2023 年 91 巻 1 号 p. 013001
A reference electrode equipped with ionic liquid salt bridge consisting of tributyl(2-methoxyethyl)phosphonium bis(pentafluoroethanesulfonyl)amide has been employed for potentiometric precipitation titration of chloride with silver ions in water at 25 °C. A model for the titration curve was regressed to experimental curves, taking into account the change in the activity coefficients of relevant ionic species in the course of the titration, to obtain the least square estimates of two adjustable parameters in the model, the solubility product (Ksp) and the analyte concentration. The least-square estimate of Ksp, (1.840 ± 0.060) × 10−10, i.e., pKsp = 9.736 ± 0.014, is in good agreement with literature data, but with higher precision.
The work described below is concerned with an application of ionic liquid salt bridge (ILSB) to potentiometric titration. In analytical chemistry and applied analysis, potentiometric titration is a matured technique that has survived to date since Robert Behrend reported it for precipitation titration of chloride and bromide in 1893.1 The longevity of this technique appears to be, in addition to the feasibility of implementation in automatic titrators, its robustness in end point detection in a variety of titrations, as introduced in textbooks of analytical chemistry.2–4 According to them, in short, as long as the potential jump in the vicinity of the equivalence point in potentiometric titration curves is large enough, the end point can be determined with good precision. While the potential at the equivalence point is susceptible to the liquid junction potential between the titrant solution, the reference electrode compartment is irrelevant to the volume of the titrant required to reach the equivalence point, and the exact knowledge of the magnitude of the liquid junction potential is insignificant to obtain the amount of unknown substance in the analyte solution.
Such a description is true in most of the cases of potentiometric titrations and highlights its unique features. But, it may sound not fully describing the virtue of potentiometry for those involved with the art of potentiometry, notably, electrometric determination of pH,5 for which most sophisticated endeavors in potentiometry have been devoted6 to proper handling of the liquid junction potential between analyte and the concentrated KCl solution.
Kolthoff, who certainly realized the significance of the concept of activity in analyzing the electrode potential,7 did not elaborate any further about it nor did consider the effect of activity coefficients on the solution equilibria in the course of titrations.
By replacing the salt bridge based on concentrated KCl with ILSB, it has been shown in the past nearly twenty years since ILSB was proposed8,9 that potentiometry allows us to more reliably estimate single ion activities, including pH defined in terms of the single ion activity of hydrogen ion.10–22 Potentiometric titration based on ILSB offers an alternative of the traditional understanding of potentiometric titration described above.
We show that least square fitting of the theoretical model taking account of the activity coefficients of ionic species to an experimental titration curve gives reasonable estimates of not only the solubility product of AgCl but other relevant parameters of the titration, such as the cell voltage and the volume of titrant at the equivalence point.
Natural extensions of the potentiometric titration with the ILSB-equipped reference electrode to different analytes will be published in a series of subsequent papers.
Tributyl(2-methoxyethyl)phosphonium chloride bis(pentafluoroethanesulfonyl)amide ([TBMOEP+][C2C2N−]) was synthesized and purified as described elsewhere.23 Sodium chloride (FUJIFILM Wako, 99.99 ± 0.01 %) was used after drying it for 24 h at 60 °C. Aliquots of 0.1 mol dm−3 silver nitrate standard solutions (Kishida Chemical, with a 4 digits factor specified) were used for titration. Milli-Q water was used for preparing salt solutions. The cell for potentiometric titration is represented as
| \begin{equation} \begin{array}{cccccccccccccccc} & \text{I} & & \text{II} & & \text{III} & & \text{IV} & & \text{V} & & \text{VI} & & \text{VII}\\ | & \text{Ag} & | & \text{AgCl} & | & \text{NaCl} & | & \text{ILSB} & | & \text{NaCl} + \text{AgNO$_{3}$} & | & \text{AgCl} & | & \text{Ag} & | \end{array} \end{equation} | (I) |
Phases III and V are the internal solution of the reference electrode and the analyte phase where the titration was made, respectively. Phases I–IV were assembled in a membrane holder made of poly(vinylidenefluoride) (PVDF) as a reference electrode with the ILSB. A gelled membrane prepared by mixing [TBMOEP+][C2C2N−] with poly(vinylidenefluoride-co-hexafluoropropylene)24 at the mixing ratio by weight of 1 : 1 was fixed with a Viton® O-ring and a screw cap at the tip of the membrane holder. In the present titration experiments, the internal solution (Phase III) filled in the membrane holder was the same as the analyte solution in Phase V, e.g., 0.01 mol dm−3 NaCl, taken from a single stock solution. Cell (I) was constructed in a tall beaker containing an analyte solution, typically 20 mL (L ≡ dm−3) NaCl, into which the reference electrode was inserted. The cell was immersed in a water bath maintained at 25.0 °C. A magnetic stirring bar at the bottom of the beaker was kept rotating for continuously stirring Phase V. Potentiometric titration curves were recorded by supplying the cell voltage of (I) to an automatic potentiometric titrator (AT-710S, Kyoto Electronic Manif. Co.). Recording one titration curve required, e.g., in the case of results in Fig. 1 below, was ca. 27 min.
