Potentiometric Determination of Ion-Pair Formation Constants of Crown Ether-Complex Ions with Some Pairing Anions in Water Using Commercial Ion-Selective Electrodes

1. Introduction

Ion-pair formation equilibrium-constants or association ones in water have been determined so far using various methods. As representative methods, one can suppose conductometry, spectrophotometry [1], potentiometry [1], voltammetry [2], calorimetry, electrophoresis [3], and so on. The conductometric measurements generally have high accuracies for their determination for metal salts (MX_z at z = 1, 2) and of their metal complex-ions (ML^z+) with pairing anions (X^z−) in water and in pure organic solvents. According to our knowledge, its experimental operation requires high experimental know-how to handle the measurements. Also, the spectrophotometric measurements require the condition that either species formed in or those consumed in the ion-pair formation are of colored at least. Solvent extraction methods are generally difficult to establish some experimental conditions, such as ionic strength (I ) of both phases and solvent compositions, compared with the above two methods. Strictly speaking, its constants are hard to recognize as thermodynamic ones.

We treat here the ion-pair formation of crown compounds (L), such as 15-crown-5 and 18-crown-6 ethers (15C5 and 18C6), with colorless alkali, alkaline-earth metal ions, and so on in water [4-8]. The methods described above are difficult to apply for the determination of the constants. For example, conductometry cannot distinguish among the metal ions M^z+, their ML^z+, and X^z− in water and ML^z+ is unstable in many cases. Also, many M^z+ and ML^z+ employed here cannot be detected spectrophotometrically. Voltammetric methods cannot apply for the determination, because working electrode suitable for M^z+ detection is difficult to get. While, polarography with DME can be effective for the measurements of such systems [2]. Unfortunately, it must use mercury and its salts which pollute the environment around us.

Thus, in order to overcome these limitations, potentiometry with ISE has been applied for the determination of the ion-pair formation constants (K_MLXz⁰) for MLX_z in water at I → 0 mol dm⁻³, although its applications are limited by kinds of commercial ISE. In the present chapter, its fundamentals and applications for the formation systems of MX_z and MLX_z in water are described. Here, the determination of K_MXz⁰, the ion-pair formation constant of MX_z in water at I → 0, is always required for that of K_MLXz⁰.

2. Emf measurements

2.1. Electrochemical cells [4-8]

Constitutions of cells employed for emf measurements of test solutions are described as follows.

As a cell with a single liquid junction

$$\text{Cell }\left(\text{A}\right):\text{ Ag}|\text{Ag}\text{Cl}|0.{\text{1 mol dm}}^{-\text{3}}{\left({\text{C}}_{\text{2}}{\text{H}}_{\text{5}}\right)}_{\text{4}}\text{NCl},\text{ LiCl},\text{ NaCl},\text{ KCl}$$E1

$$\text{or }0.0{\text{5 mol dm}}^{-\text{3}}{\text{MgCl}}_{\text{2}}|{}^{*}\text{test solution}|\text{ISE }\left[\text{4},\text{5}\right]$$E2

or as that with a double liquid junction

$$\text{Cell }\left(\text{B}\right):\text{ Ag}|\text{Ag}\text{Cl}|0.{\text{1 mol dm}}^{-\text{3}}{\left({\text{C}}_{\text{2}}{\text{H}}_{\text{5}}\right)}_{\text{4}}\text{NCl},\text{ NaCl},$$E3

$$\text{or KCl}|\text{1}{\text{ mol dm}}^{-\text{3}}{\text{KNO}}_{\text{3}}|{}^{*}\text{test solution}|\text{ISE }\left[\text{6}-\text{8}\right]$$E4

Here, the 1 mol dm⁻³ solution of KNO₃ in Cell (B) is a salt bridge, between which and the test solution, E_j estimated from the Henderson equation [2] is in the range of 1 to 3 mV in many cases [9]. Standard types for the reference electrodes of Cells (A) and (B) are $$\text{Ag}|\text{Ag}\text{Cl}|0.{\text{1 mol dm}}^{-\text{3}}$$ (C₂H₅)₄NCl and $$\text{Ag}|\text{Ag}\text{Cl}|0.{\text{1 mol dm}}^{-\text{3}}\text{KCl}|\text{1}{\text{ mol dm}}^{-\text{3}}$$ KNO₃, respectively. For Cell (A), the E_j values are corrected by the Henderson equation {see Eq. (1)}, while they are not corrected for Cell (B).

2.1.1. For ion-selective electrodes

Commercial ISEs used here are summarized in Table 1 and some comments for the present emf measurements are described.

2.1.2. Corrections of liquid junction potentials [2]

For emf measurements of the electrochemical cells, the problem of the liquid junction potentials E_j occurred at the interface marked with an asterisk cannot be avoided. Hence, correction procedures of E_j are described in this section. Here, the salt bridges with KNO₃ are experimentally used and the Henderson equation [2]

ISE	Type & Maker	Comments
Na^+-selective electrode	1512A-10C, Horiba	Glass membrane; this electrode partially responds K^+ and Rb^+.
K^+-selective electrodes	1200K, Toko	Glass membrane (not produced now); this electrode partially responded Li^+.
	8202-10C, Horiba	Liquid membrane; it is unstable for the aqueous solution of hydrophobic picrates.
Ag^+-selective electrode	8011-10C, Horiba	Solid membrane
Cd^2+-selective electrode	8007-10C, Horiba	Solid membrane; it responds Br^− partially and I^− well. See below.
Cl^−-selective electrode	8002-10C, Horiba	Solid membrane
Br^−-selective electrode	8005-10C, Horiba	Solid membrane
I^−-selective electrode	8004-10C, Horiba	Solid membrane

Table 1.

