Determination of the amount fo lead in water using double isotope dilution and inductively coupled plasma mass spectrometry
This is the example A7 of the EURACHEM / CITAC Guide "Quantifying Uncertainty in Analytical Measurement", Second Edition.
The amount content of lead in water is measured using Isotope Dilution Mass Spectrometry (IDMS) and Inductively Coupled Plasma Mass Spectrometry (ICPMS)
In this case a 'double' isotope dilution is applied. It uses a well characterised (ideally certified) material of natural isotopic composition as a primary assay standard. Two blends are then prepared: blend b, which is a blend between known masses of the sample and the enriched spike, and blend b', which is the blend between the enriched spike and the primary assay standard. The isotope ratios of the primary assay standard, the spike, the sample and the two blends are measured using ICPMS. Together with the weighing data of the blends, the amount content of lead in the sample can be calculated.
Model Equation:
{equation for the double isotope dilution}
c_{x} = (c_{z} * m_{y1} / m_{x} *m_{z} / m_{y2} * (K_{y1} * R_{y1}  K_{b1} * R_{b1}) / (K_{b1} * R_{b1}  K_{x1} * R_{x1}) * (K_{b2} * R_{b2}  K_{z1} * R_{z1}) / (K_{y1} * R_{y1}  K_{b2} * R_{b2}) / (ΣK_{zi}R_{zi}) * (ΣK_{xi}R_{xi}))  c_{blank};
ΣK_{xi}R_{xi} = K_{x1} * R_{x1} + K_{x2} * R_{x2} + K_{x3} * R_{x3} + K_{x4} * R_{x4};
ΣK_{zi}R_{zi} = K_{z1} * R_{z1} + K_{z2} * R_{z2} + K_{z3} * R_{z3} + K_{z4} * R_{z4};
{calculation of the molar mass of the lead of the primary assay standard 1}
M_{Pb}_{ }_{Assay1} = (K_{z1} * R_{z1} * M_{z1} + K_{z2} * R_{z2} * M_{z2} + K_{z3} * R_{z3} * M_{z3} + K_{z4} * R_{z4} * M_{z4}) / (ΣK_{zi}R_{zi});
{concentration of the primary assay standard z which is used for the double IDMS}
c_{z} =m_{2} / d_{2} * m_{1} * w / d_{1} / M_{Pb}_{ }_{Assay1} * k_{mol};
{calculation of the Kfactors for the various isotope ratios measured}
K_{b1} = K_{0}_{}_{b1} + K_{bias}_{}_{b1};
K_{b2} = K_{0}_{}_{b2} + K_{bias}_{}_{b2};
K_{x1} = K_{0}_{}_{x1} + K_{bias}_{}_{x1};
K_{x2} = K_{0}_{}_{x2} + K_{bias}_{}_{x2};
K_{x3} = K_{0}_{}_{x3} + K_{bias}_{}_{x3};
K_{x4} = K_{0}_{}_{x4} + K_{bias}_{}_{x4};
K_{y1} = K_{0}_{}_{y1} + K_{bias}_{}_{y1};
K_{z1} = K_{0}_{}_{z1} + K_{bias}_{}_{z1};
K_{z2} = K_{0}_{}_{z2} + K_{bias}_{}_{z2};
K_{z3} = K_{0}_{}_{z3} + K_{bias}_{}_{z3};
K_{z4} = K_{0}_{}_{z4} + K_{bias}_{}_{z4};

List of Quantities:
Quantity  Unit  Definition 

c_{x}  µmol/g  amount content of the sample x 
c_{z}  µmol/g  amount content of the primary assay standard z 
m_{y1}  g  mass of enriched spike in blend b 
m_{x}  g  mass of sample in blend b 
m_{z}  g  mass of primary assay standard in blend b' 
m_{y2}  g  mass of enriched spike in blend b' 
K_{y1}  mass bias correction of R_{y1}  
R_{y1}  measured ratio of enriched isotope to reference isotope in the enriched spike, n(^{208}Pb)/n(^{206}Pb)  
K_{b1}  mass bias correction of R_{b1}  
R_{b1}  measured ratio of blend b, n(^{208}Pb)/n(^{206}Pb)  
K_{x1}  mass bias correction of R_{x1}  
R_{x1}  measured ratio of enriched isotope to reference isotope in the sample x, n(^{208}Pb)/n(^{206}Pb)  
K_{b2}  mass bias correction of R_{b2}  
R_{b2}  measured ratio of blend b', n(^{208}Pb)/n(^{206}Pb)  
K_{z1}  mass bias correction of R_{z1}  
R_{z1}  measured ratio of enriched isotope to reference isotope in the primary assay standard z, n(^{208}Pb)/n(^{206}Pb)  
