Example A7: Determination of the Amount of Lead in Water using double Isotope Dilution and Inductively Coupled Plasma Mass Spectrometry
A7.1 Introduction
This example illustrates how the uncertainty concept can be applied to a measurement of the amount content of lead in a water sample using Isotope Dilution Mass Spectrometry (IDMS) and Inductively Coupled Plasma Mass Spectrometry (ICP-MS).
General Introduction to double IDMS
IDMS is one of the techniques that is recognised by the Comité consultatif pour la quantité de matière (CCQM) to have the potential to be a primary method of measurement, and therefore a well defined expression which describes how the measurand is calculated is available. In the simplest case of isotope dilution using a certified spike, which is an enriched isotopic reference material, isotope ratios in the spike, the sample and a blend b of known masses of sample and spike are measured. The element amount content cx in the sample is given by:
where cx and cy are element amount content in the sample and the spike respectively (the symbol c is used here instead of k for amount content1 to avoid confusion with K-factors and coverage factors k). mx and my are mass of sample and spike respectively. Rx, Ry and Rb are the isotope amount ratios. The indexes x, y and b represent the sample, the spike and the blend respectively. One isotope, usually the most abundant in the sample, is selected and all isotope amount ratios are expressed relative to it. A particular pair of isotopes, the reference isotope and preferably the most abundant isotope in the spike, is then selected as monitor ratio, e.g. n(208Pb)/n(206Pb). Rxi and Ryi are all the possible isotope amount ratios in the sample and the spike respectively. For the reference isotope, this ratio is unity. Kxi, Kyi and Kb are the correction factors for mass discrimination, for a particular isotope amount ratio, in sample, spike and blend respectively. The K-factors are measured using a certified isotopic reference material according to equation (2).
where K0 is the mass discrimination correction factor at time 0, Kbias is a bias factor coming into effect as soon as the K-factor is applied to correct a ratio measured at a different time during the measurement. The Kbias also includes other possible sources of bias such as multiplier dead time correction, matrix effects etc. Rcertified is the certified isotope amount ratio taken from the certificate of an isotopic reference material and Robserved is the observed value of this isotopic reference material. In IDMS experiments, using Inductively Coupled Plasma Mass Spectrometry (ICP-MS), mass fractionation will vary with time which requires that all isotope amount ratios in equation (1) need to be individually corrected for mass discrimination.
Certified material enriched in a specific isotope is often unavailable. To overcome this problem, 'double' IDMS is frequently used. The procedure uses a less well characterised, isotopically enriched spiking material in conjunction with a certified material (denoted z) of natural isotopic composition. The certified, natural composition material acts as the primary assay standard. Two blends are used; blend b is a blend between sample and enriched spike, as in equation (1). To perform double IDMS a second blend, b' is prepared from the primary assay standard with amount content cz, and the enriched material y. This gives a similar expression to equation (1):
where cz is the element amount content of the primary assay standard solution and mz the mass of the primary assay standard when preparing the new blend. m'y is the mass of the enriched spike solution, K'b, R'b, Kz1 and Rz1 are the K-factor and the ratio for the new blend and the assay standard respectively. The index z represents the assay standard. Dividing equation (1) with equation (3) gives
Simplifying this equation and introducing a procedure blank, cblank, we get:
This is the final equation, from which cy has been eliminated. In this measurement the number index on the amount ratios, R, represents the following actual isotope amount ratios:
R1=n(208Pb)/n(206Pb); R2=n(206Pb)/n(206Pb)
R3=n(207Pb)/n(206Pb); R4=n(204Pb)/n(206Pb)
For reference, the parameters are summarised in table A7.1.