Commercial ISEs used here^a

$$E_{\text{j}}={\varphi }^{\text{II}}-{\varphi }^{\text{I}}=-\frac{{\displaystyle \sum _j\left|z_j\right|}\frac{u_j}{z_j}(c_j^{\text{II}}-c_j^{\text{I}})}{{\displaystyle \sum _j\left|z_j\right|}{u}_j(c_j^{\text{II}}-c_j^{\text{I}})}\frac{RT}{F}\ln \frac{{\displaystyle \sum _j\left|z_j\right|}{u}_j{c}_j^{\text{II}}}{{\displaystyle \sum _j\left|z_j\right|}{u}_j{c}_j^{\text{I}}}$$E5

is analytically employed for the correction of E_j. For example, the molar concentration of j in the phase I, c_j^I, consists of c_K^I and c_Cl^I, while c_j^II does of c_Na^II, c_NaL^II, and c_Pic^II for the NaPic-L system, where ions involved in a part of the cell are expressed as

K⁺, Cl⁻ (phase I)│^* Na⁺, NaL⁺, Pic⁻ (phase II)

(see the case 1L). Also, the u_Nac_Na^II + u_NaLc_NaL^II term in Eq. (1) can be assumed to be u_Na(c_Na^II + c_NaL^II) in practice (see Table 2 for the u_Na value), because the condition of c_Na^II >> u_NaLc_NaL^II/u_Na holds in many cases with L (low stabilities of ML^z+ in water).

2.2. Emf measurements

2.2.1. Preparation of calibration curves

A representative procedure for preparing a calibration curve is described below. Using a pipette, 55 cm³ of the aqueous solution of NaCl or KCl is precisely prepared in 100 cm³ beaker, kept at 25 ± 0.3 or 0.4 ºC, and then slowly stirred with a Teflon bar containing a magnet. To this solution, the ISE corresponding to Na⁺ or K⁺ and the reference electrode are immersed. A 7 or 10 cm³ portion of pure water is added by the pipette and, after 5 minutes, emf values at a steady state are read. This operation is repeated, when the amount of the solution is reached to about 100 cm³. Consequently, 7 or 5 data are obtained in a unit operation. The thus-obtained calibration curves are shown in Fig. 1.

Figure 1.

Calibration curve of aqueous NaCl solution at 25 ºC. Emf = 55.0log [NaCl]_t − 89.4 at R = 0.999. This slope is close to 59 mV/decade, showing the Nernstian response with z = 1

A high-purity NaCl (99.98 to 99.99%) is dried about 160 ºC in an oven [12]. The purities of other standards are checked by AgNO₃ titration for which the Ag⁺ solution is standardized with the high-purity NaCl.

2.2.2. Emf measurements of test solutions

A representative procedure for the emf measurements of the test solution is described here. Using a pipette, 55 cm³ of aqueous solution of MX_z is precisely prepared in 100 cm³ beaker, kept at 25 ± 0.3 or 0.4 ºC, and then stirred with the Teflon bar. To this solution, the ISE and the reference electrode are immersed. A 7 or 10 cm³ portion of the aqueous solution of L (or pure water for the K_MXz determination) is added by a pipette and after 5 minutes, emf values at a steady state are read. This operation is repeated, when the amount of the solution is reached to about 100 cm³. Consequently, 7 or 5 data are obtained in a unit operation. Then an example for the plot is shown with that of the corresponding calibration curve of the CdSO₄-18C6 system (Fig. 2). These plots indicate that [Cd²⁺] << [CdSO₄]_t in mixtures, compared with the calibration curve depicted at open circles. All the above experiments are performed under the condition of [MX_z]_t ≈ [L]_t.

Figure 2.

Calibration curve (open circles) of aqueous CdSO₄ solution at 25 ºC. Emf = 23.8log [CdSO₄]_t − 89.4 at R = 0.999. This slope is close to 30 mV/decade, showing the Nernstian response with z = 2. Open squares show plots of mixtures of CdSO₄ with 18C6 in water at 25 ºC.

3 Theoretical treatments and data analysis

3.1. Fundamentals

3.1.1. Ionic activity coefficients [10,13]

In order to determine the ion-pair formation constants at I → 0 and 25 ºC, the ionic activity coefficients (y_j ) of ionic species j used for the activity (a_j ) calculations are evaluated from the extended Debye-Hückel equation

log y_j =$$-\frac{0.5114z_j^2\sqrt{I}}{1+0.3291a\sqrt{I}}$$

and the Davies one [13]

log y_j =$$-0.5114z_j^2\left(\frac{\sqrt{I}}{1+\sqrt{I}}-0.3I\right)$$,

where a denotes the ion size parameter (see Table 2). In general, it is mentioned that the former equation is employed in the range of less than 0.1 mol dm⁻³, while the latter one is done in that of less than about 1 mol dm⁻³. Also, the Davies equation can be used for some ions, such as ML^z+ and M^IIX⁺, or for the y_j calculations of species j, of which the ion size parameters (e.g., DDTC⁻, tfa⁻) are not available, because its equation does not involve the parameter a. However, the accuracy of its y_j will be less than that of Eq. (3). The ion size parameters of some ions in water are listed in Table 2, together with their mobility data [11] at 25 ºC.

j	a(j ) /Å	uj /10^−4		j	a(j ) /Å	uj /10^−4
H^+	9	36.25		Cl^−	3	7.912
Li^+	6	4.010		Br^−	3	8.13
Na^+	4	5.193		I^−	3	7.96
K^+	3	7.619		NO3^−	3	7.404
Ag^+	2.5	6.415		MnO4^−	3.5	6.35
Ca^2+	6	6.166		ReO4^−	3.9^c	5.697
Cd^2+	5	5.6		ClO4^−	3.5	7.05
CdCl^+	4	---^d		Pic^−	7	3.149
(C2H5)4N^+	6	3.384		BPh4^−	9^c	2.2
				SO4^2−	4	8.27
				S2O3^2−	4	8.81
				CrO4^2−	4	8.8

Table 2.