ΣK_{zi}R_{zi}  sum of all mass bias corrected ratios of the primary assay standard  
ΣK_{xi}R_{xi}  sum of all mass bias corrected ratios of the sample  
c_{blank}  µmol/g  observed amount content in procedure blank 
K_{x2}  mass bias correction of R_{x2}  
R_{x2}  measured ratio of sample, n(^{206}Pb)/n(^{206}Pb)  
K_{x3}  mass bias correction of R_{x3}  
R_{x3}  measured ratio of sample, n(^{207}Pb)/n(^{206}Pb)  
K_{x4}  mass bias correction of R_{x4}  
R_{x4}  measured ratio of sample, n(^{204}Pb)/n(^{206}Pb)  
K_{z2}  mass bias correction of R_{z2}  
R_{z2}  measured ratio of sample, n(^{206}Pb)/n(^{206}Pb)  
K_{z3}  mass bias correction of R_{z3}  
R_{z3}  measured ratio of sample, n(^{207}Pb)/n(^{206}Pb)  
K_{z4}  mass bias correction of R_{z4}  
R_{z4}  measured ratio of sample, n(^{204}Pb)/n(^{206}Pb)  
M_{Pb}_{ }_{Assay1}  g/mol  molar mass of the primary assay standard 
M_{z1}  g/mol  nuclidic mass of ^{208}Pb 
M_{z2}  g/mol  nuclidic mass of ^{206}Pb 
M_{z3}  g/mol  nuclidic mass of ^{207}Pb 
M_{z4}  g/mol  nuclidic mass of ^{204}Pb 
m_{2}  g  aliquot of the first dilution of the primary assay standard 
d_{2}  g  total mass of the second dilution of the primary assay standard 
m_{1}  g  mass of the lead piece for primary assay standard 
w  g/g  purity of the metallic lead piece, expressed as mass fraction 
d_{1}  g  total mass of first dilution of the primary assay standard 
k_{mol}  µmol/mol  conversion factor 10^{6} µmol = 1 mol 
K_{0}_{}_{b1}  mass bias correction of R_{b1} as determined at time 0  
K_{bias}_{}_{b1}  other contributions to the mass bias of R_{b1}  
K_{0}_{}_{b2}  mass bias correction of R_{b2} as determined at time 0  
K_{bias}_{}_{b2}  other contributions to the mass bias of R_{b2}  
K_{0}_{}_{x1}  mass bias correction of R_{x1} as determined at time 0  
K_{bias}_{}_{x1}  other contributions to the mass bias of R_{x1}  
K_{0}_{}_{x2}  mass bias correction of R_{x2} as determined at time 0  
K_{bias}_{}_{x2}  other contributions to the mass bias of R_{x2}  
K_{0}_{}_{x3}  mass bias correction of R_{x3} as determined at time 0  
K_{bias}_{}_{x3}  other contributions to the mass bias of R_{x3}  
K_{0}_{}_{x4}  mass bias correction of R_{x4} as determined at time 0  
K_{bias}_{}_{x4}  other contributions to the mass bias of R_{x4}  
K_{0}_{}_{y1}  mass bias correction of R_{y1} as determined at time 0  
K_{bias}_{}_{y1}  other contributions to the mass bias of R_{y1}  
K_{0}_{}_{z1}  mass bias correction of R_{z1} as determined at time 0  
K_{bias}_{}_{z1}  other contributions to the mass bias of R_{z1}  
K_{0}_{}_{z2}  mass bias correction of R_{z2} as determined at time 0  
K_{bias}_{}_{z2}  other contributions to the mass bias of R_{z2}  
K_{0}_{}_{z3}  mass bias correction of R_{z3} as determined at time 0  
K_{bias}_{}_{z3}  other contributions to the mass bias of R_{z3}  
K_{0}_{}_{z4}  mass bias correction of R_{z4} as determined at time 0  
K_{bias}_{}_{z4}  other contributions to the mass bias of R_{z4} 
m_{y1}: 
Type B normal distribution Value: 1.1360 g Expanded Uncertainty: 0.0002 g Coverage Factor: 1 
Weighings are performed by a dedicated mass metrology lab. 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. The uncertainties from the weighing certificates were treated as standard uncertainties, Type B.
