equilibrium.
F=2-p+C
=2-3+2=1
Therefore, at a given P,
all intensive variables
are fixed, and we have
a single point on a binary
Tx,x,y diagram
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Vapour-Liquid-Liquid Equilibrium (VLLE)
At a given T, we can
create Px,x,y diagrams
if we have a good
activity coefficient
model.
Note the weak
dependence of the
liquid phase
compositions on the
system pressure.
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10. Chemical Reaction Equilibrium
SVNA 15
If sufficient data exists, we can describe the equilibrium state of a
reacting system.
 If the system is able to lower its Gibbs energy through a
change in its composition, this reaction is favourable.
 However, this does not imply that the reaction will occur in a
finite period of time. This is a question of reaction kinetics.
There are several industrially important reactions that are both
rapid and “equilibrium limited”.
 Synthesis gas reaction
CH4  H2O
CO  3H2
 production of methyl-t-butyl ether (MBTE)
CH3OH  CH2  C(CH3 )2
CH3OC(CH3 )3
In these processes, it is necessary to know the thermodynamic limit
of the reaction extent under given conditions.
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Reaction Extent
Given a feed composition for a reactive system, we are most
interested in the degree of conversion of reactants into products.
 A concise measure is the reaction extent, e.
Consider the following reaction:
CH4  H2O
CO  3H2
In terms of stoichiometric coefficients:
n CH4 CH4  nH2OH2O
where,
n CO CO  nH2 H2
nCH4 = -1, nH20 = -1, nCO = 1, nH2 = 3
For any change in composition due to this reaction,
15.2
dn CH4
dn CO dnH2

 of reaction.
 de
where de is calledthe differential
extent
nCH4
nH2O
nCO
nH2
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dnH2O
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Reaction Extent
Another form of the reaction extent is:
dni  nide
(i=1,2,…,N)
15.3
The second part of our definition of reaction extent is that it equals
zero prior to the reaction.
e0
ni  nio
at
Given that we are interested in the reaction extent, and not its
differential, we integrate 15.3 from the initial, unreacted state to any
reacted state of interest:
or
ni
e
nio
o
 dn i   n ide
15.4
ni  nio  n ie
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Reaction Extent and Mole Fractions
Translating the reaction extent into mole fractions is accomplished
by calculating the total number of moles in the system at the given
state.
n   ni   (nio  nie)
 no  ne
Where,
n   ni
no   nio
n   ni
Mole fractions for all species are derived from:
15.5
ni nio  nie
yi  
n no  ne
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Multiple Reactions and the Reaction Extent
The reaction extent approach can be generalized to accommodate
two or more independent, simultaneous reactions.
For j reactions of N components:
dn i   n i, jde j
(i=1,2,…,N)
j
and the number of moles of each component for given reaction
extents is:
ni  nio   n i, je j
15.6
j
and the total number of moles in the system becomes:
n  no   (  n i, j )e j
j
where n   n
i
no   nio
i
n j   ni,j we can write:
n  no   n j e j
j
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Chemical Reaction Equilibrium Criteria
To determine the state of a
reactive system at equilibrium,
we need to relate the reaction
extent to the total Gibbs
energy, GT.
We have seen that GT of a
closed system at T,P
reaches a minimum at
an equilibrium state:
dG 
T
T,P
0
Eq. 14.4
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Reaction Extent and Gibbs Energy
For the time being, consider a single phase system in which
chemical reactions are possible.
 The changes in Gibbs energy resulting from shifts in
temperature, pressure and composition are described by the
fundamental equation:
d(nG)  (nV )dP  (nS )dT   idni
i
 At constant temperature and pressure, this reduces to:
d(nG)T,P   idni
i
and the only means the system
has to lower the Gibbs
energy is to alter the number of moles of individual
components.
 What remains is to translate changes in moles to the reaction
extent.
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Criterion for Chemical Equilibrium
For a single chemical reaction, we can apply equation 15.3 which
relates the reaction extent to the changes in the number of moles:
15.3
dni  nide
Substituting for dni in the fundamental equation yields:
d(nG)T,P   inide
i
At equilibrium, we know that d(nG)T,P, = 0. Therefore, for the above
equation to hold at any reaction extent, we require that
 i n i  0
15.8
i
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Reaction Equilibrium and Chemical Potential
We have developed a criterion for chemical equilibrium in terms the
chemical potentials of components.
 (nG ) 
0
 n i i  

