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Statistical mechanical treatment of reactive solvent extraction
Chemical Physics ELSEVIER
Chemical Physics 220 (1997) 53-61
Statistical mechanical treatment of reactive solvent extraction M. L u k h e z o
a,
L.J. D u n n e
a,b,*,
B.G. Reuben
a,
M.S. Verrall c
a School of Applied Science, South Bank University, London SE10AA, UK b School of Chemistry and Molecular Sciences, UniL'ersity of Sussex, Falmer, Brighton BN1 9Q J, UK c SmithKline Beecham Pharmaceuticals, Chemotherapeutic Research Centre, Brockham Park, Betchworth, Surrey RH3 7AJ, UK
Received 3 December 1996; in final form 18 April 1997
Abstract
A statistical mechanical lattice model of a reactive solvent extraction process involving two phase liquid-liquid ion exchange is presented. It is applied to literature data on the extraction of aliphatic carboxylic acids by quaternary ammonium salts. The Helmholtz free energy of the system is constructed and minimised subject to the constraints of mass balance and electroneutrality. The equations defining the equilibrium conditions have been solved numerically for a range of interaction parameters and hence the equilibrium concentrations of the various partitioned species are obtained. These interaction parameters are compared with experiment. The intuitive notion that the extent of partition is dominated by electrostatic sotute-extractant interactions is shown to neglect various other interactions that may, in some systems, be important. © 1997 Elsevier Science B.V.
1. Introduction
Reactive solvent extraction of high value materials is practised in the metallurgical and chemical industries, but the underlying theory is poorly understood. The development of extraction processes for complex organic compounds such as those encountered in the pharmaceutical and biotechnological industries would benefit from such a theory. Although there is a considerable body of work on conventional solvent extraction based mainly on linear solvation energy relationships [1-4], the important problem of reactive solvent extraction has received little theoretical attention [5-7]. Reactive solvent extraction usually involves a liq-
* Corresponding author. Fax: (+44-171) 815 7999.
uid-liquid ion exchange in a two-phase partitioning process. In one phase, usually aqueous, there is a solute comprising a relatively lipophilic ion and hydrophilic counter-ion. The second phase is a waterimmiscible organic solvent containing a dissolved organic extractant, which may be virtually insoluble in water and which may dissociate to produce a hydrophilic ion and lipophilic counter-ion. The extractant contributes to the partitioning of the aqueous solute by ion pairing to the lipophilic solute ion at the aqueous/organic phase interface and solubilising it into the organic phase. The electrostatic charge balance of the system is preserved by the solute counter-ion electrostatically binding reversibly to the hydrophilic extractant counter-ion and transferring it into the aqueous phase. The aim of this paper is to present a primitive lattice model of a reactive solvent extraction process. To understand the extraction of aliphatic acids by
0301-0104/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0301-0104(97)00122-5
54
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
amines is fundamentally important since it is a model of more complicated but related biochemical processes such as those involving amino acids and peptides. We have previously modelled aspects of solvent extraction and related processes [8] and interactions in electrolyte solutions [9,10], and have reported experimental work governing such systems [ 1 l-13]. The extraction of carboxylic acids (ethanoic, propanoic and butanoic) with quaternary ammonium salts is an example of a reactive solvent extraction process investigated by Yang et. al. [14] who focused on the effects of pH, extractant (the tertiary amine, trioctylamine, as well as the quaternary, methyltrioctylammonium chloride), solvent (kerosene and 2octanol), and carboxylic acid chain length. By analogy with other anion exchange and hydrometallurgical processes, the extraction should improve with increasing pH. The qualitative argument is as follows; as the protons in the system are depleted, encouraging dissociation of the carboxylic acids, more carboxyl anions are freed for ion-pairing with the quaternary ammonium cations in the organic phase. Furthermore, as the aliphatic chain length of a carboxylic acid increases, the lipophilic nature of the molecule also increases, encouraging its extraction into the organic phase. The results of Yang et al. [14] shown in Fig. 1 do not fully satisfy these expectations. The extraction of carboxylic acids indeed increases with aliphatic chain length in the order ethanoic acid < propanoic acid < butanoic acid, but the extraction profiles exhibit the opposite trend to that expected with changing pH; extraction apparently improves with falling pH. In this paper we have parameterised our model against the literature data of Yang et al. [14] shown in Fig. 1 for the reactive solvent extraction of aliphatic carboxylic acids by methyltrioctylammonium chloride. We have thereby identified the important effective interactions governing the extraction process as a part of our general effort to understand more fully effective interactions in condensed media [9]. A weakness in our approach, which we hope to remedy in further work, is that all ionic species are assumed to be dissociated. Hence the model would not be applicable to extraction by tertiary amines. The model only treats ion pairing, which is likely in the organic phase, in an approximate way. We have not found an experimental system which accurately
corresponds with our model, but the above data of Yang et al. [14] for the extraction of aliphatic carboxylic acids by quaternary ammonium salts goes a long way towards this goal. Our model is crude but provides a useful starting point for molecular theories of this important topic.
,
°
j
,
°
o
i.
o
•
i
,
,
•
¢.
,, II
EthanoicAcid
o.
i:g @ 0
*
"
'
I
O.
J
a
a
2.
I
a
t
I
J
4.
'
a
"
6.
J
a
"
l
8.
10.
Equilibrium pH o b o
Propnoic
0
Acid
0
0
0
o.
2.
4.
6.
a.
io.
Y~luilibrium p H "
'
'
'
'
'
'
"
'
'
'
"
'
"
"
'
"
'C"
Butanoic A c i d
c~
o
O.
2.
4,. E q u i l i b r i u m
6.
O.
10.
pH
Fig. 1. Experimentaldata of Yang et al. for the extraction of aliphatic carboxylic acids (ethanoic, propanoic, butanoic) by methyltrioctylammoniumchloridereproducedfrom Ref. [14]. The extraction yield rises in an acidic aqueous phase and also with chain length. CopyrightAmericanChemical Society.
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
2. The primitive model To obtain a mathematically tractable problem, the components of the system have been reduced to a minimum number, contained within a fixed and incompressible volume. These are schematically represented in Fig. 2. In this simplification, we consider an initial state of solute ions S;,q and S~q ions in the aqueous phase (assumed dissociated, but with the possibility of ion pairing to give a neutral molecule) and an ion paired extractant E+Eorg in the organic phase, which may dissociate. When these two solutions are brought into contact, an ion exchange process may occur with the S~q ions binding to E~rg to form an electrostatically neutral complex soluble in the organic phase. This is balanced by the Eorg ion migrating to the aqueous phase as Ef,q where it is solvated. The process can be written SAq -'}-E+Eorg ~- EAq + E+Sorg .
