# Combining Isotope Dilution and Standard Addition—Elemental Analysis in Complex Samples

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{i})—each consisting of roughly the same masses (m

_{x,i}) of the sample solution (x) and m

_{y,i}of a spike solution (y) plus different masses (m

_{z,i}) of a reference solution (z). Only these masses and the isotope ratios (R

_{b,i}) in the blends and reference and spike solutions have to be measured. The derivation of the underlying equations based on linear regression is presented and compared to a related concept reported by Pagliano and Meija. The uncertainties achievable, e.g., in the case of the Si blank in extremely pure TMAH of u

_{rel}(w(Si)) = 90% (linear regression method, this work) and u

_{rel}(w(Si)) = 150% (the method reported by Pagliano and Meija) seem to suggest better applicability of the new method in practical use due to the higher robustness of regression analysis.

## 1. Introduction

_{A}with the lowest associated measurement uncertainty and, after the SI revision in 2019, aimed at the dissemination of the SI units of kilogram and mole [19,20,21,22]. In that context, chemically highly pure silicon, extensively enriched in

^{28}Si, with x(

^{28}Si) > 0.9999 mol/mol, was used. For the determination of the respective molar mass M of the silicon, u

_{rel}(M) must be smaller than 1 × 10

^{−8}. This can only be achieved by reducing both the contamination and the quantification of the remaining natural silicon, which is described as an application of the new method presented.

_{x}) of sulfur in a biodiesel fuel matrix. These measurements were performed within an interlaboratory key comparison conducted by the IAWG of the CCQM (Inorganic Analysis Working Group of the Consultative Committee for Amount of Substance: Metrology in Chemistry and Biology): CCQM-K123 [23]. The mass fractions of several trace elements (impurities)—among them, sulfur—were determined. Originally, w

_{x}was determined using an established IDMS technique. In addition, one of the biodiesel fuel samples was measured using the new combined IDMS-Standard Addition approach as a complementary method (this work). This enables the comparability of the new approach with an established and validated method. The biodiesel fuel matrix is an extremely complex matrix because of its volatility and impurities. Sulfur occurs in different compounds in biodiesel fuel, and, for obtaining accurate results, the conversion to sulfate must be ensured in the digestion step. Additionally, sulfur contamination during the whole analytical procedure is an issue and must be controlled.

_{x}of elements in complex matrices. For better understanding and adaption to our notation, we have additionally derived the method reported in [1] from scratch for the three blends case: we ended up with exactly the same equation (which is Equation (9) in [1]). For the comparison, we evaluated all measurements using our new method (this work) and the one reported in [1].

## 2. Theoretical Methods

_{1}and y-intercept a

_{0}, are then used to calculate the aimed-at mass fraction w

_{x}of the analyte element in the original sample (x). The detailed derivation of Equations (1) and (2) is given in Appendix A.

_{x}is directly proportional to the mass fraction w

_{z}of a reference material (z; same element as the sample with the same isotopic composition as the analyte or, at least, very close to it); m

_{y,i}, m

_{x,i}, and m

_{z,i}are the masses of a spike solution (y; same element as x with a preferably inverse isotopic composition), sample x, and reference material solution z, respectively. A number of blends (b

_{i}; e.g., five) containing almost the same masses of x and y, respectively, and different masses of z (m

_{z,1}< m

_{z,2}< m

_{z,3}< m

_{z,4}< m

_{z,5}) are required for the experiment and evaluation (Figure 1 and Figure 2). R

_{y}, R

_{x}, and R

_{b,i}, denote isotope ratios measured in the respective spike, sample, and ith blend solution related to the isotope of major abundance (reference isotope; in the case of enriched silicon:

^{28}Si; R

_{x}= I(

^{30}Si)/I(

^{28}Si), with the intensities I of the measured isotope signals); M

_{x}and M

_{y}are the molar masses of the respective analyte and spike material, which need not be known. The only requirement is the knowledge of mass fraction w

_{z}of the element of interest in reference material z. This is given in a calibration certificate or accessible via a separate measurement (gravimetric preparation).

