DE3823433A1 - Method for the determination of a measured variable - Google Patents

Abstract

In order to minimise the expenditure on calibration measurements and to simplify the mathematical formulation in a method for the determination of a measured variable m which is connected with the measuring signal E, to a first approximation, via the relationship E = E0 + S ln(m + SIGMA Kisi> where si, with i = 0 to n, are at least partially independently determinable interference variables influencing the measuring signal E as a function of the coefficients Ki, it is proposed to extend the logarithmic term by an additive correction variable U in order to extend the measuring range into the non-linear region, so that E = E0 + S ln(m + SIGMA Kisi + U> is true, where a known or previously determined value is applied for S and Ki and U are determined from at least n + 2 calibration measurements.

Description

The invention relates to a method for determining a measurement size m , which with the measurement signal E in a first approximation of the relationship

is connected, wherein s _i with i = 0 to n are at least partially independently determinable disturbance variables influencing the measurement signal E as a function of the coefficients K _i .

The measured variable to be determined is, for example, an activity of a substance or the concentration of a measurement ion, generally a measured variable m , which ideally corresponds to the measurement signal E using the equation

E = E ₀ + S 1n m (Eq. 1)

related. For example, obey ideal ion-sensitive Electrodes (ISE) of this equation, where:

E. . . Electrode potential
E ₀. . . Electrode potential at m = 1
S. . . Slope of the electrode
m . . . Concentration of the measurement ion.

When choosing appropriate calibration solutions, the area of application and the measuring range of such electrodes can be varied within a wide range. However, one is more limited to the so-called linear area of the electrode. This is understood to mean the area in which there is a linear relationship between the logarithm of the ion activity or ion concentration m and the electrode signal E. Deviations from this linear function are observed both at extremely high concentrations and at extremely low concentrations. The non-linearities occurring at low measurement ion concentrations are particularly disruptive, since this limits the measurement range downwards.

With an ideal electrode, only the determination of E ₀ should be necessary for calibration at a known temperature, which is possible with single-point calibration. In practice, however, you need at least a 2-point calibration to adjust S , the slope of the electrode, to the real conditions. The majority of all known applications operate according to equation 1 in the linear range, shown in FIG. 1.

Deviations from equation 1 arise for various reasons. One possible reason is that the intrinsic solubility of the electroactive electrode phase - which contains the ion to be measured more or less firmly bound - in the measuring solution, the concentration of the ion to be measured in the measuring solution is significantly changed. Another practically important case is that non-linearity is caused by disturbance variables s , for example by insufficient selectivity compared to interference ions present in the measured material. Ion to be measured and interfering ions then contribute to the measurement signal, the electrode signal.

For such cases, it is known to apply Equation 2.

K _i . . . Selectivity coefficient
s _i . . . Concentration of the interference ion
i = ₀ to n
n . . . Number of interfering ions

This equation (Nikolsky-Eisenmann equation) describes the electrode potential approximately and can be used for small deviations from the linear characteristic curve, but is unsuitable for the universal and exact description of the electrical characteristic curve course, among other things because the numerical values for K _i depends on the method of determination and the selected concentrations of the interfering ions and the measuring ion.

The smaller m becomes Σ K _i · s _i , the flatter the electrode characteristic curve. Fig. 2 shows that there is a range in which changes in m do not change E - ie m is no longer measurable. In practice, however, a large part of the non-linear area causes problems. As a rule, the detection limit is the value where:

Because of the limited applicability of Equation 2, there are a number of proposals for this nonlinear range to make practical measuring purposes accessible. Common to all however, a relatively high effort regarding calibration solutions and in usually the use of complex mathematical models, e.g. B. staircase functions or non-linear compensation function nen. . .

The problem is compounded by the following fact. As can be seen from Eq. 2 can be seen, not only is the curvature of the characteristic line to be reproduced, but the characteristic line also changes its position and shape as a function of existing disturbance variables (s ₁, s ₂, s ₃), for example a possible concentration of interference ions present. An extremely complex matrix is thus obtained as a characteristic field as a function of the interference ion concentrations, as shown in FIG. 3.

The object of the present invention is a simple Ver drive to develop around the measuring range as possible much of the nonlinear area of the relationship between To expand the measurand and measuring signal and especially at small Conduct accurate measurements for values can.

This object is achieved in that the logarithmic term is expanded by an additive correction variable U to extend the measuring range into the nonlinear range, so that

applies that a known or predeterminable value is used for S , that if there are n disturbances at least n +2 calibration measurements are carried out with different, predeterminable sets of values for the measurands and disturbances and via the measurement signals F ₁, F ₂ thereby obtained. . . F _{n +2,} the coefficients K _i and the correction variable U for calculating the measured variable m are determined. Surprisingly, the method according to the invention, with the aid of the relationship between measurement variable m and measurement signal E which has been expanded by the correction variable U , has proven to be outstandingly suitable for describing complex characteristic curves or characteristic curve fields (for example FIG. 3).

Known methods aim to determine S for electrode calibration and - if necessary - to assume K _i as constant. However, measurements in the non-linear characteristic range are only possible to a very limited extent. In the method according to the invention, the K _i and U are determined during the calibration, where S can be determined, or a value to be expected for the case free of disturbances can be used. Especially in the non-linear range, K _i and U dominate , so that calibration of these parameters provides more precise values. This procedure is also applicable in the linear range, so that a single measuring method is used for the extended measuring range. In addition, the effort of calibration measurements is low, since only three calibration measurements have to be carried out in the presence of a disturbance variable, for which three calibration solutions are necessary in the case of ISE and the mathematical effort is extremely low.