(1)
The E + ions are assumed to be unable to cross the interface into the aqueous phase, and similarly the S + ions are insoluble in the organic phase. Only the S - and E - ions can cross the phase boundary, and the equilibrium concentrations of these in each phase are to be calculated by the model. The total number of each species in the system is known and fixed. We have considered here the exchange of anions but a similar treatment applies to cations, mutatis mutandis. The aqueous and organic solvents are treated here in a phenomenological way in which they provide an effective solvating medium in which the ions described above are dissolved and at this stage of development we disregard the detailed structure of
Aqueous (Aq)
Organic (Org)
@@ @ @@
55
aqueous phases at various pH values. In future work we hope to treat the aqueous phase more fundamentally. We consider four species in each phase. In the aqueous phase are Mi,Aq ions of E - , M2,Aq ions of S - , M4, Aq ions of S ÷ and MAq molecules of water. In the organic phase are Ml,org ions of E - , M2,0rg ions of S , M3,0rg ions of E ÷ and Morg molecules of organic solvent. Conditions for conservation of material are Ma, Aq ~ M2,0rg -4- M2,Aq ,
(2)
and M3,org ~- Ml,org + M1, Aq"
(3)
Since the total numbers of E + and S + ions are known, knowledge of the numbers of these species in either phase completely determines the system, so that there are two unknowns in the problem. The conditions for electroneutrality are M4,Aq ~ MZ,Aq + MI,Aq ,
(4)
and M3,org ~ Ml,org + Mz,org
(5)
Eqs. (2) to (5) serve as constraints on the total free energy minimisation to be discussed below.
3. The lattice model Lattice models of fluids have been extensively discussed in the literature [15-20]. As applied to our system, the lattice model treats each of the two phases separately. We consider a 3-dimensional cubic lattice of sites each with coordination number Z = 6 and which may be occupied by an ion or molecule of four possible different types. Nearest neighbour interactions are assumed. All the lattice sites are assumed to be occupied. The number of sites, N~, is for each phase N, = E Mi.
(6)
i Fig. 2. Schematic representation of the partitioning of species in a primitive reactive solvent extraction system. The left side denotes an aqueous phase while the right side denotes the organic phase. S- is the solute ion with an S + counter-ion while E- is the extractant anion whose counter-ion is E +.
Each of the species is assumed to occupy a single lattice site. At the beginning of an extraction experiment the aqueous phase contains the water molecules, S + ions and S - ions only. The organic phase con-
56
M. Lukhezo et a L / Chemical Physics 220 (1997) 53-61
tains the solvent, E + ions and E - ions only. No mixing of organic solvent and water is assumed to occur. The Helrnholtz free energy [21], F, of a single phase is defined as Q,
F= -kTln
(7)
where k is the Boltzmann constant, T is the absolute temperature, and Q is the canonical partition function. The total free energy of the system is the sum of the free energies of each phase, hence this equation is applied to each phase separately. The equilibrium state of the system is located by use of the method of the maximum term [21] in which the total free energy of the whole system is minimised which is achieved when chemical potentials of like species in each phase are equal. A single phase will contain four different species of molecular quantities M 1, M 2, M 3 and M 4. The canonical partition function, Q, takes into account all the possible permutations of these species on the N~ lattice sites. In a random mixing approximation the free energy is given by F = ~c - k T In
(~Mi)!
MI !Mz !M3 !M4 !
,
(8)
4. The configuration energy and interaction parameters For a three-dimensional cubic lattice, the possible types of nearest neighbour interaction are described by the parameters Ji,j, where i = 1 to 4 and j = 1 to 4. The Ji,j values in the aqueous phase will differ from those in the organic phase, due to the differences in screening environments of the different bulk media. Such effective interaction parameters are not the true bare interaction between species but mimic a complicated set of free energies of interaction in the solution and implicitly attempt to allow for all kinds of nearest neighbour interactions between species in the system. Consider as an example the interaction between species of types 1 and 2 where J~,2 is the 1,2 interaction parameter. In a random mixing approximation the energy of a type 2 species with all other type 1 species is given by ZJ,.z( M , / ~ )
,
(13)
where ( M J N s) is the probability that the neighbouring site is occupied by a type 1 species. The sign convention for the interaction parameters is a negative value for an attractive interaction, a positive value for a repulsive interaction. The total number of 1,2 type interactions is then given by
where e c is the effective configurational interaction energy discussed in the next section. Applying Stirling's theorem
ZJ1.2(M,/N~) M 2 .
In X ! = X In X - X (for large X ) ,
The number of 1,2 type interactions per site becomes
(9)
we obtain from Eq. (7)
ZJ1,2 ( ml//Ns ) ( M2//Ns ) = ZJ1,2 PI P2.
(14)
(15)
Similarly, the energy of interaction between a species i and all the other i species is + k T Y ' ~ ( M i In Mi).
(10)
i
(11)
i which may be written as the Helmholtz free energy per site, for a given phase f c = F = ~ c / N s + kTY'~ ( Pi In Pi) . i
(16)
so that the total (i,i) interaction is
Defining the variable P,. = M i / N S gives F = e c + kTY'~( M i In P i ) ,
Zli.iP i ,
(12)
ZJi.iPi( M i / 2 ) .
(17)
The factor 1 / 2 in Eq. (17) prevents interactions between any pair of the same species being counted twice. Expression (15) can be written as the energy per site given by ZJi,jPiPj ,
(18)
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
57
which from Eq. (25) gives
or
ZJi,iPi2/2,
(19)
when i = j . Combining Eq. (17) and Eq. (18) produces the general expression for the configurational (i,j) interaction energy per site given by
ZIi.jPiPJ(1 + 6i.j),
(20)
where 6i,j is the delta function. Substitution of expression (20) for ec/N s, in Eq. (12) gives the Helmholtz free energy per site, for each side
fc =kT~-,Piln Pi + ZY'~Ji,jPiPj/( 1 + ai,j)" (21) i
i<~j
~3F = ~ ( ( OFAq//OMi, Aq )
-- (OForg/OMi,Org) ) 8 Mi, Aq"
(27)
Since all the variations in the species numbers in Eq. (27) may be made independently the condition for the free energy to be at a minimum in the whole system, for all species, is expressed as
8F/SMi, Aq = 0,
(28)
and is only true if the chemical potentials of the aqueous and organic phases are equal, that is
OFAq/OMi. Aq = OForg/OMi,org
(29)
or
5. Equilibrium conditions The condition for the system at equilibrium, as mentioned earlier, is that the free energy of the total system must be at a minimum. To simplify the treatment, we shall assume both phases to have equal numbers of sites while noting that generalisation is straightforward. Any variation of the numbers of species in one phase by transfer to the other produces an incremental change in the Helmholtz free energy of the whole system 8F = 8FAq 4- 8Forg.