Potentiometric titration of 0.01 mol dm−3 NaCl with 0.01 mol dm−3 AgNO3 recorded with Cell (I) (dots) and least-square fitting curves of five models (lines).
We consider the case when an aqueous solution of NaCl with its initial concentration being $c_{\text{NaCl}}^{0}$ is titrated with an aqueous solution of AgNO3 whose concentration is $c_{\text{AgNO}_{3}}^{0}$.
The mass balance conditions during the titration are:
| \begin{equation} (V_{\text{s}} + V_{\text{t}})c_{\text{Na${^{+}}$}} = V_{\text{s}}\ c_{\text{NaCl}}^{0} \end{equation} | (1) |
| \begin{equation} (V_{\text{s}} + V_{\text{t}})c_{\text{NO${_{3}^{-}}$}} = V_{\text{t}}\ c_{\text{AgNO${_{3}}$}}^{0} \end{equation} | (2) |
The charge balance condition during the titration is
| \begin{equation} c_{\text{Na${^{+}}$}} + c_{\text{Ag${^{+}}$}} = c_{\text{Cl${^{-}}$}} + c_{\text{NO${_{3}^{-}}$}} \end{equation} | (3) |
| \begin{equation} K_{\text{sp}} = a_{\text{Ag${^{+}}$}}a_{\text{Cl${^{-}}$}} \end{equation} | (4) |
| \begin{equation} K'_{\text{sp}} = c_{\text{Ag${^{+}}$}}c_{\text{Cl${^{-}}$}} = K_{\text{sp}}/(\gamma_{\text{Ag${^{+}}$}}\gamma_{\text{Cl${^{-}}$}}) \end{equation} | (5) |
From Eqs. 1–5,
| \begin{equation} \left(\frac{\sqrt{K'_{\text{sp}}}}{c_{\text{Cl${^{-}}$}}}\right)^{2}{} - \frac{(1/\sqrt{K'_{\text{sp}}})(p - 1)}{p/c_{\text{AgNO${_{3}}$}}^{0} + 1/c_{\text{NaCl}}^{0}}\frac{\sqrt{K'_{\text{sp}}}}{c_{\text{Cl${^{-}}$}}} - 1 = 0 \end{equation} | (6) |
| \begin{equation} p = \frac{V_{\text{t}}c_{\text{AgNO${_{3}}$}}^{0}}{V_{\text{s}}c_{\text{NaCl}}^{0}} \end{equation} | (7) |
| \begin{equation} c_{\text{Cl${^{-}}$}}^{\text{V,eqv}} = \sqrt{K'_{\text{sp}}} = \sqrt{\frac{K_{\text{sp}}}{\gamma_{\text{Ag${^{+}}$}}^{\text{V,eqv}}\gamma_{\text{Cl${^{-}}$}}^{\text{V,eqv}}}} \end{equation} | (8) |
| \begin{equation} E_{\text{cell}} = -\frac{RT}{F}\ln\frac{c_{\text{Cl${^{-}}$}}^{\text{V}}\gamma_{\text{Cl${^{-}}$}}^{\text{V}}}{c_{\text{Cl${^{-}}$}}^{\text{III}}\gamma_{\text{Cl${^{-}}$}}^{\text{III}}} \end{equation} | (9) |
| \begin{equation} E_{\text{cell}}^{\text{eqv}} = -\frac{RT}{2F}\ln\left(\frac{\gamma_{\text{Cl${^{-}}$}}^{\text{V,eqv}}}{\gamma_{\text{Ag${^{+}}$}}^{\text{V,eqv}}}K_{\text{sp}}\right) + \frac{RT}{F}\ln(c_{\text{Cl${^{-}}$}}^{\text{III}}\gamma_{\text{Cl${^{-}}$}}^{\text{III}}) \end{equation} | (10) |
Each numerically calculated titration curve from Eq. 9, together with Eqs. 5–7, was fitted to an experimental curve to estimate the values of the parameters, Ksp and $c_{\text{NaCl}}^{\text{V,0}}$, by use of Microsoft Excel®. In the course of the titration, the solution composition varies at every step of the titration. At each step, the final composition of Phase V was calculated iteratively, assuming that the activity coefficients of all ionic species in V are simply represented by a function of the ionic strength in V.25,26 At each step of the regression calculation, $c_{\text{NaCl}}^{\text{III,0}}$ was renewed by setting equal to the renewed value of $c_{\text{NaCl}}^{\text{V,0}}$, because both solutions were prepared from the same stock solution and supplied to Cell (I) just before the titration.
