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Thermodynamics & Properties Lab. 1
A Statistical Associating Fluid Theoryfor Electrolyte Solutions
based on SAFT-VRE ( Gil-Villegas et al. Molecular Physics, 2001, 99, 531-546 )
Kim, Yong Soo
Chemical and Biological Engineering, Korea University
Thermodynamics & Properties Lab. 2
Introduction
Application of Electrolyte Equation of State (EoS) high pressure phase equilibria, ionic surfactant systems, gas
hydrate systems containing salts, and so on.
Approach of SAFT-VR to electrolyte systems SAFT-VR
• A second-order high-temperature perturbation expansion of a variable-ranged potential (Square well potential).
• Dispersion term : The mean-field level of van der Waals is used. Contribution due to the ion-ion interaction
• Primitive models : Debye-Hückel theory, Mean Spherical approximation (MSA)
Thermodynamics & Properties Lab. 3
Primitive Models for Electrolyte Solutions
The solvent contribution is described by a constant dielectric constants.
Interaction potential : (1) Potential
• The repulsive interactions with a hard-sphere potential.
• (2)
Electrostatic potential : (3)• The first term : the Coulomb potential between two ions.• The second term : Oppositely charged co-ions surrounding each one
of the interacting ions i and j, as well as the reaction-field potential.
)()()( rrru ijijij ψφ +=
)(rijφ
2)( where, if ,0 if ,
);( jiHSij r
rr σσσ
σσ
σφ +=
><∞
=
)(/)( rDrqqr CSijjiij ψψ +=
Thermodynamics & Properties Lab. 4
Primitive Models(PM) for Electrolyte Solutions
Expressions for An average electrostatic potential is considered as follows
(4) Poisson-boltzmann equation
(5)
Potential of mean force ( PMF ), • It is possible to obtain wij(r) by solving the Ornstein-Zernike (OZ)
equation for the interionic pair potential.• At low density, the PMF reduces to the intermolecular pair potential.
(6)
)(rijψ)(rΨ
)()( rqr jij Ψ=ψ
−−=Ψ∇j
ijjjj kTrwqD
r ]/)(exp[4)(2 ρπ
)(rwij
)()( rurw ijij =
Thermodynamics & Properties Lab. 5
Debye-Hückel Theory (I)
A low density fluid of ions of the same diameter(7)
Boundary condition Continuity of and at vanishes at infinity
Solution of average electronic potential
(8)
)()(4
]/)(exp[4)(
2
2
Ψ+=
−−=Ψ∇
jijjj
jijjjj
rrqDkT
kTruqD
r
κφρπ
ρπ
=j
jqDkT
222 4 where, ρπκ
)(rΨ )(rD Ψ∇ σ=r)(rΨ
+−−=Ψ
κσσκ
1)](exp[)( r
Drqr i
Thermodynamics & Properties Lab. 6
Debye-Hückel Theory (II)
Radial distribution function(9)
Internal energy(10)
Free energy
(11)
]exp[11
)(exp[exp)(
]//)(exp[)(
rDrkT
qqrDrkT
qqrg
kTqkTrrg
jijiHSij
jijij
κκσ
σκ
φ
−−=
+−−−=
Ψ−−=
=i j
ijijji
ION
drrgrurNkT
U )()(2 2ρρρπ
πκ
12
3
−=NkTAION
Thermodynamics & Properties Lab. 7
Debye-Hückel Theory (Summary)
The low density of ions has been assumed.
The size of the ions has been neglected.
The structure of the fluid is governed by the electrostatic interactions.
This approach is applicable up to 0.001 molal.
)()( rurw ijij =
Thermodynamics & Properties Lab. 8
Augmented Debye-Hückel Theory (I)
Electrostatic corrections Consideration for the size of the ions
(12)
(13)
Taking into account the quadratic term in (14)
(15)
+−−−=
κσσκ
1)(exp[1)( r
DrkTqq
rg jiij
+−+−=
2)1ln(
41 22
3σκκσκσ
πσNkTAION
)()(2
1)(11)( 222 rq
kTrq
kTrg jjij Ψ+Ψ−=
)(rgij
+
++−=)1(
1)1ln(24 3
2
κσκσ
πσκ
NkTAION
Thermodynamics & Properties Lab. 9
Augmented Debye-Hückel Theory (II)
Non-electrostatic corrections For higher densities of ions, the non-ionic packing information
must be considered.(16)
The radial distribution function and the free energy can be calculated in the context of perturbation theory as follows.
(17)
(18)
);();()( σεσφφ rrr HS Φ−=
221
1
)(11
)()()(
akT
akT
aNkTA
rgkT
rgrg
HSM
HSij
Mij
++=
++= ε
Thermodynamics & Properties Lab. 10
Mean Spherical Approximation (MSA)
Ornstein-Zernike (OZ) equation Linking the total correlation function to the direct
correlation function according to
(19)
MSA The direct correlation function is given by the intermolecular pair
potential outside the repulsive core of the molecule. The radial distribution function is zero inside the core.
1)()( −= rgrh)(rc
+= )()()()( 321331212 rhrcdrrcrh ρ
Thermodynamics & Properties Lab. 11
MSA-Restricted Primitive Model (MSA-RPM)
Assumption All the ions in the solution are of the same size. The pair potential is given by equations (1) to (3) and .