m_{x}: 
Type B normal distribution Value: 1.0440 g Expanded Uncertainty: 0.0002 g Coverage Factor: 1 
Weighings are performed by a dedicated mass metrology lab. 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. The uncertainties from the weighing certificates were treated as standard uncertainties, Type B.
m_{z}: 
Type B normal distribution Value: 1.1029 g Expanded Uncertainty: 0.0002 g Coverage Factor: 1 
Weighings are performed by a dedicated mass metrology lab. 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. The uncertainties from the weighing certificates were treated as standard uncertainties, Type B.
m_{y2}: 
Type B normal distribution Value: 1.0654 g Expanded Uncertainty: 0.0002 g Coverage Factor: 1 
Weighings are performed by a dedicated mass metrology lab. 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. The uncertainties from the weighing certificates were treated as standard uncertainties, Type B.
R_{y1}: 
Type A summarized Mean: 0.00064 Standard Deviation of the Mean: =0.00004/sqrt(8) Degrees of Freedom: 7 
Each ratio has been measured 8 times. The experimental uncertainty is therefore divided by sqrt(8).
R_{b1}: 
Type A summarized Mean: 0.29360 Standard Deviation of the Mean: =0.00073/sqrt(8) Degrees of Freedom: 7 
Each ratio has been measured 8 times. The experimental uncertainty is therefore divided by sqrt(8).
R_{x1}: 
Type A summarized Mean: 2.1402 Standard Deviation of the Mean: =0.0054/sqrt(8) Degrees of Freedom: 7 
Each ratio has been measured 8 times. The experimental uncertainty is therefore divided by sqrt(8).
R_{b2}: 
Type A summarized Mean: 0.5050 Standard Deviation of the Mean: =0.0013/sqrt(8) Degrees of Freedom: 7 
Each ratio has been measured 8 times. The experimental uncertainty is therefore divided by sqrt(8).
R_{z1}: 
Type A summarized Mean: 2.1429 Standard Deviation of the Mean: =0.0054/sqrt(8) Degrees of Freedom: 7 
Each ratio has been measured 8 times. The experimental uncertainty is therefore divided by sqrt(8).
c_{blank}: 
Type A summarized Mean: 4.5·10^{7} µmol/g Standard Deviation of the Mean: =4.0e7/sqrt(4) Degrees of Freedom: 3 
The procedure blank was measured using external calibration. The procedure blank was measured four times. The experimental standard deviation is divided by sqrt(4) to obtain the standard uncertainty.
R_{x2}: 
Constant Value: 1 
This is the ratio of n(^{206}Pb)/n(^{206}Pb), which is by definition equal to 1.
R_{x3}: 
Type A summarized Mean: 0.9142 Standard Deviation of the Mean: =0.0032/sqrt(8) Degrees of Freedom: 7 
Each ratio has been measured 8 times. The experimental uncertainty is therefore divided by sqrt(8).
R_{x4}: 
Type A summarized Mean: 0.05901 Standard Deviation of the Mean: =0.00035/sqrt(8) Degrees of Freedom: 7 
Each ratio has been measured 8 times. The experimental uncertainty is therefore divided by sqrt(8).
R_{z2}: 
Constant Value: 1 
This is the ratio of n(^{206}Pb)/n(^{206}Pb), which is by definition equal to 1.
R_{z3}: 
Type A summarized Mean: 0.9147 Standard Deviation of the Mean: =0.0032/sqrt(8) Degrees of Freedom: 7 
Each ratio has been measured 8 times. The experimental uncertainty is therefore divided by sqrt(8).
