Ion Concentration Measurement Device and Measurement Method

Abstract

An object of the present invention is to provide an ion concentration measurement device and a measurement method capable of precisely determining an ion concentration by clarifying a method for selecting an ion-selective electrode that has a small measurement error and hardly receives interference of a coexisting ion and using the selected ion-selective electrode. An ion concentration measurement device including two or more kinds of ion-selective electrodes that have mutually different selectivity coefficient ratios for a plurality of ions and satisfy formulas (1) and (2), wherein the two or more kinds of ion-selective electrodes are constituted by n kinds of ion-selective electrodes, and wherein n is the number of kinds of the plurality of ions and is an integer of 2 or more.

Description

BACKGROUND OF THE INVENTION

Field of the Invention

The present invention relates to an ion concentration measurement device and a measurement method.

Description of Related Art

An ion-selective electrode is generally used for measurement of an ion concentration in blood in a clinical examination. By immersing the ion-selective electrode and a reference electrode in a sample solution, a potential difference corresponding to an ion concentration of the sample solution is obtained. Therefore, by grasping a relationship between the potential difference and the concentration in advance, the ion concentration in the sample solution can be found from the potential difference. Since the ion concentration in the sample solution can be measured simply and quickly only by immersing the electrodes in the sample solution, ion concentration measurement using the ion-selective electrode is widely used currently.

A case where one kind of ion concentration is measured using an ion-selective electrode S and a reference electrode will be briefly described. A relationship between a potential difference E_S obtained when the ion-selective electrode S and the reference electrode are immersed in a sample solution containing an ion X^a and an ion concentration (C(X^a)) in the sample solution can be expressed by a Nernst equation (formula (4)).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}1\right]& \\ E_S=E_{0,S}+\frac{2.303\mathrm{RT}}{\mathrm{aF}}\times {\log }_{10}C\left(X^a\right)& \left(4\right)\end{array}$

In the formula (4),

• • E_S is a potential difference generated between the ion-selective electrode S and the reference electrode,
  • E_0,S is a standard potential difference generated between the ion-selective electrode S and the reference electrode,
  • R is a gas constant,
  • T is an absolute temperature,
  • a is a valence of the ion X^a,
  • F is a Faraday constant, and
  • C(X^a) is a concentration of the ion X^a.

By measuring a solution containing the ion X^a having a known concentration with the ion-selective electrode S, a calibration curve representing formula (4) can be created. Using the created calibration curve, an unknown sample solution is measured, and an ion concentration can be calculated from a potential difference obtained.

Note that the sample solution may contain, in addition to an ion to be measured (hereinafter, referred to as “measurement ion”), an ion other than the measurement ion (hereinafter, referred to as “coexisting ion”). Here, when a concentration of a measurement ion B1 contained in a sample solution containing n kinds of ions consisting of the measurement ion B1 and coexisting ions B2 to Bn is measured using the ion-selective electrode S and the reference electrode, there is a relationship of a Nicolsky-Eisenmann equation (formula (5)).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}2\right]& \\ E_S=E_{0,S}+\frac{2.303\mathrm{RT}}{\mathrm{aF}}\times {\log }_{10}\left[C\left(B1\right)+\sum _{j=2}^n{K}_S\left(B1,\mathrm{Bj}\right)C\left(\mathrm{Bj}\right)\right]& \left(5\right)\end{array}$

In the formula (5),

• • E_S is a potential difference generated between the ion-selective electrode S and the reference electrode,
  • E_0,S is a standard potential difference generated between the ion-selective electrode S and the reference electrode,
  • R is a gas constant,
  • T is an absolute temperature,
  • a is a valence of the ion B1,
  • F is a Faraday constant,
  • Bj represents a j-th ion among the n kinds of ions,
  • j is an integer of 2 to n, and
  • C(Bj) is a concentration of the ion Bj.

Here, K_S(B1, Bj) is an ion selectivity coefficient of the ion-selective electrode S for the ion Bj when the ion B1 is used as a reference in the ion-selective electrode S. In other words, K_S(B1, Bj) is a coefficient representing a magnitude of an influence of the ion Bj on a potential difference generated between the ion-selective electrode S and the reference electrode by a ratio when the ion B1 is used as a reference.

When the above measurement is performed on a sample solution containing the ion B1 having a concentration C(B1) but not containing the coexisting ions B2 to Bn using an ion-selective electrode S having an ion selectivity coefficient for the ion B2 when the ion B1 is used as a reference is K_S(B1, B2), the following potential difference E_B1 is obtained.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}3\right]& \\ E_{B1}=E_{0,S}+\frac{2.303\mathrm{RT}}{\mathrm{aF}}\times {\log }_{10}C\left(B1\right)& \left(6\right)\end{array}$

Furthermore, when the ion B2 having a concentration C(B2) is added to this sample solution (that is, in a case of a sample solution containing the ion B1 having the concentration C(B1) and the ion B2 having the concentration C(B2) but not containing the coexisting ions B3 to Bn), and the above measurement is performed, the following potential difference E_B1+B2 is obtained.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}4\right]& \\ E_{B1+B2}=E_{0,S}+\frac{2.303\mathrm{RT}}{\mathrm{aF}}\times {\log }_{10}\left[C\left(B1\right)+K_S\left(B1,B2\right)C\left(B2\right)\right]& \left(7\right)\end{array}$

Here, in the above formulas (6) and (7), E_0,S, R, T, a, and F are the same as those defined in the above formula (5), respectively, and E_B1 in the above formula (6) and E_B1+B2 in the above formula (7) correspond to E_S in the above formula (5).

As suggested from the above formulas (6) and (7), when the ion selectivity coefficient K_S(B1, B2) is large, B2 ion interference occurs when the ion B1 is measured, which causes an error.

Therefore, in order to measure a concentration of a specific ion, a selectivity coefficient for another ion is desirably as small as possible. It is known that, when an ion-selective membrane containing bis [(12-crown-4)methyl]2-dodecyl-2-methylmalonate [CAS No. 80403-59-4] is used for a sodium ion-selective electrode, a selectivity coefficient K_S(Na⁺, K⁺) of a potassium ion to a sodium ion is 0.01, which is put to practical use (Non Patent Document 1: Anal. Chem., 1982, 54 (7), 1224-1227).

However, depending on an ion to be measured, it may be difficult to develop an electrode having a sufficiently small ion selectivity coefficient for another ion. In particular, there are few ion-selective membranes suitable for measurement of an anion, and an ion-exchange membrane that is not sufficiently selective for an ion to be measured is often used for the measurement of an anion. For example, in order to measure a bicarbonate ion (HCO₃⁻) in blood, an electrode satisfying a selectivity coefficient K_S(HCO₃, Cl⁻) of 0.01 or less is required, but only an electrode having K_S(HCO₃, Cl⁻) of more than 1 is known. In a clinical examination, presence of an ion that responds more strongly than an ion to be measured in a sample solution is a problem in practical use of an electrode that measures a bicarbonate ion (HCO₃⁻) concentration in blood.

Therefore, a method for measuring a plurality of kinds of ion concentrations using a plurality of ion-selective electrodes having insufficient selectivity for a measurement ion has been developed. Examples thereof include Patent Document 1: JP H1-502360 A, Patent Document 2: JP S54-030094 A, and Patent Document 3: WO 2019/163281 A. There is also an example using an ion-selective electrode for removing an influence of an interfering ion in addition to an ion-selective electrode used for measurement of an ion concentration (Patent Document 4: JP H07-167818 A).

SUMMARY OF THE INVENTION

Technical Problem

As in Patent Document 2 (JP S54-030094 A), a method is known in which simultaneous equations are solved for values calculated from n measured values obtained with a plurality of (n) ion-selective electrodes having low ion selectivity, to calculate concentrations of n kinds of ions present in a sample solution. However, there is an error in a measured value in an actual measurement. In addition, when the sample solution contains n+1 or more kinds of ions, interference of a coexisting ion cannot be ignored. Therefore, as a result of studies by the present inventors, it has become clear that depending on selection of an ion-selective electrode, an ion concentration cannot be precisely determined due to a measurement error or interference of a coexisting ion.

Therefore, an object of the present invention is to provide an ion concentration measurement device and a measurement method capable of precisely determining an ion concentration by clarifying a method for selecting an ion-selective electrode that has a small measurement error and hardly receives interference of a coexisting ion and using the selected ion-selective electrode.

Solution to Problem

The present inventors have conducted intensive studies in order to solve the above problems. As a result, the present inventors have found an ion concentration measurement device and a measurement method capable of precisely determining an ion concentration by clarifying a method for selecting an ion-selective electrode that has a small measurement error and hardly receives interference of a coexisting ion and using the selected ion-selective electrode.

That is, a first aspect of the present invention provides an ion concentration measurement device including two or more kinds of ion-selective electrodes that have mutually different selectivity coefficient ratios for a plurality of ions and satisfy the following formulas (1) and (2), wherein the two or more kinds of ion-selective electrodes are constituted by n kinds of ion-selective electrodes, and wherein n is the number of kinds of the plurality of ions and is an integer of 2 or more.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}5\right]& \\ \mathrm{Matrix}A=\left(\begin{array}{cccccc}{K}_{S1}\left(B1,B1\right)& K_{S1}\left(B1,B2\right)& \dots & K_{S1}\left(B1,\mathrm{Bj}\right)& \dots & K_{S1}\left(B1,\mathrm{Bn}\right)\\ K_{S2}\left(B1,B1\right)& K_{S2}\left(B1,B2\right)& \dots & K_{S2}\left(B1,\mathrm{Bj}\right)& \dots & K_{S2}\left(B1,\mathrm{Bn}\right)\\ ⋮& ⋮& & ⋮& & ⋮\\ K_{\mathrm{Si}}\left(B1,B1\right)& K_{\mathrm{Si}}\left(B1,B2\right)& \dots & K_{\mathrm{Si}}\left(B1,\mathrm{Bj}\right)& \dots & K_{\mathrm{Si}}\left(B1,\mathrm{Bn}\right)\\ ⋮& ⋮& & ⋮& & ⋮\\ K_{\mathrm{Sn}}\left(B1,B1\right)& K_{\mathrm{Sn}}\left(B1,B2\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bj}\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bn}\right)\end{array}\right)& \left(1\right)\end{array}$ $\begin{array}{cc}A>0.1& \left(2\right)\end{array}$

In formula (1), each component of the matrix A represents an ion selectivity coefficient, wherein Bj represents a j-th ion among n kinds of ions constituting the plurality of ions,

• • Si is a symbol representing an i-th ion-selective electrode constituting the n kinds of ion-selective electrodes,
  • i and j are each independently an integer of 1 to n, and
  • K_si (B1, Bj) represents an ion selectivity coefficient for the ion Bj when an ion B1 is used as a reference in the ion-selective electrode Si.

In formula (2), ∥A∥ is an absolute value of a determinant of the matrix A.

In addition, a second aspect of the present invention provides a measurement method including a step of measuring an ion concentration in a sample solution using a plurality of ion-selective electrodes satisfying the above formulas (1) and (2).

Advantageous Effects of Invention

The present invention has clarified a method for selecting an ion-selective electrode that has a small measurement error and hardly receives interference of a coexisting ion even when the number of kinds (n+1 or more) of coexisting ions is more than the number of electrodes (n). By selecting an ion-selective electrode having a large absolute value (∥A∥) of the determinant of the matrix A, it is possible to precisely measure an ion concentration with a small relative standard deviation of a measurement result. The present invention can provide an ion concentration measurement device and a measurement method capable of precisely determining an ion concentration.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a schematic diagram illustrating an ion concentration measurement device according to an embodiment of the present invention.

DESCRIPTION OF THE INVENTION

Hereinafter, an embodiment for carrying out the present invention will be described.

[Description of Formulas (1), (2), and (3)]

First, formulas (1), (2), and (3) used in the present invention will be described. By using formulas (1) and (2), it is possible to select an ion-selective electrode that has a small measurement error and is resistant to interference of a coexisting ion. As a result, it is possible to provide an ion concentration measurement device capable of precisely determining an ion concentration, which is a characteristic of the present invention.

As a preceding stage for describing a case of measurement with n kinds of ion-selective electrodes, a case of measurement with one kind of ion-selective electrode will be described.

Case of Measurement with One Kind of Ion-Selective Electrode

When a concentration of a specific ion (ion B1) contained in a sample solution containing n kinds (n is an integer of 2 or more) of ions is measured using one kind of ion-selective electrode S (hereinafter, also referred to as “electrode S”) and a reference electrode, there is a relationship of a Nicolsky-Eisenmann equation (formula (5)). Here, (2.303RT)/aF is referred to as Slope. That is, Slope is a constant. In practice, Slope often does not present a theoretical value for various reasons such as electrical resistance in a device and a liquid junction potential. In this case, it is preferable to determine Slope by creating a calibration curve.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}6\right]& \\ \end{array}$ $\begin{array}{cccc}{E}_S& =& E_{0,S}+\frac{2.303\mathrm{RT}}{\mathrm{aF}}\times {\log }_{10}\left[C\left(B1\right)+\sum _{j=2}^n{K}_S\left(B1,\mathrm{Bj}\right)C\left(\mathrm{Bj}\right)\right]& \left(5\right)\\ =& E_{0,S}+\mathrm{Slope}\times {\log }_{10}\left[C\left(B1\right)+\sum _{j=2}^n{K}_S\left(B1,\mathrm{Bj}\right)C\left(\mathrm{Bj}\right)\right]& \left(5^{\prime }\right)\\ & & \end{array}$

• • wherein
  • E_S is a potential difference generated between the ion-selective electrode S and the reference electrode,
  • E_0,S is a standard potential difference generated between the ion-selective electrode S and the reference electrode,
  • Slope is 2.303RT/aF,
  • R is a gas constant,
  • T is an absolute temperature,
  • a is a valence of the ion B1,
  • F is a Faraday constant,
  • Bj represents a j-th ion among n kinds of ions,
  • j is an integer of 2 to n,
  • C(Bj) is a concentration of the ion Bj, and
  • K_S(B1, Bj) is a selectivity coefficient for the ion Bj when an ion B1 is used as a reference in the electrode S.