The residual sum of squares (RSS) over the entire titration curve calculated by fitting the model to a titration curve was minimized to obtain the least square estimates of $K'_{\text{sp}}$ and $c_{\text{NaCl}}^{\text{V,0}}$, in our case not in one step but by two successive Excel® Solver operations.
Figure 1 shows a typical potentiometric titration curve experimentally obtained (dots) with Cell (I) and four least-square fitting curves based on different models of the activity coefficients of monovalent ionic species, i.e., Debye-Hückel limiting law (DHLL), Güntelberg, Davies, and Bates-Guggenheim models (See Table 1), together with the case of γi ≡ 1, that is, without taking account of activity corrections to the concentration terms.
| Model | −log10γi | γi at I = 0.005 | γi at I = 0.01 |
|---|---|---|---|
| DHLL | $A\sqrt{I} $ | 0.9202 | 0.8890 |
| Güntelberg | $A\sqrt{I} /(1 + \sqrt{I} )$ | 0.9253 | 0.8986 |
| Davies | $A(\sqrt{I} /(1 + \sqrt{I} ) - 0.3I)$ | 0.9269 | 0.9018 |
| Bates-Guggenheim | $A\sqrt{I} /(1 + 1.5\sqrt{I} )$ | 0.9256 | 0.9028 |
One can see in Fig. 1 that all least-square fitted curves based on different assumptions on the activity coefficients appear to overlap with each other; all models seem to successfully explain the experimental behavior of the points. In fact, a closer look at the curves in Fig. 1 (Fig. S1 in Supporting information) reveals the difference among them in the region before the equivalence point, but the difference becomes incognizable around the equivalence point and beyond.
A more quantitative comparison of the performance of these models is given in Table 2. Note that the five digit numbers in the table are theoretical, and are shown to compare the predictions by five different models, not to claim the experimental accuracy. The regressed values of $c_{\text{s}}^{\text{V,0}}$ agree well with each other. The mutual agreements among five cases are excellent also for $E_{\text{cell}}^{\text{eqv}}$ and $V_{\text{t}}^{\text{eqv}}$. These agreements may be taken as suggesting that the present approach to find the equivalence point via Ksp is successful in the present case. Incidentally, the automatic titrator employed automatically detected the values of $c_{\text{s}}^{\text{V,0}} = 20.74$ mL and $E_{\text{cell}}^{\text{eqv}} = 0.1685$ V, which were obtained through numerical differentiation of the titration curve, are in excellent agreement with those in Table 2.
| Model | Ksp | pKsp | RSS | $E_{\text{cell}}^{\text{eqv}}$ | $c_{\text{s}}^{\text{V},0}$ | Veqv | $\gamma_{\text{i}}^{\text{eqv}}$ |
|---|---|---|---|---|---|---|---|
| ×1010 | (mV)2 | V | ×102 mol dm−3 | mL | |||
| γi ≡ 1 | 2.1465 | 9.6683 | 180 | 0.16870 | 1.0420 | 20.740 | 1 |
| B-G* | 1.7858 | 9.7482 | 209 | 0.16843 | 1.0421 | 20.739 | 0.92675 |
| Davies | 1.7820 | 9.7490 | 210 | 0.16843 | 1.0420 | 20.738 | 0.92607 |
| Güntelberg | 1.7723 | 9.7515 | 213 | 0.16840 | 1.0421 | 20.738 | 0.92439 |
| DHLL | 1.7660 | 9.7530 | 204 | 0.16854 | 1.0422 | 20.740 | 0.91920 |
*Bates-Guggenheim.
Naturally, however, five models examined are not coequal; among Ksp values in the second column in Table 2 the value, 2.1465 × 10−10, obtained assuming γi ≡ 1 is far from four others, whose average and standard deviation being (1.7765 ± 0.0090) × 10−10, though the RSS is the smallest. This peculiar behavior should be ascribed to be a kind of artifact, not fortuitously but necessarily caused by the characteristics of the titration curve and models.
