Evaluation of Uncertainty for Determination of Trace Uranium in Biology by Laser Fluorescence Method

Abstract

Large amounts of uranium-containing radioactive waste are generated in reactor operation and in the research and manufacture of nuclear fuel elements. At present, there are various uranium enrichment and separation methods such as reduction precipitation method, ion exchange method, solvent extraction method, membrane filtration method, adsorption method, and microorganism method to treat the uranium-containing radioactive waste generated in the related processes of nuclear facilities. However, the airborne effluent or liquid effluent discharged after treatment may still contain radioactive uranium. It is well known that uranium is a radioactive heavy metal element, and its radioactive and chemical toxicity cannot be ignored. Uranium in the environment enters the human body through the food route, and its long half-life can make the human body suffer from continuous radioactive internal radiation damage. As an environmental medium, organisms are closely related to the entry of uranium into the human body through food. Therefore, it is of great significance to carry out accurate measurement of uranium content in environmental-grade biological samples around nuclear facilities, however, complete and accurate measurement results include measurement data and uncertainty. Laser fluorescence method is a method for rapid analysis of uranium content in environmental samples. It has the advantages of high sensitivity, simple sample pretreatment, and wide measurement range, which has been widely used in nuclear industry, environmental monitoring and scientific research. At present, there is a lack of relevant reports on the uncertainty of the measurement of total uranium content in environmental-grade biological samples by laser fluorescence method. It is of great significance to accurately measure the uranium content in biological samples by evaluating the uncertainty of this method. In this paper, the WGJ-III trace uranium analyzer was used to analyze the uncertainty source of total uranium in environmental-grade biological samples by laser fluorescence method. The uncertainty measurement model was established, the uncertainty components were quantified, and the expanded uncertainty of the measurement of total uranium content in environmental biological samples was calculated. The evaluation results showed that the expanded uncertainty of a 0.05 g environmental biological sample is 10.8% (k = 2) without dilution, and the dominant uncertainty component is derived from the measurement uncertainty of sample fluorescence counting.

1 Introduction

Large amounts of uranium-containing radioactive waste have been generated in reactor operation and in the research and manufacture of nuclear fuel elements. At present, there are various uranium enrichment and separation methods such as reduction precipitation method, ion exchange method, solvent extraction method, membrane filtration method, adsorption method, and microorganism method to treat the uranium-containing radioactive waste generated in the related processes of nuclear facilities. However, the airborne effluent or liquid effluent discharged from the treatment may contain radioactive uranium [1]. As a radioactive heavy metal element, uranium’s radioactive toxicity and chemical toxicity cannot be ignored, uranium in the environment enters the human body through food, and its long half-life can make the human body suffer from continuous radioactive internal radiation damage [2]. As an environmental medium, organisms are closely related to the entry of uranium into the human body through food. Therefore, it is of great significance to accurately measure the total uranium content in environmental-grade biological samples around nuclear facilities, however, complete and accurate measurement results include measurement data and uncertainty.

The laser fluorescence method is famous for its rapid analysis of uranium content in environmental samples. It has the advantages of high sensitivity, simple sample pretreatment, and wide measurement range, which has been widely used in the nuclear industry, environmental monitoring and scientific research [3, 4]. As we all know, the complete measurement result of the sample should contain the measurement result data and measurement uncertainty, where the measurement uncertainty is used to indicate the reliability of the measurement result [5, 6]. After investigation, only the uncertainty evaluation method for the determination of uranium content in water, soil and uranium ore by laser fluorescence method has been reported, and there is a lack of relevant reports on the uncertainty of the measurement of total uranium content in environmental-grade biological samples. The evaluation of the uncertainty of the results obtained from the samples is of great significance for the accurate measurement of uranium content in organisms. In view of this, the laser fluorescence method has been used to measure trace uranium in organisms, and the uncertainty of the method has been evaluated in this paper.