i
 e  T,P
15.8
While this criterion is complete, it is not in a useable form.
 Recall our definition of fugacity which applies to any species
in any phase (vapour, liquid, solid)
i  Gi (T )  RT ln fˆi
In dealing with reaction equilibria, we need to pay particular
attention to the reference state, Gi(T). We can assign a standard
state, Gio, as:
o
o
Gi  Gi (T)  RT ln fi
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Standard States
4.4 SVNA
For our purposes, the Gibbs energy at standard conditions is of
greatest interest.
Gio  Gi (T)  RT ln fio
This is the molar Gibbs energy of:
 pure component i
 at the reaction temperature
 in a user-defined phase
 at a user-defined pressure (often 1 bar)
A great deal of thermodynamic data are published as the standard
properties of formation at STP (Table C.4 of the text)
 DGfo is standard Gibbs energy of formation per mole of the
compound when formed from its elements in its standard
state at 25oC.
» Gases: pure, ideal gas at 1 bar
» Liquids: pure substance at 1 bar
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Chemical Potential and Activity
Substituting our standard Gibbs energy (Gio) in the place of Gi(T),
the chemical potential of component i in our system becomes:
i 
Gio
fˆi
 RT ln o
fi
15.9
We define a new parameter, activity, to simplify this expression:
15.11
o
i  Gi  RT ln aˆ i
where,
ˆai  fˆi fio
The activity of a component is the ratio of its mixture fugacity to its
pure component fugacity at the standard state.
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Reaction Equilibrium and Activity
When a reactive system reaches an equilibrium state, we know that
the equilibrium criterion is satisfied. Recall that chemical reaction
equilibrium requires:
 n ii  0
i
where ni is the stoichiometric coefficient of component i and i is
the chemical potential of component i at the given P,T, and
composition.
Substituting our expression for chemical equilibrium into the above
equation gives us :


o
 nii   ni Gi  RT ln aˆ i  0
i
Or,
CHEE 311
i
 iniGio
 ni ln aˆ i 
RT
i
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The Equilibrium Constant
Our equilibrium expression for reactive systems can be expressed
concisely in the form:
ln  aˆ i
i
ni
 in iGio

RT
15.12
where P signifies the product over all species.
The right hand side of equation 15.12 is a function of pure
component properties alone, and is therefore constant at a given
temperature.
 The equilibrium constant, K, for the reaction is defined as:
 in iGio
n
K  exp
  aˆ i i
RT Gibbsi energies of the pure
K is calculated from the standard
15.13
components and the stoichiometric coefficients of the reaction.
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Standard Gibbs Energy Change of Reaction
The conventional means of representing the equilibrium constant
uses DGo, the standard Gibbs energy change the reaction.
DGo  ini Gio
Using this notation, our equilibrium constant assumes the familiar
form:
15.14
 DGo
K  exp
RT
When calculating an equilibrium constant (or interpreting a
literature value), pay attention to standard state conditions.
 Each Gio must represent the pure component at the
temperature of interest
 The state of the component and the pressure are arbitrary,
but they must correspond with fio used to calculate the activity
of the component in the mixture.
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Temp. Dependence of Reaction Equilibrium
Defined by the following relationship,
o
DG
  ln K
RT
the equilibrium constant is a function of temperature.
 Recall that DGo represents the standard Gibbs energy of
reaction at the specified temperature.
o
d
(
D
G
/ RT )
We know that: DHo  RT 2
dT
15.15
From which we can derive the temperature dependence of K:
d ln K DHo

dT
RT 2
15.16
If we assume that DHo is independent of temperature, we can
integrate 15.16 directly to yield:
K
DHo  1 1 
ln  
  
K1
R  T T1 
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15.17
22
K vs Temperature
Equation 15.17 predicts that ln K
versus 1/T is linear. This is based on
the assumption that DHo is a weak
function of temperature over the
range of interest.
 This is true for a number of
reactions, including those
depicted by Figure15.2.
 A rigorous development of
temperature dependence
of K may be found in the text
(Equation 15.20)
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Equilibrium State of a Reactive System
Given that an equilibrium constant for a reaction can be derived
from the standard state Gibbs energies of the pure components, we
can define the composition of the system at equilibrium.
 in iGio
n
exp
 K   aˆ i i
RT
i
15.13
Consider the gas phase reaction:
CH4  H2O
CO  3H2
The equilibrium constant gives us:
K
Or
aˆ CH4 aˆ H2O
o
( fˆCO / fCO
)( fˆH2 / fHo2 )3
K
o
( fˆCH / fCH
)( fˆH O / fHo O )
4
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aˆ COaˆ H3 2
4
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