Interpretation by the Solvophobic Theory on the ... - ACS Publications

J. Phys. Chem. B 2000, 104, 8481-8490

8481

Interpretation by the Solvophobic Theory on the Linear Additive Representation of the Logarithm of Ion-Pair Extraction Constant with Individual Contributions of Cation, Anion, and Organic Solvent Kanji Miyabe,*,† Shigeru Taguchi,‡ Issei Kasahara,‡ and Katsumi Goto§ Faculty of Education and Faculty of Science, Toyama UniVersity, 3190, Gofuku, Toyama 930-8555, Japan ReceiVed: March 9, 2000; In Final Form: June 19, 2000

A theoretical basis was given for the linear additive representation of the logarithm of ion-pair solvent extraction constant (Kex) in terms of logarithmic values of three individual extraction constants (Kcation, Kanion, and Ksolvent) for cations, anions, and organic solvents constituting ion-pair extraction systems. The differences in extractability between cations and between anions were quantitatively discussed according to the thermodynamic cycle model of 1:1 ion-pair extraction. The increment of log Kcation and log Kanion of hydrophobic ions is explained by taking the increment of the free energy change of the hydration (∆Ghyd) of the ions into account. The order of the extractability of inorganic cations and anions was also interpreted using literature data of ∆Ghyd of each inorganic ion. The free energy change of the solvation of a hypothetical ion-pair having a shape of spherical was calculated on the basis of the solvophobic theory. The difference in extracting powers of some extracting organic solvents could also quantitatively be interpreted. There are a few sets of proposed values of the individual extraction constants representing the extractability of the cations and anions, which were calculated based on different reference assumptions. The conclusions of this study are effective for the interpretation concerning the increment of the extractability of the ions irrespective of the reference assumptions for determining the individual extraction constants. It was also attempted to estimate absolute values of Kex by the theoretical approach in this study. The Kex values of a few inorganic ion-pairs thus calculated were of the same order of magnitude as those estimated from the individual extraction constants experimentally determined. It is concluded that the thermodynamic cycle model based on the solvophobic theory provides an essential framework for the theoretical interpretation of the mechanism of the ion-pair extraction, and that the validity of the linear additive representation of log Kex with the three individual extraction constants is demonstrated.

1. Introduction Solvent extraction has been used as one of the most useful and convenient techniques for both industrial and analytical scale separations. For the analytical purpose, an analyte in water is extracted into a relatively small volume of an immiscible organic solvent in many cases. Then, the analyte concentrated in the organic phase is determined by means of various analytical methods. When the analyte is positively or negatively charged, it is required at the first step to cancel its charge by using different chemical reactions or interactions because charged species cannot be extracted into the organic phase. Ion-pair formation is frequently applied for this purpose. When the analyte is a cation (C+), it should be converted into an electrically neutral ion-pair with a proper ion as a counterion. Similarly, when an anion (A-) is the component to be separated, a cation is used as the counterion having the opposite charge. The electrically neutral ion-pair can be extracted into the organic solvent. Figure 1 shows the schematic illustration of the 1:1 ion-pair extraction. It is reasonable to assume that the mechanism of the ion-pair extraction consists of two equilibrium processes. One is the formation of the ion-pair (C+A-) in * Corresponding author. E-mail: [email protected]. Fax: 81-76-445-6264. † Faculty of Education. ‡ Faculty of Science. § Professor emeritus of Toyama University.

Figure 1. Ion-pair solvent extraction equilibria. The symbols C+, A-, and C+A- represent the cation, anion, and ion-pair, respectively.

aqueous phase and the other the distribution of C+A- between the aqueous and organic phases. The cation or anion can be separated from concomitant ions in the sample water by exploiting the difference in extractabilities of different ion-pairs by using appropriate combinations of counterions and organic solvents according to the purpose and the degree of difficulty of the intended separations. To the contrary, an optimum extraction system must be selected in order to accurately determine the component of concern. However, it is hard to experimentally choose the suitable combination of the counterion and organic solvent from numerous combinations of them. A kind of theoretical approach might be effective for selecting the preferable combination of the ion-pair extraction system. Harris et al.1 studied the ionpair extraction of alkyl sulfates using d-3-methoxy-N-methylmorphinan as the countercation and chloroform or chloroformcarbon tetrachloride mixtures as the extracting organic solvents, and indicated that the free energy change (∆Gex) of the ion-

10.1021/jp000918c CCC: $19.00 © 2000 American Chemical Society Published on Web 08/10/2000

8482 J. Phys. Chem. B, Vol. 104, No. 35, 2000

Miyabe et al.

pair extraction was linearly correlated with the surface area of the alkyl sulfate anions, irrespective of their structural characteristics, that is, straight chains, branched chains, and cyclic compounds. They also suggested that the group contribution data might be useful for a priori prediction of partition coefficients and excess thermodynamic properties of organic molecules. A number of experimental studies were also made on the influence on the ion-pair extraction behavior of the type, size, and shape of both ions and functional groups involved in the ions and solvent molecules.2-5 The results of these studies led to the linear additive rule of the contributions of the cations and anions to the logarithm of the ion-pair extraction constant (Kex). Taguchi et al.6,7 and Goto8 demonstrated that Kex of 1:1 ionpair extraction systems could be accounted for by the following linear additive correlation:

log Kex ) log Kcation + log Kanion + log Ksolvent

(1)

where Kcation, Kanion, and Ksolvent are the individual extraction constants for the cation, anion, and organic solvent, respectively. Calculation using these constants allows the prediction of Kex under various experimental conditions of the ion-pair extraction. The values of Kex thus estimated properly agree with those experimentally measured, suggesting that the individual extraction constants are useful for selecting preferable combinations of the counterions and organic solvents of the ion-pair extraction system for the determination of the analyte. Effectiveness of eq 1 has experimentally been verified in many systems of ionpair extraction.6-8 Similar experimental results have also demonstrated the propriety of the concept that the extractability of the ion-pair is represented by the additive contributions of the cation and anion.5,9-12 However, absolute values of the individual extraction constants in eq 1 were not assigned to each extracting organic solvent in these experimental studies of the ion-pair extraction. Only the differences in extractability between the organic solvents had previously been reported.3,9 Harris et al.1 studied the contribution of various structural units on the organic ion to find the influence of the substituted functional groups on the extractability of the ion-pair formed. Matsunaga and Yotsuyanagi5 also deduced similar empirical parameters for estimating Kex from the experimental results concerning the ion-pair extraction equilibria of nickel(II) dithiooxalato complex with quaternary ammonium ions between water and chloroform. Motomizu12 indicated that log Kex of the ion-pair extraction using chloroform as the organic solvent was represented as the sum of two parameters (C and A), which respectively represented the contribution of the cation and of the anion to the ion-pair extractability. Despite the difference in the expressions of these extraction constants, for instance, between Kcation and C or between Kanion and A, it has been experimentally demonstrated that log Kex can be predicted with reasonable accuracy by assuming the individual contributions of the cations and anions.5-12 Although similar conclusions have been derived in the previous studies to show the validity of the assumption of the linear additivity of the individual contributions,5-12 the proposed values of the parameters for the same ions are different from each other, for example, between log Kcation and C or between log Kanion and A, because the reference assumptions for assigning these values are not identical. Kasahara et al.7 assigned the values of Kex on the assumption that Kcation for tetraphenylarsonium cation was identical to Kanion for tetraphenylborate anion,

Figure 2. Thermodynamic cycle model consisting of the hypothetical ion-pair formation in the gas phase and the solvation and transfer of all chemical species into the aqueous or extracting organic solvent phase. The symbols C+, A-, and C+A- represent the cation, anion, and ion-pair, respectively.