_{y}, for which no value measured in the matrix is available. Reasonable spike materials feature R

_{y}> 100. In this case, mixing ratio R

_{y}, measured in a matrix different from that of the sample matrix, with R

_{x}and R

_{b,i}measured in the sample matrix, changes the result (w

_{x}) well within its uncertainty and, therefore, is insignificant.

_{x}in the notation used in this work. This equation is exactly the same as Equation (9) in [1]. Additionally, mass fraction w

_{z}of the reference material used has to be known. Masses (m

_{j}

_{,i}) of components j (j = sample x, reference z, or spike y) in blends i (i = 1, 2, 3) have to be known (gravimetric preparation) as well as the respective isotope ratios, namely, the R

_{b3}reads:isotope ratio (monitor vs. reference isotope) in Blend 3, the R

_{z,2}reads:isotope ratio (monitor vs. reference isotope) in reference z, and x

_{z,1}reads:amount-of-substance fraction x of Isotope 1 (reference isotope) in reference material z. Note that the “reference” isotope is the isotope of the highest abundance in the sample, whereas “reference” material is a characterized material with (almost) the same isotopic composition as the sample. In the case of different isotopic compositions of x and z, the molar masses, M

_{x}(sample) and M

_{z}(reference), and the amount-of-substance fractions, x

_{z,1}and x

_{x,1}, have to be known. When applying Equation (3), three gravimetrically prepared blends are required.

## 3. Materials and Experimental Methods

^{30}Si dissolved in aqueous TMAH, whereas reference material z was prepared from well-characterized silicon crystals (material code: WASO04) with known natural-like isotopic composition [30]. All stock solutions, dilutions, and blends in this study were prepared gravimetrically, applying air buoyancy correction [31]. The silicon crystals used for the stock solutions of y (spike) and z (reference) were cleaned and etched prior to dissolution with the final mass fractions prior to blend preparation: w

_{z}= 4 µg/g; w

_{y}= 2 µg/g. The isotopic composition of y as well as the respective “true” isotope ratios, R

_{y,}were taken from [20]. For the determination of w

_{x}according to Equations (1) and (2), a series of five blends, b

_{i}, was prepared, each consisting of approximately m

_{x,i}= 10 g, m

_{y,i}= 22.5 g, and stepwise increasing amounts of solution z with m

_{z,1}= 0 g, m

_{z,2}= 7.5 g, m

_{z,3}= 10 g, m

_{z,4}= 14.5 g, and m

_{z,5}= 23 g (compare Figure 1).

_{3}/H

_{2}O

_{2}) using a high-pressure asher system (Anton Paar, Graz, Austria), with ϑ

_{max}≈ 300 °C and p

_{max}≈ 130 bar. After digestion, the solution was evaporated to dryness, redissolved with 2 mL HNO

_{3}(0.028 mol/L), and loaded on chromatographic columns filled with 1 mL AG 1X8 resin. The sample matrix was eluted with water, and, subsequently, the sulfur was eluted with 8 mL HNO

_{3}(0.25 mol/L). Samples were evaporated to dryness and redissolved in HNO

_{3}(0.02 kg/kg) to adjust a sulfur mass fraction of 2 mg/kg. Prior to the MC–ICP–MS measurements, sodium was added so that a final Na mass fraction of 4 mg/kg was achieved. The addition of sodium significantly increases sensitivity and prevents losses of sulfur through the membrane of the desolvating nebulizer system, presumably as sulfuric acid [32]. The complete analytical procedure was validated by applying the double IDMS calibration approach with reference material NIST SRM 2723a. The determined sulfur mass fraction was (10.94 ± 0.10) mg/kg, while the certified sulfur mass fraction was (10.90 ± 0.31) mg/kg, with k = 2 in both cases.