Table A7.1. Summary of IDMS parameters
| Parameter | Description | Parameter | Description |
| mx | mass of sample in blend b (g) | my | mass of enriched spike in blend b (g) |
| m'y | mass of enriched spike in blend b' (g) | mz | mass of primary assay standard in blend b' (g) |
| cx | amount content of the sample x (mol g-1 or  mol g-1)Note 1 |
cz | amount content of the primary assay standard z (mol g-1 or mol g-1)Note 1 |
| cy | amount content of the spike y (mol g-1 or mol g-1)Note 1 |
cblank | observed amount content in procedure blank (mol g-1 or mol g-1)Note 1 |
| Rb | measured ratio of blend b, n(208Pb)/n(206Pb) |
Kb | mass bias correction of Rb |
| R'b | measured ratio of blend b', n(208Pb)/n(206Pb) |
K'b | mass bias correction of R'b |
| Ry1 | measured ratio of enriched isotope to reference isotope in the enriched spike | Ky1 | mass bias correction of Ry1 |
| Rzi | all ratios in the primary assay standard, Rz1, Rz2 etc. |
Kzi | mass bias correction factors for Rzi |
| Rxi | all ratios in the sample | Kxi | mass bias correction factors for Rxi |
| Rx1 | measured ratio of enriched isotope to reference isotope in the sample x | Rz1 | as Rx1 but in the primary assay standard |
Note 1: Units for amount content are always specified in the text.
A7.2 Step 1: Specification
The general procedure for the measurements is shown in table A7.2. The calculations and measurements involved are described below.
Table A7.2. General procedure
| Step | Description |
| 1 | Preparing the primary assay standard |
| 2 | Preparation of blends: b' and b |
| 3 | Measurement of isotope ratios |
| 4 | Calculation of the amount content of Pb in the sample, cx |
| 5 | Estimating the uncertainty in cx |
Calculation Procedure for the Amount Content cx
For this determination of lead in water, four blends each of b', (assay + spike), and b, (sample + spike), were prepared. This gives a total of 4 values for cx. One of these determinations will be described in detail following table a7.2, steps 1 to 4. The reported value for cx will be the average of the four replicates.
Calculation of the Molar Mass
Due to natural variations in the isotopic composition of certain elements, e.g. Pb, the molar mass, M, for the primary assay standard has to be determined since this will affect the amount content cz. Note that this is not the case when cz is expressed in mol g-1. The molar mass, M(E), for an element E, is numerically equal to the atomic weight of element E, Ar(E). The atomic weight can be calculated according to the general expression:
where the values Ri are all true isotope amount ratios for the element E and M(iE) are the tabulated nuclide masses.
Note that the isotope amount ratios in equation (6) have to be absolute ratios, that is, they have to be corrected for mass discrimination. With the use of proper indexes, this gives equation (7). For the calculation, nuclide masses, M(iE), were taken from literature values2, while Ratios, Rzi, and K0-factors, K0(zi), were measured (table A7.8 more...). These values give
Measurement of K-Factors and Isotope Amount Ratios
To correct for mass discrimination, a correction factor, K, is used as specified in equation (2). The K0-factor can be calculated using a reference material certified for isotopic composition. In this case, the isotopically certified reference material NIST SRM 981 was used to monitor a possible change in the K0-factor. The K0-factor is measured before and after the ratio it will correct. A typical sample sequence is: 1. (blank), 2. (NIST SRM 981), 3. (blank), 4. (blend 1), 5. (blank), 6. (NIST SRM 981), 7. (blank), 8. (sample), etc.
The blank measurements are not only used for blank correction, they are also used for monitoring the number of counts for the blank. No new measurement run was started until the blank count rate was stable and back to a normal level. Note that sample, blends, spike and assay standard were diluted to an appropriate amount content prior to the measurements. The results of ratio measurements, calculated K0-factors and Kbias are summarised in table A7.8 more....
Preparing the Primary Assay Standard and Calculating the Amount Content, cz
Two primary assay standards were produced, each from a different piece of metallic lead with a chemical purity of w=99.999 %. The two pieces came from the same batch of high purity lead. The pieces were dissolved in about 10 ml of 1:3 w/w HNO3:water under gentle heating and then further diluted. Two blends were prepared from each of these two assay standards. The values from one of the assays is described hereafter.
0.36544 g lead, m1, was dissolved and diluted in aqueous HNO3 (0.5 mol l-1) to a total of d1=196.14 g. This solution is named Assay 1. A more diluted solution was needed and m2=1.0292 g of Assay 1, was diluted in aqueous HNO3 (0.5 mol l-1) to a total mass of d2=99.931g. This solution is named Assay 2. The amount content of Pb in Assay 2, cz, is then calculated according to equation (8)
Preparation of the Blends
The mass fraction of the spike is known to be roughly 20µg Pb per g solution and the mass fraction of Pb in the sample is also known to be in this range. Table A7.3 shows the weighing data for the two blends used in this example.