(22)
Expanding each term on the right hand side of Eq. (22), for a fixed volume system, according to the chain rule gives 8FAq = E (OFAq/OM,,Aq)~)Mi, Aq
(30)
/£i, Aq = ~Li.Org ,
(23)
i
which may be expressed as Ofc. Aq/OPi =Ofc" Org/OP, .
(31 )
Differentiation of Eq. (21) gives
Ofc/OPi=kr(1 + In Pi) +zY'~Ji.jP j.
(32)
i<~j Considering both the aqueous and organic phases of the system together, at equilibrium, Eq. (30) can be re-written as Ofc' Aq/OPi --
Ofc,org/OPi = O.
(33)
Thus a combination of pairs of equations from the system of equations (32) gives the equilibrium relation
kT In( Pi, Aq/Pi, Org) + Z E ( J i , j ; A q P j , Aq--Ji,j:orgPj, org)=O.
(34)
J
and ~gorg = E (Ogorg//OMi,org)~Mi,org , i
(24)
and substitution of these equations into Eq. (22) gives
8F = E ((OFAq/OMi,Aq)SMi, Aq i 4- (OForg/OMi, Org )~ Mi, Org )"
(25)
From Eq. (2) and Eq. (3), the material balance of any species M~ traversing the phase boundary requires
~Mi, Aq = -- 8Mi. Org'
(26)
The aim of the model is to calculate the numbers of species of the four unknown components E~,q, SAq, Eorg and Sorg at equilibrium. However, since EAq and Eorg, and Sf,q and Sorg are conjugates it is necessary to calculate the value of just one of each pair and then to apply Eq. (2) and Eq. (3). A pair of simultaneous, non-linear equations results
ln( P,,Aq/P|,org) + ( Z / k T ) E(JI,j;AqPj, Aq -- Jl,j;Org J
Pj, org) = O,
(35)
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
58
In( P:, Aq/P2, Org)
OF/Oy = ln( P2. ~q/P2, Org) + ( Z/kT)
+ ( Z / k T ) E ( J2,j;AqPj, Aq -- J2,j;orgPj,org) = O.
× E(J2,:AqPj.Aq-
J
J2.j:o, Pj.o, )
J
(36)
(43)
In order that the solutions obtained for the above equations represent a minimum there is a subsidiary condition that the Hessian matrix [24] of second mixed partial derivatives of the free energy be positive definite. That is the eigenvalues of the matrix
The charge balance constraint is implemented as
( H'I
G(P,.o,g,P2,org)=P,,org+P2,o~g-C,
H21
Hi2 ) H22
(37) '
should both be positive which was checked by straightforward manipulation and diagonalisation of the above matrix. The appropriate matrix elements are
nii= kT/Pi. Aq + kT/Pi,org + Z(Jii,Aq + Jii,o~g) (i = 1,2),
selecting P~,org and P2,org as the quantities to be solved for, which then becomes
where c is a constant and, in this case, equal from Eq. (5) to P3, org giving the last terms in the left hand side of Eq. (40) and Eq. (41) as
A(~G/~x) = A(1)G/OPI,o~g) = A,
(46)
A(OG/Oy) = A(OG/OPz,org )
(39)
to give with Eq. (45) the set of three simultaneous, non-linear equations
= A,
(47)
ln(P,,Aq/P,.o,g) + (Z/kT) Y'~ (J,,j;AqPj, Aq -- S,,j:OrgP/,org) J
+ A = 0,
(48)
ln(Pz, Aq/P2.org)
6. Electroneutrality constraints Up to this point, the material balance (Eq. (2) and Eq. (3)) has been included in the minimisation of the free energy but the charge balance (Eq. (4) and Eq. (5)) has not been included. This constraint is readily imposed using Lagrange's method of undetermined multipliers [21]. A function F(x, y) minimised subject to a constraint G(x, y) = 0 yields the equations (40)
(41) where A is an undetermined multiplier. The appropriate first terms in the left hand side of Eq. (40) and Eq. (41) are
aF/Ox = In(P,. Aq/Pl,Org)
(45)
(38)
Eq. (35) and Eq. (36) must be solved numerically subject to the electroneutrality condition described in the next section and the positive definiteness of the Hessian matrix (37).
(OF/Ox) + A(aG/Ox) = 0, (OF/Oy) + A(OG/Oy) = 0,
(44)
and
and H21 = H,2 = Z( J,2, Aq + J,2,Org) •
G(x, y) = G(P,,org, P2,org) = 0,
+ (Z/kT)
X E(JI,j;AqPj, A q - Jl,j;orgPj,org), J
(42)
+ ( Z / k T ) ~.~ (J2,j:AqPj, Aq -- J2,j;orgPj,org) J
+ A = 0,
(49)
and PI,org + P2,org - c = 0 ,
(50)
thus explicitly including the charge balance in the free energy minimisation. Eqs. 48 to 50 were solved numerically for various values of the interaction parameters to yield the various numbers of species in the organic and aqueous phases. Since the constraint (50) is linear, the positive definiteness of the Hessian matrix continues to signify a minimum [25]. An important implication of the electroneutrality constraint and the random mixing approximation is that, in the lattice model, the free energy of the system is independent of the simple electrostatic interaction between ions. Hence, if some members of
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
the set of interaction parameters {J/j} are set to zero the remaining members of the set measure deviations from Coulomb's law for idealised point charges and other contributions to the interactions such as those arising from solvation effects.
7. Results
and
59
the view that the dominant driving force for the reactive solvent extraction of carboxylic acids discussed by Yang et al. [14] is the attractive character of the effective aqueous phase interaction between the H + cation (the counterion to the aliphatic acid in the aqueous phase) and the Cl- anion (the counterion to the extractant in the organic phase), i.e. Aq J(Cl-, n +) or Aq J ( E - , S+). In acidic solution, there is an abundance of free protons available for solvating the C l - anions transferred from the organic phase. The attraction between the two is very strong and hence we predict that this process is exothermic. To balance this process, the aliphatic acid anions complex with the extractant cation and are transferred into the organic phase. Fig. 3 illustrates the derived relationship between the interaction parameter Aq J ( C l - , H + ) and equilibrium pH. The equilibrium pH is derived from Fig. 1 and the interaction parameters A q J ( C I - , H +) were calculated by fitting the numerically computed output from the model to the experimental data of Yang et al. [14] given in Fig. 1. The conditions for extraction are most favourable at low pH, and this is reflected in the generally more attractive interaction parameter values. A prediction, therefore, is that the extraction should not occur in basic solution. This is borne out by experiment. The fact that the exchange does not
discussion
The model allows investigation of the influence on the selective extraction of particular sets of interaction parameters and where the complement is set to zero. As discussed above, electroneutrality exists in the extraction system at equilibrium and, with the complement of interactions set to zero, the output of the model reflects deviations from Coulomb's law for idealised point charges such as those due to ion size effects and dipolar and dispersion contributions to the effective interactions. The effective parameters in our model mimic the true and undoubtedly complicated interactions in an acidic solution and allow us to focus on identifying the features which drive the extraction. We have chosen to isolate the dominant interactions governing the extraction and, although it may prove possible to perform ab initio calculation of interaction parameters, at this stage we have proceeded with estimates of such energies. By making a parameter search we have come to
10 4- 4-
OrgJ(R3NCH3-,RCOO-)/kT = -1 ÷
5 ÷
4-
4-
I
I
I
4-
+4-
4-
4-4-
4-
¢
I
I
0
v
*~
@.
