Phase Equilibria
Kaj Thomsen, kth@kt.dtu.dk
Associate Professor, DTU Chemical Engineering
iCAP thermodynamics workshop,
DTU Chemical Engineering, 05-05-2011
Phase Equilibria
• Vapor – liquid equilibrium
–Raoult’s law
–Henry’s law
–Equation of State
–Approach for electrolytes
• Liquid – liquid equilibrium
–General approach
–The ”mixed solvent” approach
–Approach for electrolytes
• Solid – liquid equilibrium
–Using fugacity coefficients
–Approach for electrolytes
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Raoults law
•The partial pressure of a component is often
calculated from the mole fraction in the liquid
phase and the pure component pressure
yi P = xi Pi sat
•This method assumes that the gas phase and
the liquid phase are ideal
•Positive and negative deviations from ideality
•This approach is called Raoults law and is
approximately valid for some non-polar systems
at low pressure
3
DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Txy-diagram – Raoult’s law
Toluene-Benzene , P= 1 atm
120
115
Vapor
Temperature °C
110
105
100
Two phases
95
90
Liquid
85
80
0
0.2
0.4
0.6
0.8
1
Benzene mole fraction in liquid and gas
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Pxy-diagram, Raoult’s law
Toluene - Benzene, t=100°C
1400
1300
Liquid
1200
Pressure, mmHg
1100
1000
Two phases
900
800
700
Vapor
600
500
0
0.2
0.4
0.6
0.8
1
Benzene mole fraction in liquid and gas
5
DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Deviation from Raoult’s law
•The non ideality in the gas phase can be
corrected with the fugacity coefficient. The non
ideality in the liquid phase can be corrected by
the activity coefficient
sat
ˆ
yiφi P = xiγ i Pi
•This equation is still not correct
thermodynamically
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Raoult’s law
•Raoult’s law can be applied to symmetric
systems
–Systems of components that are completely
miscible
•Systems like Nitrogen – water, CO2 – water, H2
– water are not symmetrical.
–In these systems, one component is
considered to be the solvent, the other is the
solute
•Henry’s law is an approach that can be used for
unsymmetric systems
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Henry’s law
yi P = xi H i
•Hi is the Henry’s law constant, which is
dependent of temperature and pressure
•The definition of the Henry’s law constant is:
yiφˆi P
lim
= H i (T , P0 )
xi →0
xi
•Henry’s law is defined in the limit of infinite
dilution but is often applied at other
concentrations too
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Deviation from Henry’s law
•The non ideality in the gas phase can be
corrected with the fugacity coefficient.
•At high pressure, the ”poynting factor” can be
applied:
 ( P − P 0 ) Vi ∞
yiφˆi P = xi H i (T , P0 ) exp 

RT





•This equation is called the KrichevskyKasarnovsky equation
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
N2 in water and H2 in
water
From Krichevsky
and Kasarnovsky,
J. Am. Chem. Soc.
57(1935)2168-2171
yiφˆi P
=
ln
xi
ln H i +
10
DTU Chemical Engineering, Technical University of Denmark
0
∞
P
P
V
−
(
)i
RT
iCAP Thermodynamics Workshop 05-05-2011
Deviation from Henry’s law
•The non ideality in the liquid phase can be
corrected by an activity coefficient:
 ( P − P 0 ) Vi ∞
yiφˆi P = xiγ i* H i (T , P0 ) exp 

RT





•This is called the Krichevsky-Ilinskaya equation
•The activity coefficient is marked with an *. This
is the unsymmetric activity coefficient
γ i* → 1 for xi → 0
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Unsymmetric activity coefficient
•The unsymmetric activity coefficient is derived
from the symmetric activity coefficient by
normalization:
γi
γ = ∞
γi
*
i
γ i∞ is the infinite dilution activity coefficient
of component i
•The infinite dilution activity coefficient of
component i is the symmetric activity coefficient
of component i at infinite dilution
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Gibbs energy
•At phase equilibrium, the chemical potential of
each species is the same in all phases
α
β
γ
µ=
µ
=
µ
i
i
i
•The chemical potential is defined by:
 ∂G 
 ∂H 
 ∂A 
 ∂U 
=
= 
=
µi ≡  


∂
∂
∂
∂
n
n
n
n
 i T , P ,n j  i  S , P ,n j  i T ,V ,n j  i  S ,V ,n j
•Phase equilibria can be expressed in terms of
chemical potentials
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Gibbs energy, temperature and
pressure dependence
•If the heat capacity is assumed independent of
temperature, the temperature dependency of
the Gibbs energy is:
GTref
HTref  Tref
 Cp 
 Tref
GT
=
+
− 1 +
 1 + ln 