A sample solution (calibration curve solution) having a known concentration of the ion B1 is measured using the ion-selective electrode S and the reference electrode, and a calibration curve representing formula (5′) can be created. From the calibration curve, Slope and a standard potential difference E_0,S at the time of measurement with the ion-selective electrode S can be determined.

Formula (5′) is transferred, and the following formula (8) can be derived. Using this formula (8), from E_0,S and a potential difference E_S obtained when the ion-selective electrode S and the reference electrode are immersed in the sample solution, an ion concentration when n kinds of ion concentrations in the sample solution are converted into any ion B1 concentration can be calculated.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}7\right]& \\ C\left(B1\right)+\sum _{j=2}^n{K}_S\left(B1,\mathrm{Bj}\right)C\left(\mathrm{Bj}\right)={10}^{\left(\frac{E_S-E_{0,S}}{\mathrm{Slope}}\right)}& \left(8\right)\end{array}$

Here, when n=2, the above formula (8) can be expressed as the following formula (8A).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}8\right]& \\ C\left(B1\right)+K_S\left(B1,B2\right)C\left(B2\right)={10}^{\left(\frac{E_S-E_{0,S}}{\mathrm{Slope}}\right)}& \left(8A\right)\end{array}$

Then, the above formula (8A) is transferred, and can be further transformed into the following formula (8B).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}9\right]& \\ K_S\left(B1,B2\right)=\frac{{10}^{\left(\frac{E_S-E_{0,S}}{\mathrm{Slope}}\right)}-C\left(B1\right)}{C\left(B2\right)}& \left(8B\right)\end{array}$

That is, when n=2, K_S(B1, B2), which is a selectivity coefficient for the ion B2 when the ion B1 is used as a reference in the electrode S, can be determined using the above formula (8B).

Case of Measurement with n Kinds of Ion-Selective Electrodes (Description of Formula (1))

A case where n kinds of ions (ions: B1, B2, . . . , Bj, . . . , and Bn, ion concentrations: C(B1), C(B2), . . . , C(Bj), . . . , and C(Bn)) in a sample solution are measured using n kinds (n is an integer of 2 or more) of ion-selective electrodes S1 to Sn will be described. At this time, the above formula (8) is established for each of the n kinds of ion-selective electrodes S1 to Sn. Therefore, simultaneous equations of formula (9) are established.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}10\right]& \\ C\left(B1\right)+K_{S1}\left(B1,B2\right)C\left(B2\right)+\dots +K_{S1}\left(B1,\mathrm{Bj}\right)C\left(\mathrm{Bj}\right)+\dots +K_{S1}\left(B1,\mathrm{Bn}\right)C\left(\mathrm{Bn}\right)={10}^{\left(\frac{E_{S1}-E_{0,S1}}{\mathrm{Slope}}\right)}& \left(9\right)\end{array}$ $C\left(B1\right)+K_{S2}\left(B1,B2\right)C\left(B2\right)+\dots +K_{S2}\left(B1,\mathrm{Bj}\right)C\left(\mathrm{Bj}\right)+\dots +K_{S2}\left(B1,\mathrm{Bn}\right)C\left(\mathrm{Bn}\right)={10}^{\left(\frac{E_{S2}-E_{0,S2}}{\mathrm{Slope}}\right)}$ $\begin{array}{cc}⋮& ⋮\end{array}$ $C\left(B1\right)+K_{\mathrm{Sn}}\left(B1,B2\right)C\left(B2\right)+\dots +K_{\mathrm{Sn}}\left(B1,\mathrm{Bj}\right)C\left(\mathrm{Bj}\right)+\dots +K_{\mathrm{Sn}}\left(B1,\mathrm{Bn}\right)C\left(\mathrm{Bn}\right)=\frac{{10}^{\left(\frac{E_{\mathrm{Sn}}-E_{0,\mathrm{Sn}}}{\mathrm{Slope}}\right)}}{\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}\mathrm{Si}\left(i=n\right)\end{array}}\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}\mathrm{Si}(i\mathrm{is}\mathrm{an}\\ \mathrm{integer}\mathrm{of}1\mathrm{to}n)\end{array}={10}^{\left(\frac{E_{\mathrm{Si}}-E_{0,\mathrm{Si}}}{\mathrm{Slope}}\right)}$

In formula (9),

• • Bj represents a j-th ion among the n kinds of ions,
  • Si is a symbol representing an i-th ion-selective electrode constituting the n kinds of ion-selective electrodes,
  • i and j are each independently an integer of 1 to n,
  • K_si (B1, Bj) is K_S(B1, Bj) when an ion-selective electrode Si (i-th ion-selective electrode) is used as the ion-selective electrode S, that is, an ion selectivity coefficient for the ion Bj when an ion B1 is used as a reference in the ion-selective electrode Si,
  • E_Si is a potential difference generated between the ion-selective electrode Si and a reference electrode,
  • E_0,Si is a standard potential difference generated between the ion-selective electrode Si and the reference electrode,
  • Slope is 2.303RT/aF, and
  • R, T, a, and F are the same as R, T, a, and F defined by the above formula (5), respectively. Here, the right side of formula (9) is a calculated value obtained with the ion-selective electrode Si. This value is a value calculated using the potential difference E_Si, the Slope determined by the calibration curve, and the standard potential differences E_0,Si.

When the simultaneous equations of the above formula (9) are converted into a form of a matrix, the following formula (10) is obtained.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}11\right]& \\ \frac{\left(\begin{array}{cccccc}1& K_{S1}\left(B1,B2\right)& \dots & K_{S1}\left(B1,\mathrm{Bj}\right)& \dots & K_{S1}\left(B1,\mathrm{Bn}\right)\\ 1& K_{S2}\left(B1,B2\right)& \dots & K_{S2}\left(B1,\mathrm{Bj}\right)& \dots & K_{S2}\left(B1,\mathrm{Bn}\right)\\ ⋮& ⋮& & ⋮& & ⋮\\ 1& K_{\mathrm{Sn}}\left(B1,B2\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bj}\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bn}\right)\end{array}\right)}{\mathrm{Matrix}{A}^{\prime }}\left(\begin{array}{c}C\left(B1\right)\\ C\left(B2\right)\\ ⋮\\ C\left(\mathrm{Bn}\right)\end{array}\right)=\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S2\\ ⋮\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}\mathrm{Sn}\end{array}\right)& \left(10\right)\end{array}$

A matrix on the left side of formula (10) is referred to as a matrix A′. Here, for each of the n kinds of ion-selective electrodes S1, S2, . . . , and Sn, a selectivity coefficient K_S(B1, B1) (that is, K_S1(B1, B1), K_S2(B1, B1), . . . , and K_Sn(B1, B1)) for any ion B1 itself serving as a reference is 1. Therefore, when 1 is rewritten as K_S(B1, B1) (that is, K_S1(B1, B1), K_S2 (B1, B1), . . . , and K_Sn (B1, B1) for n kinds of ion-selective electrodes S1, S2, . . . , and Sn, respectively), the matrix A′ can be transformed as in formula (11). The right side of formula (11) corresponds to the matrix A (formula (1)) of the present invention.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}12\right]& \\ \frac{\left(\begin{array}{cccccc}1& K_{S1}\left(B1,B2\right)& \dots & K_{S1}\left(B1,\mathrm{Bj}\right)& \dots & K_{S1}\left(B1,\mathrm{Bn}\right)\\ 1& K_{S2}\left(B1,B2\right)& \dots & K_{S2}\left(B1,\mathrm{Bj}\right)& \dots & K_{S2}\left(B1,\mathrm{Bn}\right)\\ ⋮& ⋮& & ⋮& & ⋮\\ 1& K_{\mathrm{Sn}}\left(B1,B2\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bj}\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bn}\right)\end{array}\right)}{\mathrm{Matrix}{A}^{\prime }}=\frac{\left(\begin{array}{cccccc}{K}_{S1}\left(B1,B1\right)& K_{S1}\left(B1,B2\right)& \dots & K_{S1}\left(B1,\mathrm{Bj}\right)& \dots & K_{S1}\left(B1,\mathrm{Bn}\right)\\ K_{S2}\left(B1,B1\right)& K_{S2}\left(B1,B2\right)& \dots & K_{S2}\left(B1,\mathrm{Bj}\right)& \dots & K_{S2}\left(B1,\mathrm{Bn}\right)\\ ⋮& ⋮& & ⋮& & ⋮\\ K_{\mathrm{Sn}}\left(B1,B1\right)& K_{\mathrm{Sn}}\left(B1,B2\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bj}\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bn}\right)\end{array}\right)}{\mathrm{Matrix}A}& \left(11\right)\end{array}$ $\begin{array}{cc}\mathrm{Matrix}A=\left(\begin{array}{cccccc}{K}_{S1}\left(B1,B1\right)& K_{S1}\left(B1,B2\right)& \dots & K_{S1}\left(B1,\mathrm{Bj}\right)& \dots & K_{S1}\left(B1,\mathrm{Bn}\right)\\ K_{S2}\left(B1,B1\right)& K_{S2}\left(B1,B2\right)& \dots & K_{S2}\left(B1,\mathrm{Bj}\right)& \dots & K_{S2}\left(B1,\mathrm{Bn}\right)\\ ⋮& ⋮& & ⋮& & ⋮\\ K_{\mathrm{Si}}\left(B1,B1\right)& K_{\mathrm{Si}}\left(B1,B2\right)& \dots & K_{\mathrm{Si}}\left(B1,\mathrm{Bj}\right)& \dots & K_{\mathrm{Si}}\left(B1,\mathrm{Bn}\right)\\ ⋮& ⋮& & ⋮& & ⋮\\ K_{\mathrm{Sn}}\left(B1,B1\right)& K_{\mathrm{Sn}}\left(B1,B2\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bj}\right)& \dots & K_{\mathrm{Sn}}\left(B1,\mathrm{Bn}\right)\end{array}\right)& \left(1\right)\end{array}$

In the matrix A,

• • Bj represents a j-th (j is an integer of 1 to n) ion among the n kinds of ions.
  • Si is a symbol representing an i-th (i is an integer of 1 to n) ion-selective electrode constituting the n kinds of ion-selective electrodes.
  • K_si (B1, Bj) is a selectivity coefficient for the ion Bj when the ion B1 is used as a reference in the ion-selective electrode Si. For example, K_S1(B1, B1) represents a selectivity coefficient for the ion B1 when the ion B1 is used as a reference in an ion-selective electrode S1, and K_S2 (B1, B3) represents a selectivity coefficient for an ion B3 when the ion B1 is used as a reference in an ion-selective electrode S2.

Description of Formula (2)

In the present application, a determinant of the matrix A is represented by |A|, and an absolute value of the determinant is represented by ∥A∥. In general, the determinant | A| can be calculated by cofactor expansion, for example.

When three kinds of ions (ions: B1, B2, and B3, ion concentrations: C(B1), C(B2), and C(B3)) in a sample solution are measured using three kinds (n=3) of ion-selective electrodes S1, S2, and S3, the matrix A represented by the above formula (1) can be represented as a 3×3 square matrix described below. In this case, a Sallas formula in the matrix can be used, and the determinant |A| corresponding to the matrix A can be determined by formula (12).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}13\right]& \\ \mathrm{Matrix}A\left(3\times 3\right)=\left(\begin{array}{ccc}{K}_{S1}\left(B1,B1\right)& K_{S1}\left(B1,B2\right)& K_{S1}\left(B1,B3\right)\\ K_{S2}\left(B1,B1\right)& K_{S2}\left(B1,B2\right)& K_{S2}\left(B1,B3\right)\\ K_{S3}\left(B1,B1\right)& K_{S3}\left(B1,B2\right)& K_{S3}\left(B1,B3\right)\end{array}\right)& \left(12\right)\end{array}$ $\mathrm{Determinant}❘A❘=K_{S1}\left(B1,B1\right)K_{S2}\left(B1,B2\right)K_{S3}\left(B1,B3\right)+K_{S1}\left(B1,B2\right)K_{S2}\left(B1,B3\right)K_{S3}\left(B1,B1\right)+K_{S1}\left(B1,B3\right)K_{S2}\left(B1,B1\right)K_{S3}\left(B1,B2\right)-K_{S1}\left(B1,B3\right)K_{S2}\left(B1,B2\right)K_{S3}\left(B1,B1\right)-K_{S1}\left(B1,B2\right)K_{S2}\left(B1,B1\right)K_{S3}\left(B1,B3\right)-K_{S1}\left(B1,B1\right)K_{S2}\left(B1,B3\right)K_{S3}\left(B1,B2\right)$

When the matrix A is an n×n matrix, the determinant |A| can be easily determined by using spreadsheet software (for example, Microsoft Excel) or numerical calculation software. When spreadsheet software Excel is used, calculation can be performed using a function such as “=MDETERM”. If the determinant |A| can be determined, an absolute value ∥A∥ of the determinant can be calculated.