I =
¢
g~
¢
-5
¢ O
2 2
4-
4-
4-4+
o
.0-
.~
0
,O
¢
0
@
O
+ ethanoic acid
I
-lO ..
--
.
.
o p r o p a n o i c acid
.
- butanoic acid
-15
0
I
2
3
4
5
6
7
8
Equilibrium pH Fig. 3. Theoretical curves showing the equilibrium pH dependence of the Aq J(E-, S +) interaction parameter.
60
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
occur in basic media, against intuitive notion, is evidence that the above hypothesis is essentially correct. Rising pH causes a lowering of the H ÷ cations available for solvating the C1- anions from the organic phase, thereby effectively decreasing the driving force for the extraction process. The relationship between the interaction parameter Aq J(CI-, H ÷) and the distribution coefficient k d is illustrated in Fig. 4. The k d data have been derived from Fig. 1 ( k d = [RCOO-]org/[RCOO-]A q) and the interaction parameter ( J / k T ) has been calculated by the model for ethanoic acid ( + ) , propanoic acid ( [ ] ) and butanoic acid ( v ) . The variable interaction parameter for the three different cases is the organic phase interaction parameter between the carboxylic acid anion and the extractant cation, i.e. Org J(R 3NCH ~, R C O O - ) or Org J(E +, S - ), and this differs by an amount of - 2 between adjacent cases thus illustrating the effect on extraction of the strength of affinity between the extractant cation and the carboxylic acid anion. Fig. 4 further illustrates the chain length dependence by the range of k 0 values covered by each carboxylic acid. Ethanoic acid ( + ) covers the lower range of k d values, butanoic acid ( v ) covers the higher range (the gap in the data ( k d : 4 to 8) coincides with the rapid drop in the (reverse) sigmoidal curve of Fig. l) and propanoic acid ([~) is 15
OrgJ(R3NCH3-,RCOO-)/=kT -1,-3&-5
lO[-~+, 0
+
ethanoic acid
///
['q : [~~i!! ~ / i
approximately in the middle; the data for all acids approximately cover the same pH range in Fig. 1. The interaction parameter (Aq J(CI-, H + )) values for the acids follows a similar trend, with ethanoic acid at the more repulsive end and butanoic acid at the more attractive end of the range of values. The chain length dependence is attributed in our model to the increased effective interaction with chain length of the R3NCH ~- and RCO 2. If the attraction roughly rises by a small number of CH 2CH 2 interactions (say 5) with an increase in chain length of one ( - C H 2 - ) moiety then the C H 2 - C H 2 interaction at equilibrium separation is estimated to be of the order of - 1/100 eV. Hence we have found that the chain length dependence of the extraction shown in Fig. 1 in the experimental data given by Yang et al. [14] provides a crude estimate (0.01 eV) of the dispersion interaction between pairs of CH 2 groups in the extractant molecule and acid anion. The above estimate is comparable with that of Salem [22] (for a review of Salem's work see Ref. [19]) and the calculations of Combs and Dunne [23] whose estimate is 0.005 eV. Thus with suitable parameterisation the model is capable of closely reproducing experimental data and aiding in their rationalisation; better availability of empirical interaction parameter data could enable it to be applied as a predictive tool in the design of
/
--%'
~ propanoi¢ acid
5
- bntanoic acid
./
,
,
,
... ..
-I0 -15 0
2
4
6
8
10
12
Distribution coefficient Kd
Fig. 4. Theoretical curves of distribution coefficient as a function of the AqJ(E-,S +) interaction parameter and the OrgJ(S-,E +) interaction parameterto account for the chain length dependence of the extraction. Each set of extraction data was obtained between pH 2 and 8 (from Fig. 1).
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
reactive solvent extraction systems. Further developm e n t o f this w o r k c a n b e h o p e d to p r o c e e d o n this line w h i c h will i m p r o v e b o t h statistical t r e a t m e n t a n d c h e m i c a l r e a l i s m o f the m o d e l .
Acknowledgements O n e o f us ( M . I . ) t h a n k s S m i t h K l i n e B e e c h a m a n d E P S R C for a C A S E a w a r d .
References [1] P. Mayer and G. Maurer, Ind. Eng. Chem. Res. 34 (1995) 373; 32 (1993) 2105. [2] M.J. Kamlet, J.M. Abboud, R.W. Taft, J. Am. Chem. 99 (1977) 6027. [3] M.J. Kamlet, J.M. Abboud, R.W. Taft, Prog. Phys. Org. Chem. 13 (1981) 485. [4] M.H. Abraham, Chem. Soc. Rev. 22 (1993) 73. [5] Y. Marcus, J. Phys. Chem. 95 (1991) 8886. [6] R.W. Taft, M.H. Abraham, R.H. Doherty, M.J. Kamlet, J. Am. Chem. Soc. 107 (1985) 3105. [7] M.H. Abraham, J. Liszi, J. Inorg. Nucl. Chem. 43 (1981) 143. [8] B.G. Reuben, S.A. Khan, S. Jones, International Solvent Extraction Conference, Munich, September 1986. [9] L.J. Dunne, P. Katbamna, M. Lukhezo, B.G. Reuben, Chem. Phys. Lett. 224 (1994) 581.