2 Measurement Method and Measurement Model Analysis

The laser fluorescence method in the standard HJ840-2017 has been referenced to determine the total uranium content in biological samples in this paper. The determination principle is roughly as follows: first, the environmental biological samples are converted into water samples to be tested through pretreatment. Next, measure the fluorescence counts of the samples in the following order: the water sample to be tested, the water sample to be tested after adding the fluorescence enhancer, and the water sample to be tested after adding the uranium standard solution. The fluorescence intensity is proportional to the uranium content, when the uranium content is within a certain range. The uranium content can be calculated by measuring the fluorescence intensity. The mathematical model for the determination of trace uranium content in biological samples by laser fluorescence method has been shown in Eq. (1):

$$ A = \left( {\frac{{N_{1} - N_{0} }}{{N_{2} - N_{1} }} - \frac{{N^{\prime}_{1} - N^{\prime}_{0} }}{{N^{\prime}_{2} - N^{\prime}_{1} }}} \right) \times \frac{{KVMC_{1} V_{1} }}{{V_{0} WY}} $$

(1)

where A represent the uranium content in biological samples (μg/kg), N₀ represent the fluorescence intensity without adding uranium fluorescence enhancer, N₁ represent the fluorescence intensity with adding uranium fluorescence enhancer, N₂ represent the fluorescence intensity with adding uranium standard working fluid, N₀′, N₁′ and N₂′ represent the corresponding instrument reading for reagent blank sample, C₁ represent the concentration of uranium standard solution for measuring N₂ (μg/kg), V₁ represent the volume of uranium standard solution for measuring N₂ (mL), V represent the constant volume of dissolved biological sample ash (mL), V₀ represent the volume of sample, M represent the ratio between ash content and fresh content (g/kg), K represent the dilution ratio of water sample, W represent the ash weight of biological sample, Y represent the overall recovery (%).

3 Evaluation of Uncertainty

The uncertainty of trace uranium content in organisms consists of the following nine components:

• (1) Uncertainty of repeatability of sample fluorescence intensity measurement (u_1rel);

• (2) Uncertainty of concentration of uranium standard solution at 1 μg/mL (u_2rel);

• (3) Uncertainty of volume of uranium standard solution at 1 μg/mL (u_3rel);

• (4) Uncertainty of constant volume (u_4rel);

• (5) Uncertainty of sample volume (u_5rel);

• (6) Uncertainty of overall chemical recovery (u_6rel);

• (7) Uncertainty of sample quality (u_7rel);

• (8) Uncertainty of the ash/fresh ratio (u_8rel);

• (9) Uncertainty of diluted multiples (u_9rel);

3.1 Uncertainty of Repeatability of Sample Fluorescence Intensity Measurement (u_1rel)

The uncertainty of repeatability of sample fluorescence intensity measurement comes from 6 aspects which include uncertainty of repeatability of measurement without adding fluorescence enhancer u_11rel, uncertainty of measurement repeatability with adding fluorescence enhancer u_12rel, uncertainty of measurement repeatability with adding uranium standard solution u_13rel, uncertainty of repeatability of blank samples without adding fluorescence enhancer u_14rel, Uncertainty of measurement repeatability of blank samples with adding fluorescence enhancer u_15rel, Uncertainty of measurement repeatability of blank samples with adding uranium standard solution u_16rel.

Six physical quantities N₀, N₁, N₂, N₀′, N₁′ and N₂′ were measured by trace uranium analyzer, and A class evaluation method was adopted. After pretreatment of a 0.05 g biological sample and a blank sample, the fluorescence intensity of the six physical quantities mentioned above was measured repeatedly for 5 times respectively to calculate the mean value, and the standard deviation of the mean value was calculated by Bessel equation. The measured readings were shown in Table 1. The calculation equation for uncertainty is shown in Eq. (2). After calculation, the value of u_11rel, u_12rel, u_13rel, u_14rel, u_15rel and u_16rel are 2.41%, 1.27%, 1.39%, 2.77%, 2.20% and1.52%, respectively.

