1
Liquid-liquid extraction of acids by a malonamide:
2
II- Anion specific effects in the aggregate-enhanced extraction isotherms
3
4
5
6
7
8
9
10
11
Sandrine Dourdain1,*, Christophe Déjugnat2, Laurence Berthon3, Véronique Dubois1,
Stéphane Pellet-Rostaing1, Jean-François Dufrêche1, Thomas Zemb1
Appendix
Theoretical model for Langmuir-like adsorption of electrolytes
12
The Gibbs free energy of adsorption can be estimated from the salt extraction curves thanks to
13
a simple adsorption model generalising the Langmuir isotherms to electrolytes described in
14
this part.
15
a- Model with a simple solute
16
In this paragraph, for the sake of simplicity, the electrolyte will be considered as a single
17
solute. The role of the dissociation will be taken into account in the next section. The organic
18
phase is modelled as a set of aggregates. From the solute point of view, there is no correlation
19
between the aggregates. Everything happens as if there were a set of Na accessible volumes Va
20
to the solutes S corresponding to the polar cores of aggregates. The role of the molecularity
21
will be analysed below. The model is globally the one represented in Figure 3.
22
The calculations are made at the McMillan-Mayer level of description11 (continuous solvent
23
model), which is formally equivalent to the calculation with an explicit solvent when exact
24
effective potentials between the solute particles are taken into account. The partition function
25
in the canonical ensemble reads:
26
Q = N! ∫ dpN dr N e−βNE0
27
where N is the number of solute particles in the organic phase, 𝛽 = 1/𝑘𝐵 𝑇, r and p are the
28
position and the corresponding momentum of the solute particles in the Na aggregates. E0 is
29
the adsorption energy. The integration over the momentum is straightforward:
1
(1)
1
𝑒 −𝛽𝑁𝐸0 ∫ 𝑑𝑟 𝑁
1
𝑄=
2
where Λ is the De Broglie length of the solutes. Within the continuous solvent model, Λ is an
3
effective quantity, which takes into account the solvation of the solute particles. The
4
configurational integral is:
5
∫ 𝑑𝑟 𝑁 = (𝑁
6
because each site can contain up to one solute particle. By derivation of the resulting free
7
energy 𝐹 = −𝑘𝐵 𝑇 ln 𝑄 together with the Stirling approximation ln 𝑁! = 𝑁 ln 𝑁 − 𝑁, we
8
finally get the chemical potential µ of the solute particles in the organic phase:
9
𝛽𝜇 = 3 ln 𝛬 + 𝛽𝐸0 − ln 𝑉𝑎 + ln (𝑁
𝛬3𝑁 𝑁!
𝑁𝑎 !
𝑎 −𝑁)!
(2)
𝑉𝑎𝑁
(3)
𝑁
𝑎 −𝑁
)
(4)
10
The adsorbed solute particles are in equilibrium with the aqueous phase. The chemical
11
potential of the solute in this water phase reads:
12
𝜇 = 𝜇 0 + 𝑘𝐵 𝑇 ln (𝐶 0 )
13
where 𝜇 0 = 𝑘𝐵 𝑇 ln(𝛬3 𝐶 0 ) is the standard chemical potential, 𝐶 0 the reference concentration
14
that defines the standard scale of the aqueous solution (typically C0 = 1.mol.L-1 = 1000 NA m-
15
3
16
chemical potential, we obtain the following adsorption curve:
17
𝜃=𝑁 =
18
The adsorption phenomenon corresponds to a Langmuir isotherm with the adsorption constant
19
𝐾 0 = 𝑉𝑎 𝑒 −𝛽𝐸
20
where
21
𝐸 ′ = 𝐸 0 + 3𝑘𝐵 𝑇 ln 𝛬 − 𝜇 0
𝐶
(5)
) and C is the concentration in the aqueous phase. Making equal the two expressions of the
𝑁
𝑎
𝐶
𝐾0 0
𝐶
𝐶
1+𝐾0 0
𝐶
′
(6)
(7)
(8)
1
In equation 8, E’ is related to the change of Gibbs free energy from the aqueous phase to the
2
organic phase, as can be understood from the following analysis. Indeed, the chemical
3