R_{z4}: 
Type A summarized Mean: 0.05870 Standard Deviation of the Mean: =0.00035/sqrt(8) Degrees of Freedom: 7 
Each ratio has been measured 8 times. The experimental uncertainty is therefore divided by sqrt(8).
M_{z1}: 
Type B normal distribution Value: 207.976636 g/mol Expanded Uncertainty: 0.000003 g/mol Coverage Factor: 1 
The nuclidic masses and their respective uncertainties are taken from literature. G. Audi and A. H. Wapstra, Nuclear Physics, A565 (1993).
M_{z2}: 
Type B normal distribution Value: 205.974449 g/mol Expanded Uncertainty: 0.000003 g/mol Coverage Factor: 1 
The nuclidic masses and their respective uncertainties are taken from literature. G. Audi and A. H. Wapstra, Nuclear Physics, A565 (1993).
M_{z3}: 
Type B normal distribution Value: 206.975880 g/mol Expanded Uncertainty: 0.000003 g/mol Coverage Factor: 1 
The nuclidic masses and their respective uncertainties are taken from literature. G. Audi and A. H. Wapstra, Nuclear Physics, A565 (1993).
M_{z4}: 
Type B normal distribution Value: 203.973028 g/mol Expanded Uncertainty: 0.000003 g/mol Coverage Factor: 1 
The nuclidic masses and their respective uncertainties are taken from literature. G. Audi and A. H. Wapstra, Nuclear Physics, A565 (1993).
m_{2}: 
Type B normal distribution Value: 1.0292 g Expanded Uncertainty: 0.0002 g Coverage Factor: 1 
Weighings are performed by a dedicated mass metrology lab. 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. The uncertainties from the weighing certificates were treated as standard uncertainties, Type B.
d_{2}: 
Type B normal distribution Value: 99.931 g Expanded Uncertainty: 0.01 g Coverage Factor: 1 
Weighings are performed by a dedicated mass metrology lab. 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. The uncertainties from the weighing certificates were treated as standard uncertainties, Type B.
m_{1}: 
Type B normal distribution Value: 0.36544 g Expanded Uncertainty: 0.00005 g Coverage Factor: 1 
Weighings are performed by a dedicated mass metrology lab. 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. The uncertainties from the weighing certificates were treated as standard uncertainties, Type B.
w: 
Type B normal distribution Value: 0.99999 g/g Expanded Uncertainty: 0.000005 g/g Coverage Factor: 1 
The purity of the metalic lead can be obtained through analysis or a supplier's certificate.
d_{1}: 
Type B normal distribution Value: 196.14 g Expanded Uncertainty: 0.03 g Coverage Factor: 1 
Weighings are performed by a dedicated mass metrology lab. 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. The uncertainties from the weighing certificates were treated as standard uncertainties, Type B.
k_{mol}: 
Constant Value: 1·10^{6} µmol/mol 
K_{0}_{_}_{b1}: 
Type A summarized Mean: 0.9987 Standard Deviation of the Mean: =0.0025/sqrt(8) Degrees of Freedom: 7 
The K_{0}'s are measured using a certfied isotopic reference material, and they are calculated according to the following equation:
K_{0} = R_{certified}/R_{observed}
When looking at the uncertainty contributions of R_{certified} and R_{observed,} it is clear that the contribution of R_{certified} can be neglected for this example. Henceforth, the uncertainties on the measured ratios, R_{observed}, are used for the uncertainties on K_{0.}
The original measurement data for the determination of K_{0} is not shown in this example.
K_{bias}_{_}_{b1}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.001 Coverage Factor: 1 
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor (these could be variations over time, as well as other sources of bias, such as multiplier dead time correction, matrix effects etc.). The values of these biases are not known in this case, but they are assumed to be around 0, therefore a value of 0 is applied. An uncertainty is associated to this bias, which is estimated from experience. In this case a standard uncertainty of 0.001 is considered to be sufficient to cover all effects.
K_{0}_{_}_{b2}: 
Type A summarized Mean: 0.9983 Standard Deviation of the Mean: =0.0025/sqrt(8) Degrees of Freedom: 7 
The K_{0}'s are measured using a certfied isotopic reference material, and they are calculated according to the following equation:
K_{0} = R_{certified}/R_{observed}
When looking at the uncertainty contributions of R_{certified} and R_{observed,} it is clear that the contribution of R_{certified} can be neglected for this example. Henceforth, the uncertainties on the measured ratios, R_{observed}, are used for the uncertainties on K_{0.}
The original measurement data for the determination of K_{0} is not shown in this example.