Ion size parameters^a and mobilities^b of some ions in water at 25 ºC

3.1.2. Model of ion-pair formation equilibria in water [4,6-8,12]

We introduce here three kinds of chemical equilibria for the ion-pair formation of single MX, MX₂, and M₂X and their mixtures with L, except for chemical ones for the mixture of M₂X with L.

Case (1). 1:1 and 2:2 electrolytes

M^z+ + X^z− $\stackrel{}{\rightleftharpoons }$ MX at z = 1, 2

Case (2). 2:1 electrolytes

M²⁺ + X⁻ $\stackrel{}{\rightleftharpoons }$ MX⁺

MX⁺ + X⁻ $\stackrel{}{\rightleftharpoons }$ MX₂

Case (3). 1:2 electrolytes

M⁺ + X²⁻ $\stackrel{}{\rightleftharpoons }$ MX⁻

M⁺ + MX⁻ $\stackrel{}{\rightleftharpoons }$ M₂X

Case (1L). 1:1 or 2:2 electrolytes with L

M^z+ + X^z−$\stackrel{}{\rightleftharpoons }$MX

M^z+ + L $\stackrel{}{\rightleftharpoons }$ML^z+

ML^z+ + X^z− $\stackrel{}{\rightleftharpoons }$MLX at z = 1, 2

Case (2L). 2:1 electrolytes with L

M²⁺ + X⁻$\stackrel{}{\rightleftharpoons }$MX⁺

MX⁺ + X⁻$\stackrel{}{\rightleftharpoons }$MX₂

M²⁺ + L $\stackrel{}{\rightleftharpoons }$ML²⁺

ML²⁺ + X⁻$\stackrel{}{\rightleftharpoons }$MLX⁺

MLX⁺ + X⁻$\stackrel{}{\rightleftharpoons }$MLX₂

Ion-pair formation of M⁺ or X²⁻ for Case (3) is omitted, because an example for the ion-pair formation of ML⁺ with X²⁻ was not found for the present experiments. First, one determine the formation constants for Case (1) or (2) and next do those for Case (1L) or (2L), using the equilibrium constants determined by analyzing Case (1) or (2), respectively. Therefore, as you know, the experimental errors of the K values obtainable from Cases (1) and (2) become influenced those in the K determination of Cases (1L) and (2L), respectively (see Table 5).

3.2. Theoretical treatments and data analysis for Case (1) [4]

Analytical equations are derived from the models represented in the section 3.1.2. Using these equations, analytical procedures are described for the cases of the metal salts MX, M₂X, MX₂, and their mixtures with L, except for those of M₂X with L. Here, M^z+ and X^z− at z = 1 and 2 denote a metal ion and a pairing (or counter) anion, respectively.

3.2.1. Mass and charge balances and the theoretical treatments

To solve the above equilibria, mass- and charge-balance equations are shown. As an example, Case (1) is described as follows [4].

[MX]_t = [M⁺] + [MX] + b_s for the species with M⁺

[MX]_t = [X⁻] + [MX] + b_s for those with X⁻

and [M⁺] = [X⁻]

For Case (1), its ion-pair formation constant (K_MX) in molar concentration unit is defined as

K_MX = $$y_{\text{M}}{y}_{\text{X}}{K}_{\text{MX}}^0=\frac{[\text{MX}]}{[{\text{M}}^{+}{\text{][X}}^{\text{-}}]}$$ (y_MX = 1)

Considering an apparent total concentration to be [MX]_t − b_s, one can express [MX] and [X⁻] as functions of [M⁺]. Thus, taking logarithms of the both sides of Eq. (13) and rearranging its equation, the following one is obtained.

$$\log \frac{K_{\text{MX}}}{y_{\pm }^2}=\log \left(K_{\text{MX}}^0+\frac{b_{\text{s}}}{a_{\text{M}}^2}\right)$$E6

with y_± = $$\sqrt{y_{+}{y}_{-}}$$

and K_MX = ([MX]_t − [M⁺])/[M⁺]²

When the [M⁺] value is determined with ISE, then Eq. (14-2) is easily obtained at a given I.

3.2.2. Data analysis [4]

Hence, one can plot log (K_MX/y_±²) against a_M² and immediately obtain the K_MX⁰ and b_s values (mol dm⁻³ unit) from analyzing its plot by a non-linear regression. Figure 3 shows the plot for the NaPic system at 25 ºC [4]. Table 3 lists the analytical equations for the other cases, together with equations expressing I which are derived from the charge-balance equations. Also, details for calculation of the parameters listed in Table 3 are summarized in Table 4 [4,6,8,12].

Case	Equation for analysis	Plot for analysis	I /mol dm^−3
(1)	$$\log \frac{K_{\text{MX}}}{y_{\pm }^2}=\log \left(K_{\text{MX}}^0+\frac{b_{\text{s}}}{a_{\text{M}}^2}\right)$$at z = 1, 2; bs/mol dm^−3: curve-fitting parameter	log (KMX/y±^2) versus aM^2	[M^+] or [X^−]
(2)	∑aj/aM = 1 + K1^0aX + K1^0K2^0(aX)^2	∑aj/aM versus aX	[M^2+] + [X^−]
(3)	∑aj/aX = 1 + K1^0aM + K1^0K2^0(aM)^2	∑aj/aX versus aM	[M^+] + [X^2−]
(1L)	$$\log \frac{K_{\text{MLX}}}{y_{\pm }^2}=\log \left(K_{\text{MLX}}^0+\frac{b_{\text{c}}}{y_{\text{ML}}([{\text{X}}^{-}]-[{\text{M}}^{+}])a_{\text{X}}}\right)$$at z = 1; bc/mol dm^−3: curve-fitting parameter	log (KMLX/y±^2) versus; yML([X^−] − [M^+])aX	[X^−]
(2L)	F(aj ) ≈ 1 + KMLX^0aX + KMLX^0KMLX2^0(aX)2; with F(aj ) = $$\frac{{\displaystyle \sum a_j-a_{\text{M}}(1+K_{\text{MX}}^0{a}_{\text{X}})}}{a_{\text{ML}}}$$	F(aj ) versus aX	[M^2+] + [ML^2+] + [X^−]

Table 3.