potential in the organic phase can also read
4
𝛽𝜇 = ln(𝐶 0 𝛬3 ) + 𝛽𝐸 0 + ln ((𝑁
5
where we explicitly introduce the standard reference of concentration 𝐶 0 . The two first terms
6
ln(𝐶 0 𝛬3 ) + 𝛽𝐸 0 correspond to the standard chemical potential term of the adsorbed solute
7
particle, which is different from the one in the bulk phase ln(𝐶 0 𝛬3 ) by the energy 𝛽𝐸 0 . The
8
next term ln (𝑉
9
of the aggregates 𝑉𝑡𝑜𝑡 = 𝑁𝑎 𝑉𝑎 . Finally the last term ln 𝛾 = ln (𝑁
𝑁
0
𝑎 𝑉𝑎 )𝐶
𝑁
𝑡𝑜𝑡 𝐶
0
∙𝑁
𝑁𝑎
𝑎 −𝑁
) = ln(𝐶 0 𝛬3 ) + 𝛽𝐸 0 + ln (𝑉
𝑁
𝑡𝑜𝑡 𝐶
0
) + ln 𝛾
(9)
) is nothing but the ideal activity of the solute particles in the total volume
𝑁𝑎
𝑎 −𝑁
) is the activity coefficient
10
arising from the saturation of the aggregates. Thus, the standard state we considered
11
corresponds to the concentration 𝐶 0 with a solute behaviour similar to an infinitely diluted
12
system. It is the commonly used standard state of solution chemistry. Then the adsorption
13
constant becomes:
14
𝐾 0 = 𝐶 0 𝑉𝑎 𝑒 −𝛽∆𝐺
15
where ∆𝐺 0 = 𝐸0 is the variation of the Gibbs free energy when a solute particle is moved
16
from the water phase to the polar core of the aggregates, i.e. the driving force for the
17
extraction of one given solute in any industrial process. It should be noted that ∆𝐺 0 does not
18
actually depend on the choice of 𝐶 0 because it corresponds to the infinite dilution limit. Thus
19
the simple model of solute adsorption in the aggregates yields to the following Langmuir
20
isotherm:
21
𝜃 = 𝑁 = 1+𝐾𝐶
22
with
23
𝐾 = 𝑉𝑎 𝑒 −𝛽∆𝐺
𝑁
𝐾𝐶
𝑎
0
0
(10)
(11)
(12)
1
for which Va is the accessible volume of one site. Thus the constant K has the dimension of
2
volume. For an homogeneous planar interface, it is simply the surface multiplied by the
3
thickness of the layer divided by the number of sites. For a spherical surface, it has to be
4
divided by a factor 3 because for a sphere of volume V, surface S, and radius R, V =1/3 S R.
5
The calculation of Va is detailed in the following.
6
b- Model with an electrolyte solute
7
If the solute is an electrolyte, the 𝑁 solute particles are actually dissociated into 𝜈+ 𝑁 cations
8
and 𝜈− 𝑁 anions because of the dissociation equilibrium:
9
𝜈+ 𝐶𝑎𝑡 + 𝜈− 𝐴𝑛𝑖 = 𝐸𝑙𝑒𝑐
(13)
10
where 𝐶𝑎𝑡 is the cation, 𝐴𝑛𝑖 is the anion, and 𝐸𝑙𝑒𝑐 is the neutral electrolyte.
11
In the bulk water phase, the chemical potential of the electrolyte reads:
12
𝑎𝑞
0
𝜇𝑒𝑙
= 𝜇𝑒𝑙
+ 𝑘𝐵 𝑇 ln (
13
In equation 14, 𝐶+ = 𝜈+ 𝐶 and 𝐶− = 𝜈− 𝐶 are the concentrations of the cations and the anions,
14
respectively, while 𝛾+ and 𝛾− represent their respective activity coefficients. The
15
concentration and the activity coefficient of the electrolyte are respectively 𝐶 and 𝛾 =
16
(𝛾+ + 𝛾−𝜈− )1/𝜈 , with 𝜈 = 𝜈+ + 𝜈− . The chemical potential of the electrolyte in the organic phase
17
has to be modified accordingly. We consider that in the adsorption model represented in
18
Figure 3, every occupied adsorption site has exactly one electrolyte particle, i.e. there is
19
exactly 𝜈+ cations and 𝜈− anions in any of the N occupied sites. The possibility of non-
20
neutral aggregates is forbidden. The resulting canonical partition function is:
21
𝑄=
22
Now the resulting configurational integral reads:
23
∫ 𝑑𝑟 𝑁 = (𝑁
𝜈
𝜈
𝜈
𝜈
𝐶++ 𝐶−− 𝛾++ 𝛾−−
(𝐶 0 )𝜈+ +𝜈−
𝜈
𝐶𝛾 𝜈
0
0
) = 𝜇𝑒𝑙
+ 𝑘𝐵 𝑇 ln (𝜈++ 𝜈−𝜈− (𝐶 0 ) ) = 𝜇𝑒𝑙
+ 𝑘𝐵 𝑇 ln 𝑎𝑒𝑙 (14)
𝜈
𝑒 −𝛽𝑁𝐸0
3N
3N
Λ+ + Λ− − (Nν+ )!(Nν− )!