Table A7.3
| Blend | b | b' | ||
| Solutions used | Spike | Sample | Spike | Assay 2 |
| Parameter | my | mx | m'y | m'z |
| Mass (g) | 1.1360 | 1.0440 | 1.0654 | 1.1029 |
Measurement of the Procedure Blank cBlank
In this case, the procedure blank was measured using external calibration. A more exhaustive procedure would be to add an enriched spike to a blank and process it in the same way as the samples. In this example, only high purity reagents were used, which would lead to extreme ratios in the blends and consequent poor reliability for the enriched spiking procedure. The externally calibrated procedure blank was measured four times, and cBlank found to be 4.5x10-7 µmol g-1, with standard uncertainty 4.0x10-7 µmol g-1 evaluated as type A.
Calculation of the Unknown Amount Content cx
Inserting the measured and calculated data (table A7.8 more...) into equation (5) gives cx=0.053738 µmol g-1. The results from all four replicates are given in table a7.4.
Table a7.4
| cx (µmol g-1) | |
| Replicate 1 (our example) | 0.053738 |
| Replicate 2 | 0.053621 |
| Replicate 3 | 0.053610 |
| Replicate 4 | 0.053822 |
| Average | 0.05370 |
| Experimental standard deviation (s) | 0.0001 |
A7.3 Steps 2 and 3: identifying and quantifying uncertainty sources
Strategy for the Uncertainty Calculation
If equations (2), (7) and (8) were to be included in the final IDMS equation (5), the sheer number of parameters would make the equation almost impossible to handle. To keep it simpler, K0-factors and amount content of the standard assay solution and their associated uncertainties are treated separately and then introduced into the IDMS equation (5). In this case it will not affect the final combined uncertainty of cx, and it is advisable to simplify for practical reasons.
For calculating the combined standard uncertainty, uc(cx), the values from one of the measurements, as described in a7.2, will be used. The combined uncertainty of cx will be calculated using the spreadsheet method described in appendix e.
Uncertainty on the K-Factors
i) Uncertainty on K0
K is calculated according to equation (2) and using the values of Kx1 as an example gives for K0:
To calculate the uncertainty on K0 we first look at the certificate where the certified ratio, 2.1681, has a stated uncertainty of 0.0008 based on a 95% confidence interval. To convert an uncertainty based on a 95% confidence interval to standard uncertainty we divide by 2. This gives a standard uncertainty of u(Rcertified)=0.0004. The observed amount ratio, Robserved=n(208Pb)/n(206Pb), has a standard uncertainty of 0.0025 (as rsd). For the K-factor, the combined uncertainty can be calculated as:
This clearly points out that the uncertainty contributions from the certified ratios are negligible. Henceforth, the uncertainties on the measured ratios, Robserved, will be used for the uncertainties on K0.
Uncertainty on Kbias
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor. As can be seen in the cause and effect diagram above, and in equation (2), there is a bias associated with every K-factor. The values of these biases are in our case not known, and a value of 0 is applied. An uncertainty is, of course, associated with every bias and this has to be taken into consideration when calculating the final uncertainty. In principle, a bias would be applied as in equation (11), using an excerpt from equation (5) and the parameters Ky1 and Ry1 to demonstrate this principle.
The values of all biases, Kbias(yi, xi, zi), are (0 ± 0.001). This estimation is based on a long experience of lead IDMS measurements. All Kbias(yi, xi, zi) parameters are not included in detail in table A7.5, table A7.8 more... or in equation (5), but they are used in all uncertainty calculations.
Table A7.5
| Value | Standard uncertainty | TypeNote 1 | |
| Kbias(zi) | 0 | 0.001 | B |
| Rz1 | 2.1429 | 0.0054 | A |
| K0(z1) | 0.9989 | 0.0025 | A |
| K0(z3) | 0.9993 | 0.0035 | A |
| K0(z4) | 1.0002 | 0.0060 | A |
| Rz2 | 1 | 0 | A |
| Rz3 | 0.9147 | 0.0032 | A |
| Rz4 | 0.05870 | 0.00035 | A |
| M1 | 207.976636 | 0.000003 | B |
| M2 | 205.974449 | 0.000003 | B |
| M3 | 206.975880 | 0.000003 | B |
| M4 | 203.973028 | 0.000003 | B |
Note 1. Type A (statistical evaluation) or Type B (other)
Uncertainty of the Weighed Masses
In this case, a dedicated mass metrology lab performed the weighings. The procedure applied was a bracketing technique using calibrated weights and a comparator. The bracketing technique was repeated at least six times for every sample mass determination. Buoyancy correction was applied. Stoichiometry and impurity corrections were not applied in this case. The uncertainties from the weighing certificates were treated as standard uncertainties and are given in table a7.8 more....