Equations and plots for the equilibrium analyses and I expressions corresponding to them [4,6,8,12]

Case	Molar concentrations of respective species at equilibrium	Remarks
(1)	[M^z+] or [X^z−]: analyzed for z = 1, 2	Ref. [4]
(3)	[M^+]: analyzed; [X^2−] =$$\frac{[{\text{M}}^{+}]}{2+K_{\text{MX}}[{\text{M}}^{+}]}$$; [MX^−] =$$\frac{K_{\text{MX}}{[{\text{M}}^{+}]}^2}{2+K_{\text{MX}}[{\text{M}}^{+}]}$$; [M2X] = [M2X]t −$$\frac{[{\text{M}}^{+}](1+K_{\text{MX}}[{\text{M}}^{+}])}{2+K_{\text{MX}}[{\text{M}}^{+}]}$$, where KMX = [MX^−]/[M^+][X^2−]	Ref. [12]
(1L)	[M^z+]: analyzed; [X^z−] =$$\frac{[{\text{M}}^{z+}](1+K_{\text{ML}}[{\text{M}}^{z+}])}{1-K_{\text{ML}}{K}_{\text{MX}}[{\text{M}}^{z+}]}$$; [ML^z+] = [X^z−] − [M^z+]; [MLX] = [MX]t − [X^z−](1 + KMX[M^z+]), where KMX = [MX]/[M^z+][X^z−] for z = 1, 2	Ref. [4]
(2L)	[X^−]: analyzed; [M^2+] =$$\frac{{[{\text{MX}}_{\text{2}}]}_{\text{t}}-{\displaystyle \sum [{\text{MLX}}_z}]}{1+K_{\text{MX}}[{\text{X}}^{-}]+K_{\text{ML}}[\text{L}]}$$; [L] =$$\frac{{[\text{L]}}_{\text{t}}-{\displaystyle \sum [{\text{MLX}}_z]}}{1+K_{\text{ML}}[{\text{M}}^{2+}]}$$; ∑[MLXz] = [MX2]t − [M^2+](1 + KMX[X^−] + KML[L]), where KMX = [MX^+]/[M^2+][X^−]	One can compute [M^2+], [L], and ∑ [MLXz] by a successive approxi-mation. See Ref. [8] for its details.

Table 4.

Equations and plots for the equilibrium analyses and I expressions corresponding to them [4,6,8,12]

Figure 3.

Plot of log (K_NaPic/y²) versus a_Na² with b_s = 5.3 × 10⁴ mol dm³ and R = 0.945 [4].

3.2.3. 2:2 electrolytes [6]

Table 3 lists the equation [6] for the equilibrium analysis and the parameters for its analytical plot corresponding to Case (1) at z = 2. Similarly, the equations for the calculation of the parameters and equilibrium constants are summarized in Table 4.

3.3. Theoretical treatments and data analysis for Case (2) [12]

From the mass- and charge-balance equations [12] of Case (2), the following equations are derived.

$$[{\text{X}}^{-}]=\frac{2[{\text{M}}^{2+}]}{1-K_{\text{MX}}[{\text{M}}^{2+}]}$$for M²⁺ determination by ISE

or

$$[{\text{M}}^{2+}]=\frac{[{\text{X}}^{-}]}{2+K_{\text{MX}}[{\text{X}}^{-}]}$$for X⁻ determination by ISE

The concentrations of other species are

$$[{\text{MX}}^{+}]=\frac{2K_{\text{MX}}{[{\text{M}}^{2+}]}^2}{1-K_{\text{MX}}[{\text{M}}^{2+}]}$$E7

and

$$[{\text{MX}}_{\text{2}}]={[{\text{MX}}_2]}_{\text{t}}-\frac{[{\text{M}}^{2+}](1+K_{\text{MX}}{[{\text{M}}^{2+}]}^2)}{1-K_{\text{MX}}[{\text{M}}^{2+}]}$$E8

On the basis of the above equations, one can immediately calculate K_MX = [MX⁺]/[M²⁺][X⁻] and K₂ = [MX₂]/[MX⁺][X⁻] for a given I. For the other cases, see Tables 3 and 4. As an example, the plot of the Na₂CrO₄ system is shown in Fig. 4. Also, Table 5 lists the K_MX⁰ and K₂⁰ values determined in the section 3.2 and the present one [4-8,12,14].

3.4. HSAB principle [15,16]

According to Pearson, the HSAB classifications of some species are as follows.

As hard acids: H⁺, Li⁺, Na⁺, K⁺, Ca²⁺, Sr²⁺ etc.

As borderline acids: Fe²⁺, Co²⁺, Ni²⁺, Cu²⁺, Zn²⁺, Pb²⁺ etc.

As soft acids: Ag⁺, Tl⁺, Pd²⁺, Cd²⁺, Hg²⁺ etc.

and

As hard bases: H₂O, OH⁻, F⁻, PO₄³⁻, SO₄²⁻, Cl⁻, ClO₄⁻, ROH, RO⁻, R₂O etc.

As borderline bases: N₃⁻, Br⁻, NO₂⁻, SO₃⁻, N₂ etc.