The “height” of a precipitation titration curve is primarily determined by the magnitude of Ksp, as easily envisaged by two titration curves for the formation of AgCl and AgBr. In determining Ksp through the regression, other factors that can affect the height, $c_{\text{t}}^{0}$ and $c_{\text{s}}^{0}$ in Eq. 6, have been taken into account in the regression calculation above.
Another factor that can change the height is the activity coefficients of the relevant species. In Table 2, the rightmost column lists the activity coefficients calculated by the four models at the equivalence point. All four values, 0.92675 (Bates-Guggenheim), 0.92607 (Davies), 0.92439 (Güntelberg), and 0.91920 (DHLL), are distinctively smaller than 1, and decreases in this order. Likewise, corresponding Ksp values in the second column are distinctively smaller than the top one, 2.1465 × 10−10 for the case of γi ≡ 1, and decrease downward. The apparent parallelism between the change in Ksp and $\gamma_{\text{i}}^{\text{eqv}}$ in Table 2 suggests that smaller the activity coefficients higher the “height” in the titration curve, i.e., smaller the $K'_{\text{sp}}$ value in Eq. 6. Because the DHLL obviously overestimates the magnitude of the activity coefficient, giving smaller γi values for 1-1 electrolyte solutions in the present concentration range, three models in the middle in Table 2 seem to give more reasonable estimates of Ksp. In the case of the γi ≡ 1 model the decrease in the height of the titration curve due to the activity coefficient effects is obtruded to Ksp, resulting in its increase to such an extent in Table 2.
Table 3 summarizes the results of potentiometric titrations with the same cell (I) conducted on different days. The curves were analyzed by least-square fitting of the model based on the Davies model for γi as described above. Upper seven entries in Table 3 are the titrations of 0.01 mol dm−3 NaCl in three different seasons. The third entry is the same data as that in Table 2.
| Date | Ksp | pKsp | RSS | n | $E_{\text{cell}}^{\text{eqv}}$ | $c_{\text{s}}^{\text{V,0}}$ | $V_{\text{t}}^{\text{eqv}}$ | $V_{\text{t}}^{\text{eqv}}$ (***) |
|---|---|---|---|---|---|---|---|---|
| ymd | ×1010 | (mV)2 | V | ×102 mol dm−3 | mL | mL | ||
| 22Jan27 | 1.879 | 9.726 | 793 | 112 | 0.1667 | 0.9980 | 19.86 | 19.89 |
| 22Jan27 | 1.846 | 9.732 | 335 | 99 | 0.1669 | 0.9982 | 19.87 | 19.87 |
| 21Sep27* | 1.782 | 9.749 | 210 | 115 | 0.1684 | 1.042 | 20.74 | 20.74 |
| 21Sep27 | 1.871 | 9.728 | 545 | 110 | 0.1679 | 1.045 | 20.79 | 20.64 |
| 20Dec4 | 1.949 | 9.710 | 242 | 90 | 0.1662 | 0.9980 | 20.00 | 20.04 |
| 20Dec4 | 1.773 | 9.752 | 456 | 94 | 0.1674 | 0.9955 | 19.95 | 19.93 |
| 20Dec4 | 1.916 | 9.718 | 189 | 94 | 0.1666 | 1.0419 | 19.98 | 19.98 |
| Average | 1.859 | 9.731 | 0.1672 | |||||
| STDVP | 0.060 | 0.014 | 0.0007 | |||||
| 21Feb15 | 1.813 | 9.742 | 360 | 71 | 0.1103 | 0.10158 | 20.34 | 20.44 |
| 21Feb15 | 1.759 | 9.756 | 433 | 72 | 0.1107 | 0.10220 | 20.33 | 20.56 |
| Pooled** Ave. | 1.840 | 9.736 | ||||||
| Pooled STDVP | 0.060 | 0.014 |
*Same data as in Fig. 1 and Table 2, 3rd entry. **Upper seven entries at $c_{\text{s}}^{\text{V,0}} \simeq 10$ mmol dm−3 and lower two entries at $c_{\text{s}}^{\text{V,0}} \simeq 1$ mmol dm−3 are averaged altogether. ***Automatic titrator determined.
The average of the seven entries for Ksp, (1.859 ± 0.060) × 10−10 in molarity scale, corresponding to pKsp of 9.731 ± 0.014. These values are comparable to, and within the range of standard deviation of, literature data, pKsp = 9.72 ± 0.0731 and 9.77 ± 0.06.32 Thus the potentiometric titration curves for AgCl precipitation were acquired with good reproducibility, and the least-square fittings were satisfactorily made.