and that Ksolvent for benzene was equal to unity. On the other hand, Matsunaga and Yotsuyanagi5 chose benzyldimethyltetradecylammonium ion as the reference compound. Motomizu12 took a hypothetical quaternary ammonium cation with no alkyl groups nor hydrogen atoms as the reference and estimated the C and A values from the experimental values of Kex in the ionpair extraction systems using chloroform as the organic solvent. However, the increment of log Kcation (∆log Kcation) for successive cations was almost the same as that of C (∆C) despite the difference in the reference assumptions. Similar situations were also observed between ∆log Kanion and ∆A. There is a general tendency to avoid the use of organic solvents because of their toxicity. Solvent extraction would be replaced by other methods such as solid-phase extraction, which uses no toxic organic solvents. However, it is still quite effective to analyze previous results concerning the ion-pair solvent extraction because a number of experimental data have so far been measured in detail and the mechanism of separation and concentration in the ion-pair solid-phase extraction is similar in principle to those in the solvent extraction of ion-pairs. Useful information about fundamentals of the solid-phase extraction would be derived from the study of the essential characteristics of the separation mechanisms in the ion-pair solvent extraction. This paper presents a theoretical consideration of the mechanism of the ion-pair solvent extraction on the basis of the thermodynamic cycle model (Figure 2). The goal of this study is to provide an interpretation for the linearly additive representation of log Kex in eq 1 with the logarithmic values of the individual extraction constants, i.e., Kcation, Kanion, and Ksolvent, and to demonstrate the effectiveness of the thermodynamic cycle model based on the solvophobic theory for the systematic understanding of the ion-pair extraction behavior. It was attempted to quantitatively explain the difference in the individual parameters, that is, ∆log Kcation, ∆log Kanion, and ∆log Ksolvent, for homologous or successive compounds, related ions, or extracting organic solvents according to the thermodynamic data cited from some literature and the solvophobic theory. Absolute values of Kex were also calculated for a few inorganic ion-pairs. 2. Thermodynamic Cycle Model of Ion-pair Extraction Based on the Solvophobic Theory The solvophobic theory has been applied to the interpretation of thermodynamic properties of a wide variety of chemical and physical phenomena.13-24 In this part, we briefly explain the fundamental concept of the theoretical approach based on the solvophobic theory. 2.1. Free Energy Change of Ion-Pair Extraction. Equilibrium constant (Kass,CA,aq) of the ion-pair formation between a

Ion-Pair Extraction Mechanism

J. Phys. Chem. B, Vol. 104, No. 35, 2000 8483

cation (C+) and anion (A-) in water is related with the free energy change (∆Gass,CA,aq) as follows:

∆Gass,CA,aq ) -RT ln Kass,CA,aq

(2)

where R is the gas constant and T the absolute temperature. The subscripts ass, CA, and aq denote the association process, the ion-pair (positive and negative charges were omitted), and the aqueous phase, respectively. As shown in Figure 2, the free energy change of the ion-pair formation between C+ and A- in water is represented as follows:

∆Gass,CA,aq ) ∆Gass,CA,gas + ∆Gsol,CA,aq - ∆Gsol,C,aq ∆Gsol,A,aq (3) where the subscripts gas, sol, C, and A denote the gas phase, the solvation process, the cation, and the anion, respectively. The positive and negative charges on the cation and anion were omitted. The 1:1 ion-pair C+A- formed in water migrates into the organic phase. Distribution of the ion-pair C+A- between water and the organic solvent is quantitatively represented by the distribution coefficient (Kd,CA), which is related with the free energy change of the transfer of C+A- from the aqueous phase to the organic one (∆Gd,CA) as follows:

∆Gd,CA ) - RT ln Kd,CA

(4)

where the subscript d denotes the distribution between the aqueous and organic phases. As shown in Figure 2, ∆Gd,CA can hypothetically be analyzed by taking two solvation processes into account. One is the transfer of the ion-pair C+A- from the gas phase into water and the other that into the organic solvent. Therefore, ∆Gd,CA is accounted for by the following equation:

∆Gd,CA ) ∆Gsol,CA,org - ∆Gsol,CA,aq

(5)

where the subscript org denotes the extracting organic solvent phase. According to the definition (Figure 1), Kex is equal to the product of Kass,CA,aq and Kd,CA. In this study, dissociation and/ or dimerization of ion-pairs in the organic phase were not taken into account. It is primarily important to comprehensively elucidate essential aspects of the separation mechanisms in the ion-pair solvent extraction.

Kex ) Kass,CA,aqKd,CA

(6)

Equation 6 indicates that ∆Gex is the sum of ∆Gass,CA,aq and ∆Gd,CA. By combining eqs 3 and 5, ∆Gex is represented as follows:

∆Gex ) ∆Gass,CA,aq + ∆Gd,CA

14,16,18,19

∆Gsol,i,j ) ∆Gcav,i,j + ∆Gvdw,i,j + ∆Ges,i,j + RT ln

(7) As shown in eq 7, ∆Gex consists of the four terms in the righthand side (RHS). On the other hand, eq 1 indicates that ∆Gex consists of three free energy changes, i.e., ∆Gcation, ∆Ganion, and ∆Gsolvent, as follows:

(8)

Because ∆Gass,CA,gas is the free energy change of the association between C+ and A- in the gaseous phase, it is probably

( )

RT (9) Vj

where subscripts i and j denote the solute and the solvent, respectively. The subscripts cav, vdw, and es represent the cavity formation, the van der Waals interaction, and the electrostatic interaction, respectively. The fourth term in the RHS of eq 9 indicates the contribution of the free volume reduction under atmospheric pressure (∆Gred,j). The free energy change of the cavity formation is expressed as follows:14,16,18,19

∆Gcav,i,j ) κei Riγj(1 - Wi)NA

(10)

where κe is the correction parameter of the difference in the enthalpy change for the formation of a curved surface (cavity) in the solvent and a plane having a same surface area, R the molecular surface area, γ the surface tension, and NA the Avogadro number. The two parameters, Wi and κei , are estimated as follows:

( )(

)

κsi d ln γj 2aiT Wi ) 1 - e + 3 κ d ln T i

) ∆Gass,CA,gas - ∆Gsol,C,aq - ∆Gsol,A,aq + ∆Gsol,CA,org

∆Gex ) ∆Gcation + ∆Ganion + ∆Gsolvent

divided into two contributions: one from C+ and the other from A-. Comparison of eqs 7 and 8 implies that ∆Gsol,C,aq and the contribution of C+ to ∆Gass,CA,gas correspond to ∆Gcation in eq 8 (log Kcation in eq 1), and ∆Gsol,A,aq and that of A- to ∆Gass,CA,gas correspond to ∆Ganion in eq 8 (log Kanion in eq 1). Additionally, it is likely that ∆Gsol,CA,org in eq 7 corresponds to ∆Gsolvent in eq 8 (log Ksolvent in eq 1) because only this term in the RHS of eq 7 is related with some physicochemical properties of the organic solvent. In the following, a series of discussions will be made along this hypothesis in order to prove the validity of the linear additive rule of the logarithmic values of the individual extraction constants as represented by eq 1. 2.2. Solvation and Transfer of a Solute from Gaseous Phase into a Solvent. As shown in Figure 2, the mechanism of the ion-pair extraction can be discussed by considering two conceptual processes based on the solvophobic theory. One is the hypothetical gas-phase association between C+ and Awithout the influence of water. The other is the solvation of all the related chemical species. The solvation process is considered to consist of two steps, namely the formation of a cavity in water or in the organic solvent and the interaction of the species placed in the cavity with surrounding solvent molecules. The interaction is assumed to consist of the two contributions: van der Waals and electrostatic interaction. The free energy change of the transfer of a component i from the gas phase into water or into the extracting organic solvent is represented as follows:

()

κei ) 1 + (κej - 1)

Vj Vi

(11)

2/3

(12)

where κs is the corresponding function for the entropy change associated with the formation of the cavity and similarly expressed by an equation similar to eq 12, a the cubic expansion coefficient, and V the molar volume. The free energy change of the interaction due to the dispersion forces is calculated by the following equation:14,16,18,19

∆Gvdw,i,j ) -0.606 ∆ijDiDj(Q + Q′)NA

(13)

8484 J. Phys. Chem. B, Vol. 104, No. 35, 2000

Miyabe et al.

where Q and Q′ are the dimensionless functions, which are obtained by integrating the effective pair potential between the solute and solvent molecules over the total volume. The contribution of Q′ might be neglected because Q′ < 0.1 Q.19 The value of Q is calculated by the following equation:

{

Q ) νi

(

)

2

t′ t′ t′ 1 t′ 1 + + + + 11 5 9 5 2 3 (r′av - lav)9 (r′av - lav)3

6

σav

2

}

(14)

where νi is the molecular volume of the solute. Some related parameters are estimated as follows:

lav t′ ) r′av - lav

(16)

li + lj 2

(17)

(18)

( )

(19)

3νi 4π

(20)

2.67 σi ) r′i 3.24 + 7ωi

(21)

where ωi is the acentric factor of the solute. The parameters for the solvent, i.e., rj, lj, and σj, are similarly estimated from eqs 19-21 using νj and ωj. Other parameters in eq 13 are calculated as follows:

∆ij )

Di )

∆Gass,CA,gas ) -

1.35IiIj Ii + Ij n2i - 1 n2i + 2

(22)

(23)

where I is the ionization potential and n the refractive index. The value of D of the solvent, i.e., Dj, is similarly estimated from eq 23 using nj. The free energy change resulting from the electrostatic forces is calculated as follows:14,16,18,19

δ+δ-e2 4π0rCA

(27)

where δ+ and δ- are the electric charges on C+ and A-, respectively, and unity in the case of the 1:1 ion-pair formation, e the elementary charge, rCA the distance between the positive and negative charge. Equation 27 is rearranged because rCA is the sum of the radii of C+ (rC) and A- (rA).

e2 4π0(rC + rA)

(28)

Equation 28 is rearranged at rC ≈ rA.

1/3

0.24 + 7ωi li ) r′i 3.24 + 7ωi

)

(26)

Ψβi νi

where ′ is the dielectric constant, 0 the free space permittivity constant, and β the polarizability. 2.3. Division of Ion-Pair Extraction Constant into Contributions of Cation, Anion, and Organic Solvent. First, the free energy change of the formation of the ion-pair C+Aresulting from the electrostatic interaction between C+ and Ain the gaseous phase (∆Gass,CA,gas) is expressed as follows:

∆Gass,CA,gas ) -

σi + σj σav ) 2 r′i ) 1.74

(

1

4π0 1 -

(15)

r′i + r′j r′av ) 2 lav )

P)

∆Gass,CA,gas ≈ -

Cg Cg e2 e2 )16π0rC 16π0rA rC rA

where Cg is the ratio e2/(16π0). Molar volumes of most cations and anions having hydrophobic groups, which were frequently used in the ion-pair extraction, are in the range between ca. 100 and 350 cm3 mol-1 except for inorganic cations and anions.7 When the shape of the ions is assumed to be sphere, the relative difference in their radius is calculated as about 50% () 3.51/3) between the large and small ions. When the radii of C+ and Aare relatively different about 50%, the relative difference between the values of ∆Gass,CA,gas calculated by eqs 28 and 29 is about 4%, suggesting the propriety of the simplification in eq 29. Equation 29 implies that ∆Gass,CA,gas is divided into two contributions, due to C+ and to A-. The following equation is derived by combining eqs 7, 9, and 29:

∆Gex ) -

(

Cg Cg rC rA ∆Gcav,C,aq + ∆Gvdw,C,aq + ∆Ges,C,aq + RT ln

(

(24)

2(′j - 1) 2′j + 1

RT Vaq

(

( )) RT Vaq

- ∆Gcav,CA,org + ∆Gvdw,CA,org + ∆Ges,CA,org +

where µi is the dipole moment of the solute. The two parameters, Ψ and P, are calculated as follows:

Ψ)

( ))

- ∆Gcav,A,aq + ∆Gvdw,A,aq + ∆Ges,A,aq + RT ln

NAµ2i ΨP ∆Ges,i,j ) 2νi

(29)

(25)

RT ln

( )) RT Vorg

(30)

Comparison of eq 8 with eq 30 implies the following relationships:

Ion-Pair Extraction Mechanism

∆Gcation ) -

(

Cg rC

J. Phys. Chem. B, Vol. 104, No. 35, 2000 8485

( )) RT Vaq

(31)

( ))

(32)

∆Gcav,C,aq + ∆Gvdw,C,aq + ∆Ges,C,aq + RT ln

∆Ganion ) -

(

Cg rA

∆Gcav,A,aq + ∆Gvdw,A,aq + ∆Ges,A,aq + RT ln

RT Vaq

∆Gsolvent ) ∆Gcav,CA,org + ∆Gvdw,CA,org + ∆Ges,CA,org + RT (33) RT ln Vorg

( )

Equations 31-33 probably verify the propriety of the assumption of the linear additive representation of log Kex (eq 1). 3. Quantitative Analysis of the Individual Contributions of the Cations, Anions, and Organic Solvents to the Free Energy Change of 1:1 Ion-Pair Extraction It was attempted to quantitatively consider the values of Kcation, Kanion, and Ksolvent experimentally determined7 in order to demonstrate the validity of the assignment of ∆Gcation, ∆Ganion, and ∆Gsolvent in eqs 31-33, namely, that of the linear additive rule of log Kex represented by eq 1. 3.1. Kcation. Organic Cations. It is well-known that log Kcation almost linearly increases with increasing number of carbon atoms (NC) in the alkyl chains of organic cations. The linear correlations between log Kcation and NC are respectively represented for a series of alkyltrimethylammonium and tetraalkylammonium cations as follows:7

log Kcation ) 0.58 NC - 5.9 (alkyltrimethylammonium cation) (34) log Kcation ) 0.45 NC - 5.2 (tetraalkylammonium cation) (35) The value of log Kcation increases by a factor of about 0.45 or 0.58 by adding one methylene unit to the cations. The slope of the linear correlation for alkyltrimethylammonium cations is slightly larger than that for tetraalkylammonium ones. Similar experimental results have been reported as ca. 0.44-0.99 for the increment of log Kex due to the addition of one methylene unit to the cations or anions having alkyl chains.1,3,5,9-12,25-32 As described earlier, the difference in the parameters representing the extractability of cations, i.e., ∆log Kcation or ∆C, was almost the same irrespective of the reference assumptions, which were selected for determining these parameters. The increment of log Kcation or C for one methylene group would be interpreted as follows. Equation 31 represents that ∆Gcation consists of two terms. One is the contribution of the cation to the free energy change of the ion-pair formation in the gaseous phase (∆Gass,CA,gas), which is represented by the first term in the RHS of eq 31. The other is that due to the transfer of the cation from the aqueous phase into the gas phase, i.e., the opposite process of the hydration. This is the sum of the contributions between the second and fifth terms in the RHS of eq 31. For the calculation of the first term in the RHS of eq 31, -Cg/rC, it is important to rigorously estimate rC. If we assume that the shape of the cation is spherical, rC can be estimated from its molecular volume or surface area, which is experi-