The most important point in the present invention is that the absolute value ∥A∥ of the determinant satisfies formula (2).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}14\right]& \\ A>0.1& \left(2\right)\end{array}$

<Significance of Formula (2)>

From formulas (10) and (11), the following formula (13) can be expressed.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}15\right]& \\ A·\left(\begin{array}{c}C\left(B1\right)\\ C\left(B2\right)\\ ⋮\\ C\left(\mathrm{Bn}\right)\end{array}\right)=\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\\ \mathrm{obtained}\mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\\ \mathrm{electrode}S2\\ ⋮\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\\ \mathrm{electrode}\mathrm{Sn}\end{array}\right)& \left(13\right)\end{array}$

When the determinant |A| of the matrix A is not 0, an inverse matrix A-1 exists which corresponds to the matrix A. When the inverse matrix A-1 is multiplied by each side of formula (13) from the left, formula (13) can be transformed into formula (14). Since A-1.A is an identity matrix, formula (14) can be transformed into formula (15) using a cofactor matrix of the matrix A expressed by formula (16).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}16\right]& \\ \end{array}$ $\begin{array}{cccc}{A}^{-1}·A·\left(\begin{array}{c}C\left(B1\right)\\ C\left(B2\right)\\ ⋮\\ C\left(\mathrm{Bn}\right)\end{array}\right)& =& A^{-1}·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S2\\ ⋮\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}\mathrm{Sn}\end{array}\right)& \left(14\right)\\ \left(\begin{array}{c}C\left(B1\right)\\ C\left(B2\right)\\ ⋮\\ C\left(\mathrm{Bn}\right)\end{array}\right)& =& A^{-1}·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S2\\ ⋮\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}\mathrm{Sn}\end{array}\right)& \left(3\right)\\ =& \frac{\tilde{A}}{❘A❘}·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S2\\ ⋮\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}\mathrm{Sn}\end{array}\right)& \left(15\right)\end{array}$ $\begin{array}{cc}{A}^{-1}=\frac{\tilde{A}}{❘A❘}& \left(16\right)\end{array}$

Ā: Cofactor Matrix of A

Usually, measurement using a single ion-selective electrode S is affected by a measurement error and an interfering ion. When a calculated value obtained with the ion-selective electrode S includes an error, the error is referred to as As. In the present invention, n kinds of ion-selective electrodes (ion-selective electrodes S1, S2, . . . , and Sn) are used as the ion-selective electrode S, and errors ΔS for these ion-selective electrodes are referred to as Δ_S1, Δ_S2, . . . , and Asn, respectively. Formula (15) can be expressed by formula (17). By separating a term indicating an error using a property of a 5 sum of matrices, formula (17) can be expressed by formula (17′).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}17\right]& \\ \left(\begin{array}{c}C\left(B1\right)\\ C\left(B2\right)\\ ⋮\\ C\left(\mathrm{Bn}\right)\end{array}\right)=\frac{\tilde{A}}{❘A❘}·\left(\begin{array}{c}\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S1\end{array}+{\Delta }_{S1}\\ \begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S2\end{array}+{\Delta }_{S2}\\ ⋮\\ \begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}\mathrm{Sn}\end{array}+{\Delta }_{\mathrm{Sn}}\end{array}\right)& \left(17\right)\end{array}$ $\begin{array}{cc}\left(\begin{array}{c}C\left(B1\right)\\ C\left(B2\right)\\ ⋮\\ C\left(\mathrm{Bn}\right)\end{array}\right)=\frac{\tilde{A}}{❘A❘}·\frac{\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}S2\\ ⋮\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\\ \mathrm{with}\mathrm{electrode}\mathrm{Sn}\end{array}\right)}{\begin{array}{c}\mathrm{Term}\mathrm{indicating}\\ \mathrm{accurate}\mathrm{concentration}\end{array}}+\frac{\frac{\tilde{A}}{❘A❘}·\left(\begin{array}{c}{\Delta }_{S1}\\ {\Delta }_{S2}\\ ⋮\\ {\Delta }_{\mathrm{Sn}}\end{array}\right)}{\begin{array}{c}\mathrm{Term}\\ \mathrm{indicating}\\ \mathrm{error}\end{array}}& \left({17}^{\prime }\right)\end{array}$

Since the determinant |A| exists in a denominator of the term indicating an error, a value of the term indicating an error decreases as the absolute value ∥A∥ of the determinant |A| increases. Specifically, when the absolute value ∥A∥ of the determinant is 0.1 or more, the value of the term indicating an error is small, and an electrode can be selected with which measurement that can withstand practical use is possible. Therefore, in the present invention, the absolute value ∥A∥ satisfies a relationship represented by the following formula (2).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}18\right]& \\ A>0.1& \left(2\right)\end{array}$

The absolute value ∥A∥ of the determinant is more preferably larger than 0.2, and still more preferably larger than 0.4. The larger the absolute value ∥A∥ of the determinant is, the better it is. Although there is no upper limit for the absolute value ∥A∥ of the determinant, it is usually a value of 100 or less.

This formula (2) represents a combination of selectivity coefficients K_S(B1, Bj) (that is, K_S1(B1, Bj), K_S2(B1, Bj), . . . , and K_Sn (B1, Bj)) with which an influence of an error is reduced in relation to n kinds of ion-selective electrodes (ion-selective electrodes S1, S2, . . . , and Sn) used as the ion-selective electrode S and n kinds of ions (ions: B1, B2, . . . , and Bn) in a sample solution. In the present invention, two or more kinds of ion-selective electrodes having mutually different selectivity coefficient ratios for a plurality of ions are used as the ion-selective electrode S, and by selecting n kinds of ion-selective electrodes satisfying formula (2) as the two or more kinds of ion-selective electrodes, it is possible to provide an ion concentration measurement device and a method having a small influence of an error.

Description of Formula (3)

When n kinds of ions in a sample solution are measured using the n kinds of ion-selective electrodes (ion-selective electrodes S1, S2, . . . , and Sn) as the ion-selective electrode S, n kinds of ion concentrations can be represented by formula (3).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}19\right]& \\ \left(\begin{array}{c}C\left(B1\right)\\ C\left(B2\right)\\ ⋮\\ C\left(Bn\right)\end{array}\right)=A^{-1}·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S2\\ ⋮\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}Sn\end{array}\right)& \left(3\right)\end{array}$

Here, A⁻¹ represents an inverse matrix of the matrix A, and C(Bj) represents a concentration of the ion Bj (j is an integer of 1 to n). A calculated value obtained with an electrode Si (i is an integer of 1 to n) indicates a value obtained by conversion of a potential difference measured with the ion-selective electrode Si according to the following formula.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}20\right]& \\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}Si={10}^{\left(\frac{E_{\mathrm{Si}}-E_{0,\mathrm{Si}}}{\mathrm{Slope}}\right)}& \end{array}$

In the above formula,

• • E_Si is a potential difference generated between the ion-selective electrode Si and a reference electrode,
  • E_0,Si is a standard potential difference generated between the ion-selective electrode Si and the reference electrode,
  • Slope is 2.303RT/aF, and
  • R, T, a, and F are the same as R, I, a, and F defined by the above formula (5), respectively.

In the present invention, the ion-selective electrode S is also referred to as an electrode S, and an i-th (i is an integer of 1 to n) ion-selective electrode Si among n kinds of ion-selective electrodes (ion-selective electrodes S1, S2, . . . , and Sn) is also referred to as an electrode Si. By using formula (3), an ion concentration of a sample solution can be determined from a calculated value obtained with each of the n kinds of ion-selective electrodes used as the electrode S.

Case of Using Standard Solution

When measurement is performed for a long time, the standard potential difference E_0,S may fluctuate depending on a change in measurement conditions such as temperature, and therefore it is preferable to correct the standard potential difference E_0,S using a standard solution.

A method for correcting the standard potential difference E_0,S using a standard solution will be described. A potential difference obtained when the ion-selective electrode S and the reference electrode are immersed in a standard solution having a known ion concentration (C_std) is referred to as E_std (formula (18)). At this time, the standard potential difference E_0,S can be expressed by formula (19).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}21\right]& \\ E_{Std}=E_{0,S}+\mathrm{Slope}\times {\log }_{10}{C}_{std}& \left(18\right)\end{array}$ $\begin{array}{cc}{E}_{0,S}=E_{Std}-\mathrm{Slope}\times {\log }_{10}{C}_{std}& \left(19\right)\end{array}$

In the above formulas (18) and (19),

• • Slope is 2.303RT/aF, and
  • R, T, a, and F are the same as R, T, a, and F defined by the above formula (5), respectively.

When the standard solution is measured and formula (19) is substituted for the standard potential difference E_0,S, a calculated value obtained with the electrode S can be expressed by formula (20).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}22\right]& \\ \begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S={10}^{\left(\frac{E_S-E_{0,S}}{\mathrm{Slope}}\right)}\\ ={10}^{\left(\frac{E_S-E_{\mathrm{Std}}}{\mathrm{Slope}}+{\log }_{10}{C}_{\mathrm{Std}}\right)}\end{array}& \left(20\right)\end{array}$

It is preferable to use formula (20) for the calculated value obtained from the electrode S because measurement can be performed while fluctuation of the standard potential difference E_0,S is corrected when measurement conditions such as temperature change or when measurement is performed for a long time.

In the present invention, n kinds of ion-selective electrodes (ion-selective electrodes S1, S2, . . . , and Sn) are used as the electrode S. Therefore, by applying the above formula (20) to each of calculated values obtained from these ion-selective electrodes, it is possible to perform measurement similarly while fluctuation of the standard potential difference E_0,S (that is, E_0,S1, E_0,S2, . . . , and E_0,Sn) for these ion-selective electrodes is corrected. For example, when the above formula (20) is applied to an i-th (i is an integer of 1 to n) ion-selective electrode Si among n kinds of ion-selective electrodes (ion-selective electrodes S1, S2, . . . , and Sn), the above formula (20) can be expressed as the following formula (20′).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}23\right]& \\ \begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}\mathrm{Si}={10}^{\left(\frac{E_{\mathrm{Si}}-E_{0,\mathrm{Si}}}{\mathrm{Slope}}\right)}\\ ={10}^{\left(\frac{E_{\mathrm{Si}}-E_{\mathrm{Stdi}}}{\mathrm{Slope}}+{\log }_{10}{C}_{\mathrm{Std}}\right)}\end{array}& \left(20\text{'}\right)\end{array}$

In the above formula (20′),

• • E_Si is a potential difference generated between the ion-selective electrode Si and a reference electrode,
  • E_0,Si is a standard potential difference generated between the ion-selective electrode Si and the reference electrode,
  • E_stdi is a potential difference generated between the ion-selective electrode Si and the reference electrode when the ion-selective electrode Si and the reference electrode are immersed in a standard solution having a known ion concentration (C_std),
  • Slope is 2.303RT/aF, and
  • R, T, a, and F are the same as R, I, a, and F defined by the above formula (5), respectively.

[Ion Concentration Measurement Device]

The ion concentration measurement device of the present invention includes two or more kinds of ion-selective electrodes that have mutually different selectivity coefficient ratios for a plurality of ions and satisfy the above formulas (1) and (2).

Here, when the number of kinds of the plurality of ions is referred to as n, the two or more kinds of ion-selective electrodes are constituted by n kinds of ion-selective electrodes, and n is an integer of 2 or more.

In addition, in the above formula (1),

• • Bj represents a j-th ion among n kinds of ions constituting the plurality of ions,
  • Si is a symbol representing an i-th ion-selective electrode constituting the n kinds of ion-selective electrodes,
  • i and j are each independently an integer of 1 to n, and
  • K_si (B1, Bj) represents an ion selectivity coefficient for the ion Bj when an ion B1 is used as a reference in the ion-selective electrode Si.

In the present invention, it is preferable to calculate an ion concentration by the above formula (3).

Here, in the present invention, an ion B1 (that is, an ion satisfying j=1 among ions Bj) is one ion appropriately selected from n kinds of ions constituting the plurality of ions. This ion B1 is usually an ion that is often selected as an ion to be measured in concentration measurement by a single ion-selective electrode.

On the other hand, ions other than the ion B1 among the n kinds of ions constituting the plurality of ions are referred to as ions B2, B3, . . . , and Bn (that is, ions satisfying j≥2).

Here, the ions B2, B3, . . . , and Bn may include an ion whose concentration is difficult to directly and selectively measure by a single ion-selective electrode.

With respect to the two or more kinds of ion-selective electrodes, in the present invention, as long as a combination of ion-selective electrodes selected as n kinds of ion-selective electrodes (ion-selective electrodes S1, S2, . . . , and Sn) satisfies the above formula (2), assignment of the symbols S1, S2, . . . , and Sn is not particularly limited, and these symbols can be appropriately assigned to these ion-selective electrodes. Here, the ion concentration measurement device of the present invention may include more than n kinds of ion-selective electrodes, and n kinds of ion-selective electrodes may be selected from these ion-selective electrodes so as to satisfy formula (2) and used for ion concentration measurement. To each of the ion-selective electrodes constituting the ion concentration measurement device of the present invention, any determined unique identifier may be attached separately from the symbol Si. In this case, a correspondence table between these identifiers and the symbols S1, S2, . . . , and Sn is prepared according to components that can be contained in a sample solution and the kinds of ion-selective electrodes, and on the basis of this correspondence table, ion-selective electrodes to be used as n kinds of ion-selective electrodes (ion-selective electrodes S1, S2, . . . , and Sn) may be selected from the ion-selective electrodes constituting the ion concentration measurement device.