Uncertainty in the Amount Content of the Standard Assay Solution, cz
i) Uncertainty in the atomic weight of Pb
First, the combined uncertainty of the molar mass of the assay solution, Assay 1, will be calculated. The values in table a7.5 are known or have been measured:
According to equation (7), the calculation of the molar mass takes this form:
To calculate the combined standard uncertainty of the molar mass of Pb in the standard assay solution, the spreadsheet model described in Appendix E was used. There were eight measurements of every ratio and K0. This gave a molar mass M(Pb, Assay 1)=207.2103 g mol-1, with uncertainty 0.0010 g mol-1 calculated using the spreadsheet method.
ii) Calculation of the combined standard uncertainty in determining cz
To calculate the uncertainty on the amount content of Pb in the standard assay solution, cz the data from a7.2 and equation (8) are used. The uncertainties were taken from the weighing certificates, see a7.3. All parameters used in equation (8) are given with their uncertainties in table A7.6.
Table A7.6.
| Value | Uncertainty | |
| Mass of lead piece, m1 (g) | 0.36544 | 0.00005 |
| Total mass first dilution, d1 (g) | 196.14 | 0.03 |
| Aliquot of first dilution, m2 (g) | 1.0292 | 0.0002 |
| Total mass of second dilution, d2 (g) | 99.931 | 0.01 |
| Purity of the metallic lead piece, w (mass fraction) | 0.99999 | 0.000005 |
| Molar mass of Pb in the Assay Material, M (g mol-1) | 207.2104 | 0.0010 |
The amount content, cz, was calculated using equation (8). Following appendix d.5 the combined standard uncertainty in cz, is calculated to be uc(cz)=0.000028. This gives cz=0.092606 µmol g-1 with a standard uncertainty of 0.000028 µmol g-1 (0.03% as %rsd).
To calculate uc(cx), for replicate 1, the spreadsheet model was applied (appendix e). The uncertainty budget for replicate 1 will be representative for the measurement. Due to the number of parameters in equation (5), the spreadsheet will not be displayed. The value of the parameters and their uncertainties as well as the combined uncertainty of cx can be seen in table a7.8 more....
A7.4 Step 4: Calculating the Combined Standard Uncertainty
The average and the experimental standard deviation of the four replicates are displayed in table a7.7. The numbers are taken from table a7.4 and Table A7.8 more....
Table a7.7
| Replicate 1 | Mean of replicates 1-4 | |||
| cx= | 0.05374 | cx= | 0.05370 | µmol g-1 |
| uc(cx)= | 0.00018 | s= | 0.00010 Note 1 | µmol g-1 |
Note 1.This is the experimental standard uncertainty and not the standard deviation of the mean.
In IDMS, and in many non-routine analyses, a complete statistical control of the measurement procedure would require limitless resources and time. A good way then to check if some source of uncertainty has been forgotten is to compare the uncertainties from the type A evaluations with the experimental standard deviation of the four replicates. If the experimental standard deviation is higher than the contributions from the uncertainty sources evaluated as type A, it could indicate that the measurement process is not fully understood. As an approximation, using data from Table 8, the sum of the type A evaluated experimental uncertainties can be calculated by taking 92.2% of the total experimental uncertainty, which is 0.00041 µmol g-1. This value is then clearly higher than the experimental standard deviation of 0.00010 µmol g-1, see table a7.7. This indicates that the experimental standard deviation is covered by the contributions from the type A evaluated uncertainties and that no further type A evaluated uncertainty contribution, due to the preparation of the blends, needs to be considered. There could however be a bias associated with the preparations of the blends. In this example, a possible bias in the preparation of the blends is judged to be insignificant in comparison to the major sources of uncertainty.
The amount content of lead in the water sample is then:
cx=(0.05370±0.00036) µmol g-1
The result is presented with an expanded uncertainty using a coverage factor of 2.
References for Example 7
- T. Cvitas, Metrologia, 1996, 33, 35-39
- G. Audi and A.H. Wapstra, Nuclear Physics, A565 (1993)