Figure 3. Free energy change of hydration of hydrocarbons as a function of the number of carbon atom involved in the alkyl chains. Solid circles and triangles indicate results for alkanes and alkylbenzenes, respectively.

mentally measured or theoretically calculated. However, various cations having long alkyl chains have usually been used in many previous studies of the ion-pair extraction, for instance, alkyltrimethylammonium and tetraalkylammonium cations. It is hard to estimate rC with reasonable accuracies in such cases because the shape of the cations is quite different from a sphere. In addition, no definitive conclusion has probably been obtained for the structural characteristics and mechanisms of the ionpair formation in water. For instance, it has been pointed out that in water might be formed various types of ion-pairs other than the contact ion-pair, including the solvent-separated ionpair, solvent-shared ion-pair, and triplet ion-pair.33 When these types of ion-pairs are simultaneously formed, the accurate estimation of rC would be difficult. It is concluded that the rigorous calculation of the first term in the RHS of eq 31 is difficult at present. When the cation is alkyltrimethylammonium ion, the ionpair formation should take place around a positively charged nitrogen atom. In such a case, an increase in the length of the alkyl chain little influence the electrostatic interaction because the steric situation around the nitrogen atom would not significantly be changed. Although the contribution of the first term in the RHS of eq 31 to ∆Gcation cannot accurately be calculated, it seems to be almost constant irrespective of the length of the alkyl chains when alkyltrimethylammonium ions of a limited range of NC are the cations. On the other hand, the free energy change of the transfer of the cation from the aqueous phase into the gas phase can in principle be estimated by using eqs 9-26. However, there is a serious problem in the calculation of ∆Gcav,C,aq and ∆Gvdw,C,aq by the solvophobic theory. In the theory, the shapes of a solute and a cavity, in which the solute molecule would be placed, are assumed to be spherical because of the convenience of the calculations.14,16,18,19 The shape and curvature of the solute (cation) and cavity surface, however, are irregular depending on the molecular properties of the solute. It was shown that the free energy change estimated for the solutes having relatively long alkyl chains included significant calculation errors.22,24 It seems to be hard to accurately calculate the sum of the contributions between the second and fifth terms in the RHS of eq 31 for the cation having long alkyl chains in the molecule. However, the influence of hydrophobic groups, such as methylene unit and phenyl group, on the free energy change of the hydration of the cation might be interpreted as follows. Figure 3 shows the correlation between the free energy change

8486 J. Phys. Chem. B, Vol. 104, No. 35, 2000

Miyabe et al.

TABLE 1: Estimation of ∆Gcation cation

rC (cm)

-Cg/rC (kJ mol-1)

-∆Gsol,C,aq (kJ mol-1)

∆Gcation (kJ mol-1)

Li+ Na+ K+

0.78 × 10-8a 0.98 × 10-8a 1.33 × 10-8a

-445 -354 -261

517b 411c 337c

72 57 76

a

Data from ref 37. b Data from ref 39. c Data from ref 38.

of the hydration (∆Ghyd) of hydrocarbons and NC in the alkyl chains. Solid circles and triangles indicate the correlations for alkanes and alkylbenzenes, respectively.34 Similar values of the thermodynamic parameters have been reported for the transfer of hydrocarbons into water.35,36 The value of ∆Ghyd reasonably increases with increasing NC in the hydrocarbons. From the slope of the two linear lines, the increment of ∆Ghyd for one methylene unit is found as about 3.1-4.4 kJ mol-1. Additionally, the difference between the two lines indicates that ∆Ghyd increases by a factor of about 10.6 kJ mol-1 by addition of one phenyl group. From the change in ∆Ghyd due to one methylene unit, the increment of log Kcation at 298 K is estimated as 0.54 () log(exp (3.1/298R))) -0.77 () log(exp (4.4/298R))) when the contribution of the first term in the RHS of eq 31 is constant despite the increase of one methylene unit. This result is consistent with the slope of the linear lines represented by eqs 34 and 35 and the experimental observations previously reported.1,3,5-12,25-32 On the other hand, the increment of log Kcation at 298 K for one phenyl group is similarly calculated as about 1.9 () log(exp (10.6/298R))). Harris et al.1 reported a similar value of the increment of ∆log Kex due to the addition of one phenyl group as 2.1. From the intercept of the straight lines represented by eqs 34 and 35, the log Kcation value for the hypothetical quaternary ammonium cation having no alkyl chains and hydrogen atoms is estimated as about -5.5. The value of log Kcation for tetraphenylammonium cation is estimated as about 2.1 () 4 × 1.9-5.5) from these estimates. This is of the same order of magnitude with log Kcation for tetraphenylarsonium cation, i.e., 2.9,7 experimentally determined although both values are not completely identical. Incidentally, Motomizu et al.9,10,12 provided two figures, i.e., 2.9 and 3.0, as the increment of C for one phenyl group. Inorganic Cations. Few experimental results have been reported for the extractability of inorganic cations. Motomizu12 measured Kex in the ion-pair extraction systems using picrate anion and ethyl acetate as the counterion and organic solvent, and reported C of Li+, Na+, and K+ as 2.72, 2.73, and 2.99, respectively. These values are almost the same, suggesting that the extractability of the inorganic cations is not so different with each other. As described earlier, it is probably difficult to estimate sufficiently rigorous values of rC of the cation constituting the various types of the ion-pairs. However, if the crystalline ion radius is used as a typical value of rC,37 the contribution of the inorganic cation to ∆Gass,CA,gas could roughly be calculated from the ratio -Cg/rC. Additionally, we can use the information about ∆Ghyd of the inorganic cations previously published.38,39 Table 1 lists the contribution to ∆Gcation of the first term in the RHS of eq 31, i.e., -Cg/rC, and the sum of those between the second and fifth terms, that is, -∆Gsol,C,aq. As a consequence, ∆Gcation was estimated as 72, 57, and 76 kJ mol-1 for Li+, Na+, and K+, respectively. These values are of the same order of magnitude, suggesting that the similarity of the extractability of the inorganic cations is confirmed. While the parameters, i.e., log Kcation and C, were estimated, their absolute values had not been determined. Derived was only the

difference in the extractabilities of the cations from that of the reference compound. It is concluded that one of the important factors influencing the extractability of hydrophobic cations is ∆Ghyd (∆Gsol,C,aq), and that the variation in log Kcation is almost quantitatively interpreted by taking the increment of ∆Ghyd into account on the assumption that the contribution of C+ to ∆Gass,CA,gas is not significantly affected by adding one functional (methylene or phenyl) group. On the other hand, for the inorganic cations discussed in this paper, their similar extractability was approximately confirmed by the calculations based on eq 31. These results suggest the validity of eq 31 and the establishment of the linear additive representation of log Kex in eq 1. 3.2. Kanion. Organic Anions. Similar to the cases of the organic cations, it has been reported that the increase in NC of the alkyl chains is accompanied by the almost linear increase in log Kanion of alkylsulfonate anions as follows:7