^{®}-DA470k/IFCC “Human Serum” (Merck KGaA, Darmstadt, Germany) was applied as the reference, and Seronorm

^{TM}Immunoprotein Lyo L-1 (Invicon GmbH, Munich, Germany), which has certified values and ranges in accordance with the Guidelines of the German Medical Association (Rili-BÄK) of 2014, was used as the sample [33]. Both materials were reconstituted according to the manufacturers’ protocols and thereafter used for sample preparation [33]. The iron saturation procedure was based on the method described by del Castillo Busto et al. and C. Frank et al. except for some optimizations [15,34]. For the individual blends, a solution containing 7.5 to 22.5 mg of an iron solution (250 µg/g in 2.5% HNO

_{3}, dilution from BAM A-primary-Fe-2; BAM, Berlin, Germany), 5 to 10 mg sodium carbonate (Na

_{2}CO

_{3}) solution (500 mmol/kg; BioUltra, Merck KGaA, Darmstadt, Germany), and 50 to 277 mg tris(hydroxymethyl)methylamine (Tris) buffer solution (12.5 mmol/kg, adjusted to pH 6.4 with acetic acid; BioUltra, Merck KGaA, Darmstadt, Germany) was mixed, and 0 to 200 mg of the reconstituted ERM

^{®}-DA470k/IFCC serum and 50 mg of the reconstituted Seronorm

^{TM}Immunoprotein Lyo L-1 serum were added. The amount of iron and Na

_{2}CO

_{3}solution was adjusted to the amount of added human serum, which increased from Blend 1 to Blend 4. All solutions were incubated at room temperature for 1.5 h. Then, 160 mg of the TRF spike was added. In Section 4.3, the added amounts of reference, sample, and spike solution for the different blends are given.

^{54}Fe spike from Trace Science International Corp. (Ontario, ON, Canada), with a certified isotopic abundance of

^{54}Fe 99.86%,

^{56}Fe 0.11%,

^{57}Fe 0.01%, and

^{58}Fe 0.02%. For this, 20 mg of apo TRF was dissolved in a solution containing 0.435 g Na

_{2}CO

_{3}(500 mmol/kg), 4.3 g Tris (12.5 mmol/kg), and 0.4 g

^{54}Fe (250 µg/g), adjusted to pH = 8 with 0.15 mmol/kg HNO

_{3}. After an incubation time of 3 h at room temperature, the solution was purified with a PD-10 desalting column (Cytiva GmbH, Freiburg im Breisgau, Germany), according to the manufacturer’s protocol. The spike solution was freeze-dried, and the solid was used for the preparation of a TRF spike solution of 2.3 mg/g, which was used to prepare the blends.

^{28}Si,

^{29}Si, and

^{30}Si) in each blend, b

_{i}(i = 1…5).

## 4. Results and Discussion

#### 4.1. Silicon in Aqueous TMAH

_{x}of the analyte in the respective sample x (TMAH

_{aq}) was derived according to Equation (2). A set of 5 blends is sufficient to obtain proper regression statistics.

_{x}(Si), obtained from six sequences (runs), are displayed in Figure 4 (left) and Table 3. They yield an average of w

_{x}(Si) = 0.081(73) µg/g. As proof of consistency of the individual measurement results, the concept of degrees of equivalence (d

_{i}) was applied. The individual d

_{i}of the respective measurements are plotted in Figure 4 (right). All data encompass the zero line with their associated expanded uncertainties, which is an approval of the complete consistency of the data.

#### 4.2. Sulfur in Biodiesel Fuel (BDF)

_{i}. The essential input and output quantities of one experimental run are given in Table 4. The corresponding linear regression curve of the respective dataset is displayed in Figure 5.

_{x}) of the analyte (sulfur) in the respective sample x (BDF), according to Equation (2).

_{x,corr}(S) = 7.36(11) µg/g. The individual results of w

_{x,corr}(S), measured in three runs, are shown in Figure 6 (left) and Table 5. Additionally, as proof of consistency, the concept of degrees of equivalence, d

_{i}, was applied. The d

_{i}of the respective measurements are plotted in Figure 6 (right). The single results encompass the zero line with their associated expanded uncertainties, which proves the complete consistency of the data. In the case of sulfur determination in this work, the respective sample was taken from a stock solution already used in the interlaboratory comparison CCQM-K123, as described in [23]. There, the sulfur determination was carried out using double IDMS, yielding w

_{x}= 7.394(46) µg/g. The corresponding result is shown in Figure 6 (left, red solid line) for comparison. Both the IDMS and the new combined IDMS/standard addition approach show excellent agreement within the limits of uncertainty.