$$ u_{11rel} = \frac{1}{N}\sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {(N_{i} - \overline{N})^{2} } }}{n - 1}} $$

(2)

Table 1. Sample measurement results of fluorescence counting

The uncertainty of repeatability of sample fluorescence intensity measurement u_1rel is shown as follows:

$$ u_{1rel} = \sqrt {u_{11rel}^{2} + u_{12rel}^{2} + u_{13rel}^{2} + u_{14rel}^{2} + u_{15rel}^{2} + u_{16rel}^{2} } = 4.92\% $$

(3)

3.2 Uncertainty of Concentration of 1.0 μg/mL Uranium Standard Solution (u_2rel)

The concentration of uranium standard solution added for the determination of fluorescence intensity N₂ was 1.0 μg/mL. The solution was prepared by moving 1 mL 100 μg/mL uranium standard solution into a 100 mL volumetric flask with constant volume to the scale line through a 1 mL pipette. The concentration of diluted solution is shown in Eq. (4).

$$ C_{2} = \frac{{C_{1} V_{2} }}{{V_{2} }} $$

(4)

where C₂ is the concentration of the diluted solution (μg/mL), V₂ is the volume of the diluted solution (mL), C₁ is the concentration of the solution before dilution (μg/mL), V₁ is the volume of the solution before dilution (mL).

As can be seen from Eq. (4), Uncertainty of 1.0 ug/mL uranium standard solution concentration u_2rel is composed of uncertainty of concentration of 100 μg/mL uranium standard solution u_21rel, uncertainty in the sample volume of 100 ug/mL uranium standard solution u_22rel, and uncertainty of diluted solution volume u_23rel.

3.2.1 Uncertainty of Concentration of 100 μg/mL Uranium Standard Solution u21_rel

The standard solution uncertainty was evaluated by class B method, and the extended uncertainty of standard solution was U = 0.2% (k = 2), u_21rel = U/k = 0.1%.

3.2.2 Uncertainty of 100 ug/mL Sample Volume of Uranium Standard Solution u_22rel

The uncertainty of 100 ug/mL sample volume of uranium standard solution u_22rel is composed of the uncertainty of pipette accuracy u_221rel, the uncertainty of temperature correction u_222rel, the uncertainty of solution removal repeatability u_223rel.

• (1) Uncertainty of pipette accuracy u_221rel

  As can be seen from the pipette surface mark, the accuracy of l mL pipette is 0.02 ml. Assuming rectangular distribution, take the inclusion factor $k = \sqrt 3$ and calculate the uncertainty u_221rel of pipette accuracy according to the evaluation method of class B uncertainty as shown in Eq. (5).

  $$ u_{221rel} = \frac{U}{K} = \frac{{0.02\,{\text{mL}}}}{{1\,{\text{mL}} \times \sqrt 3 }} = 1.15\% $$

  (5)

• (2) Uncertainty of temperature correction u_222rel

  The temperature of the laboratory varies at ±5 ℃, and the liquid volume can change under the influence of the expansion coefficient. The volumetric expansion coefficient of water is 2.1 × 10^–4 ℃⁻¹. Assuming that the distribution of temperature change is rectangular, the calculation of the uncertainty of temperature calibration u_222rel is shown in Eq. (6) according to the evaluation method:

  $$ u_{222rel} = \frac{{2.1 \times 10^{ - 4} {}^{{\text{o}}}C^{ - 1} \times 5{}^{{\text{o}}}C}}{\sqrt 3 } \times 100\% = 0.0606\% $$

  (6)

• (3) Uncertainty of solution removal repeatability u_223rel

  Sample removal was 1.00 mL, and 1.00 mL distilled water was absorbed with a 1 mL pipette, and weighed with a balance for 20 times. The deviations introduced by constant volume repeatability were determined experimentally, and the results were shown in Table 2.