The ion concentration measurement device in the present invention is not particularly limited as long as it is a device including two or more kinds of ion-selective electrodes that have mutually different selectivity coefficient ratios for a plurality of ions and satisfy formulas (1) and (2). Specific examples thereof include a device as illustrated in FIG. 1. Usually, the ion concentration measurement device includes an ion-selective electrode 1, a reference electrode 2, and a potentiometer 3. Here, each of ion-selective electrodes 1 functions as the ion-selective electrode S. The ion-selective electrodes 1 are connected to potentiometers 3 via an electric wire, respectively, and the potentiometers 3 are connected to a reference electrode 2 via an electric wire. That is, each of the n kinds of ion-selective electrodes (ion-selective electrodes S1, S2, . . . , and Sn) used as two or more kinds of ion-selective electrodes constitutes the ion-selective electrode 1 in FIG. 1. In addition, it is also possible to combine an arithmetic device capable of calculating the above-described formulas and a tank 4 to contain a solution. Here, the arithmetic device is connected to the potentiometer 3, and performs arithmetic processing such as calculation of an ion concentration on the basis of a potential difference input from the potentiometer 3. The arithmetic processing can include a step of forming the determinant |A| and a step of calculating an absolute value of the determinant |A|. The arithmetic device and the potentiometer 3 may be separate devices, or may be integrated in a single device. These are examples, and do not limit the form of the device.

The number of kinds of ion-selective electrodes 1 to be combined may be any number as long as it is two or more. The number of kinds of ion-selective electrodes 1 only needs to be appropriately adjusted according to the number of kinds of ions to be measured, but is preferably 2 or more and 10 or less from a viewpoint of measurement precision. The number of kinds of ion-selective electrodes 1 is most preferably 2 or more and 5 or less from a viewpoint of the size of the ion concentration measurement device. In other words with the n, the n is preferably 2 or more and 10 or less, and most preferably 2 or more and 5 or less. Since it is necessary to use two or more kinds of ion-selective electrodes, the number of ion-selective electrodes 1 needs to be two or more. There is no particular limitation as long as it is two or more. In an exemplary aspect of the present invention, the number of kinds of the ion-selective electrodes 1 is three (that is, n=3), but is not limited thereto. It is also possible to combine a plurality of ion-selective electrodes of the same type.

The number of the reference electrodes 2 is not particularly limited as long as it is one or more, but is usually smaller than the number of the ion-selective electrodes.

The potentiometer 3 is not limited as long as it can measure a potential difference between the two or more kinds of ion-selective electrodes and the reference electrode. A voltmeter can also be used as long as it can determine a potential difference.

The tank 4 to contain a solution can be used without particular limitation as long as the ion-selective electrode and the reference electrode can be immersed therein. A plurality of tanks 4 may be used according to the number of kinds of ion-selective electrodes, or the plurality of ion-selective electrodes may be immersed in one tank. A tank in which each electrode can be immersed in a batch manner may be used, or a channel may be formed to circulate a sample solution, for example. In addition, a sample solution may also be stirred or circulated in order to homogenize the sample solution.

Ion-Selective Electrode

The ion-selective electrode 1 functions as the ion-selective electrode S, and any ion-selective electrode that can be generally used, such as an internal liquid type ion-selective electrode or a solid electrode type ion-selective electrode, can be used without particular limitation. In the present invention, n kinds of ion-selective electrodes 1 having mutually different selectivity coefficient ratios for a plurality of ions are combined and used as the ion-selective electrodes S1, S2, . . . , and Sn.

As illustrated in FIG. 1, the internal liquid type ion-selective electrode includes an internal electrode 5, an internal liquid 6, and an ion-selective membrane 7, and the ion-selective membrane 7 and the sample solution 9 are in contact with each other.

The internal electrode 5 is not particularly limited as long as it is a usually used one. A silver/silver chloride electrode is usually used.

As the internal liquid 6, an aqueous solution of an electrolyte such as KCl or NaCl is usually used.

The ion-selective membrane 7 is not particularly limited, but a substrate of a polymer such as polyvinyl chloride is mixed with a ligand that selects an ion together with a plasticizer, and the resultant mixture is used. The ion-selective membrane 7 may contain a salt, for example. Specifically, when the ion-selective electrode 1 is a cation-selective electrode for, for example, sodium ion, potassium ion, calcium ion, or magnesium ion (that is, a cation-selective electrode that selectively reacts with a cation such as sodium ion, potassium ion, calcium ion, or magnesium ion to generate an output corresponding to a concentration (activity) of the cation), a membrane as described in Pure Appl. Chem., 2000, Vol. 72, No. 10, pp. 1851-2082 can be used as the ion-selective membrane 7. When the ion-selective electrode 1 is an anion-selective electrode for, for example, chloride ion, carbonate ion, thiocyanate ion, nitrate ion, hydroxide ion, phosphate ion, sulfate ion, or iodide ion (that is, an anion-selective electrode that selectively reacts with an anion such as chloride ion, carbonate ion, thiocyanate ion, nitrate ion, hydroxide ion, phosphate ion, sulfate ion, or iodide ion to generate an output corresponding to a concentration (activity) of the anion), a membrane as described in Pure Appl. Chem., 2002, Vol. 74, No. 6, pp. 923-994, and a silver halide such as silver chloride or silver bromide or an ion-exchange membrane (JP H10-318973 A, JP H11-132991 A, and JP 2003-207476 A) can be used as the ion-selective membrane 7. A membrane as described above can be prepared by a user and used, or a commercially available membrane can be used. That is, the ion-selective membrane 7 may be prepared by a user or may be a commercially available membrane as long as the ion-selective membrane 7 is a membrane as described above. Examples of the commercially available membrane include Neoceptor (registered trademark) CSE (registered trademark) (manufactured by ASTOM CORPORATION) and Neoceptor ASE (manufactured by ASTOM CORPORATION).

Selection of Ion-Selective Electrode

In the present invention, it is necessary to select two or more kinds of ion-selective electrodes satisfying formulas (1) and (2) as an ion-selective electrode constituting the ion concentration measurement device. Whether formulas (1) and (2) are satisfied is determined by a selectivity coefficient K_S(B1, Bj) of each of the ion-selective electrodes.

The ion selectivity coefficient K_S(B1, Bj) can be generally calculated by a single solution method or a mixed solution method. In the mixed solution method, a potential difference for a mixed solution containing an ion B1 having a known concentration and an ion Bj for which a selectivity coefficient is to be determined is measured, and the ion selectivity coefficient K_S(B1, Bj) is calculated.

An example of such calculation is described with a specific numerical value when a mixed solution containing the ion B1 and a second ion B2 is measured using two kinds of ion-selective electrodes (a first ion-selective electrode S1 and a second ion-selective electrode S2) as the ion-selective electrode S. It is assumed that ion selectivity coefficients K_S(B1, B2) for the ion B2 when the ion B1 is used as a reference in the ion-selective electrodes S1 and S2 are K_S1(B1, B2)=0.5 and K_S2 (B1, B2)=0.1, respectively. Here, the ion selectivity coefficients K_S(B1, B1) for the ion B1 itself when the ion B1 is used as a reference for the ion-selective electrodes S1 and S2 can be regarded as K_S1(B1, B1)=1 and K_S2 (B1, B1)=1, respectively. The matrix A and the determinant |A| can be expressed by the following formulas using these selectivity coefficients, respectively. In this case, the determinant |A|=−0.4 can be obtained by calculation.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}24\right]& \\ \mathrm{Matrix}A=\left(\begin{array}{cc}{K}_{S1}\left(B1,B1\right)& K_{S1}\left(B1,B2\right)\\ K_{S2}\left(B1,B1\right)& K_{S2}\left(B1,B2\right)\end{array}\right)& \end{array}$ $\begin{array}{c}\mathrm{Determinant}❘A❘=K_{S1}\left(B1,B1\right)K_{S2}\left(B1,B2\right)-K_{S1}\left(B1,B2\right)K_{S2}\left(B1,B1\right)\\ =K_{S2}\left(B1,B2\right)-K_{S1}\left(B1,B2\right)\\ =0.1-0.5=-0.4\end{array}$

Reference electrode

The reference electrode 2 is not particularly limited as long as it is a usually used one. Note that, in a typical aspect, as illustrated in FIG. 1, the reference electrode 2 includes the internal electrode 5, the internal liquid 6, and a liquid junction 8. Examples of the reference electrode 2 that can be used in the present invention include a silver-silver chloride reference electrode using a 3 M aqueous potassium chloride solution as the internal liquid 6, using a silver-silver chloride electrode as the internal electrode 5, and having the liquid junction 8 between a sample solution and the internal liquid 6. Note that the reference electrode 2 may be another reference electrode, for example, a saturated calomel electrode.

Ion

Any ion can be used as long as it can prepare a solution of the ion and can be measured by a device. The ion may be an anion or a cation, and examples of the anion include a chloride ion, a bicarbonate ion, an anion of an organic acid, a nitrate ion, and a sulfate ion, and examples of the cation include a potassium ion, a silver ion, and a copper ion. More preferable examples thereof include a plurality of ions (a chloride ion, a bicarbonate ion, and an anion of an organic acid such as lactic acid, pyruvic acid, propionic acid, fatty acid, or amino acid, for example) contained in blood from a viewpoint that the characteristics of the present device can be most utilized.

Here, the “plurality of ions” used in the present invention may be two or more kinds of anions or two or more kinds of cations. When the “plurality of ions” are anions, the “plurality of ions” may be two or more selected from the anions exemplified above, and for example, may contain a chloride ion and a bicarbonate ion, preferably a chloride ion, a bicarbonate ion, and an anion of an organic acid. In one preferable and exemplary aspect of the present invention, the “plurality of ions” are a chloride ion, a bicarbonate ion, and an acetate ion. Note that the “plurality of ions” used in the present invention are not limited to those exemplified above, and may be appropriately selected according to a measurement target.

Here, in the present invention, when evaluation using the above formulas (1) and (2) is performed, one ion which is the ion B1 (that is, an ion satisfying j=1 among ions Bj) is selected from n kinds of ions constituting the “plurality of ions”, and the other n-1 kinds of ions are defined as ions B2, B3, . . . , and Bn (that is, ions satisfying j≥2 among ions Bj).

Among these ions, the ion B1 is usually an ion whose concentration can be directly measured using one type of ion-selective electrode having an extremely small selectivity coefficient for another ion singly. Examples of an ion that can be the ion B1 include a chloride ion and a sodium ion.

On the other hand, the ions other than the ion B1 (that is, ions B2, B3, . . . and Bn) may be ions whose concentration can be directly measured by one single type of ion-selective electrode, or may be ions whose concentrations are difficult to directly measure by one single type of ion-selective electrode. Examples of an ion that can be an ion other than the ion B1 include a bicarbonate ion and an anion of an organic acid, such as an acetate ion.

According to the present invention, it is possible to measure a concentration of even an ion for which it is difficult to obtain an electrode having a sufficiently small ion selectivity coefficient for another ion, such as a bicarbonate ion. Based on this fact, in one preferable and exemplary aspect of the present invention, the ion B1 is a chloride ion, and ions other than the ion B1 include a bicarbonate ion.

[Ion Concentration Measurement Method]

An ion concentration measurement method of the present invention includes a step of measuring an ion concentration in a sample solution using a plurality of ion-selective electrodes having mutually different selectivity coefficient ratios for a plurality of ions and satisfying the above formulas (1) and (2).

Here, when the number of kinds of the plurality of ions is referred to as n, the two or more kinds of ion-selective electrodes are constituted by n kinds of ion-selective electrodes, and n is an integer of 2 or more.

In addition, in the above formula (1),

• • Bj represents a j-th ion among n kinds of ions constituting the plurality of ions,
  • Si is a symbol representing an i-th ion-selective electrode constituting the n kinds of ion-selective electrodes,
  • i and j are each independently an integer of 1 to n, and
  • K_si (B1, Bj) represents an ion selectivity coefficient for the ion Bj when an ion B1 is used as a reference in the ion-selective electrode Si.

In the present invention, it is preferable to calculate an ion concentration by the above formula (3).

The ion concentration measurement method of the present invention is not particularly limited regarding, for example, a tool or a device to be used as long as it is an ion concentration measurement method using a plurality of ion-selective electrodes having mutually different selectivity coefficient ratios for a plurality of ions and satisfying the above formulas (1) and (2), but for example, it is convenient to use an ion concentration measurement device as described above.

A concentration of an ion (the ion B1) to be measured is not particularly limited, but is preferably 0 mM to 1 M. The concentration is more preferably 0 mM to 300 mM.

When measurement is performed for a long time, the standard potential difference E_0,S may fluctuate depending on a change in measurement conditions such as temperature, and therefore it is preferable to correct the standard potential difference E_0,S using a standard solution.

First, the sample solution and the standard solution will be described.

Sample Solution

<Sample>

A sample is not particularly limited, and any sample containing an ion when the sample is made into a solution can be measured. In particular, the effect of the present invention is high in a sample containing a plurality of kinds of ions, such as a biological sample (blood, serum, plasma, for example). A sample may be measured as it is depending on a state or a concentration of the sample and a coexisting substance, but preferably, the sample is diluted with a buffer, water, an organic solvent, for example, in order to suppress fluctuation of pH and to eliminate a liquid junction potential, and measured as a sample solution. A sample can be used after being pretreated with, for example, a column or a filter depending on a coexisting substance.

<Buffer>

It is not necessary to use a buffer as long as an ion concentration can be stably measured, but it is preferable to use the buffer because fluctuation of pH is suppressed and a liquid junction potential is eliminated.

As the buffer, a generally known buffer can be used without limitation as long as the above object can be achieved. Specific examples thereof include a type of buffer called Good's Buffer (a HEPES buffer, a MES buffer, and an ADA buffer) and a Tris buffer. For example, in a case of measurement of a bicarbonate ion, a HEPES buffer is preferable from a viewpoint that a buffer that makes a response inhibition of an ion-selective electrode to an ion small is preferable. The buffer has pH of about 6.0 to 8.0 in a case of Good's Buffer. A concentration of the buffer is not particularly limited, but is preferably 1 mM to 1 M. A buffer having a concentration of 1 mM to 500 mM is more preferable. A buffer having a concentration of 5 mM to 300 mM is still more preferable.