#### 4.3. Transferrin in Human Serum

_{i}. The essential input and output quantities of one experimental run are given in Table 6. The corresponding linear regression curve of the respective dataset is displayed in Figure 7.

_{x}of the analyte TRF in the respective sample x (human serum) according to Equation (2).

_{x}(TRF), obtained from two independent measurement sequences, are displayed in Figure 8 (left). Both values, w

_{x}(TRF) = 2.128(45) mg/g and w

_{x}(TRF) = 2.340(76) mg/g for k = 1, are consistent with the certified target value and its allowed range [33]. As proof of consistency of the individual measurement results, the concept of degrees of equivalence, d

_{i}, was applied. The individual d

_{i}of the respective measurements are plotted in Figure 8 (right). All data encompass the zero line with their associated expanded uncertainties.

#### 4.4. Comparison of Linear Regression (This Work) and Analytical Solution (Pagliano and Meija)

_{1}, b

_{2}, b

_{3}, b

_{4}, b

_{5}). Figure 9 shows the results of w

_{x}(Si), determined in six sequences (runs), using five blends (this work) and three blends (b

_{2}, b

_{3}, b

_{4}; approach of [1] using Equation (3)). At first glance, the associated uncertainties of the results determined in this work (black circles) are significantly smaller than the respective uncertainties of the results determined using the approach of [1] (red circles).

_{1}, b

_{2}, b

_{3}or b

_{2}, b

_{3}, b

_{4}, or b

_{3}, b

_{4}, b

_{5}each yielded extremely different results. The analytical approach (Equation (9) of [1], which is equal to Equation (3) in this publication) is based on a set of three blends only, without the need for the measurement of R

_{y}, which appears initially as a benefit. However, the analysis shows that the three-blend application leads, in some cases, to unrealistic (not accurate) results (results of triples b

_{1}, b

_{2}, b

_{3}and b

_{3}, b

_{4}, b

_{5}). Equation (9) in [1] reacts extremely sensitively towards that kind of input data. In our case, the three inner blends yield the most reasonable (“accurate”) result, e.g., the first blend, b

_{1}, includes no reference material z (m

_{z}= 0 g). This suggests a rather unstable and sensitive applicability of Equation (9) of [1] compared to the linear regression approach of this work. Moreover, for practical applications, Equations (1) and (2) are much simpler and user-friendly, and the respective evaluations and measurements are much easier than in the case of the approach used in [1], although the latter is correct from the mathematical point of view.

_{x}, which is the second-most important outcome of the respective approach, we chose the results of the blend triple b

_{2}, b

_{3}, b

_{4}of the six sequences. As can be seen in Figure 9, the uncertainties obtained using the approach of this work are significantly smaller than those obtained using Equation (9) from [1].

_{b2}, R

_{b1}, and R

_{b3}, range in the 10

^{1}-10

^{2}region. This seems rather high and is mainly responsible for the elevated uncertainty compared to the smaller uncertainty obtained using linear regression in the approach of this work.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Derivation of Combined IDMS and Standard Addition Method**

_{b,i)}in the blends (i) containing the analyte can be expressed as the ratio of the sums of the amount-of-substances (n) of the respective components of x (sample), y (spike), and z (reference). In our notation, Subscript 1 denotes the reference and Subscript 2 the monitor (spike) isotope.

_{b,i}can be rewritten accordingly

_{0}and slope a

_{1}yield the mass fraction (w

_{x}) of the analyte element in the sample

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**Figure 1.**Schematic of a set of 5 blends (b

_{i}) gravimetrically prepared from approximately the same masses (m

_{x,i}) of analyte sample x, same masses (m

_{y,i}) of spike solution y, and different masses (m

_{z,i}) of reference solution z (m

_{z,1}< m

_{z,2}< m

_{z,3}< m

_{z,4}< m

_{z,5}; in this work, m

_{z,1}= 0 g).

