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

Written By

Yoshihiro Kudo

Submitted: November 14th, 2011 Published: February 20th, 2013

DOI: 10.5772/48206

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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 (MXz at z = 1, 2) and of their metal complex-ions (MLz+) with pairing anions (Xz) 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 Mz+, their MLz+, and Xz in water and MLz+ is unstable in many cases. Also, many Mz+ and MLz+ employed here cannot be detected spectrophotometrically. Voltammetric methods cannot apply for the determination, because working electrode suitable for Mz+ 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 (KMLXz0) for MLXz 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 MXz and MLXz in water are described. Here, the determination of KMXz0, the ion-pair formation constant of MXz in water at I → 0, is always required for that of KMLXz0.


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

Cell (A): Ag|AgCl|0.1 mol dm3(C2H5)4NCl, LiCl, NaCl, KClE1
or 0.05 mol dm3MgCl2|*test solution|ISE [4,5]E2

or as that with a double liquid junction

Cell (B): Ag|AgCl|0.1 mol dm3(C2H5)4NCl, NaCl,E3
or KCl|1 mol dm3KNO3|*test solution|ISE [68]E4

Here, the 1 mol dm−3 solution of KNO3 in Cell (B) is a salt bridge, between which and the test solution, Ej 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 Ag|AgCl|0.1 mol dm3 (C2H5)4NCl and Ag|AgCl|0.1 mol dm3KCl|1 mol dm3 KNO3, respectively. For Cell (A), the Ej 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 Ej occurred at the interface marked with an asterisk cannot be avoided. Hence, correction procedures of Ej are described in this section. Here, the salt bridges with KNO3 are experimentally used and the Henderson equation [2]

ISEType & MakerComments
Na+-selective electrode1512A-10C, HoribaGlass membrane; this electrode partially responds K+ and Rb+.
K+-selective electrodes1200K, TokoGlass membrane (not produced now); this electrode partially responded Li+.
8202-10C, HoribaLiquid membrane; it is unstable for the aqueous solution of hydrophobic picrates.
Ag+-selective electrode8011-10C, HoribaSolid membrane
Cd2+-selective electrode8007-10C, HoribaSolid membrane; it responds Br partially and I well. See below.
Cl-selective electrode8002-10C, HoribaSolid membrane
Br-selective electrode8005-10C, HoribaSolid membrane
I-selective electrode8004-10C, HoribaSolid membrane

Table 1.

Commercial ISEs used herea


is analytically employed for the correction of Ej. For example, the molar concentration of j in the phase I, cjI, consists of cKI and cClI, while cjII does of cNaII, cNaLII, and cPicII 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 uNacNaII + uNaLcNaLII term in Eq. (1) can be assumed to be uNa(cNaII + cNaLII) in practice (see Table 2 for the uNa value), because the condition of cNaII >> uNaLcNaLII/uNa holds in many cases with L (low stabilities of MLz+ 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 MXz 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 KMXz 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 [MXz]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 (yj ) of ionic species j used for the activity (aj ) calculations are evaluated from the extended Debye-Hückel equation

log yj =0.5114zj2I1+0.3291aI

and the Davies one [13]

log yj =0.5114zj2(I1+I0.3I),

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 MLz+ and MIIX+, or for the yj 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 yj 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.

ja(j ) /Åuj /10−4ja(j ) /Åuj /10−4
Li+6 4.010Br3 8.13
Na+4 5.193I3 7.96
K+3 7.619NO33 7.404
Ag+2.5 6.415MnO43.5 6.35
Ca2+6 6.166ReO43.9c 5.697
Cd2+5 5.6ClO43.5 7.05
CdCl+4---dPic7 3.149
(C2H5)4N+63.384BPh49c 2.2
SO42−4 8.27
S2O32−4 8.81

Table 2.

Ion size parametersa and mobilitiesb 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

Mz+ + Xz MX at z = 1, 2

Case (2). 2:1 electrolytes

M2+ + X MX+

MX+ + X MX2

Case (3). 1:2 electrolytes

M+ + X2− MX

M+ + MX M2X

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

Mz+ + XzMX

Mz+ + L MLz+

MLz+ + Xz MLX at z = 1, 2

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

M2+ + XMX+

MX+ + XMX2

M2+ + L ML2+

ML2+ + XMLX+


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, Mz+ and Xz 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] + bs for the species with M+

[MX]t = [X] + [MX] + bs for those with X

and [M+] = [X]

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

KMX = yMyXKMX0=[MX][M+][X-] (yMX = 1)

Considering an apparent total concentration to be [MX]tbs, 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.


with y± = y+y

and KMX = ([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 (KMX/y±2) against aM2 and immediately obtain the KMX0 and bs 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].