K_{bias}_{_}_{b2}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.001 Coverage Factor: 1 
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor (these could be variations over time, as well as other sources of bias, such as multiplier dead time correction, matrix effects etc.). The values of these biases are not known in this case, but they are assumed to be around 0, therefore a value of 0 is applied. An uncertainty is associated to this bias, which is estimated from experience. In this case a standard uncertainty of 0.001 is considered to be sufficient to cover all effects.
K_{0}_{_}_{x1}: 
Type A summarized Mean: 0.9992 Standard Deviation of the Mean: =0.0025/sqrt(8) Degrees of Freedom: 7 
The K_{0}'s are measured using a certfied isotopic reference material, and they are calculated according to the following equation:
K_{0} = R_{certified}/R_{observed}
When looking at the uncertainty contributions of R_{certified} and R_{observed,} it is clear that the contribution of R_{certified} can be neglected for this example. Henceforth, the uncertainties on the measured ratios, R_{observed}, are used for the uncertainties on K_{0.}
The original measurement data for the determination of K_{0} is not shown in this example.
K_{bias}_{_}_{x1}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.001 Coverage Factor: 1 
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor (these could be variations over time, as well as other sources of bias, such as multiplier dead time correction, matrix effects etc.). The values of these biases are not known in this case, but they are assumed to be around 0, therefore a value of 0 is applied. An uncertainty is associated to this bias, which is estimated from experience. In this case a standard uncertainty of 0.001 is considered to be sufficient to cover all effects.
K_{0}_{_}_{x2}: 
Constant Value: 1 
This mass bias correction refers to the ratio of n(^{206}Pb)/n(^{206}Pb), which is by definition equal to 1 and not measured. Therefore no mass bias correction is needed, the factor is equal to 1.
K_{bias}_{_}_{x2}: 
Constant Value: 0 
This mass bias correction refers to the ratio of n(^{206}Pb)/n(^{206}Pb), which is by definition equal to 1 and not measured. Therefore no mass bias correction is needed, this factor is equal to 0.
K_{0}_{_}_{x3}: 
Type A summarized Mean: 1.0004 Standard Deviation of the Mean: =0.0035/sqrt(8) Degrees of Freedom: 7 
The K_{0}'s are measured using a certfied isotopic reference material, and they are calculated according to the following equation:
K_{0} = R_{certified}/R_{observed}
When looking at the uncertainty contributions of R_{certified} and R_{observed,} it is clear that the contribution of R_{certified} can be neglected for this example. Henceforth, the uncertainties on the measured ratios, R_{observed}, are used for the uncertainties on K_{0.}
The original measurement data for the determination of K_{0} is not shown in this example.
K_{bias}_{_}_{x3}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.001 Coverage Factor: 1 
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor (these could be variations over time, as well as other sources of bias, such as multiplier dead time correction, matrix effects etc.). The values of these biases are not known in this case, but they are assumed to be around 0, therefore a value of 0 is applied. An uncertainty is associated to this bias, which is estimated from experience. In this case a standard uncertainty of 0.001 is considered to be sufficient to cover all effects.
K_{0}_{_}_{x4}: 
Type A summarized Mean: 1.001 Standard Deviation of the Mean: =0.006/sqrt(8) Degrees of Freedom: 7 
The K_{0}'s are measured using a certfied isotopic reference material, and they are calculated according to the following equation:
K_{0} = R_{certified}/R_{observed}
When looking at the uncertainty contributions of R_{certified} and R_{observed,} it is clear that the contribution of R_{certified} can be neglected for this example. Henceforth, the uncertainties on the measured ratios, R_{observed}, are used for the uncertainties on K_{0.}
The original measurement data for the determination of K_{0} is not shown in this example.