**Figure 2.**Relationship of the initial components (sample x, spike y, and reference z) and the respective blends (b

_{i}), indicating (a) the abundances in which the components were blended and (b) the quantities, which have to be measured to be able to calculate analyte mass fraction w

_{x}in the sample.

**Figure 3.**Linear regression evaluation of the Si mass fraction in TMAH

_{aq}according to Equation (2). Dataset in Table 2.

**Figure 4.**(

**Left**): Mass fraction w

_{x}(Si) in TMAH

_{aq}. Error bars denote combined uncertainties (k = 1). The red dashed line indicates the average value. (

**Right**): Degrees of equivalence d

_{i}of the respective measurement results. Error bars indicate expanded uncertainties (k = 2) associated with d

_{i}. All single results are consistent with the average value since the respective uncertainties encompass the red dashed zero line.

**Figure 5.**Linear regression of the S mass fraction in biodiesel fuel according to Equation (2). Dataset in Table 4.

**Figure 6.**(

**Left**) Black circles: corrected mass fractions w

_{x,corr}(S) of sulfur in BDF and the associated uncertainties (k = 1); black dashed line: average of three single runs. Data from the new combined IDMS/standard addition approach (this work). Red solid line: average value of w(S) from BAM, applying IDMS and using the same solution [23]. Upper and lower associated uncertainties: red dotted lines. (

**Right**) Degrees of equivalence (d

_{i}) of the respective measurement results. Error bars indicate expanded uncertainties (k = 2) associated with d

_{i}. All single results are consistent with the average value since the respective uncertainties encompass the red dashed zero line.

**Figure 7.**Linear regression evaluation of the TRF mass fraction in human serum according to Equation (2). Dataset in Table 6.

**Figure 8.**(

**Left**) Black circles: mass fractions w

_{x}(TRF) of transferrin in human serum and the associated uncertainties (k = 1); dashed black line: certified value of the Seronorm sample (for instrument, see Beckmann, AU). Data from the new combined IDMS/standard addition approach (this work). Red dashed lines: allowed target range according to Rili-BÄK [33]. (

**Right**) Degrees of equivalence, d

_{i}, of the respective measurement results. Error bars indicate expanded uncertainties (k = 2) associated with d

_{i}. All single results are consistent with the certified value since the respective uncertainties encompass the red dashed zero line.

**Figure 9.**Black circles (error bars): mass fractions w

_{x}(Si) and the associated uncertainties (k = 1) of six runs determined using the approach of this work. Red circles (error bars): results of the same input data (blends b

_{2}, b

_{3}, b

_{4}only) using the approach of [1].

**Table 1.**Operation parameters of the mass spectrometric isotope ratio measurements applied for the three sample/matrix systems. (SP = spike; K = K factor).

Sample/Matrix | Silicon/TMAH | Sulfur/Biodiesel Fuel | TRF/Human Serum |
---|---|---|---|

Laboratory | PTB | BAM | PTB |

Instrument | Thermo MC–ICP–MS Neptune | Thermo MC-ICP-MS Neptune Plus | Agilent 8900 ICP-QQQ-MS |

Sample Introduction | PFA nebulizer 100 µL/min PEEK/PFA cyclonic + Scott chamber sapphire torch + injector BN shield Ni sampler + Ni X-skimmer | Aridus II desolvating system PFA nebulizer 100 µL/min Aridus PFA spray chamber standard torch and injector quartz shieldNi sampler + Ni H-skimmer | PFA MicroFlow nebulizer 700 µL, Scott chamber at 3 °C torch with 1 mm injector Pt shield Pt sampler and skimmer |

Gas Flow Rates (Ar) | cooling: 16 L min^{−1}auxiliary: 0.8 L min ^{−1}sample: 1.0 L min ^{−1} | cooling: 16 L min^{−1}auxiliary: 0.9 L min ^{−1}sample: 0.85 L min ^{−1} | cooling: 15 L min^{−1}auxiliary: 0.9 L min ^{−1}nebulizer gas: 0.8 L min ^{−}^{1}reaction gas (H _{2}): 6.1 mL min^{−}^{1} |