As soft bases: R₂S, RSH, RS⁻, I⁻, SCN⁻, CN⁻ etc. with R = aryl or alkyl group

These species are best classified by using the following criteria. Class (a) acids (hard ones) form more stable complexes with ligands having the Y donor atoms in the order Y = N >> P > As > Sb; O >> S > Se > Te; F > Cl > Br > I [15]. On the other hand, Class (b) acids (soft ones) form in the order N << P > As > Sb; O << S < Se ~ Te; F < Cl < Br < I [15]. So, what criteria do ML^z+ classify? What criteria do L classify? For some ions and crown ethers, these HSAB classifications are going to be examined below (see 4.2) on the basis of the K_MXz⁰ and K_MLXz⁰ values at z = 1 and 2.

Figure 4.

Plot of ∑a_j/a_CrO4 versus a_Na with R = 0.994

3.5. Theoretical treatments and data analysis for mixtures with L

3.5.1. For Cases (1L) and (2L) [4,8]

As similar to the section 3.2, Table 3 summarizes the analytical equations and their plot types. Examples of the plots for Cases (1L) and (2L) are shown in Figs. 5 and 6, respectively. The plot in Fig. 5 is similar to that in Fig. 3. A fitting curve of log (K_MLX/y_±²) versus a_MLa_X is depicted with a solid line in Fig. 5, where a_ML = y_ML([X⁻] − [M⁺]) [4]. The former parameter obviously corresponds to log (K_MX/y_±²) in Case (1) and the latter one to a_M² {see the section 3.2.1. Table 5 lists the K_MLX⁰ (and β₂⁰) values thus determined at 25 ºC. From this table, one can easily see that the K_MLX⁰ values are larger than the K_MX⁰ ones, except for several cases. These results indicate that M^z+ dehydrates in the complex formation with L in water and thereby increases its hydrophobic property.

Figure 5.

Plot of log (K_NaLDDTC/y_±²) versus a_NaLa_DDTC with R = 0.973 for L = 15C5.

Figure 6.

Plot of F(a_j ) versus a_Br with R = 0.484 for the CdBr₂-18C6 system.

4 Ion-pair formation for MX_z with L in water

4.1. Dependence of the ion-pair formation constants on some factors [4-8]

4.1.1. Effect of sizes of anions

Effect of sizes of anions X^z− on the K_MLX⁰ values is described and thereby its cause is examined. Effective ionic radii or sizes estimated from the Van der Waals (vdw) volumes are on the order X^z− = Cl⁻ < Br⁻ < tfa⁻ (estimated from vdw vol.) ≤ I⁻ < MnO₄⁻ < ReO₄⁻ < DDTC⁻ (from vdw vol.) < (SO₄²⁻ <) Pic⁻ < BPh₄⁻ [8]. The K_MLX⁰ values in Table 5 are on the orders

X⁻ = Pic⁻ ≤ MnO₄⁻ < BPh₄⁻ for Na(15C5)⁺ (O1)

MnO₄⁻ < Pic⁻ < BPh₄⁻ for Na(B15C5)⁺ (O2)

and

DDTC⁻ < Pic⁻ < MnO₄⁻ < ReO₄⁻ ≤ tfa⁻ < BPh₄⁻ for Na(18C6)⁺ (O3)

DDTC⁻ < ReO₄⁻ ≤ tfa⁻ < Pic⁻ < MnO₄⁻ < BPh₄⁻ for Na(B18C6)⁺. (O4)

The two orders, (O1) and (O2), suggest hydrophobic interactions between BPh₄⁻ and Na(15C5)⁺ or Na(B15C5)⁺ and those between Pic⁻ and Na(B15C5)⁺. That is, the former two cases reflect the interaction between the large ions, while the latter case does that between a benzene ring in Pic⁻ and a benzo group in Na(B15C5)⁺. Other K_MLX⁰ orders seem to have the characteristics that DDTC⁻ << BPh₄⁻ (is due to the hydrophobic interaction), Pic⁻ < MnO₄⁻, and ReO₄⁻ ≤ tfa⁻ (are due to the electrostatic one).

4.1.2. Effect of a benzo group added to L skeleton

Effect of a benzo group of L on the K_MLX⁰ values is described and thereby its cause is discussed. Table 5 reveals relations

L = 15C5 ≤ B15C5 for NaPic, NaMnO₄, NaBPh₄, and AgPic (R5)

15C5 > B15C5 for KPic and KMnO₄ (R6)

and

18C6 < B18C6 for NaPic, KPic, NaMnO₄, and NaDDTC (R7)

18C6 ≥ B18C6 for KMnO₄, Natfa, NaReO₄, NaBPh₄, and AgPic. (R8)

Considering the electrostatic interaction between ML⁺ and X⁻ and water molecules hydrated to M⁺ to be basic interactions, these relations of Inequalities (R5) to (R8) may be changed into the following expression: Na⁺(15C5)- ≤ (B15C5)Na⁺-X⁻ and wAg⁺(15C5)- ≤ (B15C5)Ag⁺-Pic⁻ for Inequality (R5); (15C5)K⁺w- > wK⁺(B15C5)-X⁻ for (R6) and wNa⁺(18C6)- < (B18C6)Na⁺-X⁻ and K⁺(18C6)- < (B18C6)K⁺-Pic⁻ for (R7); (18C6)K⁺- ≥ K⁺(B18C6)-MnO₄⁻ and (18C6)M⁺w- ≥ M⁺(B18C6)-X⁻ for (R8). Here, w denotes the water molecules which hydrate to M⁺ and act as Lewis base. Also, we simply define the following sequence

wM⁺L-X⁻ < M⁺L-X⁻ < LM⁺w-X⁻ < LM⁺-X⁻ (SO9)

as a measure for the strength of the interaction between ML^z+ and X^z− at z = 1. In other words, the standard order (SO9) can be interpreted as L-separated ion pair with water molecule(s) < L-separated one < w-shared one < contact one. When a cavity size of L is smaller than a size of M⁺, we will assume an opposite relation of LM⁺w-X⁻ < M⁺L-X⁻; namely, M⁺L-X⁻ approaches to LM⁺-X⁻. The relations, (R5) and (R7), seem to reflect the hydrophobic properties of ML⁺. The others can reflect simply effects of the sizes of the L skeletons with benzo groups.