61
[10] L.J. Dunne, B.G. Reuben, Chem. Phys. Lett. 242 (1995) 560. [11] B.G. Reuben, N. Ibrahim, A. McGillivary, L.J. Dunne, in: Separations for Biotechnology 3, D.L. Pyle (Ed.), SCI/RSC, London 1994, p. 221. [12] B.G. Reuben, S.A. Khan, T.J.A. Harris, in: Proceedings of the First International Conference on Pharmaceutical Ingredients and Intermediates, B.G. Reuben (Ed.), Frankfurt 1990, Expoconsult, Maarsen 1991, p. 147. [13] M. Lukhezo, N.A. Kelly, B.G. Reuben, L.J. Dunne, M.S. Verrall, in: Proceedings of 1SEC '96, Vol. 1, D.C. Shallcross, R. Paimin, L.M. Prvcic (Eds.), University of Melbourne, 1996, p. 87. [14] S.-T. Yang, S.A. White, S.-T. Hsu, Ind. Eng. Chem. Res. 30 (1991) 13335. [15] C.N. Yang, T.D. Lee, Phys. Rev. 87 (1952) 404. [16] T.D. Lee, C.N. Yang, Phys. Rev. 87 (1952) 410. [17] L.J. Dunne, G.M. Bell, J. Physics C. 12 (1980) 4137. [18] L.J. Dunne, G.M. Bell, J. Chem. Soc. Faraday Trans. II 76 (1980) 431. [19] G.M. Bell, L.L. Combs, L.J. Dunne, Chem. Rev. 81 (1981) 15. [20] L.J. Dunne, Phys. Status Solidi 178 (1993) 335. [21] T.L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley, 1960. [22] L. Salem, J. Chem. Phys 37 (1962) 2100. [23] L.L. Combs, L.J. Dunne, J. Mol. Struct. 87 (1982) 375. [24] A.C. Bajpai, L.R. Mustoe, D. Walker, Advanced Engineering Mathematics, 2nd Edn., Wiley, 1990. [25] B.D. Bunday, Basic Optimisation Methods, Edward Arnold, 1984.
Chemical Physics 220 (1997) 53-61
Statistical mechanical treatment of reactive solvent extraction M. L u k h e z o
a,
L.J. D u n n e
a,b,*,
B.G. Reuben
a,
M.S. Verrall c
a School of Applied Science, South Bank University, London SE10AA, UK b School of Chemistry and Molecular Sciences, UniL'ersity of Sussex, Falmer, Brighton BN1 9Q J, UK c SmithKline Beecham Pharmaceuticals, Chemotherapeutic Research Centre, Brockham Park, Betchworth, Surrey RH3 7AJ, UK
Received 3 December 1996; in final form 18 April 1997
Abstract
A statistical mechanical lattice model of a reactive solvent extraction process involving two phase liquid-liquid ion exchange is presented. It is applied to literature data on the extraction of aliphatic carboxylic acids by quaternary ammonium salts. The Helmholtz free energy of the system is constructed and minimised subject to the constraints of mass balance and electroneutrality. The equations defining the equilibrium conditions have been solved numerically for a range of interaction parameters and hence the equilibrium concentrations of the various partitioned species are obtained. These interaction parameters are compared with experiment. The intuitive notion that the extent of partition is dominated by electrostatic sotute-extractant interactions is shown to neglect various other interactions that may, in some systems, be important. © 1997 Elsevier Science B.V.
1. Introduction
Reactive solvent extraction of high value materials is practised in the metallurgical and chemical industries, but the underlying theory is poorly understood. The development of extraction processes for complex organic compounds such as those encountered in the pharmaceutical and biotechnological industries would benefit from such a theory. Although there is a considerable body of work on conventional solvent extraction based mainly on linear solvation energy relationships [1-4], the important problem of reactive solvent extraction has received little theoretical attention [5-7]. Reactive solvent extraction usually involves a liq-
* Corresponding author. Fax: (+44-171) 815 7999.
uid-liquid ion exchange in a two-phase partitioning process. In one phase, usually aqueous, there is a solute comprising a relatively lipophilic ion and hydrophilic counter-ion. The second phase is a waterimmiscible organic solvent containing a dissolved organic extractant, which may be virtually insoluble in water and which may dissociate to produce a hydrophilic ion and lipophilic counter-ion. The extractant contributes to the partitioning of the aqueous solute by ion pairing to the lipophilic solute ion at the aqueous/organic phase interface and solubilising it into the organic phase. The electrostatic charge balance of the system is preserved by the solute counter-ion electrostatically binding reversibly to the hydrophilic extractant counter-ion and transferring it into the aqueous phase. The aim of this paper is to present a primitive lattice model of a reactive solvent extraction process. To understand the extraction of aliphatic acids by
0301-0104/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0301-0104(97)00122-5
54
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
amines is fundamentally important since it is a model of more complicated but related biochemical processes such as those involving amino acids and peptides. We have previously modelled aspects of solvent extraction and related processes [8] and interactions in electrolyte solutions [9,10], and have reported experimental work governing such systems [ 1 l-13]. The extraction of carboxylic acids (ethanoic, propanoic and butanoic) with quaternary ammonium salts is an example of a reactive solvent extraction process investigated by Yang et. al. [14] who focused on the effects of pH, extractant (the tertiary amine, trioctylamine, as well as the quaternary, methyltrioctylammonium chloride), solvent (kerosene and 2octanol), and carboxylic acid chain length. By analogy with other anion exchange and hydrometallurgical processes, the extraction should improve with increasing pH. The qualitative argument is as follows; as the protons in the system are depleted, encouraging dissociation of the carboxylic acids, more carboxyl anions are freed for ion-pairing with the quaternary ammonium cations in the organic phase. Furthermore, as the aliphatic chain length of a carboxylic acid increases, the lipophilic nature of the molecule also increases, encouraging its extraction into the organic phase. The results of Yang et al. [14] shown in Fig. 1 do not fully satisfy these expectations. The extraction of carboxylic acids indeed increases with aliphatic chain length in the order ethanoic acid < propanoic acid < butanoic acid, but the extraction profiles exhibit the opposite trend to that expected with changing pH; extraction apparently improves with falling pH. In this paper we have parameterised our model against the literature data of Yang et al. [14] shown in Fig. 1 for the reactive solvent extraction of aliphatic carboxylic acids by methyltrioctylammonium chloride. We have thereby identified the important effective interactions governing the extraction process as a part of our general effort to understand more fully effective interactions in condensed media [9]. A weakness in our approach, which we hope to remedy in further work, is that all ionic species are assumed to be dissociated. Hence the model would not be applicable to extraction by tertiary amines. The model only treats ion pairing, which is likely in the organic phase, in an approximate way. We have not found an experimental system which accurately
corresponds with our model, but the above data of Yang et al. [14] for the extraction of aliphatic carboxylic acids by quaternary ammonium salts goes a long way towards this goal. Our model is crude but provides a useful starting point for molecular theories of this important topic.
,
°
j
,
°
o
i.
o
•
i
,
,
•
¢.
,, II
EthanoicAcid
o.
i:g @ 0
*
"
'
I
O.