Through this procedure, K _i and U can be calibrated continuously very easily, which is particularly advantageous for electrodes with a rapid change in selectivity. The lower concentration limit, which is still measurable, can be extended almost to the completely horizontal part of the electrode characteristic.

The method according to the invention is also still practicable if the interference ion concentrations approach zero as a limit case. The term Σ K _i · s _i in Eq. 2 is then omitted. In addition, only 2 calibration measurements are necessary.

In a development of the invention it is provided that the measured variable m from the difference signal

is determined, where E ₁ is the measurement signal of a calibration measurement with known values s _i ₁ of the disturbance variables s _i and known value m ₁ for the measurement variable m , and that the current measurement values of the disturbance variables s _{i are} determined by an independent measurement method. This procedure eliminates the determination of F ₀.

The method of operation of the method according to the invention is described below using the example of an ion-selective electrode, the simplest case being a system with louder monovalent ions, in which the measurement ion and an interfering ion s are located. It is provided according to the invention that when there is a single disturbance variable s for determining the coefficient K , in this case the selectivity coefficient, and the correction variable U, the value sets (m ₁, s ₁), (m ₁, s ₂), (m ₂ , s ₁) for the measurand and disturbance variable (measurement and Störio concentration). Thus, two different values m ₁, m ₂ or s ₁, s ₂ are sufficient for the concentration values of the measurement or interference ion in the calibration solutions, the abovementioned sets of values being used to determine the measurement signals F ₁, F ₂ and F ₃ will.

If there is only one interfering ion, the equation can be

simplify:

E = E ₀ + S 1n (m + Ks + U) (Eq.3a)

By measuring the electrode potential in these three calibration solutions, the three associated measurement signals F ₁, F ₂ and F ₃ are obtained. If one uses the predetermined or known value for S , three equations result from inserting the corresponding values in equation 3a. By transforming and simply combining these equations, we get:

or 4 in 5 used

This allows K to be calculated and then also U using equation 4. If you now determine the potential of the measuring electrode for any sample - and you know the concentration of the interference ion, e.g. B. by parallel determination using an electrode that is selective for the interference ion, the measurement ion concentration can be calculated according to equation 7:

m = m ₁exp ( (E - E ₁) / S) + K (s ₁exp ( (E - E ₁) / S) - s) + U (exp ( (E - E ₁) / S) - 1) (Eq. 7)

m . . . sought ion concentration of the sample
E. . . measured electrode potential
s . . . Interference ion concentration of the sample

The accuracy of this procedure is shown in the below playful Table 1, where lithium sensitive electrodes, which show a pronounced sensitivity to sodium, over a lithium range of 0.1 to 10 mmol1 / 1 and with natri conversions of 125/150/175 mmol1 / 1 were examined. Lithium- and sodium electrode were simultaneously in the  Measuring solution so that the two values are determined simultaneously could. The lithium electrode was up to Li concentrations used at which the electrode only still approx. 2% of the theoretical and in the linear range of Shows the achievable slope.

Finally, according to the invention it is provided in the measurement of the ion concentration that for the steepness S the theoretically determinable value S = R · T / z · F with

R = general gas constant
T = absolute temperature
z = valence of the measurement ion
F = Faraday constant

is used.

If the method is used for multiple interfering ions, at least one additional calibration solution is required for each fault ion does. It is also obvious that the process is identical also using activities instead of con concentrations can be used.

In the method according to the invention, the correction variable U does not have the meaning of a detection limit or a sum of unknown interference. Thus, in the case of a lithium electrode with sodium interference cited in Table 1, the typical value for U is approximately 0.5 mmol / 1, the detection limit L being much higher at approximately 2.5 mmol / 1.

Table 1

Comparison of measurement results of lithium ion sensitive Electrodes - achieved with the described method with Er Results achieved with a flame photometer in the area of 0.1-10 mmol / 1 lithium with different interference ion concentrations trations (sodium)

Claims (5)

1. Method for determining a measured variable m which is based on the relationship with the measurement signal E n is connected, where s _i with i = 0 to n at least partially independently determinable disturbing variables influencing the measurement signal E as a function of the coefficient K _i , characterized in that the logarithmic term is extended by an additive to expand the measurement range in the nonlinear range Correction variable U is expanded so that applies that a known or anticipated determinable value is used for S, that in the presence of n disturbances least n +2 calibration measurements with different, against predeterminable value sets for the measurement and disturbances Runaway leads and the gained measured signals E ₁, E ₂ . . E _{n + 2,} the coefficients K _i and the correction variable U for calculating the measured variable m are determined. 2. The method according to claim 1, characterized in that the measured variable m from the difference signal is determined, where E _{i is} the measurement signal of a calibration measurement with known values S _i ₁ of the disturbance variables s _i and known value m ₁ for the measurement variable m , and that the current measurement values of the disturbance variables s _{i are} determined by an independent measuring method. 3. The method according to claim 1 or 2, characterized in that the disturbance variables s _i and the measured variable m are determined simultaneously. 4. The method according to any one of claims 1 to 3, characterized in that in the presence of a single disturbance variable s for determining the coefficient K and the correction variable U, the value sets (m ₁, s ₁), (m ₁, s ₂), (m ₂, s ₁) can be specified for the measured and disturbance variables. 5. The method according to any one of claims 1 to 4, wherein the measurement variable m represent the concentration of a measurement ion to be measured and s _{i represent} the known or previously determinable concentrations of interference ions, characterized in that for the steepness S the theoretically determinable value S = R · T / z · F with R = general gas constant
T = absolute temperature
z = valence of the measurement ion
F = Faraday constant is used.