1 Liquid-liquid extraction of acids by a malonamide:
2 II- Anion specific effects in the aggregate-enhanced extraction isotherms
7
8
9
10
11
12
3
4
5
6
13
14
Sandrine Dourdain
1,* , Christophe Déjugnat 2
, Laurence Berthon
3 , Véronique Dubois 1
,
Stéphane Pellet-Rostaing 1 , Jean-François Dufrêche 1 , Thomas Zemb 1
Appendix
Theoretical model for Langmuir-like adsorption of electrolytes
15 The Gibbs free energy of adsorption can be estimated from the salt extraction curves thanks to
16 a simple adsorption model generalising the Langmuir isotherms to electrolytes described in
17 this part.
18 aModel with a simple solute
19 In this paragraph, for the sake of simplicity, the electrolyte will be considered as a single
20 solute. The role of the dissociation will be taken into account in the next section. The organic
21 phase is modelled as a set of aggregates. From the solute point of view, there is no correlation
22 between the aggregates. Everything happens as if there were a set of N a
accessible volumes V a
23 to the solutes S corresponding to the polar cores of aggregates. The role of the molecularity
24
25 will be analysed below. The model is globally the one represented in Figure 3.
The calculations are made at the McMillan-Mayer level of description
11
(continuous solvent
26 model), which is formally equivalent to the calculation with an explicit solvent when exact
27 effective potentials between the solute particles are taken into account. The partition function
28
29 in the canonical ensemble reads:
Q =
1
N!
∫ dp
N dr N e −βNE
0 (1)

1 where N is the number of solute particles in the organic phase, 𝛽 = 1/𝑘
𝐵
𝑇 , r and p are the
2 position and the corresponding momentum of the solute particles in the N a
aggregates. E
0
is
3 the adsorption energy. The integration over the momentum is straightforward:
4 𝑄 =
1
𝛬 3𝑁 𝑁!
𝑒 −𝛽𝑁𝐸
0 ∫ 𝑑𝑟 𝑁
(2)
5 where Λ is the De Broglie length of the solutes. Within the continuous solvent model, Λ is an
6 effective quantity, which takes into account the solvation of the solute particles. The
7 configurational integral is:
8 ∫ 𝑑𝑟 𝑁 =
𝑁 𝑎
!
(𝑁 𝑎
−𝑁)!
𝑉 𝑎
𝑁
(3)
9 because each site can contain up to one solute particle. By derivation of the resulting free
10
11 energy 𝐹 = −𝑘
𝐵
𝑇 ln 𝑄 together with the Stirling approximation ln 𝑁! = 𝑁 ln 𝑁 − 𝑁 , we finally get the chemical potential
µ
of the solute particles in the organic phase:
12 𝛽𝜇 = 3 ln 𝛬 + 𝛽𝐸
0
− ln 𝑉 𝑎
+ ln (
𝑁 𝑎
𝑁
−𝑁
) (4)
13 The adsorbed solute particles are in equilibrium with the aqueous phase. The chemical
14 potential of the solute in this water phase reads:
15
16
17
18 𝜇 = 𝜇 0 + 𝑘
𝐵
𝑇 ln (
𝐶
𝐶 0
) (5) where 𝜇 0 = 𝑘
𝐵
𝑇 ln(𝛬 3 𝐶 0 ) is the standard chemical potential, 𝐶 0 the reference concentration that defines the standard scale of the aqueous solution (typically C 0 = 1.mol.L
-1 = 1000 N
A
m -
3
) and C is the concentration in the aqueous phase. Making equal the two expressions of the
19 chemical potential, we obtain the following adsorption curve:
20