K_{bias}_{_}_{x4}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.001 Coverage Factor: 1 
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor (these could be variations over time, as well as other sources of bias, such as multiplier dead time correction, matrix effects etc.). The values of these biases are not known in this case, but they are assumed to be around 0, therefore a value of 0 is applied. An uncertainty is associated to this bias, which is estimated from experience. In this case a standard uncertainty of 0.001 is considered to be sufficient to cover all effects.
K_{0}_{_}_{y1}: 
Type A summarized Mean: 0.9999 Standard Deviation of the Mean: =0.0025/sqrt(8) Degrees of Freedom: 7 
The K_{0}'s are measured using a certfied isotopic reference material, and they are calculated according to the following equation:
K_{0} = R_{certified}/R_{observed}
When looking at the uncertainty contributions of R_{certified} and R_{observed,} it is clear that the contribution of R_{certified} can be neglected for this example. Henceforth, the uncertainties on the measured ratios, R_{observed}, are used for the uncertainties on K_{0.}
The original measurement data for the determination of K_{0} is not shown in this example.
K_{bias}_{_}_{y1}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.001 Coverage Factor: 1 
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor (these could be variations over time, as well as other sources of bias, such as multiplier dead time correction, matrix effects etc.). The values of these biases are not known in this case, but they are assumed to be around 0, therefore a value of 0 is applied. An uncertainty is associated to this bias, which is estimated from experience. In this case a standard uncertainty of 0.001 is considered to be sufficient to cover all effects.
K_{0}_{_}_{z1}: 
Type A summarized Mean: 0.9989 Standard Deviation of the Mean: =0.0025/sqrt(8) Degrees of Freedom: 7 
The K_{0}'s are measured using a certfied isotopic reference material, and they are calculated according to the following equation:
K_{0} = R_{certified}/R_{observed}
When looking at the uncertainty contributions of R_{certified} and R_{observed,} it is clear that the contribution of R_{certified} can be neglected for this example. Henceforth, the uncertainties on the measured ratios, R_{observed}, are used for the uncertainties on K_{0.}
The original measurement data for the determination of K_{0} is not shown in this example.
K_{bias}_{_}_{z1}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.001 Coverage Factor: 1 
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor (these could be variations over time, as well as other sources of bias, such as multiplier dead time correction, matrix effects etc.). The values of these biases are not known in this case, but they are assumed to be around 0, therefore a value of 0 is applied. An uncertainty is associated to this bias, which is estimated from experience. In this case a standard uncertainty of 0.001 is considered to be sufficient to cover all effects.
K_{0}_{_}_{z2}: 
Constant Value: 1 
This mass bias correction refers to the ratio of n(^{206}Pb)/n(^{206}Pb), which is by definition equal to 1 and not measured. Therefore no mass bias correction is needed, the factor is equal to 1.
K_{bias}_{_}_{z2}: 
Constant Value: 0 
This mass bias correction refers to the ratio of n(^{206}Pb)/n(^{206}Pb), which is by definition equal to 1 and not measured. Therefore no mass bias correction is needed, this factor is equal to 0.
K_{0}_{_}_{z3}: 
Type A summarized Mean: 0.9993 Standard Deviation of the Mean: =0.0035/sqrt(8) Degrees of Freedom: 7 
The K_{0}'s are measured using a certfied isotopic reference material, and they are calculated according to the following equation:
K_{0} = R_{certified}/R_{observed}
When looking at the uncertainty contributions of R_{certified} and R_{observed,} it is clear that the contribution of R_{certified} can be neglected for this example. Henceforth, the uncertainties on the measured ratios, R_{observed}, are used for the uncertainties on K_{0.}
The original measurement data for the determination of K_{0} is not shown in this example.
K_{bias}_{_}_{z3}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.001 Coverage Factor: 1 
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor (these could be variations over time, as well as other sources of bias, such as multiplier dead time correction, matrix effects etc.). The values of these biases are not known in this case, but they are assumed to be around 0, therefore a value of 0 is applied. An uncertainty is associated to this bias, which is estimated from experience. In this case a standard uncertainty of 0.001 is considered to be sufficient to cover all effects.