4.1.3. Effect of ring sizes of L

Effect of the ring sizes of L in MLX is described and thereby its cause is examined. Also, this means an increase in the number of the O donor atoms in the ring. These K_MLX⁰ relations in Table 5 are

L = 15C5 < 18C6 and B15C5 < B18C6 for NaPic, KPic, NaMnO₄, and NaBPh₄ (R10)

15C5 > 18C6 and B15C5 > B18C6 for KMnO₄ and AgPic. (R11)

Inequality (R10) can be interpreted as the following interactions of ML⁺ with X⁻: Na⁺(15C5)- < (18C6)Na⁺w-Pic⁻; wK⁺(15C5)- < (18C6)K⁺-Pic⁻; Na⁺(15C5)- < (18C6)K⁺w-X⁻ at X⁻ = MnO₄⁻ and BPh₄⁻. Similarly, interactions for (R10) may be Na⁺(B15C5)- < (B18C6)Na⁺-X⁻ at X⁻ = Pic⁻ and MnO₄⁻; wK⁺(B15C5)- < (B18C6)K⁺-Pic⁻. On the other hand, Inequality (R11) seems to simply reflect a coulombic interaction between ML⁺ with X⁻.

4.1.4. Effect of sizes of the central M^z+ in/on L rings

Effective ionic radii of M⁺ are on the order M⁺ = Li⁺ (0.76 Å) < Na⁺ (1.02) < Ag⁺ (1.15) < K⁺ (1.38) [17]. Also, the hydration free energies (−ΔG_h⁰) of M⁺ are on the order K⁺ (304 kJ mol⁻¹) < Na⁺ (375) < Li⁺ (481) << Ag⁺ (1856) [18]. The K_MLX⁰ orders are

M = K < Na < Ag for M(15C5)Pic (O12)

K < Li < Na < Ag for M(B15C5)Pic (O13)

Li ≤ Na < Ag < K for M(18C6)Pic (O14)

Ag < Na < K for M(B18C6)Pic (O15)

and

Na < K for M(15C5)MnO₄ (R16)

K < Na for M(B15C5)MnO₄, M(18C6)MnO₄, and M(B18C6)MnO₄. (R17)

If the coulombic force is simply effective for these ion-pair formation, then the K_MLX⁰ order can be K⁺ < Ag⁺ < Na⁺ < Li⁺. Similarly, if the dehydration of M⁺ is effective for the formation, then the order can be Ag⁺ << Li⁺ < Na⁺ < K⁺. Comparison of these orders with (O12) to (O14) shows the complications of their ion-pair formation.

The orders or relations can be changed into wK⁺(15C5)- & (15C5)K⁺w- < Na⁺(15C5)- < (15C5)Ag⁺w- & wAg⁺(15C5)-Pic⁻ for Inequality (O12); wK⁺(B15C5)- < Li⁺(B15C5)- < Na⁺(B15C5)- & (B15C5)Na⁺- < (B15C5)Ag⁺-Pic⁻ for (O13); wLi⁺(18C6)- ≤ wNa⁺(18C6)- & (18C6)Na⁺w- < (18C6)Ag⁺w- & wAg⁺(18C6)- < (18C6)K⁺- & K⁺(18C6)-Pic⁻ for (O14); Ag⁺(B18C6)- < (B18C6)Na⁺- < K⁺(B18C6)-Pic⁻ for (O15) and Na⁺(15C5)- < (15C5)K⁺w-MnO₄⁻ for (R16); wK⁺(B15C5)- & (B15C5)K⁺w- < Na⁺(B15C5)- & (B15C5)Na⁺-MnO₄⁻; K⁺(18C6)- and (18C6)K⁺- < (18C6)Na⁺w- & wNa⁺(18C6)-MnO₄⁻; K⁺(B18C6)- < (B18C6)Na⁺-MnO₄⁻ for (R17). The above expression “M⁺L-X⁻ & LM⁺-X⁻” means that the fraction of the left M⁺L-X⁻ is major in comparison with that of the right LM⁺-X⁻.

In (O12), the inverse between K(15C5)⁺ and Na(15C5)⁺ is due to the fact that K(15C5)⁺ satisfies the condition that the cavity size of 15C5 < the size of K⁺ (see 4.1.2); that between Na(15C5)⁺ and Ag(15C5)⁺ suggests that a fraction of (15C5)Ag⁺w-Pic⁻ is dominant. Further, the Na(B15C5)⁺ < Ag(B15C5)⁺ relation in (O13) suggests that Na⁺ in the former complex ion is more shielded by B15C5 than Ag⁺ in the latter ion. The same can be true of the relation of Na(18C6)⁺ < Ag(18C6)⁺ in (O14).

In the present section 4.1, the orders in magnitude among the K_MLX⁰ values are interpreted by supposing the shapes of the MLX ion pairs based on Inequality (SO9). Of course, validity of such interpretations has to be clarified experimentally.