Machine Parameters | high resolution (M/∆M = 8000) RF power 1180 W integration time 4.194 s idle time 3 s number of blocks 6 cycles/block 3 rotating amplifiers: yes Faraday cups: L3( ^{28}Si), C(^{29}Si), H3(^{30}Si) | high resolution (M/∆M = 8000) RF power 1200 W integration time 4.194 s idle time 3 s number of blocks 1 cycles/block 40 rotating amplifiers: no Faraday cups: L3( ^{32}S), C(^{33}S), H3(^{34}S) | MS/MS mode RF power 1550 W Sample depth 8.0 mm x-lens configuration integration time 0.1 s m/z 53, 54, 56, 57, 58, 60 |

Sequence Settings | rinse time 120 s take-up time 60 s measured samples/sequence b1, b2, b3, b4, b5 (4 times each) | rinse time 30 s take-up time 80 s measured samples/sequence b1, b2, b3, b4, b5 (3 times each, separated by a block of 5 standards) | rinse time + take-up not applicable: HPLC separationmeasured samples/sequence b1, b2, b3, b4, blank, SP, K, blank (4 times) |

Separation Settings | Agilent Bioinert 1260 HPLC system Column: MonoQ ^{®} GL 5/50 from GE Healthcare (Uppsala, Sweden)Mobile phase A: 12.5 mmol/L Tris at pH = 6.4 Mobile phase B: 12.5 mmol/L Tris + 125 mmol/L NH _{4}Ac at pH = 6.4Flow: 0.5 mL min ^{−1}Gradient: 0 min → 0% B, 20 min → 100% B, 27 min → 100% B Column oven 30 °C Injection volume: 10 µL MWD 254 nm, 280 nm |

**Table 2.**Determination of w

_{x}(Si) in TMAH

_{aq}. The relevant input and output data of the linear regression analysis are given for a single representative dataset (Sequence 1), with w

_{z}= 4.0069 µg/g.

x | z | y | ||||||
---|---|---|---|---|---|---|---|---|

b_{i} | TMAH_{aq} | WASO04 | “Si30” | R_{b,i} | R_{x,i} | R_{y,i} | ||

i | m_{x,i} | m_{z,i} | m_{y,i} | I(^{30}Si)/I(^{28}Si) | x_{i} | y_{i} | I(^{30}Si)/I(^{28}Si) | I(^{30}Si)/I(^{28}Si) |

g | g | g | V/V | g/g | g/g | V/V | mol/mol | |

1 | 10.0863 | 0.0000 | 22.8557 | 113.77732 | 0.0000 | 1.80 | 0.03353 | 204.19578 |

2 | 9.7836 | 7.8858 | 22.7577 | 1.59655 | 0.8060 | 301.51 | 0.03353 | 204.19578 |

3 | 9.6700 | 10.5255 | 22.3198 | 1.18847 | 1.0885 | 405.71 | 0.03353 | 204.19578 |

4 | 11.3192 | 15.3000 | 22.4864 | 0.83531 | 1.3517 | 503.86 | 0.03353 | 204.19578 |

5 | 10.0440 | 22.4061 | 22.7651 | 0.61405 | 2.2308 | 794.84 | 0.03353 | 204.19578 |

a_{1} | a_{0} | w_{x} | ||||||

(g/g)/(g/g) | (g/g) | µg/g | ||||||

356.10062 | 11.474 | 0.13 |

**Table 3.**Mass fractions w

_{x}(Si) of silicon in TMAH

_{aq}and the associated uncertainties (k = 1).

Run | w_{x} (Si) | u(w_{x} (Si)) |
---|---|---|

µg/g | µg/g | |

1 | 0.13 | 0.11 |

2 | 0.12 | 0.10 |

3 | 0.12 | 0.10 |

4 | 0.044 | 0.012 |

5 | 0.040 | 0.011 |

6 | 0.028 | 0.007 |

average | 0.081 | 0.073 |

**Table 4.**Determination of w

_{x}in BDF. The relevant input and output data of linear regression analysis are given for a single representative dataset (1st run), with w

_{z}= 16.145 µg/g. A procedural blank was subtracted from the result. The procedural blank was determined during an external measurement, as described in [23].