𝑁𝑎 !
𝑎 −𝑁)!𝑁!
∫ 𝑑𝑟 𝑁
(15)
(ν+ +ν− )N
(Nν+ )! (Nν− )! Va
(16)
(ν+ +ν− )N
1
because it is the configurational integral of one site Va
2
of possibilities of choosing the 𝑁 occupied sites among 𝑁𝑎 and by the number of possibilities
3
of distributing the 𝜈+ 𝑁 cations and 𝜈− 𝑁 anions. Similarly to the last paragraph we finally get
4
the following expression of the chemical potential of the electrolyte in the organic phase:
5
𝛽𝜇𝑒𝑙 = 3 ln Λ ++ Λ𝜈−− + 𝛽𝐸0 − (ν+ + ν− ) ln Va + ln (N
6
Identifying the two expressions of the chemical potential of the electrolyte in the two phases
7
and considering the definition of the ideal term
8
0
𝛽𝜇𝑒𝑙
= 3 ln((𝐶 0 Λ3+ )𝜈+ (𝐶 0 Λ3− )𝜈− )
9
we obtain the following Langmuir isotherm equation for the adsorption of the electrolyte:
10
11
𝜈
𝑜𝑟𝑔
N
multiplied by the total number
)
(17)
a −N
(18)
𝐾0,𝑐 𝑎𝑐
𝑁
𝜃 = 𝑁 = 1+ 𝐾0,𝑐𝑒𝑙𝑎𝑐
𝑎
(19)
𝑒𝑙
where the electrolyte activity in water is defined in the standard scale of concentrations:
12
𝜈
𝐶𝛾 𝜈+ +𝜈−
13
𝑐
𝑎𝑒𝑙
= 𝜈++ 𝜈−𝜈− (𝐶 0 )
14
The resulting equilibrium constant is expressed as:
15
𝐾 0,𝑐 = (𝐶 0 𝑉𝑎 )𝜈++𝜈− 𝑒 −𝛽Δ𝐺
16
Thus, the standard equilibrium constant is a pure number, because the volume of the site Va is
17
compensate by the reference concentration of the standard state C0 (generally C0 corresponds
18
to 1 mol.L-1 = 1000 NA m-3). The practical constant 𝐾 = 𝑉𝑎 𝜈++𝜈− 𝑒 −𝛽Δ𝐺 is proportional to a
19
volume to the power the total valency of the electrolyte. Δ𝐺 0 = 𝐸0 is the variation of the
20
Gibbs free energy when an electrolyte “molecule” is moved from the water phase to the polar
21
core of the aggregates. If the activity scale of the water phase is defined as a function of the
22
molality 𝑚, the adsorption equation still corresponds to a Langmuir law:
23
𝜃 = 𝑁 = 1+ 𝐾0,𝑚𝑒𝑙𝑎𝑚
(20)
0
(21)
0
𝑁
𝑎
𝐾0,𝑚 𝑎𝑚
𝑒𝑙
(22)
1
𝑐
where the activity of the electrolyte in the standard scale of concentrations 𝑎𝑒𝑙
is replaced by
2
𝑚
the activity in the standard scale of molalities 𝑎𝑒𝑙
:
3
𝑚
𝑎𝑒𝑙
= 𝜈++ 𝜈−𝜈− (𝑚0 )
𝑚𝛾 𝜈+ +𝜈−
𝜈
(23)
4
where 𝑚0 is the reference molality that defines the standard scale (typically 𝑚0 = 1 mol.kg-1).
5
The corresponding equilibrium constant reads:
6
𝐾 0,𝑚 = (𝑚0 𝜌0 𝑉𝑎 )𝜈++𝜈− 𝑒 −𝛽Δ𝐺
7
where 𝜌0 is the mass density of the pure solvent. In the case of water at 25 °C, 𝑚0 𝜌0 = 𝐶 0
8
and the two scales are the same (𝐾 0,𝑚 = 𝐾 0,𝑐 ).
0
(24)
9
10
11
Calculation of accessible volumes Va
12
Va represents the volume of one site, which corresponds to the electrolyte accessible domain.