𝑁 𝜃 =
𝑁 𝑎
=
𝐾
0 𝐶
𝐶0
1+𝐾 0 𝐶
𝐶0
(6)
21
22
23
The adsorption phenomenon corresponds to a Langmuir isotherm with the adsorption constant
𝐾 0 = 𝑉 𝑎 𝑒 −𝛽𝐸
′
(7) where

1
2
𝐸 ′ = 𝐸 0 + 3𝑘
𝐵
𝑇 ln 𝛬 − 𝜇 0
(8)
In equation 8,
E’
is related to the change of Gibbs free energy from the aqueous phase to the
3 organic phase, as can be understood from the following analysis. Indeed, the chemical
4 potential in the organic phase can also read
5
6 𝛽𝜇 = ln(𝐶 0 𝛬 3 ) + 𝛽𝐸 0 + ln (
(𝑁 𝑎
𝑁
𝑉 𝑎
)𝐶 0
∙
𝑁 𝑎
𝑁 𝑎
−𝑁
) = ln(𝐶 0 𝛬 3 ) + 𝛽𝐸 0 + ln (
𝑁
𝑉 𝑡𝑜𝑡
𝐶 0
) + ln 𝛾 (9) where we explicitly introduce the standard reference of concentration 𝐶 0
. The two first terms
7
8 ln(𝐶 0 𝛬 3 ) + 𝛽𝐸 0
correspond to the standard chemical potential term of the adsorbed solute particle, which is different from the one in the bulk phase ln(𝐶 0 𝛬 3 ) by the energy 𝛽𝐸 0
. The
9
10 next term ln (
𝑁
𝑉 𝑡𝑜𝑡
𝐶 0
) is nothing but the ideal activity of the solute particles in the total volume of the aggregates 𝑉 𝑡𝑜𝑡
= 𝑁 𝑎
𝑉 𝑎
. Finally the last term ln 𝛾 = ln (
𝑁 𝑎
𝑁 𝑎
−𝑁
) is the activity coefficient
11
12 arising from the saturation of the aggregates. Thus, the standard state we considered corresponds to the concentration 𝐶 0
with a solute behaviour similar to an infinitely diluted
13 system. It is the commonly used standard state of solution chemistry. Then the adsorption
14
15
16 constant becomes:
𝐾 0 = 𝐶 0 𝑉 𝑎 𝑒 −𝛽∆𝐺 0
(10) where ∆𝐺 0 = 𝐸
0
is the variation of the Gibbs free energy when a solute particle is moved
17
18
19 from the water phase to the polar core of the aggregates, i.e. the driving force for the extraction of one given solute in any industrial process. It should be noted that ∆𝐺 0
does not actually depend on the choice of 𝐶 0
because it corresponds to the infinite dilution limit. Thus
20 the simple model of solute adsorption in the aggregates yields to the following Langmuir
21 isotherm:
22 𝜃 =
𝑁
𝑁 𝑎
=
𝐾𝐶
1+𝐾𝐶
23
24 with
𝐾 = 𝑉 𝑎 𝑒 −𝛽∆𝐺
0
(11)
(12)

1 for which V a
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 bModel 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
+
𝐶 𝑎𝑡
+ 𝜈
−
𝐴 𝑛𝑖
= 𝐸 𝑙𝑒𝑐
10
11
12
(13) where 𝐶 𝑎𝑡
is the cation, 𝐴 𝑛𝑖
is the anion, and 𝐸 𝑙𝑒𝑐
is the neutral electrolyte.
In the bulk water phase, the chemical potential of the electrolyte reads: 𝜇 𝑎𝑞 𝑒𝑙
= 𝜇 0 𝑒𝑙
+ 𝑘
𝐵
𝑇 ln (
𝐶 𝜈+
+
(𝐶
𝐶 𝜈−
− 𝛾 𝜈+
+ 𝛾 𝜈−
−
0 ) 𝜈++𝜈−
) = 𝜇 0 𝑒𝑙
+ 𝑘
𝐵
𝑇 ln (𝜈 𝜈
+
+ 𝜈 𝜈
−
− (
𝐶𝛾
𝐶 0
) 𝜈
) = 𝜇 0 𝑒𝑙
+ 𝑘
𝐵
𝑇 ln 𝑎 𝑒𝑙
(14)
13
14
15
16
In equation 14, 𝐶
+
= 𝜈
+
𝐶 and 𝐶
−
= 𝜈
−
𝐶 are the concentrations of the cations and the anions, respectively, while 𝛾
+
and 𝛾
−
represent their respective activity coefficients. The concentration and the activity coefficient of the electrolyte are respectively 𝐶 and 𝛾 =
(𝛾 𝜈
+
+ 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
20 exactly 𝜈
+
cations and 𝜈
−
anions in any of the N occupied sites. The possibility of nonneutral aggregates is forbidden. The resulting canonical partition function is:
21 𝑄 = 𝑒
−𝛽𝑁𝐸0
Λ
3N+
+
Λ
3N−
−
(Nν
+
)!(Nν
−
)!