log Kanion ) 0.64 NC - 7.9 (alkylsulfonate anion) (36) Likewise as the results for the hydrophobic cations, the slope of the linear relationship is 0.64. As described earlier, corresponding experimental results have similarly been reported as ca. 0.44-0.99 for the slope.1,3,5-12,25-32 The increment of the parameters, i.e., ∆log Kanion or ∆A, due to the addition of one methylene unit was almost the same irrespective of the reference assumptions.7,12 The similar interpretation is possible for the increment of log Kanion or A as described in the previous subsection, 3.1. For alkylsulfonate anions, the contribution of the first term in the RHS of eq 32 cannot be accurately calculated because it is difficult to obtain the extremely rigorous value of rA. The shape of the anions is also quite different from a sphere in this case. Actual situation and structure of the ion-pair consisting of alkylsulfonate anions are not clarified either. However, the positive charge on a counterion should interact with the negative charge located around the sulfonate group on the anions. In such a case, similar to the ion-pair formation with alkyltrimethylammonium ions as cations, an increase in the length of the alkyl chain in the alkylsulfonate anions might have little influence on the electrostatic interaction of the ion-pair formation. Irrespective of the length of the alkyl chain, the steric situation around the sulfonate group would not so significantly be changed. In conclusion, although the contribution of the first term in the RHS of eq 32 to ∆Ganion cannot be accurately calculated, it again might be almost constant irrespective of the length of the alkyl chains. Similar to the interpretation for the change in log Kcation or C, the increment of log Kanion or A due to the addition of methylene or phenyl group can probably be accounted for by considering the influence of these hydrophobic groups on ∆Ghyd (∆Gsol,A,aq). Again the number preceding NC in eq 36 is in the range from 0.54 to 0.77, which is calculated earlier from the increment of ∆Ghyd due to the addition one methylene unit (3.14.4 kJ mol-1) shown in Figure 3. In addition, the value of log Kanion for dodecylbenzenesulfonate anion is calculated as 1.7 () -0.2 + 1.9) from that for dodecylsulfonate anion, i.e., -0.2,7 and the predictable increment of log Kanion due to the addition of one benzene unit, i.e., 1.9 () log(exp (10.6/298R))). This is appropriately in agreement with the value of log Kanion for dodecylbenzenesulfonate anion, i.e., 1.5, experimentally determined.7 It has also been experimentally confirmed that the extractability of sulfonate anion is larger than that of carbonate anion. Motomizu et al.11 reported that the extractability is 1.30 log

Ion-Pair Extraction Mechanism

J. Phys. Chem. B, Vol. 104, No. 35, 2000 8487

TABLE 2: Estimation of ∆Ganion anion

rA (cm)

FClBrIOHNO3ClO4-

1.33 × 10-8a 1.81 × 10-8a 1.96 × 10-8a 2.20 × 10-8a 1.40 × 10-8a 2.40 × 10-8b 2.68 × 10-8b

-Cg/rA -∆Gsol,A,aq ∆Ganion log Kanion (kJ mol-1) (kJ mol-1) (kJ mol-1) (calc) -261 -192 -177 -158 -248 -145 -130

434c 317c 303c 257c 379d 270e 190e

173 125 126 99.2 131 125 60.4

-30.3 -21.9 -22.1 -17.4 -23.0 -22.0 -10.6

a Data from ref 37. b Data from ref 41. c Data from ref 38. d Data from ref 39. e Data from ref 40.

unit larger for sulfonate than for carbonate group. Negative charges on sulfonate and carbonate anions may be distributed on three S-O and on two C-O bonds, respectively. The density of negative charges is probably lower around a sulfonate group than around a carbonate group, suggesting that the number of water molecules attracted to a carbonate group is higher than that to a sulfonate group. In such cases, it is likely that the substantial radius of hydrated anions (rA) in the first term in the RHS of eq 32 is smaller for sulfonate anion than for carbonate anion. In conclusion, due to the contribution of the first term, ∆Ganion for sulfonate anion may be smaller than that for carbonate anion even under the conditions that both anions have the same alkyl chain in their molecules. This is probably the reason the extractability is larger for sulfonate group than for carbonate one. Equation 32 is useful for the quantitative or qualitative analysis of the difference in extractability of various anions. Inorganic Anions. There are several experimental results for the extractability of inorganic anions. The order of the extractability, i.e., log Kanion or A, was F- < OH- < Cl- < NO3- < Br- < I- < ClO4-. Kasahara et al.7 reported log Kanion as -5.6 and -3.1 for Br- and I-, respectively. Motomizu12 estimated A as follows: -11.06 for F-, -9.07 for OH-, -8.08 for Cl-, -7.05 for NO3-, -6.89 for Br-, -5.32 for I-, and -5.03 for ClO4-. According to eq 32, ∆Ganion was roughly calculated using the crystalline ion radius (rA)37 and ∆Ghyd (∆Gsol,A,aq) of each inorganic anion38-40 listed in Table 2. The radius of NO3- and ClO4- was estimated from the ionic partial molal volume.41 The results of calculation are also listed in Table 2. The order of the calculated values of ∆Ganion is nearly consistent with that of the extractability of the inorganic anions. The value of log Kanion(cal) was derived from ∆Ganion at 298 K. The difference of log Kanion(cal) between two anions does not completely agree with that observed experimentally. However, it is worthwhile that the order of log Kanion or A can be quantitatively interpreted. It is concluded that ∆Ghyd has an important influence on the extractability of hydrophobic anions and that the variation in log Kanion or A due to the addition of one methylene or phenyl group is almost quantitatively interpreted by considering the increment of ∆Ghyd on the assumption that the contribution of A- to ∆Gass,CA,gas is approximately constant, irrespective of the length of the alkyl chains. On the other hand, for the inorganic anions, the order of their extractability was interpreted by eq 32, although the calculated values of log Kanion did not sufficiently agree with those experimentally measured. These results seem to verify the propriety of eq 32 and imply the establishment of the linear additive representation of log Kex in eq 1. 3.3. Ksolvent. The ion-pair formed in water is extracted into an immiscible organic phase. The extractability of the ion-pair depends on the type and characteristics of the organic solvent. It has empirically been known that the extractability of the ion-

pair is increased with an increase in the dielectric constant or polarity of the organic solvent. Motomizu et al.3,9 studied the relation between a few parameters of the organic solvents and the extractability of the ion-pair of univalent anions with ethyl violet or crystal violet. They reported that the difference between log Kex for several extracting solvents and that for benzene was almost linearly correlated with the ET parameter of the organic solvents. However, there is no theoretical explanation for the linear relationship between ∆log Kex and the ET value. In this study, we attempted to provide a theoretical interpretation for the difference in extractability of the organic solvents characterized by the individual extraction constant for the organic solvents, Ksolvent, in eq 1. Only in the previous research works by Taguchi et al.6,7 and Goto,8 the individual extraction constants were assigned to extracting organic solvents such as as cations and anions. As shown in eq 33, ∆Gsolvent might consist of four contributions. The first one is the free energy change for the generation of the cavity in the bulk organic solvent to accommodate the ion-pair formed in the aqueous phase. The second and third contributions are respectively attributed to the van der Waals and electrostatic interaction of the ion-pair with the surrounding organic solvent molecules. The last one results from the reduction of the free volume. As described earlier, the model used for the calculation in the solvophobic theory is based on the assumption that the shape of the solute, solvent, and cavity is spherical for the sake of the simplification of the calculations. It is unlikely that the first and second terms in the RHS of eq 33 can accurately be estimated for the ion-pair consisting of cations and/or anions having long alkyl chains. In this study, a quantitative interpretation was attempted for the solvation and transfer of a hypothetical ion-pair (C+A-) having a spherical shape from the gas phase into the organic solvents. Some physicochemical properties of the hypothetical ion-pair, which were necessary for the calculations of ∆Gcav,CA,org and ∆Gvdw,CA,org by eqs 10-23, were assumed on the basis of those of many organic compounds.42-46 The molar volume (VCA) of the hypothetical ion-pair was assumed to range from 300 to 700 cm3 mol-1 because that of the hydrophobic cations and anions usually lies between 100 and 350 cm3 mol-1.7 The acentric factor (ωCA) of the hypothetical ion-pair was assumed to be about 0.2-0.4 because many nonpolar organic compounds of symmetrical shapes have ω values in the region. Besides the parameters described above, the refractive index (nCA) and ionization potential (ICA) of the hypothetical ion-pair were respectively taken as 1.5 and 10 eV. On the basis of these assumptions, ∆Gsolvent was calculated according to eq 33. Table 3 lists the calculated values of ∆Gsolvent, Ksolvent, and individual contributions of the four terms in the RHS of eq 33 to ∆Gsolvent at VCA ) 500 cm3 mol-1 and ωCA ) 0.3. The values of ∆Gcav, ∆Gvdw, ∆Ges, and ∆Gred in Table 3 indicate the calculated values of the free energy change due to the cavity formation, the van der Waals interaction, the electrostatic interaction, and the reduction of free volume of the organic solvent, respectively. These correspond to the solvation and transfer of the hypothetical ion-pair into the extracting organic phase. The value of log Ksolvent(cal) for each organic solvent at 298 K was calculated from ∆Gsolvent. Figure 4 illustrates the correlation between the difference in log Ksolvent(cal) calculated with benzene as reference and log Ksolvent(exp) experimentally measured. Solid circles represent the calculated values at VCA ) 500 cm3 mol-1 and ωCA ) 0.3. Although Figure 4 shows some scatter, the tendency of ∆log Ksolvent(cal) is the same as that of log Ksolvent(exp) on the whole, suggesting that the

