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Beginners Guide to ISE measurement. Chapter 11.

ACCURACY AND PRECISION OF SAMPLE MEASUREMENTS

a) General Discussion

The accuracy (how close the result is to the true value) and precision (= reproducibility; i.e. how close are a series of measurements on the same sample to each other) of ISE measurements can be highly variable and are dependent on several factors. The concentration is proportional to the measured voltage and so any error in measurement will cause an error in the concentration, but this is not directly proportional. It is a logarithmic relationship which depends on the slope of the calibration line. For mono-valent ions with a calibration slope of around 55 millivolts per decade of concentration, an error of 1 mV in measuring the electrode potential will cause approximately 4% error in the concentration, whereas for di-valent ions, with a slope of around 26, the error will be more like 8% per mV. It must also be noted that the slope becomes less at the lower end of the concentration range, in the non-linear area, and hence the error per mV can be even greater at low concentrations. Thus it is important to use a meter which is capable of measuring the millivolts accurately and precisely. With modern meter technology this is not normally the limiting factor, although for the most precise work it can be beneficial to adopt multiple-sampling techniques (i.e. by using an integrating voltmeter or computer interface) in order to ensure the most reliable voltage measurements.

Apart from the accuracy and precision of the measuring device (meter or computer interface), the most important factors in achieving the most precise results is controlling the electrode drift and hysteresis (or memory), and limiting the variability in the Liquid Junction Potential of the reference electrode, so that the measured voltage is reproducible. The amount of the drift and hysteresis effects can vary significantly between different ions and different electrode types, with crystal membranes being generally more stable than PVC - techniqes for controlling or minimising drift and hysteresis are described elsewhere in this work (Chapter 9).

The most effective way of minimising the variation in LJP is by using Standard Addition or Sample Addition Techniques (see later - section e). Alternatively, but less effectively, by using reference electrodes with nearly equi-transferrent filling solutions (in which both ions have the same mobility when diffusing through the ceramic tip) such as KNO3 or Li Acetate - but this is not always possible (depending on likely interference effects).

The accuracy of the results is affected by several other factors:
1) The presence of interfering ions.
2) Any difference in ionic strength between the sample and standard solutions.
3) Any difference in temperature between sample and standards - A re-calibration should be made if the sample temperature changes by more than 1 degree C from the calibration temperature.
4) Any variation in the electrode slope in different parts of the curve. Although the calibration graph may show a straight line over several decades of concentration with an average slope of say 54.5 ± 2 mV/dec. it is highly unlikely that this will be exactly the same across the whole of this range. If separate two-point calibrations are made between two more closely spaced points at different concentration ranges then there may be a variation of several millivolts between the individual slopes. Thus, if samples are calculated using the overall slope then they will give results which will differ in concentration from those calculated using the appropriate individual slope by 4% times the difference in mV between the two slopes.

Therefore, for the most accurate results, it is recommended that the electrode slope is determined using two standards which closely span the expected range of the samples. It must be noted, however, that it is not beneficial to have standards too close together because the measured slope is dependent on the difference in voltages. So, for example, if the difference in mV is 50 then a 1 mV error in measurement will only cause a 2% error in the slope but if the difference is only 10 mV then the same measurement error will result in a 10% error in the slope. Thus it is normally recommended that calibration standards are about an order of magnitude different in concentration and should not be less than 20 mV difference in reading.

Nevertheless, whichever slope is used, the reproducibility of replicate measurements of the same sample should be the same.

By taking special precautions to overcome drift problems (such as frequent recalibration and ensuring that you wait for stable readings, or read after a regular time interval), and by adding special ISABs to equalise activity effects and remove interfering ions, direct potentiometry can give very reasonable results (reproducibility of ± 2 or 3%, one standard deviation, and accurate within these precision limits). Even without taking these precautions, it is possible to achieve satisfactory reproducibility and accuracy (± 10 to 15%) for many applications where the highest accuracy is not necessary and ionic strength and interfering ions are not a problem.

b) Reproducibility Experiments using an Ammonium Electrode.