K_{0}_{_}_{z4}: 
Type A summarized Mean: 1.0002 Standard Deviation of the Mean: =0.006/sqrt(8) Degrees of Freedom: 7 
The K_{0}'s are measured using a certfied isotopic reference material, and they are calculated according to the following equation:
K_{0} = R_{certified}/R_{observed}
When looking at the uncertainty contributions of R_{certified} and R_{observed,} it is clear that the contribution of R_{certified} can be neglected for this example. Henceforth, the uncertainties on the measured ratios, R_{observed}, are used for the uncertainties on K_{0.}
The original measurement data for the determination of K_{0} is not shown in this example.
K_{bias}_{_}_{z4}: 
Type B normal distribution Value: 0 Expanded Uncertainty: 0.001 Coverage Factor: 1 
This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor (these could be variations over time, as well as other sources of bias, such as multiplier dead time correction, matrix effects etc.). The values of these biases are not known in this case, but they are assumed to be around 0, therefore a value of 0 is applied. An uncertainty is associated to this bias, which is estimated from experience. In this case a standard uncertainty of 0.001 is considered to be sufficient to cover all effects.
Interim Results:
Quantity  Value 
Standard Uncertainty 

c_{z}  0.0926048 µmol/g  27.8·10^{6} µmol/g 
K_{y1}  0.99990  1.33·10^{3} 
K_{b1}  0.99870  1.33·10^{3} 
K_{x1}  0.99920  1.33·10^{3} 
K_{b2}  0.99830  1.33·10^{3} 
K_{z1}  0.99890  1.33·10^{3} 
ΣK_{zi}R_{zi}  4.11331  3.90·10^{3} 
ΣK_{xi}R_{xi}  4.11212  3.90·10^{3} 
K_{x2}  1.0  not valid! 
K_{x3}  1.00040  1.59·10^{3} 
K_{x4}  1.00100  2.35·10^{3} 
K_{z2}  1.0  not valid! 
K_{z3}  0.99930  1.59·10^{3} 
K_{z4}  1.00020  2.35·10^{3} 
M_{Pb}_{ }_{Assay1}  207.210345 g/mol  665·10^{6} g/mol 
Uncertainty Budgets:
c_{x}: amount content of the sample x
Quantity  Value 
Standard Uncertainty 
Distribution 
Sensitivity Coefficient 
Uncertainty Contribution 
Index 

m_{y1}  1.136000 g  200·10^{6} g  normal  0.047  9.5·10^{6} µmol/g  0.3 % 
m_{x}  1.044000 g  200·10^{6} g  normal  0.051  10·10^{6} µmol/g  0.3 % 
m_{z}  1.102900 g  200·10^{6} g  normal  0.049  9.7·10^{6} µmol/g  0.3 % 
m_{y2}  1.065400 g  200·10^{6} g  normal  0.050  10·10^{6} µmol/g  0.3 % 
R_{y1}  640.0·10^{6}  14.1·10^{6}  normal  0.077  1.1·10^{6} µmol/g  0.0 % 
R_{b1}  0.293600  258·10^{6}  normal  0.21  55·10^{6} µmol/g  9.3 % 
R_{x1}  2.14020  1.91·10^{3}  normal  0.016  31·10^{6} µmol/g  2.9 % 
R_{b2}  0.505000  460·10^{6}  normal  0.14  64·10^{6} µmol/g  12.7 % 
R_{z1}  2.14290  1.91·10^{3}  normal  0.020  38·10^{6} µmol/g  4.4 % 
c_{blank}  450·10^{9} µmol/g  200·10^{9} µmol/g  normal  1.0  200·10^{9} µmol/g  0.0 % 
R_{x2}  1.0  