Two lower entries in Table 3 were the results from titrations of 1 mmol m−3 NaCl with 1 mmol dm−3 AgNO3. Averaging of Ksp and pKsp values over all nine entries were (1.840 ± 0.060 S.D.) × 10−10 and 9.736 ± 0.014, respectively. Comparing the corresponding values from the upper seven entries, there seems to be no reason to distinguish the lower two where the analyte was 1 mmol dm−3 NaCl. It is seen that Ksp is independent of the analyte concentration, and hence the ionic strength in the range lower than 10 mmol dm−3. In view of Eq. 10, it means that $\gamma_{\text{Cl}^{ - }}^{\text{V,eqv}}/\gamma_{\text{Ag}^{ + }}^{\text{V,eqv}}$ is very close to unity in the concentration range studied, and Ksp and pKsp values obtained in Table 3 are likely to be of thermodynamic significance.
To put it the other way around, the results in Table 3 may be seen as evidence that the ILSB is stable and reproducible to the level of standard deviation of $E_{\text{cell}}^{\text{eqv}}$, 0.7 mV, in Table 3.
A new method has been proposed for determining the solubility product of silver chloride based on potentiometric titration by use of ILSB-equipped reference electrode. Some years ago, James N. Butler wrote “Solubility products do not grow in table from spores. They depend on a great deal of tedious work”.33 In the present method, one titration curve can be obtained within half an hour by an automatic titrator.
In the method of analysis proposed, the first quantity we obtain is a least-square estimate of Ksp. Other characteristic properties including $E_{\text{cell}}^{\text{eqv}}$ and $V_{\text{t}}^{\text{eqv}}$ were derived based on the estimate of Ksp. This work flow of data analysis may look opposite to the traditional methods of analyzing potentiometric titration curves, where the $V_{\text{t}}^{\text{eqv}}$, ideally, or the corresponding quantity at the end point is identified, first.
What is described above may be seen as a way of back to the future in potentiometric titrations. In seeking for a method of potentiometric titration of dilute solutions, Kolthoff once expected the possibility of using the equivalence potential, but concluded “the exact determination of the equivalence potential is impossible, owing to an unknown liquid junction potential”.34 Seventy years later, the present study suggests his expectation may be realized on the basis of ILSB-based titrations.
The authors thank Kyoto Electronic Manif. Co. for the use of automatic potentiometric titrator. The authors’ thanks go to Kenji Chayama, Ryo Murakami, Hiroshi Danjo, and Yuki Kitazumi for synthesis and purification of ionic liquids as well as insightful discussion. The research reported in this paper was supported by JSPS KAKENHI 21245021, 15K05552 (T.K.) and 18K05307 (M.Y.).
The data that support the findings of this study are openly available under the terms of the designated Creative Commons License in J-STAGE Data at https://doi.org/10.50892/data.electrochemistry.21614745.
Takashi Kakiuchi: Conceptualization (Lead), Data curation (Lead), Formal analysis (Lead), Funding acquisition (Lead), Investigation (Lead), Methodology (Lead), Project administration (Lead), Software (Lead), Supervision (Lead), Validation (Lead), Writing – original draft (Lead), Writing – review & editing (Lead)
Ryunosuke Tanigo: Data curation (Lead), Formal analysis (Lead), Investigation (Equal)
Atsushi Tani: Data curation (Lead), Formal analysis (Lead), Investigation (Equal)
Takeshi Yamazaki: Data curation (Lead), Formal analysis (Lead), Investigation (Equal)
Kohta Komatsubara: Data curation (Lead), Formal analysis (Lead), Investigation (Equal)
Keiji Nakano: Conceptualization (Lead), Data curation (Lead), Formal analysis (Lead), Investigation (Equal)
Masahiro Yamamoto: Conceptualization (Lead), Formal analysis (Lead), Funding acquisition (Lead), Investigation (Lead), Methodology (Equal), Project administration (Lead), Resources (Lead), Supervision (Lead), Validation (Lead), Writing – review & editing (Lead)
The authors declare no conflict of interest in the manuscript.
Japan Society for the Promotion of Science: 21245021
Japan Society for the Promotion of Science: 15K05552
Japan Society for the Promotion of Science: 18K05307
T. Kakiuchi: ECSJ Senior Member
M. Yamamoto: ECSJ Active Member