Standard Solution

The standard solution is an ion solution having a known concentration, used to correct the standard potential difference E_0,S. An ion used for the standard solution is not particularly limited as long as a potential difference (E_std) can be measured. Two or more kinds of ions can be mixed and used for the standard solution. Nevertheless, it is usually convenient to select one kind from measurement ions and use it for the standard solution, which is preferable.

A solvent used for the standard solution is not limited as long as a potential difference can be measured, and examples thereof include water and an organic solvent. Two or more kinds of solvents may be mixed and used. Water is preferably used as the solvent from a viewpoint of stability of pH. A buffer is more preferably used in order to suppress fluctuation of pH and to eliminate a liquid junction potential. As the buffer, those that can be used for the sample solution as described above can be used.

An ion concentration (C_std) of the standard solution is not particularly limited, and only needs to be appropriately determined according to a measurement purpose. Usually, the ion concentration (C_std) is 0.1 to 10 mM. More preferably, the ion concentration is 0.5 to 3 mM.

Measurement of Potential Difference and Calculation of Ion Concentration

An ion concentration measurement method will be described, but it is an example and is not particularly limited.

In general, a potential difference E_S is measured, values of E_0,S and Slope are determined, and a calculated value obtained with the electrode S is determined from these values. Then, an ion concentration is calculated from the calculated value obtained with the electrode S.

Calculation of an ion concentration in a case of using the standard solution will also be described, but it is one of means for performing measurement with high precision, and does not limit the present invention.

<Measurement of Potential Difference E_S of Sample Solution>

The potential difference E_S is measured by immersing an ion-selective electrode and a reference electrode in a sample solution and measuring a potential difference with a potentiometer. Measurement conditions such as a measurement temperature and a measurement time are not particularly limited as long as the potential difference E_S can be measured. The measurement temperature is preferably 20° C. to 45° C., and the measurement time is preferably 2 seconds to 15 minutes from a viewpoint that a stable potential difference can be measured.

<Creation of Calibration Curve (Calculation of Values of E_0,S and Slope)>

By measuring a potential difference E_S of a known solution containing a certain ion B1 at an ion concentration C(B1), a calibration curve is created from a Nernst equation, and values of E_0,S, and Slope can be determined.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}25\right]& \\ E_S=E_{0,S}+\mathrm{Slope}\times {\log }_{10}C\left(B1\right)& \end{array}$

In the above formula,

• • Slope is 2.303RT/aF, and
  • R, T, a, and F are the same as R, I, a, and F defined by the above formula (5), respectively.

As an ion used for the calibration curve, any one type of ion among ions to be measured is usually used. Here, the ion used for the calibration curve is described as B1.

A solution having a known concentration for creating the calibration curve preferably has a composition close to that of a sample solution to be measured from a viewpoint of increasing precision of the calibration curve. Therefore, a solvent of the sample solution and a solvent of the solution used for the calibration curve are preferably the same. An ion concentration C(B1) only needs to be appropriately determined according to the sample solution to be measured. The calibration curve can be created by measuring ion concentrations at two different points, but the calibration curve is created by measuring the ion concentration C(B1) at three or more points from a viewpoint of increasing the precision.

<Calculation of Calculated Value Obtained with Electrode S and Ion Concentration>

Using a potential difference E_S between an ion-selective electrode and a reference electrode, obtained by immersing them in a sample solution, and the calibration curve as described above, a calculated value obtained with the electrode S can be calculated by the following formula using the determined values of E_0,S and Slope.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}26\right]& \\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S={10}^{\left(\frac{E_S-E_{0,S}}{\mathrm{Slope}}\right)}& \end{array}$

For n kinds of ion-selective electrodes S1 to Sn used as the electrode S, the “calculated values obtained with the electrode S” are determined and defined as “a calculated value obtained with the electrode S1”, “a calculated value obtained with the electrode S2”, . . . , and “a calculated value obtained with the electrode Sn”, respectively. By using these calculated values and an inverse matrix A⁻¹ of a matrix A constituted by selectivity coefficients of the ion-selective electrodes used, ion concentrations (C(B1), C(B2), . . . , and C(Bn)) of n kinds of ions (ions: B1, B2, . . . , and Bn) can be determined.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}27\right]& \\ \left(\begin{array}{c}C\left(B1\right)\\ C\left(B2\right)\\ ⋮\\ C\left(Bn\right)\end{array}\right)=A^{-1}·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S2\\ ⋮\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}Sn\end{array}\right)& \left(3\right)\end{array}$

<Calculation of Ion Concentration in Case of Using Standard Solution>

When the standard solution is used, the ion-selective electrode and the reference electrode are immersed in the standard solution, the potential difference E_std is determined, and the standard potential difference E_0,S is corrected. A tool or a device, for example, to be used is not particularly limited as long as the potential difference E_std can be measured, but it is convenient to use a similar device to that for measuring the sample solution.

In measurement of the standard solution, a measurement temperature is preferably 20° C. to 45° C., and a measurement time is preferably 2 seconds to 15 minutes from a viewpoint that a stable potential difference can be measured.

By once measuring the potential difference E_std of the standard solution, correction can be performed. However, in order to determine the calculated value obtained with the electrode S with higher precision, it is preferable to repeatedly measure the potential difference E_std between measurements of the sample solution. An interval at which the potential difference E_std of the standard solution is repeatedly measured only needs to be appropriately determined. When the measurement is repeatedly performed, a potential difference E_std at any time point may be adopted, and correction may be performed. Potential differences E_std at several points may be averaged and used for correction. It is preferable to adopt a potential difference E_std measured 2 seconds to 1 hour before measuring the potential difference E_S of the target sample solution from a viewpoint of precision.

Using the potential difference E_std obtained when the electrode S is immersed in the standard solution and the ion concentration (C_std) of the standard solution, a calculated value obtained with the electrode S can be calculated by formula (20).

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}28\right]& \\ \begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S={10}^{\left(\frac{E_S-E_{0,S}}{\mathrm{Slope}}\right)}\\ ={10}^{\left(\frac{E_S-E_{\mathrm{Std}}}{\mathrm{Slope}}+{\log }_{10}{C}_{\mathrm{Std}}\right)}\end{array}& \left(20\right)\end{array}$

For n kinds of ion-selective electrodes S1 to Sn used as the electrode S, the “calculated values obtained with the electrode S” are determined and defined as “a calculated value obtained with the electrode S1”, “a calculated value obtained with the electrode $2”, . . . , and “a calculated value obtained with the electrode Sn”, respectively. By using these calculated values and an inverse matrix A⁻¹ of a matrix A constituted by selectivity coefficients of the ion-selective electrodes used, ion concentrations (C(B1), C(B2), . . . , and C(Bn)) of n kinds of ions (ions: B1, B2, . . . , and Bn) can be determined.

EXAMPLES

Hereinafter, the present invention will be described in more detail with reference to Examples, but the present invention is not limited in any way by these Examples.

[Configuration of Ion Concentration Measurement Device]

An ion concentration measurement device used in the following Examples and Comparative Example includes an ion-selective electrode, a reference electrode, a potentiometer, a tank to contain a solution, and an arithmetic device.

The ion-selective electrode is connected to the potentiometer via an electric wire, and the potentiometer is connected to the reference electrode via an electric wire. In addition, the potentiometer is connected to the arithmetic device. In the ion concentration measurement, a solution to be measured (for example, a sample solution, a standard solution, or a calibration curve solution described later) is introduced into the “tank to contain a solution”, and the ion concentration measurement is performed in a state where a portion having an ion-selective membrane in the ion-selective electrode and a portion having a liquid junction in the reference electrode are immersed in the “solution to be measured”.

In the following Examples and Comparative Example, as the reference electrode, the potentiometer, the tank to contain a solution, and the arithmetic device, those described below were used unless otherwise specified. Note that the ion-selective electrodes used in the following Examples and Comparative Example will be described later in the following Examples and Comparative Example.

(Reference Electrode)

As the reference electrode, a commercially available silver-silver chloride reference electrode (218087) manufactured by A & T Co., Ltd. was used.

(Potentiometer, Arithmetic Device, and Tank to Contain a Solution)

As the potentiometer, the arithmetic device, and the tank to contain a solution, those attached to an electrolyte analyzer ELA08 manufactured by A & T Co., Ltd. were used.

[Preparation of Ion-Selective Electrode]

As the ion-selective electrode, an electrode obtained by modifying a commercially available chloride ion electrode (manufactured by A & T Co., Ltd.) such that an ion-selective membrane was replaceable was used.

The ion-selective membrane was prepared by changing a blending ratio of an amine and a halide and a reaction time as presented in Table 1 on the basis of the method described in Example 1 of JP 2018-004633 A. At this time, trimethylamine was used as a tertiary amine. The reaction time was appropriately adjusted between 12 hours and 24 hours, to give seven kinds of ion-selective membranes having mutually different ion selectivities.

Ion-selective electrodes obtained using these seven kinds of ion-selective membranes (represented by electrode symbols EL1 to EL7, respectively) were used for ion concentration measurement in the following Examples and Comparative Example. Here, in the following description, the ion-selective electrodes of the electrode symbols EL1 to EL7 may be referred to as electrodes EL1 to EL7, respectively. Note that the following electrode symbols EL1 to EL7 correspond to unique identifiers separately determined from the symbols S1 to Sn of the ion-selective electrode for the matrix A.

	TABLE 1
	Electrode	Ratio of	Reaction time
	symbol	amine:halide	(hour)
	EL1	1:9	12
	EL2	1:1	24
	EL3	2:1	15
	EL4	1:2	12
	EL5	1:1	20
	EL6	2:1	24
	EL7	1:2	24

Example 1

1-1 Configuration of Ion Concentration Measurement Device

In the present Example, as an ion concentration measurement device, a device having the configuration described in the above “Configuration of ion concentration measurement device” was used.

(Ion-Selective Electrode)

As ion-selective electrodes, the ion-selective electrode of the electrode symbol EL1 (“electrode EL1”), the ion-selective electrode of the electrode symbol EL2 (“electrode EL2”), and the ion-selective electrode of the electrode symbol EL3 (“electrode EL3”) obtained in the above “Preparation of ion-selective electrode” were used.

(Reference Electrode, Potentiometer, Arithmetic Device, and Tank to Contain a Solution)

As a reference electrode, a potentiometer, an arithmetic device, and a tank to contain a solution, those described in the above “Configuration of ion concentration measurement device” were used.

In the device used in Examples, potential differences E_S of the plurality of ion-selective electrodes with respect to a solution introduced into the tank to contain a solution can be simultaneously measured in parallel. Therefore, in Examples, the potential differences of the plurality of ion-selective electrodes with respect to a solution were simultaneously measured, but this does not limit the present invention. Any method capable of obtaining a potential difference between a plurality of ion-selective electrodes with respect to the same solution can be used without particular limitation, and the potential differences obtained for the ion-selective electrodes may be measured simultaneously or in any order.

1-2 Sample Solution

Since anions in blood are Cl⁻>HCO₃⁻> an ion of an organic acid (lactic acid, pyruvic acid, propionic acid, fatty acid, amino acid, for example) (here, an acetate ion (AcO⁻) is used instead) in descending order, these three anions were used as ions to be measured. Since blood contains a contaminant ion in addition to these anions, a solution obtained by adding a phosphate ion, a bromine ion, and a thiocyanate ion was used as a sample solution. A salt containing an ion to be measured and a contaminant substance ion (contaminant ion) presented in Table 2 was dissolved in a HEPES buffer (100 mM, pH 8.0), to give a sample solution X.

	TABLE 2
	Sample solution/mM
	Measurement	NaHCO3	24
	target	NaCl	105
		NaOAc	6
	Contaminant	NaH2PO4	1
	substance	NaBr	1
		NaSCN	2

A solvent is a HEPES buffer (100 mM, pH 8.0) 1-3 Standard solution

In 1 L of HEPES buffer (100 mM, pH 8.0), 0.58 g of sodium chloride (NaCl) was dissolved, to give a 1 mM (=C_std) sodium chloride (NaCl)/HEPES buffer.

1-4 Calibration Curve Solution

Sodium chloride (NaCl) was dissolved in a HEPES buffer (100 mM, pH 8.0) to give 80, 100, and 120 mM sodium chloride (NaCl)/HEPES buffers, and these buffers were used as calibration curve solutions.

1-5 Creation of Calibration Curve (Calculation of Values of E_std and Slope)

A calibration curve was created as follows.

In a vial, 2 ml of a calibration curve solution was put, the vial was set in an electrolyte analyzer ELA08 manufactured by A & T Co., Ltd., and measurement was performed at room temperature. The measurement was performed by first measuring a potential difference E_std of the standard solution prepared in the above section 1-3 for one calibration curve solution using the ion-selective electrode (the electrode EL1, the electrode EL2, or the electrode EL3) and the reference electrode connected to each other via the potentiometer, and then measuring a potential difference E_S of the calibration curve solution. Such a series of operations was performed for each of the three kinds of calibration curve solutions. Here, when the standard solution is set in a predetermined place, the standard solution is automatically circulated, and the potential difference E_std of the standard solution is measured as needed.

Table 3 presents a potential difference E_S when each of the calibration curve solutions of 80, 100, and 120 mM (═C(Cl⁻)) prepared in the above section 1-4 was measured with each of the electrodes EL1, EL2, and EL3, and a potential difference E_std of the standard solution.

A value of Slope was determined as follows.