4.2. Try to understand the ion-pair formation based on the HSAB principle [8]

According to the HSAB classification, Pic⁻, MnO₄⁻, ReO₄⁻, tfa⁻, and BPh₄⁻ were reported to be hard bases [8]. Also, DDTC⁻ has been classified as a soft base [19]. Then the K_MLX⁰ values are on the orders

X = DDTC⁻ ≤ Pic⁻ < MnO₄⁻ < ReO₄⁻ ≤ tfa⁻ < BPh₄⁻ for Na(18C6)⁺ (O3)

DDTC⁻ < ReO₄⁻ ≤ tfa⁻ < Pic⁻ < MnO₄⁻ < BPh₄⁻ for Na(B18C6)⁺ (O4)

Here, only DDTC⁻ has S donor atoms in it, while the other X⁻ does O donor atoms in them. The sequence of donor atoms in X⁻ obviously show S ≤ O < O < O < O (or F) for (O3) and S < O < O (or F) < O < O for (O4), except for X⁻ = BPh₄⁻. Comparing the orders with those described in 3.4, they indicate at least that Na(18C6)⁺ and Na(B18C6)⁺ have the higher affinities for the O donor atoms in their X⁻ than the S donor ones in DDTC⁻. This fact suggests that both the NaL⁺ are classified as hard acids [8].

Similarly, from the following orders, readers can see that the same is true of the K_CdLX⁰ values for the CdX⁺-L systems with X⁻ = Cl⁻, Br⁻, and I⁻.

X⁻ = I⁻ < Br⁻ < Cl⁻ (≤ Pic⁻) for Cd(18C6)X⁺ and Cd(B18C6)X⁺ (O18)

That is, it is suggested that both the complex ions are hard acids [8], although Cd²⁺ is classified as a soft acid (see 3.4); K_CdX2⁰ is on the order X⁻ = Cl⁻ ≤ (Pic⁻ ≤) Br⁻ < I⁻ (see Table 5). As can be seen from the section 3.4 and the above, the facts that Cl⁻ and Pic⁻ are the hard bases support this suggestion

MXz or M2X	KMX^0 [K2^0], (std)^a	MLXz	KMLXz^0 {β2^0}, (std)^a
LiPic	10.9(1.8)	Li(B15C5)Pic	205(111)
		Li(18C6)Pic	52(---^b)
NaPic	4.2(0.7)	Na(15C5)Pic	34(6)
		Na(B15C5)Pic	517(91)
		Na(18C6)Pic	62(18)
		Na(B18C6)Pic	642(96)
KPic	5.8(1.8)	K(15C5)Pic	6.0(3.9)
		K(B15C5)Pic	12(---^b)
		K(18C6)Pic	738(210)
		K(B18C6)Pic	1.37(0.12) × 10^3
AgPic	2.8(0.3)	Ag(15C5)Pic	556(68)
		Ag(B15C5)Pic	1.58(0.21) × 10^3
		Ag(18C6)Pic	191(19)
		Ag(B18C6)Pic	157(22)
NaMnO4	8.0(0.9)	Na(15C5)MnO4	38(21)
		Na(B15C5)MnO4	354(75)
		Na(18C6)MnO4	231(30)
		Na(B18C6)MnO4	1.63(0.41) × 10^3
KMnO4	9.1(2.3)	K(15C5)MnO4	239(80)
		K(B15C5)MnO4	137(110)
		K(18C6)MnO4	93(22)
		K(B18C6)MnO4	72(12)
Natfa	4.0(0.8)	Na(18C6)tfa	384(67)
		Na(B18C6)tfa	201(25)
NaReO4	4.1(0.7)	Na(18C6)ReO4	340(66)
		Na(B18C6)ReO4	188(112)
NaDDTC	32.8(2.7)	Na(18C6)DDTC	48(23)
		Na(B18C6)DDTC	100(25)
NaBPh4	14.3(1.5)	Na(15C5)BPh4	7.36(1.51) × 10^3
		Na(B15C5)BPh4	9.07(6.30) × 10^3
		Na(18C6)BPh4	2.9(2.0) × 10^5
		Na(B18C6)BPh4	1.24(1.02) × 10^5
CaCl2 [6]	40(7), 41(3)
Ca(Pic)2 [6]	88(58)
CdCl2	86(30), 92(4); [8.7(7.5), 13(5)]	Cd(18C6)Cl2	3.8 × 10^4; {1.39(1.00) × 10^6}
		Cd(B18C6)Cl2	1.7 × 10^4,; 2.70(2.48) × 10^6
CdBr2	118(19) [25(19)]	Cd(18C6)Br2	0.57(0.13)
		Cd(B18C6)Br2	1.32(0.05)
CdI2	308(5) [40(3)]	Cd(18C6)I2, Cd(B18C6)I2	< 1^c
Cd(Pic)2	108(11), 107(17)	Cd(18C6)(Pic)2	3.3 × 10^4; {1.77(1.62) × 10^7}
		Cd(B18C6)(Pic)2	6.5 × 10^4; {2.29(0.29) × 10^7}
CdSO4	221(31)	Cd(18C6)SO4	4.38(0.68) × 10^4
		Cd(B18C6)SO4	1.83(0.51) × 10^4
Na2SO4	14(0.7) [7.4(1.0)]
NaS2O3	14(0.6) [3.2(0.6)]
Na2CrO4	12(2) [7.9(2.8)]

Table 5.

Ion-pair formation constants of MX_z, M₂X, and MLX_z in water at I → 0 mol dm⁻³ and 25 ºC [4-8,12,14]

5. Topics relevant to the ion-pair formation in water

5.1. Application to solvent extraction [20]

Using the thus-determined formation constants for M^IX and M^ILX in water, overall extraction equilibrium-constants are resolved into the following component ones [21].