RT RTref Tref R  T
 R 
 T
 Tref 

−
T


•Pressure dependency of Gibbs energy:
=
GP GPref
P
+ RT ln
Pref
GP = GPref + V ( P − Pref )
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DTU Chemical Engineering, Technical University of Denmark
-For ideal gas
-For incompressible liquid
iCAP Thermodynamics Workshop 05-05-2011
Component i in a symmetrical
mixture
l
=
µ
µ
•Liquid mixture: i
i , P + RT ln xi γ i
g
•Gas mixture: =
µ µ + RT ln y φˆ
i
i,P
i i
•Pressure dependence of the standard state
chemical potential:
l
l
l
µ
=
µ
+
V
i , P0
i ( P − P0 )
•Liquid mixture: i , P
P
g
ig
•Gas mixture:
µ=
µi , P0 + RT ln
i,P
P0
•At equilibrium between species i in liquid and
gas at T and P, the chemical potential of
component i is identical in the two phases:
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Equilibrium between phases I
µiig, P + RT ln
0
P
+ RT ln yiφˆi = µil, P0 + Vi l ( P − P0 ) + RT ln xiγ i
P0
•The terms of the equation are reordered to give:
−
µiig, P − µil, P
0
0
RT
Vi l ( P − P0 )
yiφˆi P
+
=
ln
RT
xiγ i P0
•For pure component i it gives:
−
µiig, P − µil, P
0
RTb
0
φisat P sat
Vi l ( P sat − P0 )
+
=
ln
RTb
P0
•Raoults law appears by combining the two right
hand sides. It is only correct at Tb and Psat:
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φˆi
yi sat P = xiγ i Pi sat
φi
DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Equilibrium between phases II
µiig, P + RT ln
0
P
+ RT ln yiφˆi = µil, P0 + Vi l ( P − P0 ) + RT ln xiγ i
P0
•The terms of the equation are reordered to give:
−
µiig, P − µil, P
0
RT
0
Vi l ( P − P0 )
yiφˆi P
+
=
ln
RT
xiγ i P0
•The terms on the left hand side are evaluated as
a function of temperature and pressure, and the
equation is solved
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Activity and fugacity coefficient
•The equation used to express VLE with yi = xi :
µiig,P + RT ln
0
P
+ RT ln xiφˆi = µil,P0 + Vi l ( P − P0 ) + RT ln xiγ i
P0
•The same equation for pure component i:
µiig, P + RT ln
0
P
+ RT ln φi =µil, P0 + Vi l ( P − P0 )
P0
•By subtraction, the relation between activity and
fugacity coefficient is obtained:
φˆi
= γi
φi
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Component i in an unsymmetrical mixture
=
µi
•Liquid mixture:
•Gas mixture:
µi , P (aq) + RT ln miγ i ,m
=
µi µ
g
i,P
+ RT ln yiφˆi
•mi is the molality, mol/(kg water)
•Pressure dependence of the standard state
chemical potential:
( aq )
(
)
(
)
µ
aq
=
µ
aq
+
V
( P − P0 )
i , P0
i
•Liquid mixture: i , P
P
•Gas mixture:
g
ig
µ=
µi , P0 + RT ln
i,P
P0
•At equilibrium between species i in liquid and
gas at T and P, the chemical potential of
component i is identical in the two phases:
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Equilibrium between two phases
P
yiφˆi µi , P0 (aq ) + Vi ( aq ) ( P − P0 ) + RT ln miγ i ,m
+ RT ln =
P0
µiig, P + RT ln
0
•Re-writing this equation:
−
µiig, P − µi , P (aq) Vi ( aq ) ( P − P0 )
0
0
RT
H i ,m (T , P )
yiφi P
+ = ln

→ ln
m →0
RT
miγ i ,m P0
P0
This is inserted in the above equation:
( aq )

V
( P − P0 ) 
i
ˆ
yiφi P = miγ i ,m H i ,m (T , P0 ) exp 

RT


•This is the Krichewski-Ilinskaya equation!
•Theoretically limited to Tc, but practically used
beyond
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Gamma-phi method
•The use of activity coefficients (gamma) as well
as fugacity coefficients (phi) in these methods
have given them the name ”gamma-phi method”
for vapor-liquid equilibrium.
•The gamma – phi method is commonly used in
systems that require different models for the gas
phase and the liquid phase.
•The gamma – phi method Krichevsky-Ilinskaya
is often used for VLE calculations for electrolyte
systems such as CO2 capture using aqueous
alkanolamines.
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Equations of State
•Fugacity coefficients and fugacities can be
calculated from equations of state
•Fugacity coefficients can be used for calculating
the residual Gibbs energy
µi =µ + µ
ig
i
res
i
=µ