8488 J. Phys. Chem. B, Vol. 104, No. 35, 2000

Miyabe et al.

TABLE 3: Estimation of Ksolventa solvent

∆Gcavb (kJ mol-1)

∆Gvdwb (kJ mol-1)

∆Gesb (kJ mol-1)

∆Gredb (kJ mol-1)

∆Gsolvent (kJ mol-1)

log Ksolvent (calc)c

∆log Ksolvent (calc)d

log Ksolvent (exp)e

benzene carbon tetrachloride o-xylele toluene chlorobenzene dichloromethane chloroform 1,2-dichloroethane o-dichlorobenzene 2-hexanone

72.7 68.0 76.6 71.9 85.0 70.0 68.6 82.0 94.2 65.5

-162 -162 -187 -172 -182 -178 -180 -206 -197 -186

-13.1 -12.8 -14.5 -13.6 -21.5 -24.6 -20.4 -24.5 -24.4 -25.6

13.9 13.7 13.1 13.5 13.6 14.7 14.2 14.2 13.3 13.1

-88.2 -92.8 -112 -100 -105 -118 -118 -135 -114 -133

15.5 16.3 19.6 17.6 18.3 20.6 20.7 23.6 19.9 23.2

0.8 4.1 2.1 2.9 5.2 5.2 8.2 4.5 7.8

0 -1.3 -0.8 -0.6 2.3 4.1 3.4-4.2 4.5 4.6 6.5

a Equilibrium constant for the solvation and transfer of the hypothetical ion-pair having a shape of spherical into the organic solvent. Physicochemical properties of the hypothetical ion-pair were assumed as follows: acentric factor (ωCA) ) 0.3, ionization potential (ICA) ) 10 eV, refractive index (nCA) ) 1.5. b Physicochemical properties of the solvents required to the calculations were cited from refs 42-46. c Calculated from ∆Gsolvent at 298 K. d Difference in log Ksolvent(cal) between each solvent and benzene. e Data from ref 7.

Figure 4. Comparison of ∆log Ksolvent(cal) calculated with log Ksolvent(exp) experimentally determined. The value of ∆log Ksolvent(cal) was calculated by taking benzene as a reference solvent. Symbols indicate the calculated values at VCA ) 300 cm3 mol-1, ωCA ) 0.3 (solid triangle); at VCA ) 500 cm3 mol-1, ωCA ) 0.3 (solid circle); and at VCA ) 700 cm3 mol-1, ωCA ) 0.3 (solid square). Vertical lines indicate error ranges corresponding to (5% of ∆Gvdw calculated at VCA ) 500 cm3 mol-1, ωCA ) 0.3 (solid circle).

Figure 5. Comparison of ∆log Ksolvent(cal) calculated with log Ksolvent(exp) experimentally determined. The value of ∆log Ksolvent(cal) was calculated by taking benzene as a reference solvent. Symbols indicate the calculated values at VCA ) 500 cm3 mol-1, ωCA ) 0.2 (solid triangle); at VCA ) 500 cm3 mol-1, ωCA ) 0.3 (solid circle); and at VCA ) 500 cm3 mol-1, ωCA ) 0.4 (solid square). Vertical lines indicate error ranges corresponding to (5% of ∆Gvdw calculated at VCA ) 500 cm3 mol-1, ωCA ) 0.3 (solid circle).

difference in extractability of the organic solvents can roughly be interpreted by eq 33. However, relatively large discrepancies are observed between the calculated and experimental values of log Ksolvent for toluene (log Ksolvent(exp) ) -0.6) and o-xylene (log Ksolvent(exp) ) -0.8). As a consequence of the systematic study of the extractability of alkylbenzene derivatives used as the extracting solvent, it has been clarified that log Ksolvent decreases with an increase in the length of the alkyl chain attached to the benzene ring, and that the π-π interaction between the ion-pair and organic solvent has an important contribution to Kex.7,47 The calculation in this study cannot explain the influence of the substituted alkyl groups on Ksolvent because of the simplification of the solvation model in the solvophobic theory. The calculated values in Figure 4 probably include some errors. When ∆Gcav,CA,org was calculated by eqs 10-12, the values of κie and Wi were respectively assumed to be unity and zero because Vi was several times larger than Vorg and because ke values of the organic solvents were usually close to unity, for instance, ke of benzene was reported as 0.629 by Halicioglu and Sinanoglu.16 Similar situation might be observed for κis. The value of ks of benzene was also reported as 0.469.16 In addition, when ∆Gvdw,CA,org was calculated by eqs 13-23, the value of Q′ was neglected because of its small contribution to ∆Gvdw,CA,org. The neglect of the contribution due to Q′ might provide the calculation error to ∆Gvdw,CA,org about 10% at

maximum. Because ∆Gvdw,CA,org was estimated between about -160 and -210 kJ mol-1 and has the largest influence on ∆Gsolvent as listed in Table 3, the error range of about (1.4 () log(exp (8/298R))) to (1.8 () log(exp (10.5/298R))) logarithmic unit was indicated by the vertical lines in Figure 4. As described above, the value of VCA and ωCA of the hypothetical ion-pair was assumed to range between 300 and 700 cm3 mol-1 and between 0.2 and 0.4, respectively. We first studied the influence of VCA on the calculation of Ksolvent. Similar calculations were made at ωCA ) 0.3 while changing VCA. Solid triangles and squares in Figure 4 represent the calculated results at VCA ) 300 cm3 mol-1 and ωCA ) 0.3 and at VCA ) 700 cm3 mol-1 and ωCA ) 0.3, respectively. It is indicated that the results of the calculation are in agreement with each other irrespective of VCA. On the other hand, Figure 5 shows the influence of the variation in ωCA on the calculation of Ksolvent. The symbols of solid triangle and square represent the calculated values at VCA ) 500 cm3 mol-1 and ωCA ) 0.2 and at VCA ) 500 cm3 mol-1 and ωCA ) 0.4, respectively. The discrepancy between the calculated and experimental values of log Ksolvent increases with increasing ωCA in most cases. Although the plots of solid triangle, circle, and square scatter in Figure 5, the tendency of the extractability of the several organic solvents is interpreted on the whole regardless of the value of ωCA used in this study. These results probably allow us to conclude that eq 33 provides

Ion-Pair Extraction Mechanism

J. Phys. Chem. B, Vol. 104, No. 35, 2000 8489

TABLE 4: Estimation of the Sum ∆Gcation + ∆Ganion ion-pair

VCA (cm3 mol-1)

ACA (cm2 mol-1)

µCAa (C m)

rC-Aa (cm)

∆Gass,CA,gas (kJ mol-1)

-∆Gsol,C,aqb (kJ mol-1)

-∆Gsol,A,aqb (kJ mol-1)

∆Gcation+∆Ganion (kJ mol-1)

KCl KBr KI

26.5 33.7 45.4

3.63 × 109 4.26 × 109 5.20 × 109

3.43 × 10-29 3.55 × 10-29 3.61 × 10-29

2.67 × 10-8 2.82 × 10-8 3.05 × 10-8

-521 -492 -456

337 337 337

317 303 257

133 148 138

a

Data from ref 45. b Data from ref 38.