CaseEquation for analysisPlot for analysisI /mol dm−3
(1)logKMXy±2=log(KMX0+bsaM2)at z = 1, 2;
bs/mol dm−3: curve-fitting parameter
log (KMX/y±2) versus aM2[M+] or [X]
(2)∑aj/aM = 1 + K10aX + K10K20(aX)2∑aj/aM versus aX[M2+] + [X]
(3)∑aj/aX = 1 + K10aM + K10K20(aM)2∑aj/aX versus aM[M+] + [X2−]
(1L)logKMLXy±2=log(KMLX0+bcyML([X][M+])aX)at z = 1;
bc/mol dm−3: curve-fitting parameter
log (KMLX/y±2) versus
yML([X] − [M+])aX
(2L)F(aj ) ≈ 1 + KMLX0aX + KMLX0KMLX20(aX)2
with F(aj ) = ajaM(1+KMX0aX)aML
F(aj ) versus aX[M2+] + [ML2+] + [X]

Table 3.

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

CaseMolar concentrations of respective species at equilibriumRemarks
(1)[Mz+] or [Xz−]: analyzed for z = 1, 2Ref. [4]
(3)[M+]: analyzed; [X2−] =[M+]2+KMX[M+]; [MX] =KMX[M+]22+KMX[M+];
[M2X] = [M2X]t[M+](1+KMX[M+])2+KMX[M+], where KMX = [MX]/[M+][X2−]
Ref. [12]
(1L)[Mz+]: analyzed; [Xz−] =[Mz+](1+KML[Mz+])1KMLKMX[Mz+]; [MLz+] = [Xz−] − [Mz+];
[MLX] = [MX]t − [Xz−](1 + KMX[Mz+]), where KMX = [MX]/[Mz+][Xz−] for z = 1, 2
Ref. [4]
(2L)[X]: analyzed; [M2+] =[MX2]t[MLXz]1+KMX[X]+KML[L]; [L] =[L]t[MLXz]1+KML[M2+];
[MLXz] = [MX2]t − [M2+](1 + KMX[X] + KML[L]), where KMX = [MX+]/[M2+][X]
One can compute [M2+], [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 (KNaPic/y2) versus aNa2 with bs = 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.

[X]=2[M2+]1KMX[M2+]for M2+ determination by ISE
[M2+]=[X]2+KMX[X]for X determination by ISE

The concentrations of other species are


On the basis of the above equations, one can immediately calculate KMX = [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 KMX0 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.


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 MLz+ 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 KMXz0 and KMLXz0 values at z = 1 and 2.

Figure 4.

Plot of ∑aj/aCrO4 versus aNa 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 (KMLX/y±2) versus aMLaX is depicted with a solid line in Fig. 5, where aML = yML([X] − [M+]) [4]. The former parameter obviously corresponds to log (KMX/y±2) in Case (1) and the latter one to aM2 {see the section 3.2.1. Table 5 lists the KMLX0 (and β20) values thus determined at 25 ºC. From this table, one can easily see that the KMLX0 values are larger than the KMX0 ones, except for several cases. These results indicate that Mz+ dehydrates in the complex formation with L in water and thereby increases its hydrophobic property.

Figure 5.

Plot of log (KNaLDDTC/y±2) versus aNaLaDDTC with R = 0.973 for L = 15C5.

Figure 6.

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

4 Ion-pair formation for MXz 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 Xz on the KMLX0 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 Xz = Cl < Br < tfa (estimated from vdw vol.) ≤ I < MnO4 < ReO4 < DDTC (from vdw vol.) < (SO42− <) Pic < BPh4 [8]. The KMLX0 values in Table 5 are on the orders

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

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


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 KMLX0 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 KMLX0 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)


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 MLz+ and Xz 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 KMLX0 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 Mz+ 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 (−ΔGh0) of M+ are on the order K+ (304 kJ mol−1) < Na+ (375) < Li+ (481) << Ag+ (1856) [18]. The KMLX0 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)


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 KMLX0 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 KMLX0 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 KMLX0 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 KCdLX0 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); KCdX20 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 M2XKMX0 [K20], (std)aMLXzKMLXz0 {β20}, (std)a
LiPic 10.9(1.8)Li(B15C5)Pic205(111)
K(B18C6)Pic1.37(0.12) × 103
AgPic 2.8(0.3)Ag(15C5)Pic556(68)
Ag(B15C5)Pic1.58(0.21) × 103
Na(B18C6)MnO41.63(0.41) × 103
K(18C6)MnO4 93(22)
Natfa 4.0(0.8)Na(18C6)tfa384(67)
NaReO4 4.1(0.7)Na(18C6)ReO4340(66)
NaDDTC 32.8(2.7)Na(18C6)DDTC48(23)
NaBPh414.3(1.5)Na(15C5)BPh47.36(1.51) × 103
Na(B15C5)BPh49.07(6.30) × 103
Na(18C6)BPh42.9(2.0) × 105
Na(B18C6)BPh41.24(1.02) × 105
CaCl2 [6]40(7), 41(3)
Ca(Pic)2 [6]88(58)
CdCl286(30), 92(4)
[8.7(7.5), 13(5)]
Cd(18C6)Cl23.8 × 104
{1.39(1.00) × 106}
Cd(B18C6)Cl21.7 × 104,
2.70(2.48) × 106
CdBr2118(19) [25(19)]Cd(18C6)Br2 0.57(0.13)
Cd(B18C6)Br2 1.32(0.05)
CdI2308(5) [40(3)]Cd(18C6)I2, Cd(B18C6)I2< 1c
Cd(Pic)2108(11), 107(17)Cd(18C6)(Pic)23.3 × 104
{1.77(1.62) × 107}
Cd(B18C6)(Pic)26.5 × 104
{2.29(0.29) × 107}
CdSO4221(31)Cd(18C6)SO44.38(0.68) × 104
Cd(B18C6)SO41.83(0.51) × 104
Na2SO414(0.7) [7.4(1.0)]
NaS2O314(0.6) [3.2(0.6)]
Na2CrO4 12(2) [7.9(2.8)]

Table 5.