L $\stackrel{}{\rightleftharpoons }$L_o

M⁺ + X⁻$\stackrel{}{\rightleftharpoons }$MX

M⁺ + L$\stackrel{}{\rightleftharpoons }$ML⁺

ML⁺ + X⁻$\stackrel{}{\rightleftharpoons }$MLX

MLX$\stackrel{}{\rightleftharpoons }$MLX_o

and

MLX_o$\stackrel{}{\rightleftharpoons }$ML⁺_o + X⁻_o

Here, the subscript “o” denotes an organic phase. The reactions (19-1) to (19-5) correspond to the extraction into low polar diluents (or organic solvents) and the reaction (19-6) will be added to the cases of the extraction into high polar diluents. In general, the equilibrium constants of the process (19-1) and the reaction (19-3) can be determined by separate experiments. Then, the overall extraction equilibria are characterized on the basis of their component ones, either (19-1) to (19-5) or to (19-6). One can clearly see that, for the above extraction model, the ion-pair formation, (19-2) and (19-4), is important. The NaMnO₄ extraction by B15C5 into 1,2-dichloroethane (DCE) and nitrobenzene (NB) are analyzed as the results in Table 6. There are remarked differences in K_D,NaLX, K_D,X, and K_NaLX^org between DCE and NB. According to the relation K_ex = K_MLK_MLXK_D,MLX/K_D,L [21], a difference in K_ex between NB and DCE mainly comes from that between K_D,MLX. Similarly, that between K_ex± does from the difference between K_MLX^org in addition to K_D,MLX, because K_ex± = K_ex/K_MLX^org.

5.2. Development into study of the abnormal potential responses of ISEs [9]

Abnormal potential responses, Λ-shaped potential ones [9,12] of emf-versus-log [CdX₂]_t plots (X = I, Br), of the commercial Cd²⁺-selective electrode listed in Table 1 have been observed and then an answer is described using a model for the potential response with interactions of the electrode surface with counter X⁻, in addition to that with Cd²⁺. Its processes are

Cd²⁺ + Y²⁻_s$\stackrel{}{\rightleftharpoons }$CdY_s (subscript “s”: solid phase)

CdY_s + 2X⁻$\stackrel{}{\rightleftharpoons }$X₂CdY_s

Cd²⁺ + X⁻$\stackrel{}{\rightleftharpoons }$CdX⁺,

where Y²⁻_s is a functional group of the electrode material and X⁻ denotes halide ions. Applying electrochemical potentials [2] for the electrode processes (20-1) and (20-2) and introducing mass-balance equations in the overall process, we obtain the following equation as an expression of emf.

emf = A + Blog [Cd]_t + Clog {1+ 4K_s([Cd]_t)²}

Here, A, [Cd]_t, B, C, and K_s refer to a constant (V versus Ag│AgCl electrode), the total concentration of CdX₂ in the test solution, values (V) corresponding to −2.3RT/2F, and value (mol⁻³ dm⁹ unit) being inversely proportional to the solubility product of CdX₂, respectively. One can immediately obtain these values, analyzing the plot of emf versus log [Cd]_t by non-linear regression: K_s(CdCl₂) (not be determined) < K_s(CdBr₂) (= 10^4.2 mol⁻³ dm⁹) < K_s(CdI₂) (= 10^6.98) [8]. It is obvious that the larger K_s is, the more easy CdX₂ interacts with the electrode and accordingly the larger the interference of X⁻ against the electrode response becomes. Similar tendencies have been obtained in a commercial Cu²⁺-selective electrode with a solid membrane and a Ca²⁺-selective one with a liquid membrane.

Diluent	log Kex^a	log Kex±^b	log KML^c	log KMLX; (I )^d	log KMX	log KD,MLX^e	log KD,L^f	log KD,X^g	log KMLXorg h; (Io)i
DCE	2.24	−3.75	0.45	2.47; (0.0084)	0.54	1.23	1.910	−3.3	6.0; (4.5 × 10^−6)
NB	3.79	−0.23	0.45	2.51; (0.0015)	0.59	2.4	1.6	−1.7	4.0; (3.5 × 10^−5)

Table 6.

Extraction constants and their component equilibrium-constants for the NaMnO₄-B15C5 extraction systems into DCE and NB at 25 ºC [20]

6. Summary

In this chapter, the determination procedures of the K_MXz⁰, K_M2X⁰, and K_MLXz⁰ values in water were discussed and their some applications were described. The present potentiometric procedures will support other methods, such as conductometric, spectrophotometric, solvent extraction ones, and so on. Their applications have been limited to some ions and hydrophilic L, because kinds of commercial ISE is limited and low solubility of hydrophobic L to water limits its emf measurements; the solvent extraction method [21] is going to compensate for the latter limitation. Instead of such situations, the present procedures are useful for the K_MLXz⁰ determination. As the pH values are measured at many laboratories in the world, thus handy procedures, compared with the other methods, will take effect at the determination of K_MXz and K_MLXz at I → 0.

7. Appendix

1. Symbols and abbreviations used in this chapter

2. Symbol and abbreviationMeaning

3. a(j)Ion size parameter of j

4. B15C5Benzo-15-crown-5 ether

5. B18C6Benzo-18-crown-6 ether

6. cj^I or cj^IIMolar concentration of j in the phase I or II

7. DDTC⁻Diethyldithiocarbamate ion

8. DMEDropping mercury electrode

9. EjLiquid junction potential at an interface

10. I → 0Ionic strength at infinite dilution

11. [ j ]Molar concentration of j at equilibrium

12. [ j ]tTotal concentration of j

13. KThe equilibrium constant

14. Pic⁻Picrate ion

15. RCorrelation coefficient

16. RT/F0.02569 V at 298 K

17. tfaTrifluoroacetate ion

18. ujMobility of j in water

19. zFormal charge of an ion or a number expressing the composition of a salt or an ion pair

20. zjFormal charge of j

21. $${\varphi }^{\text{I}}\text{ or }{\varphi }^{\text{II}}$$Inner potential in the phase I or II

22. 2.3RT/F0.05916 V at 298 K