TABLE 5: Estimation of Kexa ion-pair

ωCA

∆Gcavb (kJ mol-1)

∆Gvdwb (kJ mol-1)

∆Gesb (kJ mol-1)

∆Gredb (kJ mol-1)

∆Gsolvent (kJ mol-1)

∆Gcation+∆Ganion (kJ mol-1)

∆Gex (kJ mol-1)

log Kex (calc)c

log Kex (pred)d

KCl KBr KI KCl KCl

0.3 0.3 0.3 0.2 0.4

6.7 8.4 11.1 6.7 6.7

-36.1 -42.0 -50.5 -28.0 -44.3

-33.1 -27.9 -21.5 -33.1 -33.1

13.9 13.9 13.9 13.9 13.9

-48.6 -47.6 -47.0 -40.6 -56.8

133 148 138 133 133

84.5 100 91.3 92.6 76.3

-14.8 -17.5 -16.0 -16.2 -13.4

-11.6 -10.4 -7.9 -11.6 -11.6

a Equilibrium constant for the ion-pair extraction of the inorganic ion-pairs into benzene. b Physicochemical properties of the inorganic ion-pairs required to the calculations were assumed as follows: acentric factor (ωCA) ) 0.2-0.4, ionization potential (ICA) ) 11 eV, refractive index (nCA) ) 1.5. Physicochemical properties of benzene were cited from refs 42-46. c Calculated from ∆Gex at 298 K. d Log Kex predicted from experimental data of refs 7 and 12.

a basic aspect for interpreting the contribution of the organic solvents to Kex, and that the validity of the linear additive rule of the individual extraction constants as indicated in eq 1 is fundamentally demonstrated. 4. Estimation of Absolute Value of Kex by the Solvophobic Theory As described in the previous subsections, it is likely that the difference in each individual extraction constant, i.e., Kcation, Kanion, and Ksolvent, between homologous series of ions and different organic solvents can almost quantitatively be interpreted. In this part, it was attempted to confirm the accuracy of the estimation of Kex by the theoretical approach interpreted in this paper. The inorganic ion-pairs of K+ with Cl-, Br-, and Iextracted into benzene were taken as examples because the estimation of ∆Gcav and ∆Gvdw brought about significantly large calculation errors when the cation, anion, ion-pair, and extracting organic solvent had long alkyl chains. Some items of information about the intrinsic physicochemical properties of the inorganic cation, anions, and ion-pairs and benzene could also be cited from some literature. Benzene is used as one of the most popular extracting organic solvents. Absolute values of ∆Gex were estimated using the information about the thermodynamic properties of the hydration of the ions and ion-pairs. The value of Kex derived from ∆Gex was compared with that predicted from the summation of the logarithmic values of Kcation, Kanion, and Ksolvent, which were experimentally determined.7 Table 4 lists the results of calculation relating to ∆Gcation and ∆Ganion. The value of ∆Gass,CA,gas was calculated by eq 27. The internuclear distance (rC-A)45 was used as rCA. The sum ∆Gcation + ∆Ganion was calculated from ∆Gass,CA,gas, ∆Gsol,C,aq,38 and ∆Gsol,A,aq.38 Table 5 lists the results concerning ∆Gsolvent and Kex. The free energy change due to the cavity formation in benzene (∆Gcav in Table 5) was estimated by using eq 10, in which Wi was assumed to be zero. Equation 12 was used for estimating κie in eq 10. As described above, κe of benzene was reported as 0.629.16 The free energy change of the van der Waals interaction (∆Gvdw in Table 5) was calculated by eqs 13-23. According to the data of physicochemical properties published in some literature,42-46 the acentric factor (ωCA), the ionization potential (ICA), and the refractive index (nCA) of the inorganic ion-pairs were assumed as 0.3, 11 eV, and 1.5, respectively. Other information necessary for the estimation of ∆Gcav in Table

5 was cited from the references.42-46 The free energy change of the electrostatic interaction (∆Ges in Table 5) was calculated by eqs 24-26. The value of P was assumed to be equal to 1/(4π0) because Ψ was calculated as 0.46 for benzene and the polarizability (β) of various inorganic compounds was usually of the order of 1 × 10-24 cm3, which was about 1 order of magnitude smaller than ν of the inorganic ion-pairs listed in Table 5. The value of ∆Gsolvent was calculated by summing ∆Gcav, ∆Gvdw, ∆Ges, and ∆Gred in Table 5. According to eq 8, the sum of ∆Gcation, ∆Ganion, and ∆Gsolvent should be equal to ∆Gex, from which log Kex was respectively calculated as -14.8, -17.5, and -16.0 for KCl, KBr, and KI at ωCA ) 0.3. Kasahara et al.7 indicated log Kcation for bis[2-(5-methyl-2pyridylazo)-5-diethylaminophenolato]cobalt(III) cation ([Co(5Me-PADAP)2]+) as 4.7. On the other hand, Motomizu12 reported C of 2.99 and 12.5 for K+ and [Co(5-Me-PADAP)2]+, respectively. From these results, log Kcation of K+ based on the reference assumptions proposed by Kasahara et al.7 is estimated as -4.8. Additionally, for inorganic anions, Kasahara et al.7 indicated log Kanion for Br- and I- as -5.6 and -3.1, respectively. As described earlier, Motomizu12 reported that A values for Cl- and Br- as -8.08 and -6.89, respectively. From these experimental results, log Kanion for Cl- would be calculated as -6.8. According to eq 1, log Kex for the extraction of the inorganic ion-pairs into benzene is estimated as -11.6, -10.4, and -7.9 for KCl, KBr, and KI, respectively, because Ksolvent of benzene is assigned as unity.7 Table 5 also shows the results for KCl by assuming ωCA ) 0.2 and 0.4. According to the variation in ωCA, log Kex(cal) changes to a certain extent. However, the calculated values are of the same order of magnitude irrespective of ωCA. The values of log Kex(cal) for the inorganic ion-pairs estimated using the theoretical approach in this study are not in very good agreement with those of log Kex(pred) calculated by summing up the logarithms of the individual extraction constants experimentally determined.7,12 However, both values are almost of the same order of magnitude as will be seen in Table 5. Considering the uncertainty involving in the estimation of the related parameters and the simplification of the calculation model of the solvophobic theory, it might be concluded that the thermodynamic cycle model based on the solvophobic theory as indicated in Figure 2 provides an essential framework for the comprehensive interpretation of the mechanism of the ionpair extraction.

8490 J. Phys. Chem. B, Vol. 104, No. 35, 2000

Miyabe et al.