From formulas (18′) and (19), the following formula (X1) is obtained. The above values were substituted into this formula (X1) to calculate a potential difference (E_S−E_std) and (log₁₀C(Cl⁻)−log₁₀C_std), and a calibration curve was prepared to determine Slope. Table 3 presents Slope of each of the electrodes.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}29\right]& \\ E_S=E_{0,S}+\mathrm{Slope}\times {\log }_{10}C\left({\mathrm{Cl}}^{-}\right)& \left(18\text{'}\right)\end{array}$ $\begin{array}{cc}{E}_{0,S}=E_{Std}-\mathrm{Slope}\times {\log }_{10}{C}_{std}& \left(19\right)\end{array}$ $\begin{array}{cc}{E}_S+E_{\mathrm{Std}}=\mathrm{Slope}\times \left({\log }_{10}C\left({\mathrm{Cl}}^{-}\right)-{\log }_{10}{C}_{\mathrm{std}}\right)& \left(X1\right)\end{array}$

TABLE 3
Potential difference and Slope value obtained by measuring calibration
curve solution using electrode EL1, EL2, or EL3 as electrode S
	Electrode symbol
Calibration	EL1	EL2	EL3
curve solution	Es/mV	Estd/mV	Es/mV	Estd/mV	Es/mV	Estd/mV
Cl− 80 mM	52.31	146.27	51.85	144.55	50.37	144.33
Cl− 100 mM	46.77	145.51	46.42	143.84	44.83	143.57
Cl− 120 mM	42.73	145.38	42.52	143.80	40.81	143.46
Slope/(mV/dec)	−49.37	−48.71	−49.37

1-6 Calculation of Selectivity Coefficient and Creation of matrix A for ion-selective electrode

(Calculation of Selectivity Coefficient)

For each of the electrodes (electrodes EL1 to EL3) of the electrode symbols EL1 to EL3, a selectivity coefficient for an ion to be measured was determined. Here, an example in which a chloride ion (Cl⁻) is adopted as the ion to be measured is described.

The selectivity coefficient was calculated as follows.

(1) Calculation of Bicarbonate Ion (HCO₃⁻) Selectivity Coefficient

First, a bicarbonate ion (HCO₃⁻) selectivity coefficient calculating solution containing an ion (ion to be measured) (chloride ion (Cl⁻)) as a reference for creating a calibration curve and an ion (bicarbonate ion (HCO₃⁻)) whose selectivity coefficient was to be calculated was prepared.

Sodium bicarbonate (NaHCO₃) and sodium chloride (NaCl) were dissolved in a HEPES buffer (100 mM, pH 8.0), to give a 10 mM sodium bicarbonate (NaHCO₃) and 100 mM sodium chloride (NaCl)/HEPES buffer, and the 10 mM sodium bicarbonate (NaHCO₃) and 100 mM sodium chloride (NaCl)/HEPES buffer was used as the bicarbonate ion (HCO₃⁻) selectivity coefficient calculating solution.

In a vial, 2 ml of the bicarbonate ion (HCO₃⁻) selectivity coefficient calculating solution was put, the vial was set in an electrolyte analyzer ELA08 manufactured by A & T Co., Ltd., and measurement was performed at room temperature. The measurement was performed by first measuring a potential difference E_std of the standard solution for one selectivity coefficient calculating solution using the ion-selective electrode (the electrode EL1, the electrode EL2, or the electrode EL3) and the reference electrode connected to each other via the potentiometer, and then measuring a potential difference E_S of the selectivity coefficient calculating solution.

Here, into the following formula (19),

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}30\right]& \\ E_{0,S}=E_{Std}-\mathrm{Slope}\times {\log }_{10}{C}_{std}& \left(19\right)\end{array}$

a potential difference E_std at the time of measuring the standard solution, a value of Slope described in the above Table 3, obtained in the above “1-5 Creation of calibration curve”, and an ion concentration C_std of the standard solution are substituted to determine a standard potential difference E_0,S.

Furthermore, into the following formula (8B),

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{Formula}31\right]& \\ K_S\left(B1,B2\right)=\frac{{10}^{\left(\frac{E_S-E_{0,S}}{\mathrm{Slope}}\right)}-C\left(B1\right)}{C\left(B2\right)}& \left(8B\right)\end{array}$ $\begin{array}{cc}=\frac{{10}^{\left(\frac{E_S-E_{\mathrm{Std}}}{\mathrm{Slope}}+{\log }_{10}{C}_{\mathrm{Std}}\right)}-C\left(B1\right)}{C\left(B2\right)}& \left(8B\text{'}\right)\end{array}$

the standard potential difference E_0,S, the potential difference E_S, the value of Slope, and an ion concentration (A concentration of a chloride ion (Cl⁻) is represented as C(B1), and a concentration of a bicarbonate ion (HCO₃⁻) is represented as C(B2).) in the bicarbonate ion (HCO₃⁻) selectivity coefficient calculating solution are substituted, whereby a selectivity coefficient (K_S(Cl⁻, HCO₃⁻)) for the bicarbonate ion (HCO₃⁻) when the chloride ion (Cl⁻) is used as an ion to be measured is obtained.

Table 4 presents the potential difference E_S, the potential difference E_std of the standard solution, and the selectivity coefficient (K_S(Cl⁻, HCO₃)) when the bicarbonate ion (HCO₃⁻) selectivity coefficient calculating solution is measured with each of the electrodes EL1, EL2, and EL3.

TABLE 4
Measured values of potential difference ES and potential difference
EStd of standard solution, and bicarbonate ion (HCO3−) selectivity
coefficient when electrode EL1, EL2, or EL3 is used as electrode S
	Electrode symbol
	EL1	EL2	EL3
	Es/mV	Estd/mV	Es/mV	Estd/mV	Es/mV	Estd/mV
HCO3−	47.07	146.02	45.93	144.50	43.35	143.88
selectivity
coefficient
calculating
solution
Ks (Cl−, HCO3−)	0.10	0.56	0.87

(2) Calculation of Acetate Ion (AcO⁻) Selectivity Coefficient

In a similar manner to the above (1), an acetate ion (AcO⁻) selectivity coefficient calculating solution was prepared.

Sodium acetate (NaOAc) and sodium chloride (NaCl) were dissolved in a HEPES buffer (100 mM, pH 8.0), to give a 10 mM sodium acetate (NaOAc) and 100 mM sodium chloride (NaCl)/HEPES buffer, and the 10 mM sodium acetate (NaOAc) and 100 mM sodium chloride (NaCl)/HEPES buffer was used as the acetate ion (AcO⁻) selectivity coefficient calculating solution.

In a vial, 2 ml of the acetate ion (AcO⁻) selectivity coefficient calculating solution was put, the vial was set in an electrolyte analyzer ELA08 manufactured by A & T Co., Ltd., and measurement was performed at room temperature. The measurement was performed by first measuring a potential difference E_std of the standard solution for one selectivity coefficient calculating solution using the ion-selective electrode (the electrode EL1, the electrode EL2, or the electrode EL3) and the reference electrode connected to each other via the potentiometer, and then measuring a potential difference E_S of the selectivity coefficient calculating solution.

Here, by substituting various parameters into the above formulas (19) and (8B) in a similar manner to the above “(1) Calculation of bicarbonate Ion (HCO₃⁻) selectivity coefficient” except that the above potential difference E_S is used as the potential difference E_S, and concentrations of a chloride ion (Cl⁻) and an acetate ion (AcO⁻) in the acetate ion (AcO⁻) selectivity coefficient calculating solution are used as C(B1) and C(B2), respectively, a selectivity coefficient (K_S(Cl⁻, AcO⁻)) for the acetate ion (AcO⁻) when the chloride ion (Cl⁻) is used as an ion to be measured is obtained.

Table 5 presents the potential difference E_S, the potential difference E_std of the standard solution, and the selectivity coefficient (K_S(Cl⁻, AcO⁻)) when the acetate ion (AcO⁻) selectivity coefficient calculating solution is measured with each of the electrodes EL1, EL2, and EL3.

TABLE 5
Measured values of potential difference ES and potential difference
EStd of standard solution, and acetate ion (AcO−) selectivity
coefficient when electrode EL1, EL2, or EL3 is used as electrode S
	Electrode symbol
	EL1	EL2	EL3
	Es/mV	Estd/mV	Es/mV	Estd/mV	Es/mV	Estd/mV
AcO−	46.66	145.55	44.78	143.75	44.30	143.72
selectivity
coefficient
calculating
solution
Ks (Cl−, AcO−)	0.07	0.76	0.32

(Creation of Matrix A)

A matrix A was created from the values of the selectivity coefficients obtained in the above “Calculation of selectivity coefficient” for the electrodes EL1, EL2, and EL3.

The following Table 6 collectively presents the selectivity coefficients obtained in the above “Calculation of selectivity coefficient”. Here, the selectivity coefficient (K_S(Cl⁺, Cl⁻)) for an ion to be measured (chloride ion (Cl⁻)) itself is set to 1.

TABLE 6
Selectivity coefficient of each ion-selective
electrode for ion to be measured
	Electrode symbol	EL1	EL2	EL3
	Selectivity	Ks (Cl−, Cl−)	1	1	1
	coefficient	Ks (Cl−, HCO3−)	0.10	0.56	0.87
		Ks (Cl−, AcO−)	0.07	0.76	0.32

A matrix A corresponding to the selectivity coefficients presented in Table 6 above can be expressed as follows. Here, for determining the following matrix A, the electrodes EL1, EL2, and EL3 were defined as a first ion-selective electrode (ion-selective electrode S1), a second ion-selective electrode (ion-selective electrode S2), and a third ion-selective electrode (ion-selective electrode S3), respectively. In the following matrix A, selectivity coefficients K_S(Cl⁻, Cl⁻), K_S(Cl⁻, HCO₃⁻), and K_S(Cl⁻, AcO⁻) for an i-th (i is 1 or more and 3 or less) ion-selective electrode among these three ion-selective electrodes are represented by K_si (Cl⁻, Cl⁻), K_si (Cl⁻, HCO₃), and K_si (Cl⁻, AcO⁻), respectively.

$\begin{array}{cc}\left[\mathrm{Mathematical}\mathrm{formula}32\right]& \\ \mathrm{Matrix}A=\left(\begin{array}{ccc}{K}_{S1}\left({\mathrm{Cl}}^{-},{\mathrm{Cl}}^{-}\right)& K_{S1}\left({\mathrm{Cl}}^{-},{\mathrm{HCO}}_3^{-}\right)& K_{S1}\left({\mathrm{Cl}}^{-},{\mathrm{AcO}}^{-}\right)\\ K_{S2}\left({\mathrm{Cl}}^{-},{\mathrm{Cl}}^{-}\right)& K_{S2}\left({\mathrm{Cl}}^{-},{\mathrm{HCO}}_3^{-}\right)& K_{S2}\left({\mathrm{Cl}}^{-},{\mathrm{AcO}}^{-}\right)\\ K_{S3}\left({\mathrm{Cl}}^{-},{\mathrm{Cl}}^{-}\right)& K_{S3}\left({\mathrm{Cl}}^{-},{\mathrm{HCO}}_3^{-}\right)& K_{S3}\left({\mathrm{Cl}}^{-},{\mathrm{AcO}}^{-}\right)\end{array}\right)=\left(\begin{array}{ccc}1& 0.1& 0.07\\ 1& 0.56& 0.76\\ 1& 0.87& 0.32\end{array}\right)& \end{array}$

This matrix A was input to a cell (A1: C3) of spreadsheet software Excel, and an absolute value ∥A∥ of the determinant was determined using a function of “=MDETERM (A1: C3)” and found to be 0.42.

Here, the calculation of the absolute value ∥A∥ in Excel can be specifically performed as follows. First, 1 is input to a cell A1, 1 is input to a cell A2, 1 is input to a cell A3, 0.10 is input to a cell B1, 0.56 is input to a cell B2, 0.87 is input to a cell B3, 0.07 is input to a cell C1, 0.76 is input to a cell C2, and 0.32 is input to a cell C3. Next, when “=MDETERM (A1: C3)” is input to any cell other than these cells, a calculated value of “=MDETERM (A1: C3)” can be obtained as a value of the determinant |A| (in this case, −0.42). Then, by applying an ABS function to the value of the determinant |A|, an absolute value of the determinant |A| can be obtained, and this value can be adopted as the absolute value ∥A∥.

Here, in the above Tables 4 to 6 and the above determinant A, the values of K_S(Cl⁻, HCO₃) and K_S(Cl⁻, AcO⁻) for each ion-selective electrode are shown in a form of being rounded off to the third decimal place for convenience of documentation. Note that, in calculation of an inverse matrix A⁻¹ and calculation of a concentration of each ion, which will be described later in the following “1-8 Calculation of ion concentration of sample solution”, values were used in a manner that their third decimal places and subsequent decimal places were taken into consideration, instead of adopting values obtained by rounding off these values to the third decimal place.

1-7 Measurement of Potential Difference E_S of Sample Solution

In a vial, 2 ml of a sample solution X was put, the vial was set in an electrolyte analyzer ELA08 manufactured by A & T Co., Ltd., and measurement was performed at room temperature. The measurement was performed by first measuring a potential difference E_std of the standard solution using the ion-selective electrode (the electrode EL1, the electrode EL2, or the electrode EL3) and the reference electrode connected to each other via the potentiometer, and then measuring a potential difference E_S of the sample solution X. Table 7 presents measured values and calculated values obtained from the measured values (calculated values obtained with the electrode S). Correction was performed with E_std in Table 7 using the values of Slope in Table 3, determined in the above “1-5 Creation of calibration curve (calculation of values of E_std and Slope) “.