=
µ + RT ln 

0,ig
i
P
+ RT ln   + RT ln yiφi
 P0 
yiφi P 
fˆi
0,ig
µi + RT ln
=
P0 
P0
0,ig
i
•EOS use same standard state for liquid and gas
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Equilibrium calculation with EOS
•The same standard state is used for all phases
and therefore cancel:
l
v
ˆ
ˆ
f
f
µil =
µi0,ig + RT ln i =
µi0,ig + RT ln i =
µig
P0
P0
•The iso-fugacity criterion can be used for
determining phase equilibrium
•If the same component appears in the vapor,
liquid, and solid phase, the criterion for phase
equilibrium is:
v
l
ˆ=
ˆs
fˆ=
f
f
i
i
i
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
VLE calculation for electrolytes
•One of the gamma – phi methods are usually
used
•A speciation equilibrium calculation has to be
performed simultaneously
•Speciation is the distribution of species when
electrolytes are dissolved in a polar medium
•Example: a mixture of H2O, NaOH, and CO2
•The amounts of the following species are
determined in the speciation equilibrium
calculation:
•H2O, CO2, HCO3-, CO32-, OH-, H+, and Na+
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Speciation equilibrium calculation
•The following equilibria need to be considered:
H 2O(l ) ↔ H + (aq ) + OH − (aq )
CO2 (aq ) + OH − (aq ) ↔ HCO3− (aq )
HCO3− (aq ) ↔ CO32− (aq ) + H + (aq )
•The chemical potential of each species is
determined. Example:
(
)
*
*
*
*
µ HCO =
µ HCO
+ RT ln xHCO γ HCO
=
µ HCO
+ RT ln aHCO
−
3
−
3
−
3
−
3
−
3
−
3
•The asterisk (*) signifies un-symmetric
convention a* is the activity
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Speciation equilibrium calculation
•The equilibrium criterion is equality of chemical
potential for each of the three equilibria:
µ H0 O (l ) + RT ln aH O =µ H*
2
2
+
*
*
*
ln
ln
RT
a
RT
a
µ
+
+
+
+
−
( aq )
H ( aq )
OH ( aq )
OH − ( aq )
*
*
*
ln
RT
a
µCO
µ
+
+
( aq )
CO ( aq )
OH
2
2
*
µ HCO
−
3 ( aq )
*
*
*
ln
ln
RT
a
RT
a
µ
+
=
+
−
( aq )
OH − ( aq )
HCO − ( aq )
HCO − ( aq )
3
*
*
*
*
*
+
RT
ln
a
ln
RT
a
µ
µ
+ RT ln aHCO
=
+
+
−
+
2−
2−
( aq )
H + ( aq )
CO ( aq )
CO ( aq )
H ( aq )
3
3
3
•All equations are brought on a form similar to
the VLE equation. All equations are solved
simultaneously:
∆G 0
νi
−
=
ln
a
∑ i
RT
26
3
DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Application of Extended UNIQUAC model for
VLE calculation:
CO2 solubility in K2CO3 solution
3
130°C
3.1 molal K2CO3
Partial pressure of CO2 / bar
2.5
Tosh et al. (1959)
Extended UNIQUAC
2
1.5
110°C
90°C
1
70°C
0.5
0
27
0
0.2
DTU Chemical Engineering, Technical University of Denmark
0.4
0.6
Loading mol CO2/mol K2CO3
0.8
1
iCAP Thermodynamics Workshop 05-05-2011
Liquid-liquid equilibrium
•General approach:
•fi’=fi’’
•Often an activity coefficient model is used and
fugacities are replaced by activities:
•ai’=ai’’
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DTU Chemical Engineering, Technical University of Denmark
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The mixed solvent approach
•Usually applied to solutions with electrolytes
•A mixed solvent can be a water – ethanol
mixture.
•The solvent is modeled separately.
•Electrolyte interactions are influenced by the
dielectric properties of the solvent
•Each solvent composition is a ”new solvent”
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Conventional and ”Mixed solvent”
approach
µi =
µ + RT ln xi + RT ln γ
*
i
*
i