J
a
a
2.
I
a
t
I
J
4.
'
a
"
6.
J
a
"
l
8.
10.
Equilibrium pH o b o
Propnoic
0
Acid
0
0
0
o.
2.
4.
6.
a.
io.
Y~luilibrium p H "
'
'
'
'
'
'
"
'
'
'
"
'
"
"
'
"
'C"
Butanoic A c i d
c~
o
O.
2.
4,. E q u i l i b r i u m
6.
O.
10.
pH
Fig. 1. Experimentaldata of Yang et al. for the extraction of aliphatic carboxylic acids (ethanoic, propanoic, butanoic) by methyltrioctylammoniumchloridereproducedfrom Ref. [14]. The extraction yield rises in an acidic aqueous phase and also with chain length. CopyrightAmericanChemical Society.
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
2. The primitive model To obtain a mathematically tractable problem, the components of the system have been reduced to a minimum number, contained within a fixed and incompressible volume. These are schematically represented in Fig. 2. In this simplification, we consider an initial state of solute ions S;,q and S~q ions in the aqueous phase (assumed dissociated, but with the possibility of ion pairing to give a neutral molecule) and an ion paired extractant E+Eorg in the organic phase, which may dissociate. When these two solutions are brought into contact, an ion exchange process may occur with the S~q ions binding to E~rg to form an electrostatically neutral complex soluble in the organic phase. This is balanced by the Eorg ion migrating to the aqueous phase as Ef,q where it is solvated. The process can be written SAq -'}-E+Eorg ~- EAq + E+Sorg .
(1)
The E + ions are assumed to be unable to cross the interface into the aqueous phase, and similarly the S + ions are insoluble in the organic phase. Only the S - and E - ions can cross the phase boundary, and the equilibrium concentrations of these in each phase are to be calculated by the model. The total number of each species in the system is known and fixed. We have considered here the exchange of anions but a similar treatment applies to cations, mutatis mutandis. The aqueous and organic solvents are treated here in a phenomenological way in which they provide an effective solvating medium in which the ions described above are dissolved and at this stage of development we disregard the detailed structure of
Aqueous (Aq)
Organic (Org)
@@ @ @@
55
aqueous phases at various pH values. In future work we hope to treat the aqueous phase more fundamentally. We consider four species in each phase. In the aqueous phase are Mi,Aq ions of E - , M2,Aq ions of S - , M4, Aq ions of S ÷ and MAq molecules of water. In the organic phase are Ml,org ions of E - , M2,0rg ions of S , M3,0rg ions of E ÷ and Morg molecules of organic solvent. Conditions for conservation of material are Ma, Aq ~ M2,0rg -4- M2,Aq ,
(2)
and M3,org ~- Ml,org + M1, Aq"
(3)
Since the total numbers of E + and S + ions are known, knowledge of the numbers of these species in either phase completely determines the system, so that there are two unknowns in the problem. The conditions for electroneutrality are M4,Aq ~ MZ,Aq + MI,Aq ,
(4)
and M3,org ~ Ml,org + Mz,org
(5)
Eqs. (2) to (5) serve as constraints on the total free energy minimisation to be discussed below.
3. The lattice model Lattice models of fluids have been extensively discussed in the literature [15-20]. As applied to our system, the lattice model treats each of the two phases separately. We consider a 3-dimensional cubic lattice of sites each with coordination number Z = 6 and which may be occupied by an ion or molecule of four possible different types. Nearest neighbour interactions are assumed. All the lattice sites are assumed to be occupied. The number of sites, N~, is for each phase N, = E Mi.
(6)
i Fig. 2. Schematic representation of the partitioning of species in a primitive reactive solvent extraction system. The left side denotes an aqueous phase while the right side denotes the organic phase. S- is the solute ion with an S + counter-ion while E- is the extractant anion whose counter-ion is E +.
Each of the species is assumed to occupy a single lattice site. At the beginning of an extraction experiment the aqueous phase contains the water molecules, S + ions and S - ions only. The organic phase con-
56
M. Lukhezo et a L / Chemical Physics 220 (1997) 53-61
tains the solvent, E + ions and E - ions only. No mixing of organic solvent and water is assumed to occur. The Helrnholtz free energy [21], F, of a single phase is defined as Q,
F= -kTln
(7)
where k is the Boltzmann constant, T is the absolute temperature, and Q is the canonical partition function. The total free energy of the system is the sum of the free energies of each phase, hence this equation is applied to each phase separately. The equilibrium state of the system is located by use of the method of the maximum term [21] in which the total free energy of the whole system is minimised which is achieved when chemical potentials of like species in each phase are equal. A single phase will contain four different species of molecular quantities M 1, M 2, M 3 and M 4. The canonical partition function, Q, takes into account all the possible permutations of these species on the N~ lattice sites. In a random mixing approximation the free energy is given by F = ~c - k T In
(~Mi)!
MI !Mz !M3 !M4 !
,
(8)
4. The configuration energy and interaction parameters For a three-dimensional cubic lattice, the possible types of nearest neighbour interaction are described by the parameters Ji,j, where i = 1 to 4 and j = 1 to 4. The Ji,j values in the aqueous phase will differ from those in the organic phase, due to the differences in screening environments of the different bulk media. Such effective interaction parameters are not the true bare interaction between species but mimic a complicated set of free energies of interaction in the solution and implicitly attempt to allow for all kinds of nearest neighbour interactions between species in the system. Consider as an example the interaction between species of types 1 and 2 where J~,2 is the 1,2 interaction parameter. In a random mixing approximation the energy of a type 2 species with all other type 1 species is given by ZJ,.z( M , / ~ )
,
(13)
where ( M J N s) is the probability that the neighbouring site is occupied by a type 1 species. The sign convention for the interaction parameters is a negative value for an attractive interaction, a positive value for a repulsive interaction. The total number of 1,2 type interactions is then given by
where e c is the effective configurational interaction energy discussed in the next section. Applying Stirling's theorem
ZJ1.2(M,/N~) M 2 .
In X ! = X In X - X (for large X ) ,
The number of 1,2 type interactions per site becomes
(9)
we obtain from Eq. (7)
ZJ1,2 ( ml//Ns ) ( M2//Ns ) = ZJ1,2 PI P2.
(14)
(15)
Similarly, the energy of interaction between a species i and all the other i species is + k T Y ' ~ ( M i In Mi).
(10)
i
(11)
i which may be written as the Helmholtz free energy per site, for a given phase f c = F = ~ c / N s + kTY'~ ( Pi In Pi) . i
(16)
so that the total (i,i) interaction is
Defining the variable P,. = M i / N S gives F = e c + kTY'~( M i In P i ) ,
Zli.iP i ,
(12)
ZJi.iPi( M i / 2 ) .