Ion-pair formation constants of MXz, M2X, and MLXz 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 MIX and MILX in water, overall extraction equilibrium-constants are resolved into the following component ones [21].

L Lo

M+ + XMX

M+ + LML+


MLXoML+o + Xo

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 KD,NaLX, KD,X, and KNaLXorg between DCE and NB. According to the relation Kex = KMLKMLXKD,MLX/KD,L [21], a difference in Kex between NB and DCE mainly comes from that between KD,MLX. Similarly, that between Kex± does from the difference between KMLXorg in addition to KD,MLX, because Kex± = Kex/KMLXorg.

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−sCdYs (subscript “s”: solid phase)

CdYs + 2XX2CdYs

Cd2+ + XCdX+,

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+ 4Ks([Cd]t)2}

Here, A, [Cd]t, B, C, and Ks 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: Ks(CdCl2) (not be determined) < Ks(CdBr2) (= 104.2 mol−3 dm9) < Ks(CdI2) (= 106.98) [8]. It is obvious that the larger Ks 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.

Diluentlog Kexalog Kex±blog KMLclog KMLX
(I )d
log KMXlog KD,MLXelog KD,Lflog KD,Xglog KMLXorg h
(4.5 × 10−6)
NB 3.79−0.23 0.452.51
0.59 2.4 1.6−1.74.0
(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 KMXz0, KM2X0, and KMLXz0 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 KMLXz0 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 KMXz and KMLXz 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. cjI or cjIIMolar concentration of j in the phase I or II

  7. DDTCDiethyldithiocarbamate 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. PicPicrate 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. ϕI or ϕIIInner potential in the phase I or II

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


  1. 1. ChristianG. D.Analytical“.Chemistry”5th.edJohnWiley.SonsNew.York, 1994
  2. 2. BardA. J.FaulknerL. R.Electrochemical“.MethodsFundamentals.Applications”2nd.edJohnWiley.SonsNew.York, 2001
  3. 3. KatsutaS.WakabayashiH.TamaruM.KudoY.TakedaY.SolutionJ.Chem2007
  4. 4. KudoY.WakasaM.ItoT.UsamiJ.KatsutaS.TakedaY.AnalBioanal.Chem2005
  5. 5. KudoY.FijiharaR.OhtakeT.WakasaM.KatsutaS.TakedaY.ChemJ.EngData.2006
  6. 6. KudoY.TakeuchiS.KobayashiY.KatsutaS.TakedaY.ibid2007
  7. 7. KudoY.KobayashiY.KatsutaS.TakedaY.MolJ.Liquids2009
  8. 8. KudoY.KoideT.ZhaoY.KatsutaS.TakedaY.AnalSci.2011
  9. 9. KudoY.TodorokiD.SuzukiN.HoriuchiN.KatsutaS.TakedaY.AmerJ.AnalChem.2011
  10. 10. KiellandJ.AmJ.ChemSoc.1937
  11. 11. VanýsekP.Ionic“.ConductivityDiffusionat.InfiniteDilution”.in“. C. R. C.HandBook.ofChemistry.Physics”ed.D. R. Lide, 74th ed., CRC Press, Boca Raton, 1993
  12. 12. KudoY.TodorokiD.HoriuchiN.KatsutaS.TakedaY.ChemJ.EngData.2010
  13. 13. R. de Levie, “Aqueous Acid-Base Equilibria and Titration”, ed. R. G. Compton, Oxford Chemistry Primers 80, Oxford University Press, New York, 1999.
  14. 14. KudoY.UsamiJ.KatsutaS.TakedaY.MolJ.Liquids2006
  15. 15. PearsonR. G.ChemJ.Educ1968
  16. 16. PearsonR. G.InorgChim.Acta1995
  17. 17. ShannonR. D.ActaCrystallogr.1976A32, 751.
  18. 18. MarcusY.Ion“.Properties”Marcel.DekkerInc.NewYork, 1997
  19. 19. KawamotoH.AkaiwaH.ChemLett.1990
  20. 20. KudoY.HarashimaK.KatsutaS.TakedaY.InternationalJ.Chem2011
  21. 21. TakedaY.TaguchiR.KatsutaS.MolJ.Liquids2004

Written By

Yoshihiro Kudo

Submitted: November 14th, 2011 Published: February 20th, 2013