 


ideal
excess
"Mixed solvent" approach:
µi = µ
+ RT ln xi + RT ln γ

 
Mixed solvent
i
ideal
Mixed solvent
i
excess
In the ”Mixed solvent” approach, the standard
state chemical potential of solute i is a
function of the solvent composition – is this
allowed?
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Extended UNIQUAC approach
•Water is the only solvent
•Same standard state used for all solutes
•Advantages:
–Standard state chemical potentials cancel out:
µ + ν RT ln ( x γ
*
s
I
±
*, I
±
µ + ν RT ln ( x
)=
*
s
II
±
γ
*, II
±
)
–No need for a separate model for the mixed
solvent
–More precise representation of experimental
data
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
LLE and SLE in the Na2SO4 - iso-propanol –
water system at 35°C
100
0
Water
Thomsen et al.,
Chem. Eng. Sci.
59(2004)3631-3647
35°C
90
10
80
20
70
30
60
40
50
50
40
60
Extended UNIQUAC
Lynn et al. (1996)
70
30
20
80
10
90
Na2SO4
100
0
10
20
30
40
50
60
70
80
90
0
100
i-propanol
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Solubility of NaCl in ethanol solution
30%
Experimental
ElecNRTL
ElecNRTL optimized
OLI MSE
Extended UNIQUAC
20%
Wt% NaCl
Lin et al.,
AIChE Journal,
56(2010)13341351
25%
15%
10%
5%
T = 15 °C
0%
0%
20%
40%
60%
80%
Wt% ethanol, Saltfree
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100%
Solid-liquid equilibrium (EOS)
•This equation is often used for solid liquid
equilibrium in non-electrolyte systems
•The right hand side is the temperature and
pressure dependency of the Gibbs energy
change. The melting point temperature Tm is
used as reference:
µi0,l + RT ln ( xiγ i ) =
µi0, s
 µi0,l − µi0, s
=
xiγ i exp  −
RT

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DTU Chemical Engineering, Technical University of Denmark

 evaluated at T and P

iCAP Thermodynamics Workshop 05-05-2011
Solid-liquid equilibrium (EOS)
•The symmetric activity coefficient is calculated
from the corresponding fugacity coefficient:
 µi0,l − µi0,s 
φˆi
exp  −
x=

i
RT 
φi

•The temperature and pressure dependence of
the standard state chemical potential is
determined.
•The fusion temperature is often used as
reference temperature
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Solid-liquid equilibrium (EOS)
•For a composite solid, such as a dihydrate, the
equilibrium equation is:
µi0,l + RT ln ( xiγ i ) + 2 µ w0 + 2 RT ln ( xwγ w ) =
µi0,s
0
0
0


2
µ
+
µ
−
µ
2
i ,l
w
i ,s
exp  −
xiγ i ( xwγ=

w)
RT


•The temperature and pressure dependence of
the standard state chemical potential is
determined like before. The right hand side will
therfore be the same expression as before.
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
SLE calculations, electrolytes
•A speciation calculation needs to be performed
at the same time as the SLE calculation.
•The procedure is almost the same as outlined
before
•Solid salts consist of minimum two ions and
often some hydrate water, like Glauber salt,
Na2SO4·10H2O
•The equilibrium equation express that the
chemical potential of solid salt is equal to the
sum of chemical potentials of the parts, the salt
was made from (e.g. 2Na+, SO42-, 10H2O)
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
SLE calculations, electrolytes
•It is not practical to use the fusion temperature
and fusion enthalpy as reference for salts
•Formation enthalpies of ions and salts at T0 =
298.15 K are found in many tables.
•The properties of salts and ions are calculated at
the relevant temperature by integration from T0
−
*
*
0
0
2 µ Na
+
µ
+
10
µ
−
µ
+
2−
H 2O ( l )
Na2 SO4 ·10 H 2O (s )
SO ( aq )
( aq )
4
RT
(
)
2


*
*
10
ln  a Na + ( aq ) aSO
a
=
2−
H 2O ( l ) 
4 ( aq )


•A pressure term is often relevant to add at
pressures above 50 bar
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DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
SLE calculations, electrolytes
•The equations for the speciation calculation and
for the solid-liquid equilibrium calculation are all
on the form:
∆G 0
νi
ln
−
=
a
∑
i
RT
•The equations are solved simultaneously, just
like it was done in VLE calculations
39
DTU Chemical Engineering, Technical University of Denmark
iCAP Thermodynamics Workshop 05-05-2011
Solubility of potassium carbonate in water
130
110
Extended UNIQUAC
Experimental data
90
Temperature °C
70
50
2K2CO3·3H2O
30
10
-10
Ice
K2CO3·6H2O
-30
-50
0
10
20
30
40
50
60
70
80
Mass % K2CO3
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iCAP Thermodynamics Workshop 05-05-2011