(17)
The factor 1 / 2 in Eq. (17) prevents interactions between any pair of the same species being counted twice. Expression (15) can be written as the energy per site given by ZJi,jPiPj ,
(18)
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
57
which from Eq. (25) gives
or
ZJi,iPi2/2,
(19)
when i = j . Combining Eq. (17) and Eq. (18) produces the general expression for the configurational (i,j) interaction energy per site given by
ZIi.jPiPJ(1 + 6i.j),
(20)
where 6i,j is the delta function. Substitution of expression (20) for ec/N s, in Eq. (12) gives the Helmholtz free energy per site, for each side
fc =kT~-,Piln Pi + ZY'~Ji,jPiPj/( 1 + ai,j)" (21) i
i<~j
~3F = ~ ( ( OFAq//OMi, Aq )
-- (OForg/OMi,Org) ) 8 Mi, Aq"
(27)
Since all the variations in the species numbers in Eq. (27) may be made independently the condition for the free energy to be at a minimum in the whole system, for all species, is expressed as
8F/SMi, Aq = 0,
(28)
and is only true if the chemical potentials of the aqueous and organic phases are equal, that is
OFAq/OMi. Aq = OForg/OMi,org
(29)
or
5. Equilibrium conditions The condition for the system at equilibrium, as mentioned earlier, is that the free energy of the total system must be at a minimum. To simplify the treatment, we shall assume both phases to have equal numbers of sites while noting that generalisation is straightforward. Any variation of the numbers of species in one phase by transfer to the other produces an incremental change in the Helmholtz free energy of the whole system 8F = 8FAq 4- 8Forg.
(22)
Expanding each term on the right hand side of Eq. (22), for a fixed volume system, according to the chain rule gives 8FAq = E (OFAq/OM,,Aq)~)Mi, Aq
(30)
/£i, Aq = ~Li.Org ,
(23)
i
which may be expressed as Ofc. Aq/OPi =Ofc" Org/OP, .
(31 )
Differentiation of Eq. (21) gives
Ofc/OPi=kr(1 + In Pi) +zY'~Ji.jP j.
(32)
i<~j Considering both the aqueous and organic phases of the system together, at equilibrium, Eq. (30) can be re-written as Ofc' Aq/OPi --
Ofc,org/OPi = O.
(33)
Thus a combination of pairs of equations from the system of equations (32) gives the equilibrium relation
kT In( Pi, Aq/Pi, Org) + Z E ( J i , j ; A q P j , Aq--Ji,j:orgPj, org)=O.
(34)
J
and ~gorg = E (Ogorg//OMi,org)~Mi,org , i
(24)
and substitution of these equations into Eq. (22) gives
8F = E ((OFAq/OMi,Aq)SMi, Aq i 4- (OForg/OMi, Org )~ Mi, Org )"
(25)
From Eq. (2) and Eq. (3), the material balance of any species M~ traversing the phase boundary requires
~Mi, Aq = -- 8Mi. Org'
(26)
The aim of the model is to calculate the numbers of species of the four unknown components E~,q, SAq, Eorg and Sorg at equilibrium. However, since EAq and Eorg, and Sf,q and Sorg are conjugates it is necessary to calculate the value of just one of each pair and then to apply Eq. (2) and Eq. (3). A pair of simultaneous, non-linear equations results
ln( P,,Aq/P|,org) + ( Z / k T ) E(JI,j;AqPj, Aq -- Jl,j;Org J
Pj, org) = O,
(35)
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
58
In( P:, Aq/P2, Org)
OF/Oy = ln( P2. ~q/P2, Org) + ( Z/kT)
+ ( Z / k T ) E ( J2,j;AqPj, Aq -- J2,j;orgPj,org) = O.
× E(J2,:AqPj.Aq-
J
J2.j:o, Pj.o, )
J
(36)
(43)
In order that the solutions obtained for the above equations represent a minimum there is a subsidiary condition that the Hessian matrix [24] of second mixed partial derivatives of the free energy be positive definite. That is the eigenvalues of the matrix
The charge balance constraint is implemented as
( H'I
G(P,.o,g,P2,org)=P,,org+P2,o~g-C,
H21
Hi2 ) H22
(37) '
should both be positive which was checked by straightforward manipulation and diagonalisation of the above matrix. The appropriate matrix elements are
nii= kT/Pi. Aq + kT/Pi,org + Z(Jii,Aq + Jii,o~g) (i = 1,2),
selecting P~,org and P2,org as the quantities to be solved for, which then becomes
where c is a constant and, in this case, equal from Eq. (5) to P3, org giving the last terms in the left hand side of Eq. (40) and Eq. (41) as
A(~G/~x) = A(1)G/OPI,o~g) = A,
(46)
A(OG/Oy) = A(OG/OPz,org )
(39)
to give with Eq. (45) the set of three simultaneous, non-linear equations
= A,
(47)
ln(P,,Aq/P,.o,g) + (Z/kT) Y'~ (J,,j;AqPj, Aq -- S,,j:OrgP/,org) J
+ A = 0,
(48)
ln(Pz, Aq/P2.org)
6. Electroneutrality constraints Up to this point, the material balance (Eq. (2) and Eq. (3)) has been included in the minimisation of the free energy but the charge balance (Eq. (4) and Eq. (5)) has not been included. This constraint is readily imposed using Lagrange's method of undetermined multipliers [21]. A function F(x, y) minimised subject to a constraint G(x, y) = 0 yields the equations (40)
(41) where A is an undetermined multiplier. The appropriate first terms in the left hand side of Eq. (40) and Eq. (41) are
aF/Ox = In(P,. Aq/Pl,Org)
(45)
(38)
Eq. (35) and Eq. (36) must be solved numerically subject to the electroneutrality condition described in the next section and the positive definiteness of the Hessian matrix (37).
(OF/Ox) + A(aG/Ox) = 0, (OF/Oy) + A(OG/Oy) = 0,
(44)
and
and H21 = H,2 = Z( J,2, Aq + J,2,Org) •
G(x, y) = G(P,,org, P2,org) = 0,
+ (Z/kT)
X E(JI,j;AqPj, A q - Jl,j;orgPj,org), J
(42)
+ ( Z / k T ) ~.~ (J2,j:AqPj, Aq -- J2,j;orgPj,org) J
+ A = 0,
(49)
and PI,org + P2,org - c = 0 ,
(50)
thus explicitly including the charge balance in the free energy minimisation. Eqs. 48 to 50 were solved numerically for various values of the interaction parameters to yield the various numbers of species in the organic and aqueous phases. Since the constraint (50) is linear, the positive definiteness of the Hessian matrix continues to signify a minimum [25]. An important implication of the electroneutrality constraint and the random mixing approximation is that, in the lattice model, the free energy of the system is independent of the simple electrostatic interaction between ions. Hence, if some members of
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
the set of interaction parameters {J/j} are set to zero the remaining members of the set measure deviations from Coulomb's law for idealised point charges and other contributions to the interactions such as those arising from solvation effects.