x | z | y | ||||||
---|---|---|---|---|---|---|---|---|

b_{i} | BDF | NIST SRM 3154 | BAM S-34 | R_{b,i} | R_{x,i} | R_{y,i} | ||

i | m_{x,i} | m_{z,i} | m_{y,i} | I(^{32}S)/I(^{34}S) | x_{i} | y_{i} | I(^{32}S)/I(^{34}S) | I(^{32}S)/I(^{34}S) |

g | g | g | V/V | g/g | g/g | V/V | mol/mol | |

1 | 0.23748 | 0.00000 | 0.09670 | 0.20030 | 0.00000 | 0.00387 | 21.16643 | 0.00099 |

2 | 0.24149 | 0.10125 | 0.09843 | 0.36731 | 0.41927 | 0.00718 | 21.16643 | 0.00099 |

3 | 0.23819 | 0.20828 | 0.10899 | 0.48453 | 0.87443 | 0.01070 | 21.16643 | 0.00099 |

4 | 0.25126 | 0.30375 | 0.10631 | 0.64994 | 1.20891 | 0.01339 | 21.16643 | 0.00099 |

5 | 0.24311 | 0.40679 | 0.09966 | 0.83887 | 1.67328 | 0.01690 | 21.16643 | 0.00099 |

a_{1} | a_{0} | w_{x} | w_{x,corr} | |||||

(g/g)/(g/g) | g/g | µg/g | µg/g | |||||

0.007797 | 0.003894 | 8.063 | 7.36 |

**Table 5.**Blank corrected mass fractions w

_{x,corr}(S) of sulfur in biodiesel fuel (uncertainties with k = 1).

Run | w_{x,corr}(S) | u(w_{x,corr}(S)) |
---|---|---|

µg/g | µg/g | |

1 | 7.36 | 0.13 |

2 | 7.36 | 0.13 |

3 | 7.358 | 0.079 |

average | 7.36 | 0.11 |

**Table 6.**Determination of w

_{x}(TRF) in human serum. The relevant input and output data of linear regression analysis are given for a single representative dataset (M21-3), with w

_{z}= 2.296 mg/g.

x | z | y | ||||||
---|---|---|---|---|---|---|---|---|

b_{i} | Seronorm^{TM} Immuno-Protein Lyo L-1 | ERM^{®}-DA470k/IFCC | In-House Prepared TRF Spike | R_{b,i} | R_{x,i} | R_{y,i} | ||

i | m_{x,i} | m_{z,i} | m_{y,i} | R(^{54}Fe/^{56}Fe) | x_{i} | y_{i} | R(^{54}Fe/^{56}Fe) | R(^{54}Fe/^{56}Fe) |

g | g | g | mol/mol | g/g | g/g | mol/mol | mol/mol | |

1 | 0.04848 | 0.00000 | 0.15104 | 3.01104 | 0.00000 | 262.4 | 0.063703 | 251.22 |

2 | 2.99958 | 0.00000 | 263.4 | |||||

3 | 2.99154 | 0.00000 | 264.1 | |||||

4 | 2.98887 | 0.00000 | 264.4 | |||||

5 | 0.04923 | 0.05834 | 0.15123 | 1.31282 | 1.18510 | 614.6 | ||

6 | 1.32126 | 1.18510 | 610.5 | |||||

7 | 1.31656 | 1.18510 | 612.8 | |||||

8 | 1.31365 | 1.18510 | 614.2 | |||||

9 | 0.04910 | 0.07424 | 0.14909 | 1.15279 | 1.51220 | 697.3 | ||

10 | 1.15426 | 1.51220 | 696.3 | |||||

11 | 1.15733 | 1.51220 | 694.4 | |||||

12 | 1.14892 | 1.51220 | 699.8 | |||||

13 | 0.04923 | 0.14561 | 0.14951 | 0.74959 | 2.95788 | 1109.1 | ||

14 | 0.74722 | 2.95788 | 1113.0 | |||||

15 | 0.74440 | 2.95788 | 1117.6 | |||||

16 | 0.74494 | 2.95788 | 1116.7 | |||||

a_{1} | a_{0} | w_{x} | ||||||

(g/g)/(g/g) | g/g | mg/g | ||||||

287.078 | 266.04 | 2.128 |

**Table 7.**A representative result of mass fractions w

_{x}(Si) determined using the approach of this work (using all five blends) and the approach of [1] (using the three triple blends: b

_{1}, b

_{2}, b

_{3}; b

_{2}, b

_{3}, b

_{4}; and b

_{3}, b

_{4}, b

_{5}). Only the combination of b

_{2}, b

_{3}, b

_{4}(the inner blends) yielded a reasonable numerical result.