MSA expressions with RPM
0=CSijψ
(21) for )(
(20) for 1)(
σ
σ
>−=
<−=
rDkTr
qqrc
rrh
jiij
ij
Thermodynamics & Properties Lab. 12
MSA-Restricted Primitive Model (MSA-RPM)
The radial distribution function (Blum et al., 1970)
Where,
The electrostatic residual free energy
(23) )21(
(22) ),1()()(
2
2/12
2
xxxxxτ
DkTrqq
gg jiHSijij
+−+=
−−= τσσ
κσ=x
(24) 12
)21(22633
2/32
πρσxxx
NkTAION +−++−=
Thermodynamics & Properties Lab. 13
MSA-Primitive Model (MSA-PM)
Anions and cations are not restricted to be of the same size. The structural and thermodynamic properties are written as
functions of a characteristic inverse length, Γ.
==
=
=
−=ΔΓ+Δ
+=Ω
Γ+Ω=
Γ+Δ−=
=Γ
n
iiii
n
i ii
ii
n
i ii
iiiin
ii
niiii
n
iii
qPPqQ
QDkT
1
3
1
3
1
2
2/1
1
2
61 ,
121
11 ,
1)2/(
(25)
σρπσ
σπσ
σρσ
σπ
ρπ
Thermodynamics & Properties Lab. 14
MSA-Primitive Model (MSA-PM)
The radial distribution function (Blum et al., 1975)
The residual free energy
Where
(26) )1(2
31
)( 23
2
3 DkTQQ
g jijjiiijijij −
−+
−=
ςσσς
ςσ
σσ
(27) 321
1 3
1
22
ρππ
σρ
ρΓ+
Ω
Δ+
Γ+Γ−=
=
n
in
ii
iiION
PqDkTNkT
A
. and 2/)( 1 ==+= ni
liiiljiij σρςσσσ
Thermodynamics & Properties Lab. 15
SAFT-VRE Approach
SAFT-VRE
Ideal free energy of the mixture
where, Λi is the thermal de Broglie wavelength of species i.
(28) NkTA
NkTA
NkTA
NkTA
NkTA
NkTA IONASSOCCHAINMIDEAL
++++=
(29) 1ln1
3 −Λ= =
n
iiii
IDEAL
xNkT
A ρ
Thermodynamics & Properties Lab. 16
SAFT-VRE Approach
The monomer-monomer interaction
is the hard-sphere free energy of the mixture are the first two terms of the perturbation expansion
associated with the attractive energy.
(30) )(
11221 a
kTa
kTaa
NkTA HSM
M
++==
HSa
21 and aa
(31) ];[1 1
31 1
11 = == =
−==n
i
n
j
effij
HSij
VDWijji
n
i
n
j
ijji gxxaxxa ςσαρ
(32) 3/)1(2 33 −= ijijijVDWij λσπεα
(33) 2 1 1
1
1 122
= == = ∂∂==
n
i
n
j
ij
ijHS
ji
n
i
n
j
ijji
aKxxaxxaρ
ερ
Thermodynamics & Properties Lab. 17
SAFT-VRE Approach
Association term Water-water ( hydrogen bonding ), ion-water ( solvation ), ion-ion
(ion-pairing) interactions.
The effect of the association interactions depends on the range of densities, temperatures, and the nature of the solvent.
Solvation : the formation of hydration shells in aqueous solutions
Ion-pairing is not important in aqueous solutions of strong electrolytes at ambient temperature.
(34) 22
ln1
,,
1
+
−= ==
iS
a
iiaia
n
ii
ASSOC sXXx
NkTA
Thermodynamics & Properties Lab. 18
Model parameter study for a model solution of NaCl
Effect of the dielectric constant Determining the nature of the
solvent The higher the dielectric con-
stant the better the ionic solvent
Using DH and MSA-RPM theories
D=1
D=∞
Thermodynamics & Properties Lab. 19
Solvent-ion attractive interactions : dispersive type Interactions through the
repulsive and Coulombic potentials only.
The vapor pressure is under-predicted as the solvent-ion interaction is strong.
For greater λ, a less deep well and higher vapor pressure.
α1j /α11=0.31For λ1j =1.2
α1j /α11=1
For α1j /α11=1
λ1j =1.2
λ1j =1.8
3/)1(2 33 −= ijijijVDWij λσπεα
Thermodynamics & Properties Lab. 20
Solvent-ion attractive interactions : associative type To describe the solvent-ion
interactions (solvation shell)
Adjustable parameter• Water-ion interations :
• The number of attractive site :][/1 KkHB
jε
iS
For 6 sites
For 12 sites
For 6 sites
For 12 sites
1366K
2448K1224K
1366K
1366K
1224K
1366K2448K
Thermodynamics & Properties Lab. 21
Ion pairing ion-ion interactions
• Strong electrolytes are fully dissociated.
• The ion pairing occur around the critical region.
Three models• Coulombic interaction• Anion-cation attractive square-
well interaction• Association via short-ranged
attractive sites
For supplementing solvent-ion attractive square-well interactions+(a)
Coulombic interaction only(1)Ion-ion association+(1)
Attractive ion-ion square-well interaction+(1)+(2)
Thermodynamics & Properties Lab. 22
Conclusion
Several levels of approximation have been considered for primitive model theories.
The solvent-ion interaction is modeled via an attractive potential of variable-range.
Associative sites of ion-water interactions have been considered.
SAFT-Electrolyte EoS is applicable to phase equilibria containing strong electrolyte.