7. Results
and
59
the view that the dominant driving force for the reactive solvent extraction of carboxylic acids discussed by Yang et al. [14] is the attractive character of the effective aqueous phase interaction between the H + cation (the counterion to the aliphatic acid in the aqueous phase) and the Cl- anion (the counterion to the extractant in the organic phase), i.e. Aq J(Cl-, n +) or Aq J ( E - , S+). In acidic solution, there is an abundance of free protons available for solvating the C l - anions transferred from the organic phase. The attraction between the two is very strong and hence we predict that this process is exothermic. To balance this process, the aliphatic acid anions complex with the extractant cation and are transferred into the organic phase. Fig. 3 illustrates the derived relationship between the interaction parameter Aq J ( C l - , H + ) and equilibrium pH. The equilibrium pH is derived from Fig. 1 and the interaction parameters A q J ( C I - , H +) were calculated by fitting the numerically computed output from the model to the experimental data of Yang et al. [14] given in Fig. 1. The conditions for extraction are most favourable at low pH, and this is reflected in the generally more attractive interaction parameter values. A prediction, therefore, is that the extraction should not occur in basic solution. This is borne out by experiment. The fact that the exchange does not
discussion
The model allows investigation of the influence on the selective extraction of particular sets of interaction parameters and where the complement is set to zero. As discussed above, electroneutrality exists in the extraction system at equilibrium and, with the complement of interactions set to zero, the output of the model reflects deviations from Coulomb's law for idealised point charges such as those due to ion size effects and dipolar and dispersion contributions to the effective interactions. The effective parameters in our model mimic the true and undoubtedly complicated interactions in an acidic solution and allow us to focus on identifying the features which drive the extraction. We have chosen to isolate the dominant interactions governing the extraction and, although it may prove possible to perform ab initio calculation of interaction parameters, at this stage we have proceeded with estimates of such energies. By making a parameter search we have come to
10 4- 4-
OrgJ(R3NCH3-,RCOO-)/kT = -1 ÷
5 ÷
4-
4-
I
I
I
4-
+4-
4-
4-4-
4-
¢
I
I
0
v
*~
@.
I =
¢
g~
¢
-5
¢ O
2 2
4-
4-
4-4+
o
.0-
.~
0
,O
¢
0
@
O
+ ethanoic acid
I
-lO ..
--
.
.
o p r o p a n o i c acid
.
- butanoic acid
-15
0
I
2
3
4
5
6
7
8
Equilibrium pH Fig. 3. Theoretical curves showing the equilibrium pH dependence of the Aq J(E-, S +) interaction parameter.
60
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
occur in basic media, against intuitive notion, is evidence that the above hypothesis is essentially correct. Rising pH causes a lowering of the H ÷ cations available for solvating the C1- anions from the organic phase, thereby effectively decreasing the driving force for the extraction process. The relationship between the interaction parameter Aq J(CI-, H ÷) and the distribution coefficient k d is illustrated in Fig. 4. The k d data have been derived from Fig. 1 ( k d = [RCOO-]org/[RCOO-]A q) and the interaction parameter ( J / k T ) has been calculated by the model for ethanoic acid ( + ) , propanoic acid ( [ ] ) and butanoic acid ( v ) . The variable interaction parameter for the three different cases is the organic phase interaction parameter between the carboxylic acid anion and the extractant cation, i.e. Org J(R 3NCH ~, R C O O - ) or Org J(E +, S - ), and this differs by an amount of - 2 between adjacent cases thus illustrating the effect on extraction of the strength of affinity between the extractant cation and the carboxylic acid anion. Fig. 4 further illustrates the chain length dependence by the range of k 0 values covered by each carboxylic acid. Ethanoic acid ( + ) covers the lower range of k d values, butanoic acid ( v ) covers the higher range (the gap in the data ( k d : 4 to 8) coincides with the rapid drop in the (reverse) sigmoidal curve of Fig. l) and propanoic acid ([~) is 15
OrgJ(R3NCH3-,RCOO-)/=kT -1,-3&-5
lO[-~+, 0
+
ethanoic acid
///
['q : [~~i!! ~ / i
approximately in the middle; the data for all acids approximately cover the same pH range in Fig. 1. The interaction parameter (Aq J(CI-, H + )) values for the acids follows a similar trend, with ethanoic acid at the more repulsive end and butanoic acid at the more attractive end of the range of values. The chain length dependence is attributed in our model to the increased effective interaction with chain length of the R3NCH ~- and RCO 2. If the attraction roughly rises by a small number of CH 2CH 2 interactions (say 5) with an increase in chain length of one ( - C H 2 - ) moiety then the C H 2 - C H 2 interaction at equilibrium separation is estimated to be of the order of - 1/100 eV. Hence we have found that the chain length dependence of the extraction shown in Fig. 1 in the experimental data given by Yang et al. [14] provides a crude estimate (0.01 eV) of the dispersion interaction between pairs of CH 2 groups in the extractant molecule and acid anion. The above estimate is comparable with that of Salem [22] (for a review of Salem's work see Ref. [19]) and the calculations of Combs and Dunne [23] whose estimate is 0.005 eV. Thus with suitable parameterisation the model is capable of closely reproducing experimental data and aiding in their rationalisation; better availability of empirical interaction parameter data could enable it to be applied as a predictive tool in the design of
/
--%'
~ propanoi¢ acid
5
- bntanoic acid
./
,
,
,
... ..
-I0 -15 0
2
4
6
8
10
12
Distribution coefficient Kd
Fig. 4. Theoretical curves of distribution coefficient as a function of the AqJ(E-,S +) interaction parameter and the OrgJ(S-,E +) interaction parameterto account for the chain length dependence of the extraction. Each set of extraction data was obtained between pH 2 and 8 (from Fig. 1).
M. Lukhezo et al. / Chemical Physics 220 (1997) 53-61
reactive solvent extraction systems. Further developm e n t o f this w o r k c a n b e h o p e d to p r o c e e d o n this line w h i c h will i m p r o v e b o t h statistical t r e a t m e n t a n d c h e m i c a l r e a l i s m o f the m o d e l .
Acknowledgements O n e o f us ( M . I . ) t h a n k s S m i t h K l i n e B e e c h a m a n d E P S R C for a C A S E a w a r d .
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