∫ 𝑑𝑟 𝑁
22 Now the resulting configurational integral reads:
23 ∫ 𝑑𝑟 𝑁 =
𝑁 𝑎
!
(𝑁 𝑎
−𝑁)!𝑁!
(Nν
+
)! (Nν
−
)! V
(ν
+ a
+ν
−
)N
(15)
(16)

1
2 because it is the configurational integral of one site V
(ν a
+
+ν
−
)N
multiplied by the total number of possibilities of choosing the 𝑁 occupied sites among 𝑁 𝑎
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 𝛽𝜇 𝑜𝑟𝑔 𝑒𝑙
= 3 ln Λ 𝜈
+
+ Λ 𝜈
−
− + 𝛽𝐸
0
− (ν
+
+ ν
−
) ln V a
+ ln (
N a
N
−N
) (17)
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 𝛽𝜇 0 𝑒𝑙
= 3 ln((𝐶 0 Λ 3
+
) 𝜈
+ (𝐶 0 Λ 3
−
) 𝜈
− ) (18)
9
10 we obtain the following Langmuir isotherm equation for the adsorption of the electrolyte:
𝑁 𝜃 =
𝑁 𝑎
=
𝐾
0,𝑐
1+ 𝐾 𝑎
0,𝑐 𝑐 𝑒𝑙 𝑎 𝑐 𝑒𝑙
(19)
11 where the electrolyte activity in water is defined in the standard scale of concentrations:
12
13 𝑎 𝑐 𝑒𝑙
= 𝜈 𝜈
+
+ 𝜈 𝜈
−
− (
𝐶𝛾
𝐶 0
) 𝜈
+
+𝜈
−
(20)
14
15
16
17
18
19
20
The resulting equilibrium constant is expressed as:
𝐾 0,𝑐 = (𝐶 0 𝑉 𝑎
) 𝜈
+
+𝜈
− 𝑒 −𝛽Δ𝐺
0
(21)
Thus, the standard equilibrium constant is a pure number, because the volume of the site V a
is compensate by the reference concentration of the standard state C
0
(generally C
0
corresponds to 1 mol.L
-1
= 1000 N
A
m
-3
). The practical constant 𝐾 = 𝑉 𝑎 𝜈
+
+𝜈
− 𝑒 −𝛽Δ𝐺 0
is proportional to a volume to the power the total valency of the electrolyte. Δ𝐺 0 = 𝐸
0
is the variation of the
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
23 molality 𝑚 , the adsorption equation still corresponds to a Langmuir law: 𝜃 =
𝑁
𝑁 𝑎
=
𝐾
0,𝑚 𝑎 𝑚 𝑒𝑙
1+ 𝐾 0,𝑚 𝑎 𝑚 𝑒𝑙
(22)

1
2
3
4 where the activity of the electrolyte in the standard scale of concentrations 𝑎 𝑐 𝑒𝑙
is replaced by the activity in the standard scale of molalities 𝑎 𝑚 𝑒𝑙
: 𝑎 𝑚 𝑒𝑙
= 𝜈 𝜈
+
+ 𝜈 𝜈
−
− ( 𝑚𝛾 𝑚 0
) 𝜈
+
+𝜈
−
(23)
where 𝑚 0
is the reference molality that defines the standard scale (typically 𝑚 0
= 1 mol.kg
-1
).
5 The corresponding equilibrium constant reads:
6
7
8
𝐾 0,𝑚 = (𝑚 0 𝜌 0 𝑉 𝑎
) 𝜈
+
+𝜈
− 𝑒 −𝛽Δ𝐺
0
(24) where 𝜌 0
is the mass density of the pure solvent. In the case of water at 25 °C, 𝑚 0 𝜌 0 = 𝐶 0 and the two scales are the same ( 𝐾 0,𝑚 = 𝐾 0,𝑐
).