TABLE 7
Measured values of potential difference Es and potential difference
ESta of standard solution, and calculated value obtained with electrode
S when electrode EL1, EL2, or EL3 is used as electrode S
Electrode			Calculated value obtained
symbol	Es/mV	Estd/mV	with electrode S
EL1	45.68	146.21	108.72
EL2	42.37	144.49	125.10
EL3	39.85	144.26	130.24

$\left[\mathrm{Mathematical}\mathrm{formula}33\right]$ $\begin{array}{cc}\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S={10}^{\left(\frac{E_s-E_{0,S}}{\mathrm{Slope}}\right)}\\ ={10}^{\left(\frac{E_s-E_{\mathrm{Std}}}{\mathrm{Slope}}+{\log }_{10}{C}_{\mathrm{Std}}\right)}\end{array}& \left(20\right)\end{array}$

1-8 Calculation of Ion Concentration of Sample Solution

An ion concentration of the sample solution (sample solution X) was calculated as follows.

First, an inverse matrix A⁻¹ of the matrix A obtained in the above “1-6 Calculation of selectivity coefficient and creation of matrix A for ion-selective electrode” is determined. This matrix A is obtained by using the electrodes EL1, EL2, and EL3 as the ion-selective electrode S1, the ion-selective electrode S2, and the ion-selective electrode S3, respectively. The inverse matrix A⁻¹ was determined by inputting the matrix A to a cell (A1: C3) of spreadsheet software Excel and using a function of “=MINVERSE (A1: C3) “.

Here, the calculation of the inverse matrix A⁻¹ in Excel can be performed by inputting “=MINVERSE (A1: C3)” to any cell away from the cells A1 to C3 input in the above “1-6 Calculation of selectivity coefficient and creation of matrix A for ion-selective electrode”, and a group of calculated values displayed in the cell input as “=MINVERSE (A1: C3)” and its neighboring cells can be adopted as the inverse matrix A⁻¹.

$\left[\mathrm{Mathematical}\mathrm{formula}34\right]$ $A^{-1}=\left(\begin{array}{ccc}1.16& -0.07& -0.09\\ -1.06& -0.6& 1.66\\ -0.74& 1.85& -1.1\end{array}\right)$

By applying the inverse matrix A⁻¹ to the “calculated value obtained with electrode S” described in Table 7, obtained in the above “1-7 Measurement of potential difference E_S of sample solution”, a concentration C(Cl⁻) of a chloride ion (Cl⁻), a concentration C(HCO₃⁻) of a bicarbonate ion (HCO₃⁻), and a concentration C(AcO⁻) of an acetate ion (AcO⁻) were determined. Here, the “calculated value obtained with electrode S” for the electrode EL1 was used as “calculated value obtained with electrode S1”, the “calculated value obtained with electrode S” for the electrode EL2 was used as “calculated value obtained with electrode S2”, and the “calculated value obtained with electrode S” for the electrode EL3 was used as “calculated value obtained with electrode S3”. As a result, C(Cl⁻)=105.7 mM, C(HCO₃⁻)=26.0 mM, and C(AcO⁻)=6.09 mM.

$\left[\mathrm{Mathematical}\mathrm{formula}35\right]$ $\left(\begin{array}{c}C\left(C{1}^{-}\right)\\ C\left({\mathrm{HCO}}_3^{-}\right)\\ C\left({\mathrm{AcO}}^{-}\right)\end{array}\right)=A^{-1}·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S2\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S3\end{array}\right)=\left(\begin{array}{ccc}1.16& -0.07& -0.09\\ -1.06& -0.6& 1.66\\ -0.74& 1.85& -1.1\end{array}\right)·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S2\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S3\end{array}\right)$

Here, the inverse matrix A⁻¹, the values of K_S(Cl⁻, HCO₃⁻) and K_S(Cl⁻, AcO⁻) for each ion-selective electrode determined in the above “1-6 Calculation of selectivity coefficient and creation of matrix A for ion-selective electrode”, and the value of” calculated value obtained with electrode S″ obtained in the above “1-7 Measurement of potential difference E_S of sample solution” are shown in a form of being rounded off to the third decimal place for convenience of documentation. Note that, in calculation of a concentration of each ion, values were used in a manner that their third decimal places and subsequent decimal places were taken into consideration, instead of adopting values obtained by rounding off these values to the third decimal place. Then, the processing of rounding off a numerical value was not performed in the middle of calculating the concentration of each ion, but was performed only for a finally obtained concentration value.

A concentration of each ion in the following Example 2 and Comparative Example 1 was calculated similarly.

Note that, due to an error caused by such processing of rounding off a numerical value, a calculation result for a concentration of each ion determined in the present Example and the following Example 2 and Comparative Example 1 may have a slight deviation from a concentration value determined by using, as values of “calculated value obtained with electrode S”, values rounded off to the third decimal place and using, as an inverse matrix A⁻¹, an inverse matrix including, as a value of each element, values rounded off to the third decimal place, and simply combining them.

The operations of the above “1-7 Measurement of potential difference E_S of sample solution” and the above “1-8 Calculation of ion concentration of sample solution” were repeated six times, and a relative standard deviation (percentage of a value obtained by dividing a standard deviation by an average value) (%) was determined for each of a chloride ion, a bicarbonate ion, and an acetate ion.

Table 18 presents the results.

Example 2

2-1 Configuration of Ion Concentration Measurement Device

As an ion concentration measurement device, a device having the configuration described in the above “Configuration of ion concentration measurement device” was used as in Example 1. Here, as an ion-selective electrode, a reference electrode, a potentiometer, a tank to contain a solution, and an arithmetic device, those described below were used.

(Ion-Selective Electrode)

As ion-selective electrodes, the ion-selective electrode of the electrode symbol EL1 (“electrode EL1”), the ion-selective electrode of the electrode symbol EL2 (“electrode EL2”), and the ion-selective electrode of the electrode symbol EL4 (“electrode EL4”) obtained in the above “Preparation of ion-selective electrode” were used.

(Reference Electrode, Potentiometer, Arithmetic Device, and Tank to Contain a Solution)

As a reference electrode, a potentiometer, an arithmetic device, and a tank to contain a solution, those described in the above “Configuration of ion concentration measurement device” were used as in Example 1.

2-2 Sample Solution

A sample solution same as the sample solution X used in Example 1 was used.

2-3 Standard Solution

A standard solution same as that in Example 1 was used.

2-4 Calibration Curve Solution

A calibration curve solution same as that in Example 1 was used.

2-5 Creation of Calibration Curve (Calculation of Values of E_std and Slope)

Creation of a calibration curve and calculation of a value of Slope were performed in the same manner as in Example 1 except that the electrode EL1, the electrode EL2, or the electrode EL4 was used as the ion-selective electrode. Table 8 presents a determined value of Slope.

TABLE 8
Potential difference and Slope value obtained by measuring calibration
curve solution using electrode EL1, EL2, or EL4 as electrode S
	Electrode symbol
Calibration	EL1	EL2	EL4
curve solution	Es/mV	Estd/mV	Es/mV	Estd/mV	Es/mV	Estd/mV
Cl− 80 mM	52.40	146.36	51.93	144.63	37.39	144.44
Cl− 100 mM	47.54	146.28	47.14	144.56	31.84	144.34
Cl− 120 mM	43.66	146.31	43.31	144.59	27.43	144.38
Slope/(mV/dec)	−49.37	−48.71	−56.25

2-6 Calculation of Selectivity Coefficient and Creation of Matrix a for Ion-Selective Electrode

(Calculation of Selectivity Coefficient)

For each of the electrodes (electrodes EL1, EL2, and EL4) of the electrode symbols EL1, EL2, and EL4, a selectivity coefficient for an ion to be measured was determined. The calculation of the selectivity coefficient was performed in the same manner as in Example 1. Tables 9 and 10 present the measured values and the determined selectivity coefficient values.

TABLE 9
Measured values of potential difference ES and potential difference EStd
of standard solution, and bicarbonate ion (HCO3−) selectivity coefficient
of electrode S when electrode EL1, EL2, or EL4 is used as electrode S
	Electrode symbol
	EL1	EL2	EL4
	Es/mV	Estd/mV	Es/mV	Estd/mV	Es/mV	Estd/mV
HCO3−	47.16	146.11	45.97	144.54	29.77	143.31
selectivity
coefficient
calculating
solution
Ks (Cl−, HCO3−)	0.10	0.56	0.44

TABLE 10
Measured values of potential difference ES and potential difference
EStd of standard solution, and acetate ion (AcO−) selectivity coefficient
of electrode S when electrode EL1, EL2, or EL4 is used as electrode S
	Electrode symbol
	EL1	EL2	EL4
	Es/mV	Estd/mV	Es/mV	Estd/mV	Es/mV	Estd/mV
AcO−	46.66	145.55	44.78	143.75	30.88	143.72
selectivity
coefficient
calculating
solution
Ks (Cl−, AcO−)	0.07	0.76	0.14

(Creation of Matrix A)

A matrix A was created from the values of the selectivity coefficients obtained in the above “Calculation of selectivity coefficient” for the electrodes EL1, EL2, and EL4.

The following Table 11 collectively presents the selectivity coefficients obtained in the above “Calculation of selectivity coefficient”. Here, the selectivity coefficient (K_S(Cl⁻, Cl⁻)) for an ion to be measured (chloride ion (Cl⁻)) itself is set to 1.

TABLE 11
Selectivity coefficient of each ion-selective
electrode for ion to be measured
	Electrode symbol	EL1	EL2	EL4
	Selectivity	Ks (Cl−, Cl−)	1	1	1
	coefficient	Ks (Cl−, HCO3−)	0.10	0.56	0.44
		Ks (Cl−, AcO−)	0.07	0.76	0.14

A matrix A corresponding to the selectivity coefficients presented in Table 11 above can be expressed as follows. Here, for determining the following matrix A, the electrodes EL1, EL2, and EL4 were defined as a first ion-selective electrode (ion-selective electrode S1), a second ion-selective electrode (ion-selective electrode S2), and a third ion-selective electrode (ion-selective electrode S3), respectively. In the following matrix A, selectivity coefficients K_S(Cl⁻, Cl⁻), K_S(Cl⁻, HCO₃⁻), and K_S(Cl⁻, AcO⁻) for an i-th (i is 1 or more and 3 or less) ion-selective electrode among these three ion-selective electrodes are represented by K_si (Cl⁻, Cl⁻), K_si (Cl⁻, HCO₃⁻), and K_si (Cl⁻, AcO⁻), respectively.

$\left[\mathrm{Mathematical}\mathrm{formula}36\right]$ $\mathrm{Matrix}A=\left(\begin{array}{ccc}{K}_{S1}\left(C{1}^{-},{C1}^{-}\right)& K_{S1}\left({C1}^{-},{\mathrm{HCO}}_3^{-}\right)& K_{S1}\left(C{1}^{-},{\mathrm{AcO}}^{-}\right)\\ K_{S2}\left(C{1}^{-},C{1}^{-}\right)& K_{S2}\left(C{1}^{-},{\mathrm{HCO}}_3^{-}\right)& K_{S2}\left(C{1}^{-},{\mathrm{AcO}}^{-}\right)\\ K_{S3}\left(C{1}^{-},C{1}^{-}\right)& K_{S3}\left({C1}^{-},{\mathrm{HCO}}_3^{-}\right)& K_{S3}\left(C{1}^{-},{\mathrm{AcO}}^{-}\right)\end{array}\right)=\left(\begin{array}{ccc}1& 0.1& 0.07\\ 1& 0.56& 0.76\\ 1& 0.44& 0.14\end{array}\right)$

This matrix A was input to a cell (A1: C3) of spreadsheet software Excel, and an absolute value ∥A∥ of the determinant was determined using a function of “=MDETERM (A1: C3)” and found to be 0.20.

2-7 Measurement of Potential Difference E_S of Sample Solution

Measurement of a potential difference E_S of a sample solution was performed in the same manner as in Example 1 except that the electrode EL1, the electrode EL2, or the electrode EL4 was used as the ion-selective electrode. Table 12 presents measured values and calculated values obtained from the measured values (calculated values obtained with the electrode S). Correction was performed with E_std in Table 12 using the values of Slope in Table 8, determined in the same manner as in Example 1.

TABLE 12
Measured values of potential difference ES and potential difference
ESta of standard solution, and calculated value obtained with electrode
S when electrode EL1, EL2, or EL4 is used as electrode S
Electrode			Calculated value obtained
symbol	Es/mV	Estd/mV	with electrode S
EL1	44.54	145.14	109.02
EL2	41.35	143.49	125.01
EL4	26.57	143.22	118.51

2-8 Calculation of Ion Concentration of Sample Solution

An ion concentration of the sample solution was calculated in the same manner as in Example 1.

First, an inverse matrix A⁻¹ of the matrix A obtained in the above “2-6 Calculation of selectivity coefficient and creation of matrix A for ion-selective electrode” is determined in the same manner as in Example 1. This matrix A is obtained by using the electrodes EL1, EL2, and EL4 as the ion-selective electrode S1, the ion-selective electrode S2, and the ion-selective electrode S3, respectively. The inverse matrix A⁻¹ was determined by inputting the matrix A to a cell (A1: C3) of spreadsheet software Excel and using a function of “=MINVERSE (A1: C3)”.

$\left[\mathrm{Mathematical}\mathrm{formula}37\right]$ $A^{-1}=\left(\begin{array}{ccc}1.27& -0.09& -0.18\\ -3.09& -0.32& 3.41\\ 0.61& 1.66& -2.27\end{array}\right)$

By applying the inverse matrix A⁻¹ to the “calculated value obtained with electrode S” described in Table 12, obtained in the above “2-7 Measurement of potential difference E_S of sample solution”, a concentration C(Cl⁻) of a chloride ion (Cl⁻), a concentration C(HCO₃⁻) of a bicarbonate ion (HCO₃⁻), and a concentration C(AcO⁻) of an acetate ion (AcO⁻) were determined. Here, the “calculated value obtained with electrode S” for the electrode EL1 was used as “calculated value obtained with electrode S1”, the “calculated value obtained with electrode S” for the electrode EL2 was used as “calculated value obtained with electrode S2”, and the “calculated value obtained with electrode S” for the electrode EL4 was used as “calculated value obtained with electrode S3”. As a result, C(Cl⁻)=105.7 mM, C(HCO₃⁻)=27.6 mM, and C(AcO⁻)=5.21 mM.