R_{x3}  0.91420  1.13·10^{3}  normal  0.013  15·10^{6} µmol/g  0.7 % 
R_{x4}  0.059010  124·10^{6}  normal  0.013  1.6·10^{6} µmol/g  0.0 % 
R_{z2}  1.0  
R_{z3}  0.91470  1.13·10^{3}  normal  0.013  15·10^{6} µmol/g  0.7 % 
R_{z4}  0.058700  124·10^{6}  normal  0.013  1.6·10^{6} µmol/g  0.0 % 
M_{z1}  207.97663600 g/mol  3.00·10^{6} g/mol  normal  130·10^{6}  400·10^{12} µmol/g  0.0 % 
M_{z2}  205.97444900 g/mol  3.00·10^{6} g/mol  normal  63·10^{6}  190·10^{12} µmol/g  0.0 % 
M_{z3}  206.97588000 g/mol  3.00·10^{6} g/mol  normal  58·10^{6}  170·10^{12} µmol/g  0.0 % 
M_{z4}  203.97302800 g/mol  3.00·10^{6} g/mol  normal  3.7·10^{6}  11·10^{12} µmol/g  0.0 % 
m_{2}  1.029200 g  200·10^{6} g  normal  0.052  10·10^{6} µmol/g  0.3 % 
d_{2}  99.9310 g  0.0100 g  normal  540·10^{6}  5.4·10^{6} µmol/g  0.0 % 
m_{1}  0.3654400 g  50.0·10^{6} g  normal  0.15  7.4·10^{6} µmol/g  0.2 % 
w  0.99999000 g/g  5.00·10^{6} g/g  normal  0.054  270·10^{9} µmol/g  0.0 % 
d_{1}  196.1400 g  0.0300 g  normal  270·10^{6}  8.2·10^{6} µmol/g  0.2 % 
k_{mol}  1.0·10^{6} µmol/mol  
K_{0}_{}_{b1}  0.998700  884·10^{6}  normal  0.062  55·10^{6} µmol/g  9.4 % 
K_{bias}_{}_{b1}  0.0  1.00·10^{3}  normal  0.062  62·10^{6} µmol/g  12.1 % 
K_{0}_{}_{b2}  0.998300  884·10^{6}  normal  0.070  62·10^{6} µmol/g  12.0 % 
K_{bias}_{}_{b2}  0.0  1.00·10^{3}  normal  0.070  70·10^{6} µmol/g  15.3 % 
K_{0}_{}_{x1}  0.999200  884·10^{6}  normal  0.034  30·10^{6} µmol/g  2.8 % 
K_{bias}_{}_{x1}  0.0  1.00·10^{3}  normal  0.034  34·10^{6} µmol/g  3.6 % 
K_{0}_{}_{x2}  1.0  
K_{bias}_{}_{x2}  0.0  
K_{0}_{}_{x3}  1.00040  1.24·10^{3}  normal  0.012  15·10^{6} µmol/g  0.7 % 
K_{bias}_{}_{x3}  0.0  1.00·10^{3}  normal  0.012  12·10^{6} µmol/g  0.4 % 
K_{0}_{}_{x4}  1.00100  2.12·10^{3}  normal  770·10^{6}  1.6·10^{6} µmol/g  0.0 % 
K_{bias}_{}_{x4}  0.0  1.00·10^{3}  normal  770·10^{6}  770·10^{9} µmol/g  0.0 % 
K_{0}_{}_{y1}  0.999900  884·10^{6}  normal  49·10^{6}  44·10^{9} µmol/g  0.0 % 
K_{bias}_{}_{y1}  0.0  1.00·10^{3}  normal  49·10^{6}  49·10^{9} µmol/g  0.0 % 
K_{0}_{}_{z1}  0.998900  884·10^{6}  normal  0.042  37·10^{6} µmol/g  4.3 % 
K_{bias}_{}_{z1}  0.0  1.00·10^{3}  normal  0.042  42·10^{6} µmol/g  5.5 % 
K_{0}_{}_{z2}  1.0  
K_{bias}_{}_{z2}  0.0  
K_{0}_{}_{z3}  0.99930  1.24·10^{3}  normal  0.012  15·10^{6} µmol/g  0.7 % 
K_{bias}_{}_{z3}  0.0  1.00·10^{3}  normal  0.012  12·10^{6} µmol/g  0.4 % 
K_{0}_{}_{z4}  1.00020  2.12·10^{3}  normal  750·10^{6}  1.6·10^{6} µmol/g  0.0 % 
K_{bias}_{}_{z4}  0.0  1.00·10^{3}  normal  750·10^{6}  750·10^{9} µmol/g  0.0 % 
c_{x}  0.053737 µmol/g  180·10^{6} µmol/g 
Results:
Quantity  Value 
Expanded Uncertainty 
Coverage factor 
Coverage 

c_{x}  0.05374 µmol/g  180·10^{6} µmol/g  1.00  manual 