  Table 2. 1 mL pipette absorbed 1.00 mL distilled water quality change

  According to the evaluation method of class A uncertainty, the uncertainty of constant volume repeatability u_233rel is shown in Eq. (7).

  $$ u_{223rel} = \frac{1}{{\overline{m}}}\sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {(m_{i} - \overline{m})^{2} } }}{n - 1}} = 0.30\% $$

  (7)

  Therefore, the synthesis expression of uncertainty of 100 ug/mL sampling volume of uranium standard solution u_22rel is shown in Eq. (8).

  $$ u_{22rel} = \sqrt {u_{221rel}^{2} + u_{222rel}^{2} + u_{223rel}^{2} } = 1.19\% $$

  (8)

3.2.3 Uncertainty of Constant Volume of Diluted Solution u_23rel

The uncertainty of constant volume of diluted solution u_23rel consists of the uncertainty of volumetric flask accuracy u_231rel, the uncertainty of temperature correction u_232rel and the uncertainty of constant volume repeatability u_233rel.

• (1) Uncertainty of volumetric flask accuracy u_231rel

  A 100.00 mL volumetric flask with a precision of 0.10 mL was used to volume the solution to 100.00 mL. Assuming from triangular distribution, the inclusion factor $k = \sqrt 6$ was taken. According to the evaluation method of class B uncertainty, the volumetric flask accuracy uncertainty u_231rel was 0.04%.

• (2) Uncertainty of temperature correction u_232rel

  The laboratory temperature varies at ±5 ℃, and the calculation method of uncertainty of temperature correction u_232rel is the same as that of temperature correction uncertainty u_222rel in Sect. 3.2.2 is consistent, and u_232rel is 0.0606%.

• (3) Uncertainty of constant volume repeatability u_233rel

  The 100.00 mL volumetric bottle was fixed volume with distilled water, and the net weight of distilled water was weighed by a balance. The number n was 20 times. The deviation introduced by the constant volume repeatability was determined in the experiment, and detailed data results were shown in Table 3. According to the evaluation method of class A uncertainty, the uncertainty of constant volume repeatability u_232rel was 0.15%.

  Table 3. Quality change of distilled water in 100 mL volumetric bottle with constant volume

  Therefore, the synthesis expression of the uncertainty of constant volume of diluted solution u_23rel is shown in Eq. (9).

  $$ u_{23rel} = \sqrt {u_{231rel}^{2} + u_{232rel}^{2} + u_{233rel}^{2} } = 0.167\% $$

  (9)

  Overall, the synthesis expression of the Uncertainty of concentration of 1.0 μg/mL uranium standard solution u_2rel is shown in Eq. (10).

  $$ u_{2rel} = \sqrt {u_{21rel}^{2} + u_{22rel}^{2} + u_{23rel}^{2} } = 1.21\% $$

  (10)

3.3 Uncertainty of 1.0 μg/mL Uranium Standard Solution Volume (u3_rel)

The 1.0 pg/mL uranium standard solution was moved by a micro sampler, and u_3rel was composed of the uncertainty of measuring accuracy of micro sampler u_31rel, and the uncertainty of temperature correction u_32rel.

3.3.1 Uncertainty of Measurement Accuracy of Micro Sampler u_31rel

The accuracy of 10 μL micro sampler is 0.1 μL, assuming the rectangular distribution is followed, take the inclusion factor $k = \sqrt 3$ in Eq. (11):

$$ u_{31rel} = \frac{{0.1\,{\mu L}}}{{5\,{\mu L} \times \sqrt 3 }} = 1.15\% $$

(11)

3.3.2 Uncertainty of Temperature Correction u_32rel

The temperature of the laboratory varies at ±5 ℃, and the calculation method of the temperature correction uncertainty u_32rel is consistent with that of the temperature correction uncertainty u_222rel in Sect. 3.2.2, and the calculated u_32rel is 0.0606%.