$\left[\mathrm{Mathematical}\mathrm{formula}38\right]$ $\left(\begin{array}{c}{C\left(C1\right)}^{-})\\ C\left({\mathrm{HCO}}_3^{-}\right)\\ C\left({\mathrm{AcO}}^{-}\right)\end{array}\right)=A^{-1}·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S2\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S3\end{array}\right)=\left(\begin{array}{ccc}1.27& -0.09& -0.18\\ -3.09& -0.32& 3.41\\ -0.61& 1.66& -2.27\end{array}\right)·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S2\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S3\end{array}\right)$

The operations of the above “2-7 Measurement of potential difference E_S of sample solution” and the above “2-8 Calculation of ion concentration of sample solution” were repeated six times, and a relative standard deviation (%) was determined in the same manner as in Example 1. Table 18 presents the results.

Comparative Example 1

C1-1 Configuration of Ion Concentration Measurement Device

As an ion concentration measurement device, a device having the configuration described in the above “Configuration of ion concentration measurement device” was used as in Example 1. Here, as an ion-selective electrode, a reference electrode, a potentiometer, a tank to contain a solution, and an arithmetic device, those described below were used.

(Ion-Selective Electrode)

As ion-selective electrodes, the ion-selective electrode of the electrode symbol EL5 (“electrode EL5”), the ion-selective electrode of the electrode symbol EL6 (“electrode EL6”), and the ion-selective electrode of the electrode symbol EL7 (“electrode EL7”) obtained in the above “Preparation of ion-selective electrode” were used.

(Reference Electrode, Potentiometer, Arithmetic Device, and Tank to Contain a Solution)

As a reference electrode, a potentiometer, an arithmetic device, and a tank to contain a solution, those described in the above “Configuration of ion concentration measurement device” were used as in Example 1.

C1-2 Sample Solution

A sample solution same as the sample solution X used in Example 1 was used.

C1-3 Standard Solution

A standard solution same as that in Example 1 was used.

C1-4 Calibration Curve Solution

A calibration curve solution same as that in Example 1 was used.

C1-5 Creation of Calibration Curve (Calculation of Values of E_std and Slope)

Creation of a calibration curve and calculation of a value of Slope were performed in the same manner as in Example 1 except that the electrode EL5, the electrode EL6, or the electrode EL7 was used as the ion-selective electrode. Table 13 presents a determined value of Slope.

TABLE 13
Potential difference and Slope value obtained by measuring calibration
curve solution using electrode EL5, EL6, or EL7 as electrode S
	Electrode symbol
Calibration	EL5	EL6	EL7
curve solution	Es/mV	Estd/mV	Es/mV	Estd/mV	Es/mV	Estd/mV
Cl− 80 mM	71.04	145.47	71.27	143.80	70.67	143.54
Cl− 100 mM	66.96	145.18	67.30	143.52	66.66	143.24
Cl− 120 mM	63.86	145.18	64.27	143.51	63.63	143.24
Slope/(mV/dec)	−39.11	−38.11	−38.29

C1-6 Calculation of Selectivity Coefficient and Creation of Matrix a for Ion-Selective Electrode

(Calculation of Selectivity Coefficient)

For each of the electrodes (electrodes EL5, EL6, and EL7) of the electrode symbols EL5, EL6, and EL7, a selectivity coefficient for an ion to be measured was determined. The calculation of the selectivity coefficient was performed in the same manner as in Example 1. Tables 14 and 15 present the measured values and the determined selectivity coefficient values.

TABLE 14
Measured values of potential difference ES and potential difference
EStd of standard solution, and bicarbonate ion (HCO3−) selectivity
coefficient when electrode EL5, EL6, or EL7 is used as electrode S
	Electrode symbol
	EL5	EL6	EL7
	Es/mV	Estd/mV	Es/mV	Estd/mV	Es/mV	Estd/mV
HCO3−	66.05	145.03	65.99	143.48	65.80	143.11
selectivity
coefficient
Ks (Cl−, HCO3−)	0.46	0.80	0.45

TABLE 15
Measured values of potential difference ES and potential difference
EStd of standard solution, and acetate ion (AcO−) selectivity coefficient
of electrode S when electrode EL5, EL6, or EL7 is used as electrode S
	Electrode symbol
	EL5	EL6	EL7
	Es/mV	Estd/mV	Es/mV	Estd/mV	Es/mV	Estd/mV
AcO−	66.16	145.05	65.97	143.33	66.44	143.58
selectivity
coefficient
Ks (Cl−, AcO−)	0.40	0.71	0.34

(Creation of Matrix A)

A matrix A was created from the values of the selectivity coefficients obtained in the above “Calculation of selectivity coefficient” for the electrodes EL5, EL6, and EL7.

The following Table 16 collectively presents the selectivity coefficients obtained in the above “Calculation of selectivity coefficient”. Here, the selectivity coefficient (K_S(Cl⁻, Cl⁻)) for an ion to be measured (chloride ion (Cl⁻)) itself is set to 1.

TABLE 16
Selectivity coefficient of each ion-selective
electrode for ion to be measured
	Electrode symbol	EL5	EL6	EL 7
	Selectivity	Ks (Cl−, Cl−)	1	1	1
	coefficient	Ks (Cl−, HCO3−)	0.46	0.80	0.45
		Ks (Cl−, AcO−)	0.40	0.71	0.34

A matrix A corresponding to the selectivity coefficients presented in Table 16 above can be expressed as follows.

Here, for determining the following matrix A, the electrodes EL5, EL6, and EL7 were defined as a first ion-selective electrode (ion-selective electrode S1), a second ion-selective electrode (ion-selective electrode S2), and a third ion-selective electrode (ion-selective electrode S3), respectively. In the following matrix A, selectivity coefficients K_S(Cl⁻, Cl⁻), K_S(Cl⁻, HCO₃⁻), and K_S(Cl⁻, AcO⁻) for an i-th (i is 1 or more and 3 or less) ion-selective electrode among these three ion-selective electrodes are represented by K_si (Cl⁻, Cl⁻), K_si (Cl⁻, HCO₃⁻), and K_si (Cl⁻, AcO⁻), respectively.

$\left[\mathrm{Mathematical}\mathrm{formula}39\right]$ $\mathrm{Matrix}A=\left(\begin{array}{ccc}{K}_{S1}\left(C{1}^{-},{C1}^{-}\right)& K_{S1}\left(C{1}^{-},{\mathrm{HCO}}_3^{-}\right)& K_{S1}\left({C1}^{-},{\mathrm{AcO}}^{-}\right)\\ K_{S2}\left(C{1}^{-},{C1}^{-}\right)& K_{S2}\left({C1}^{-},{\mathrm{HCO}}_3^{-}\right)& K_{S2}\left(C{1}^{-},{\mathrm{AcO}}^{-}\right)\\ K_{S3}\left({C1}^{-},{C1}^{-}\right)& K_{S3}\left(C{1}^{-},{\mathrm{HCO}}_3^{-}\right)& K_{S3}\left(C{1}^{-},{\mathrm{AcO}}^{-}\right)\end{array}\right)=\left(\begin{array}{ccc}1& 0.46& 0.4\\ 1& 0.8& 0.71\\ 1& 0.45& 0.34\end{array}\right)$

This matrix A was input to a cell (A1: C3) of spreadsheet software Excel, and an absolute value ∥A∥ of the determinant was determined using a function of “=MDETERM (A1: C3)” and found to be 0.02.

C1-7 Measurement of Potential Difference E_S of Sample Solution

Measurement of a potential difference E_S of a sample solution was performed in the same manner as in Example 1 except that the electrode EL5, the electrode EL6, or the electrode EL7 was used as the ion-selective electrode. Table 17 presents measured values and calculated values obtained from the measured values (calculated values obtained with the electrode S). Correction was performed with E_std in Table 17 using the values of Slope in Table 13, determined in the same manner as in Example 1.

[Table 17]

Table 17 Measured values of potential difference E_S and potential difference E_std of standard solution, and calculated value obtained with electrode S when electrode EL5, EL6, or EL7 is used as electrode S

TABLE 17
Measured values of potential difference ES and potential difference
EStd of standard solution, and calculated value obtained with electrode
S when electrode EL5, EL6, or EL7 is used as electrode S
Electrode			Calculated value obtained
symbol	Es/mV	Estd/mV	with electrode S
EL5	62.93	144.36	120.82
EL6	60.63	141.75	134.44
EL7	62.26	141.96	120.61

C1-8 Calculation of Ion Concentration of Sample Solution

An ion concentration of the sample solution was calculated in the same manner as in Example 1.

First, an inverse matrix A⁻¹ of the matrix A obtained in the above “C1-6 Calculation of selectivity coefficient and creation of matrix A for ion-selective electrode” is determined in the same manner as in Example 1. This matrix A is obtained by using the electrodes EL5, EL6, and EL7 as the ion-selective electrode S1, the ion-selective electrode S2, and the ion-selective electrode S3, respectively. The inverse matrix A⁻¹ was determined by inputting the matrix A to a cell (A1: C3) of spreadsheet software Excel and using a function of “=MINVERSE (A1: C3)”.

$\left[\mathrm{Mathematical}\mathrm{formula}40\right]$ $A^{-1}=\left(\begin{array}{ccc}2.73& -1.36& -0.37\\ -20.62& 3.45& 17.18\\ 19.4& -0.55& -18.84\end{array}\right)$

By applying the inverse matrix A⁻¹ to the “calculated value obtained with electrode S” described in Table 17, obtained in the above “C1-7 Measurement of potential difference E_S of sample solution”, a concentration C(Cl⁻) of a chloride ion (Cl⁻), a concentration C(HCO₃) of a bicarbonate ion (HCO₃⁻), and a concentration C(AcO⁻) of an acetate ion (AcO⁻) were determined. Here, the “calculated value obtained with electrode S” for the electrode EL5 was used as “calculated value obtained with electrode S1”, the “calculated value obtained with electrode S” for the electrode EL6 was used as “calculated value obtained with electrode S2”, and the “calculated value obtained with electrode S” for the electrode EL7 was used as “calculated value obtained with electrode S3”. As a result, C(Cl⁻)=102.3 mM, C(HCO₃)=43.2 mM, and C(AcO⁻)=−3.46 mM.

$\left[\mathrm{Mathematical}\mathrm{formula}41\right]$ $\left(\begin{array}{c}{C\left(C1\right)}^{-})\\ C\left({\mathrm{HCO}}_3^{-}\right)\\ C\left({\mathrm{AcO}}^{-}\right)\end{array}\right)=A^{-1}·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S2\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S3\end{array}\right)=\left(\begin{array}{ccc}2.73& -1.36& -0.37\\ -20.62& 3.45& 17.18\\ 19.4& -0.55& -18.84\end{array}\right)·\left(\begin{array}{c}\mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S1\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S2\\ \mathrm{Calculated}\mathrm{value}\mathrm{obtained}\mathrm{with}\mathrm{electrode}S3\end{array}\right)$

The operations of the above “C1-7 Measurement of potential difference E_S of sample solution” and the above “C1-8 Calculation of ion concentration of sample solution” were repeated six times, and a relative standard deviation (%) was determined in the same manner as in Example 1. Table 18 presents the results.

[Table 18]

TABLE 18
Summary of results
Sample solution	Measurement results
	Concen-		Example	Example	Comparative
	tration/		1	2	Example 1
Ion	mM	||A||	0.42	0.20	0.02
Cl−	105	Measurement	105.9	105.7	102.9
		average
		value/mM
		Relative	0.1	0.2	0.4
		standard
		deviation/%
HCO3−	24	Measurement	25.6	27.6	39.4
		average
		value/mM
		Relative	1.1	2.0	9.9
		standard
		deviation/%
AcO−	6	Measurement	6.2	5.2	−0.1
		average
		value/mM
		Relative	3.6	5.9	100 or
		standard			more
		deviation/%

Reference Signs List

• • 1 Ion-selective electrode
  • 2 Reference electrode
  • 3 Potentiometer
  • 4 Tank to contain a solution
  • 5 Internal electrode
  • 6 Internal liquid
  • 7 Ion-selective membrane
  • 8 Liquid junction
  • 9 Sample solution

Claims

1. An ion concentration measurement device comprising two or more kinds of ion-selective electrodes that have mutually different selectivity coefficient ratios for a plurality of ions and satisfy the following formulas (1) and (2), wherein the two or more kinds of ion-selective electrodes are constituted by n kinds of ion-selective electrodes, andwherein n is the number of kinds of the plurality of ions and is an integer of 2 or more:

2. The ion concentration measurement device according to claim 1, wherein an ion concentration is calculated by the following formula (3):

3. An ion concentration measurement method comprising a step of measuring an ion concentration in a sample solution using two or more kinds of ion-selective electrodes that have mutually different selectivity coefficient ratios for a plurality of ions and satisfy the following formulas (1) and (2),wherein the two or more kinds of ion-selective electrodes are constituted by n kinds of ion-selective electrodes, andwherein n is the number of kinds of the plurality of ions and is an integer of 2 or more:

4. The ion concentration measurement method according to claim 3, wherein an ion concentration is calculated by the following formula (3):