5. Conclusion

References and Notes

By applying the solvophobic theory, the usefulness of the linear additive representation of log Kex (eq 1) was theoretically demonstrated using the three individual extraction constants, that is, Kcation, Kanion, and Ksolvent. On the basis of the thermodynamic cycle model of the 1:1 ion-pair extraction (Figure 2), we first indicated that ∆Gex was divided into ∆Gcation, ∆Ganion, and ∆Gsolvent, which were assigned to the cations, anions, and extracting organic solvents, respectively. Then, an attempt was made for the quantitative interpretation of the individual extraction constants representing the extractability of the cations, anions, and organic solvents. It seems that ∆Ghyd significantly influences the extractability of hydrophobic cations and anions. The increment of log Kcation and log Kanion of the ions having alkyl chains and/or phenyl groups is almost quantitatively explained by considering the increment of ∆Ghyd per structural unit. On the other hand, for inorganic cations and anions, log Kcation and log Kanion were estimated from ∆Ghyd data and the crystalline ion radius of the inorganic ions. Although the values of log Kcation and log Kanion thus estimated were not sufficiently in agreement with those experimentally determined, the magnitude or the order of the extractabilities of the inorganic cations and anions could roughly be interpreted. Many experimental studies have been carried out extensively on the ion-pair formation in water.48 More accurate explanations of Kcation and Kanion would be derived by investigating in detail the experimental results. This is one of subsequent research subjects. In addition, it was attempted to quantitatively interpret the difference between the extractabilities of some extracting organic solvents. The conceptual solvation of the hypothetical spherical ion-pair and transfer from the gaseous phase into the bulk organic solvent was discussed based on the solvophobic theory. As a consequence, the value of log Ksolvent could be almost quantitatively interpreted. Other than the three individual extraction constants, i.e., Kcation, Kanion, and Ksolvent,6-8 some pieces of information about the parameters representing the extractability of the 1:1 ionpair extraction have also been reported.1,3,5,9-12,48 The proposed values of these parameters for the same ions are different because the reference assumptions are not identical. Nevertheless, the increments of log Kcation or log Kanion per structural unit of successive cations or anions are almost the same as those of the other parameters. Because the increment of the extractability parameters between ions and between organic solvents was studied, the results in this paper are useful for interpreting the extractability parameters irrespective of the reference assumptions. Finally, we additionally estimated the absolute value of Kex based on the theoretical approaches described in this study. The Kex values of a few inorganic ion-pairs were calculated as examples. It was also indicated that the absolute value of Kex was roughly estimated. In conclusion, we demonstrated the propriety of the linear additive representation of log Kex with the three individual extraction constants respectively assigned to cations, anions, and organic solvents (eq 1). The thermodynamic cycle model based on the solvophobic theory is very useful for the consistent interpretation of the mechanism of the ion-pair extraction.

(1) Harris, M. J.; Higuchi, T.; Rytting, J. H. J. Phys. Chem. 1973, 77, 2694. (2) Taguchi, S.; Kasahara, I.; Goto, K. Bunseki Kagaku 1981, 30, 513. (3) Motomizu, S.; Fujiwara, S.; Toei, K. Anal. Chim. Acta 1981, 128, 185. (4) Iwamoto, E.; Ito, K.; Yamamoto, Y. J. Phys. Chem. 1981, 85, 894. (5) Matsunaga, H.; Yotsuyanagi, T. Nippon Kagaku Kaishi 1982, 785. (6) Taguchi, S.; Nakamura, N.; Hiraide, T.; Goto, K. Bunseki Kagaku 1982, 31, 548. (7) Kasahara, I.; Ohgaki, Y.; Matsui, K.; Kano, K.; Taguchi, S.; Goto, K. Nippon Kagaku Kaishi 1986, 894. (8) Goto, K. Dojin News 1995, No.74, 3. (9) Motomizu, S.; Fujiwara, A.; Toei, K. Bunseki Kagaku 1983, 32, 91. (10) Motomizu. S.; Hamada, S.; Toei, K. Bunseki Kagaku 1983, 32, 648. (11) Motomizu. S.; Hamada, S.; Toei, K. Bunseki Kagaku 1983, 32, 703. (12) Motomizu, S. Bunseki Kagaku 1984, 33, 31. (13) Sinanoglu, O. Intermolecular Forces in Liquids; Hirschfelder, J. O., Ed.; Wiley (Interscience): New York, 1967; Vol. 12, p 283. (14) Sinanoglu, O. Molecular Associations in Biology; Pullman B., Ed.; Academic Press: New York, 1968; p 427. (15) Sinanoglu, O. Molecular Interactions; Ratajczak, H., OrvilleThomas, W. J., Eds.; J. Wiley: New York, 1982; Vol. 3, p 283. (16) Halicioglu, T.; Sinanoglu, O. Ann. N. Y. Acad. Sci. 1969, 158, 308. (17) Belfort, G. EnViron. Sci. Technol. 1979, 13, 939. (18) Belfort, G.; Altshuler, G. L.; Thallam, K. K.; Feerick, C. P., Jr.; Woodfield, K. L. AIChE J. 1984, 30, 197. (19) Horva´th, C.; Melander, W.; Molnar, I. J. Chromatogr. 1976, 125, 129. (20) Vailaya, A.; Horva´th, C. J. Phys. Chem. B 1997, 101, 5875. (21) Vailaya, A.; Horva´th, C. J. Chromatogr. A 1998, 829, 1. (22) Miyabe, K.; Suzuki, M. AIChE J. 1995, 41, 536. (23) Miyabe, K.; Takeuchi, S. J. Chem. Eng. Jpn. 1997, 30, 1047. (24) Miyabe, K.; Guiochon, G. AdVances in Chromatography 2000, 40, 1. (25) Biles, J. A.; Plakogannis, F. M.; Wong, B. J.; Biles, P. M. J. Pharm. Sci. 1966, 55, 909. (26) Yeh, K. C.; Higuchi, W. I. J. Pharm. Sci. 1972, 61, 1648. (27) Gustavii, K. Acta Pharm. Suec. 1967, 4, 233. (28) Eskborg, S.; Schill, G. Anal. Chem. 1973, 45, 2092. (29) Schill, G. Acta Pharm. Suec. 1965, 2, 13. (30) Modin, R.; Schill, G. Acta Pharm. Suec. 1967, 4, 301. (31) Modin, R.; Schill, G. Acta Pharm. Suec. 1970, 7, 585. (32) Modin, R.; Schill, G. Talanta 1975, 22, 1017. (33) Marcus, Y. Introduction to Liquid-State Chemistry; John Wiley & Sons: New York, 1977; Chapter 6. (34) Seno M. Entropy; Kyouritsu Shuppan: Tokyo, Japan, 1997; p 95. (35) Kauzmann, W. AdV. Protein Chem. 1959, 14, 1. (36) Moore, W. J. Basic Physical Chemistry; Prentice Hall: Englewood Cliffs, New Jersey, 1983; Chapter 10. (37) Landolt-Bornstein Zahlenwerte und Funktionen; I Band, Atom- und Molekularphysik; 4 Teil; Kristalle; Springer-Verlag: 1955; p 522. (38) Rosseinsky D. R. Chem. ReV. 1965, 65, 467. (39) Latimer, W. M. Oxidation Potential; Prentice Hall: Englewood Cliffs, New Jersey, 1952. (40) Vasil’ev, V. P.; Zolotasev, E. K.; Kapustinskii, A. F.; Mishchenko, K. P.; Podgornaya, E. A.; Yatsimirskii, K. B. Zh. Fiz. Khim. 1960, 34, 1763. (41) Zana, R.; Yeager, E. J. Phys. Chem. 1967, 71, 521. (42) CRC Handbook of Chemistry and Physics; Lide, D. R., Ed.; CRC Press: Boca Raton, 1997. (43) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987. (44) Kagaku Kogaku Binran; Maruzen: Tokyo, Japan, 1984. (45) Kagaku Binran; Maruzen: Tokyo, Japan, 1984. (46) Yozai Handbook; Kodansha: Tokyo, Japan, 1976. (47) Kasahara, I. Ph.D. Dissertation, 1989, Hokkaido University, Sapporo, Japan. (48) Motomizu, S. Bunseki Kagaku 1999, 48, 151.