9
10
11
12
13 Calculation of accessible volumes V a
14 V a represents the volume of one site, which corresponds to the electrolyte accessible domain.
15 It is the volume occupied by one acid molecule and its hydration water. Considering that the
16 hydration state of each acid is the same in the organic phase than in aqueous phase, V a
can be
17 estimated from the amount of extracted water by one acid molecule:
18 𝑉 𝑎
= 𝑉 𝑎𝑐𝑖𝑑
+ 𝑁 𝑤
∙ 𝑉 𝑤
(27)
19 where V acid
and V w
are the partial molar volumes of the acid and water, respectively (taken at
20 the concentrations in the aggregates). N w
is the number of water molecules co-extracted with
21 one acid molecule at saturation of isotherm:
22 𝑁 𝑤
=
[𝐻
2
𝑂] 𝑜𝑟𝑔@𝑠𝑎𝑡
[𝐴𝑐𝑖𝑑] 𝑜𝑟𝑔@𝑠𝑎𝑡
(28)
23
24
The amount (in moles) of extracted acid in 1 kg of extracted H
2
O is: 𝑛 𝑎𝑐𝑖𝑑
=
1
𝑁 𝑤
∙
𝑀(𝐻2𝑂)
1000
25 And the corresponding mass (in g) of extracted acid in 1 kg of extracted H
2
O is:
(29)

1 𝑚 𝑎𝑐𝑖𝑑
=
𝑀(𝑎𝑐𝑖𝑑)
𝑁 𝑤
∙
𝑀(𝐻2𝑂)
1000
2 Therefore the total mass (in g) of extracted solute containing 1 kg H
2
O is:
3 𝑚 𝑠𝑜𝑙𝑢𝑡𝑒
= 𝑚
𝐻
2
𝑂
+ 𝑚 𝑎𝑐𝑖𝑑
= 1000 +
𝑀(𝑎𝑐𝑖𝑑)
𝑁 𝑤
∙
𝑀(𝐻2𝑂)
1000
(30)
(31)
4 The corresponding volume of extracted solute containing 1 kg H
2
O is:
5 𝑉 𝑠𝑜𝑙𝑢𝑡𝑒
= 𝑚 𝑠𝑜𝑙𝑢𝑡𝑒
/𝜌 𝑠𝑜𝑙𝑢𝑡𝑒
(32)
6 where ρ solute
is the solute’s density in the aggregate’s core. It can be determined from
7 density/concentration tables, knowing the concentration of acid in the cores. For 1 mol acid,
8 this concentration is:
9 𝐶 𝑎𝑐𝑖𝑑
= 𝑛(1 𝑚𝑜𝑙 𝑎𝑐𝑖𝑑)
𝑉(1 𝑚𝑜𝑙 𝑎𝑐𝑖𝑑)+𝑁 𝑤
∙𝑉(1 𝑚𝑜𝑙 𝐻
2
𝑂)
=
1
𝑀(𝑎𝑐𝑖𝑑) 𝜌(𝑝𝑢𝑟𝑒 𝑎𝑐𝑖𝑑)
+𝑁 𝑤
∙
𝑀(𝐻2𝑂) 𝜌(𝐻2𝑂)
(33)
10 Finally V a
(volume of acid and water per one acid molecule) can be determined. In an
11
12 extracted solute containing 1 kg H
2
O:
𝑉 𝑎
=
𝑉 𝑠𝑜𝑙𝑢𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑐𝑖𝑑 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠
= 𝑚 𝑠𝑜𝑙𝑢𝑡𝑒 𝜌 𝑠𝑜𝑙𝑢𝑡𝑒
∙𝑁 𝑎
∙𝑛 𝑎𝑐𝑖𝑑
= 𝜌 𝑠𝑜𝑙𝑢𝑡𝑒 𝑚
∙𝑁 𝑠𝑜𝑙𝑢𝑡𝑒
1000 𝑎
∙
𝑁𝑤∙𝑀(𝐻2𝑂)
13 This finally leads to:
14 𝑉 𝑎
= (1 +
𝑀(𝑎𝑐𝑖𝑑)
𝑁 𝑤
∙𝑀(𝐻
2
𝑂)
) ∙ (
𝑁 𝑤
∙𝑀(𝐻
2 𝜌 𝑠𝑜𝑙𝑢𝑡𝑒
𝑂)
)
∙𝑁 𝑎
15
(34)
(35)