Some of the suggestions in the foregoing discussion, and the levels of accuracy and precision achievable with careful work, can be illustrated with the results of some experiments conducted by the author. Reproducibility tests were carried out using an ‘ELIT’ 8 mm diameter, solid-state ammonium electrode (PVC membrane) with a lithium acetate double junction reference electrode and pure ammonium solutions (no ISAB). Standard solutions containing 1 ppm and 10 ppm NH₄⁺ were used for calibration and a 5 ppm solution was used as the test sample. Measurements were made after immersing the electrodes in approximately 50 mls of solution in a 100 ml beaker, swirling the solution for 5 secs. then leaving to stand for 20 secs. Each millivolt measurement was the average of ten readings taken at one second intervals. The electrodes were rinsed with a jet of de-ionised water, then soaked in a beaker of water for twenty seconds, then dabbed dry with a low-lint tissue between each measurement. The solutions were measured in the sequence 1 ppm, 5 ppm, 10 ppm, and this pattern was repeated six times. The data were obtained using a meterless PC interface and specially written software.

For this experiment, the concentration results were calculated with an EXCEL spreadsheet using the Nernst equation in the standard form for a straight line: y = mx + c.

Where:

y is the measured voltage,

m is the electrode slope

(calculated from the two-point calibration data: (V1-V2)/ ((Log ppm1) - (Log ppm2))),

x is the logarithm of the concentration in the sample,

c, the intercept on the y axis, is E^o.

The experimental data were processed in several different ways:

1) Using only the first measurement of the two standards to define the slope and intercept, six measurements of the 5 ppm sample, taken over approximately half an hour, gave an average of 4.71 ± 0.14 ppm (±2.96% one standard deviation). However it was noticeable that successive measurements gave progessively lower values due to electrode drift after calibration (causing a difference of nearly 8% between the highest and lowest results).

2) The drift effect was compensated for by recalculating each result using different values for the slope and intercept as defined by the standards measured immediately adjacent to each sample measurement. This produced a significant improvement in the reproducibility and only a random variation in the results rather than a progressive drift downwards. This clearly demonstrates the importance of measuring samples soon after calibration. The average concentration this time was 4.90 ± 0.06 ppm (±1.20%, 1 S.D.) Although remarkably precise and very close to the true value, the accuracy of this average is not quite within the precision limits. As noted above, this can probably be explained by variation in the electrode slope and this suggestion is supported by examining the individual slope values which can be calculated from the various measurements. The average value for six determinations of the slope between 1 and 5 ppm was 55.92 ± 0.92 whereas that between 5 and 10 ppm was 58.21 ± 0.78; i.e. there is a significant difference in slope between the two adjacent ranges.

3) A third method of calculating these results, using the slope defined by the first calibration for all samples but a different intercept value as given by each successive two-point calibration, was less satisfactory and gave 4.87 ± 0.12 ppm (± 2.35%) which is only slightly better than the results using only a single calibration at the beginning. Thus these data would appear to suggest that the effect of electrode drift is more significant in producing changes in the measured slope between different sample measurements rather than producing a change in the calculated value for the intercept. This conclusion is also borne out by examining the individual calibration data. Whereas the average slope between 1 and 10 ppm was 56.74 ± 0.51 (± 0.90%) for six successive measurements and these showed a gradual drift downwards (57.49, 57.04, 56.84, 56.64, 56.49, 55.98) the associated intercept calculations showed a more random distribution and gave a much more precise average value of 346.18 ± 0.26 mV (± 0.07%).

c) Reproducibility of Chloride Measurements.

A second experiment using the same techniques as above, but with a chloride (crystal membrane) electrode and calibration standards of 25 and 250 ppm also yielded very impressive results. Eight measurements of a 100 ppm test solution gave an average of 95.4 ± 0.6 ppm (± 0.63%) when two-point calibrations were made immediately prior to each sample measurement.

d) Conclusions from the Experimental Data.

e) Standard Addition and Sample Addition Methods