13
It is the volume occupied by one acid molecule and its hydration water. Considering that the
14
hydration state of each acid is the same in the organic phase than in aqueous phase, Va can be
15
estimated from the amount of extracted water by one acid molecule:
16
𝑉𝑎 = 𝑉𝑎𝑐𝑖𝑑 + 𝑁𝑤 ∙ 𝑉𝑤
17
where Vacid and Vw are the partial molar volumes of the acid and water, respectively (taken at
18
the concentrations in the aggregates). Nw is the number of water molecules co-extracted with
19
one acid molecule at saturation of isotherm:
20
𝑁𝑤 = [𝐴𝑐𝑖𝑑]
21
The amount (in moles) of extracted acid in 1 kg of extracted H2O is:
22
𝑛𝑎𝑐𝑖𝑑 =
23
And the corresponding mass (in g) of extracted acid in 1 kg of extracted H2O is:
24
𝑚𝑎𝑐𝑖𝑑 =
[𝐻2 𝑂]𝑜𝑟𝑔@𝑠𝑎𝑡
𝑜𝑟𝑔@𝑠𝑎𝑡
1
𝑁𝑤 ∙
𝑀(𝐻2 𝑂)
1000
𝑀(𝑎𝑐𝑖𝑑)
𝑁𝑤 ∙
𝑀(𝐻2 𝑂)
1000
(27)
(28)
(29)
(30)
1
Therefore the total mass (in g) of extracted solute containing 1 kg H2O is:
2
𝑚𝑠𝑜𝑙𝑢𝑡𝑒 = 𝑚𝐻2 𝑂 + 𝑚𝑎𝑐𝑖𝑑 = 1000 +
3
The corresponding volume of extracted solute containing 1 kg H2O is:
4
𝑉𝑠𝑜𝑙𝑢𝑡𝑒 = 𝑚𝑠𝑜𝑙𝑢𝑡𝑒 /𝜌𝑠𝑜𝑙𝑢𝑡𝑒
5
where ρsolute is the solute’s density in the aggregate’s core. It can be determined from
6
density/concentration tables, knowing the concentration of acid in the cores. For 1 mol acid,
7
this concentration is:
8
𝐶𝑎𝑐𝑖𝑑 = 𝑉(1 𝑚𝑜𝑙 𝑎𝑐𝑖𝑑)+𝑁
9
Finally Va (volume of acid and water per one acid molecule) can be determined. In an
𝑀(𝑎𝑐𝑖𝑑)
𝑁𝑤 ∙
(32)
𝑛(1 𝑚𝑜𝑙 𝑎𝑐𝑖𝑑)
𝑤 ∙𝑉(1 𝑚𝑜𝑙 𝐻2 𝑂)
=
10
extracted solute containing 1 kg H2O:
11
𝑉𝑎 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑒
=𝜌
𝑎𝑐𝑖𝑑 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠
12
This finally leads to:
13
𝑉𝑎 = (1 + 𝑁
𝑉
𝑀(𝑎𝑐𝑖𝑑)
(31)
𝑀(𝐻2 𝑂)
1000
1
(33)
𝑀(𝑎𝑐𝑖𝑑)
𝑀(𝐻2 𝑂)
+𝑁𝑤 ∙
𝜌(𝑝𝑢𝑟𝑒 𝑎𝑐𝑖𝑑)
𝜌(𝐻2 𝑂)
𝑚𝑠𝑜𝑙𝑢𝑡𝑒
𝑠𝑜𝑙𝑢𝑡𝑒 ∙𝑁𝑎 ∙𝑛𝑎𝑐𝑖𝑑
=
𝑚𝑠𝑜𝑙𝑢𝑡𝑒
𝜌𝑠𝑜𝑙𝑢𝑡𝑒 ∙𝑁𝑎 ∙
(34)
1000
𝑁𝑤 ∙𝑀(𝐻2 𝑂)
𝑁 ∙𝑀(𝐻2 𝑂)
) ∙ ( 𝜌𝑤
𝑤 ∙𝑀(𝐻2 𝑂)
𝑠𝑜𝑙𝑢𝑡𝑒 ∙𝑁𝑎
)
(35)
14
15
M(acid)
Nw
msolute
[acid]@100%
[acid]core
ρacid@100%
g/mol
HCOOH
ρsolute
Va (nm3)
17.95
1.16
0.091
1.5
10.79
1.17
0.149
16.68
2.01
13.32
1.76
0.118
4.50
23.87
1.5
16.70
1.42
0.095
4.02
19.04
1.87
11.78
1.57
0.138
@sat
(kg/kg H2O)
(mol/L)
(mol/L)
46
1.0
3.55
26.52
1.22
HCl
36.5
3.8
1.53
41.1
HClO4
100.5
1.4
4.98
HNO3
63
1.0
H3PO4
98
1.8
16
17
Appendix Table. Numerical values required for the calculation of Va, the accessible volume
18
to each anion in the aggregate