This Work | Approach of [1] |
---|---|

blends | blends |

b_{1}, b_{2}, b_{3}, b_{4}, b_{5} | b_{1}, b_{2}, b_{3} |

w_{x}(Si) | w_{x}(Si) |

µg/g | µg/g |

0.1292 | −0.5004 |

**Table 8.**A representative uncertainty budget of mass fraction w

_{x}(Si) determined using the approach of [1] (using the blends b

_{2}, b

_{3}, b

_{4}). The main contributions originate from R

_{b2}, R

_{b1}, and R

_{b3}, with the largest absolute values of respective sensitivity coefficients.

Quantity | Unit | Best Estimate (Value) | Standard Uncertainty | Sensitivity Coefficient | Index |
---|---|---|---|---|---|

X_{i} | [X_{i}] | x_{i} | u(x_{i}) | c_{i} | |

w_{z} | µg/g | 4.00694 | 6.01 × 10^{−3} | 0.031 | 0.0% |

m_{y1} | g | 22.49660 | 1.00 × 10^{−3} | 2.2 | 0.0% |

m_{z2} | g | 10.31270 | 1.00 × 10^{−3} | 11 | 0.0% |

m_{y3} | g | 22.26850 | 1.00 × 10^{−3} | 2.8 | 0.0% |

R_{b2} | V/V | 1.23933 | 3.62 × 10^{−3} | 92 | 74.4% |

R_{z2} | V/V | 0.033527 | 335 × 10^{−6} | ||

R_{b3} | V/V | 0.88472 | 1.50 × 10^{−3} | −73 | 8.0% |

R_{b1} | V/V | 1.68846 | 5.46 × 10^{−3} | −30 | 17.5% |

m_{y2} | g | 22.43470 | 1.00 × 10^{−3} | −5.0 | 0.0% |

m_{z3} | g | 14.57170 | 1.00 × 10^{−3} | −4.2 | 0.0% |

m_{z1} | g | 7.46840 | 1.00 × 10^{−3} | −6.4 | 0.0% |

m_{x2} | g | 9.22730 | 1.00 × 10^{−3} | 0.33 | 0.0% |

R_{x2} | V/V | 0.033527 | 335 × 10^{−6} | 10 | 0.0% |

m_{x3} | g | 9.79000 | 1.00 × 10^{−3} | −0.13 | 0.0% |

m_{x1} | g | 9.46490 | 1.00 × 10^{−3} | −0.20 | 0.0% |

w_{x} | g | 0.125 | 0.388 |

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**MDPI and ACS Style**

Brauckmann, C.; Pramann, A.; Rienitz, O.; Schulze, A.; Phukphatthanachai, P.; Vogl, J. Combining Isotope Dilution and Standard Addition—Elemental Analysis in Complex Samples. *Molecules* **2021**, *26*, 2649.
https://doi.org/10.3390/molecules26092649

**AMA Style**

Brauckmann C, Pramann A, Rienitz O, Schulze A, Phukphatthanachai P, Vogl J. Combining Isotope Dilution and Standard Addition—Elemental Analysis in Complex Samples. *Molecules*. 2021; 26(9):2649.
https://doi.org/10.3390/molecules26092649

**Chicago/Turabian Style**

Brauckmann, Christine, Axel Pramann, Olaf Rienitz, Alexander Schulze, Pranee Phukphatthanachai, and Jochen Vogl. 2021. "Combining Isotope Dilution and Standard Addition—Elemental Analysis in Complex Samples" *Molecules* 26, no. 9: 2649.
https://doi.org/10.3390/molecules26092649