Overall, the synthesis expression of the Uncertainty of 1.0 μg/mL uranium standard solution volume u3rel is shown in Eq. (12).

$$ u_{3rel} = \sqrt {u_{31rel}^{2} + u_{32rel}^{2} } = 1.15\% $$

(12)

3.4 Uncertainty of Sample Constant Volume (u_4rel)

The uncertainty of sample constant volume u_4rel was composed of uncertainty of measurement accuracy of volumetric flask u_41rel, uncertainty of temperature correction u_42rel and uncertainty of constant volume repeatability u_43rel.

3.4.1 Uncertainty of Measurement Accuracy of Volumetric Flask u_41rel

A 25.00 mL volumetric bottle with a precision of 0.03 mL was used to keep the volume from 25.00 mL to 25.00 mL. Assuming from triangular distribution, the inclusion factor $k = \sqrt 6$ was taken. According to the evaluation method of class B uncertainty, the measurement accuracy uncertainty of the volumetric bottle u_41rel was calculated as shown in Eq. (13):

$$ u_{41rel} = \frac{{0.03\,{\text{mL}}}}{{25\,{\text{mL}} \times \sqrt 3 }} = 0.05\% $$

(13)

3.4.2 Uncertainty of Temperature Correction u_42rel

The temperature in the laboratory was changing at ±5 ℃, and the calculation method of the uncertainty of temperature correction u_42rel was consistent with the uncertainty of temperature correction u_222rel in Sect. 3.2.2. The u_42rel calculated was 0.0606%.

3.4.3 Uncertainty of Constant Volume Repeatability u_43rel

The 25.00 mL volumetric bottle was fixed volume with distilled water, and the net weight of distilled water was weighed by a balance. The number n was 20 times. The deviation introduced by constant volume repeatability was determined in the experiment, and detailed data results were shown in Table 4. According to the evaluation method of class A uncertainty, the uncertainty of constant volume repeatability u_43rel was 0.32%.

Table 4. Quality change of distilled water from constant volume to 25.00 mL volumetric flask

Overall, the synthesis expression of the uncertainty of sample constant volume u_4rel is shown in Eq. (14).

$$ u_{4rel} = \sqrt {u_{41rel}^{2} + u_{42rel}^{2} + u_{43rel}^{2} } = 0.34\% $$

(14)

3.5 Uncertainty of Sample Volume (u_5rel)

The uncertainty of sample volume u_5rel is composed the uncertainty of pipette measurement accuracy u_51rel, the uncertainty of temperature correction u_52rel and the uncertainty of transport repeatability u_53rel.

3.5.1 Uncertainty of Pipette Measurement Accuracy u_51rel

The uncertainty of pipette measurement accuracy u_51rel can be seen from the pipette surface mark that the accuracy of 5 mL pipette is 0.03 mL. Assuming rectangular distribution, take the inclusion factor $k = \sqrt 3$ and calculate the uncertainty u_51rel of 5 mL according to the evaluation method of class B uncertainty as shown in Eq. (15):

$$ u_{51rel} = \frac{{0.03\,{\text{mL}}}}{{5\,{\text{mL}} \times \sqrt 3 }} = 0.35\% $$

(15)

3.5.2 Uncertainty of Temperature Correction u_52rel

The temperature in the laboratory varies at ±5 ℃, and the calculation method of the uncertainty of temperature correction u_52rel is consistent with the uncertainty of temperature correction in Sect. 3.2.2, and the calculated u_52rel is 0.0606%.

3.5.3 Uncertainty of Transport Repeatability u_53rel

Sample removal was 5.00 mL, 5.00 mL water sample was absorbed with a 5 mL pipette and weighed with a balance. The number of times n = 20 was used to determine the deviation introduced by the removal repeatability. The results were shown in Table 5.

Table 5. 5 mL pipette absorbed 5.00mL water sample quality change

According to class A uncertainty evaluation method, u_53rel can be calculated according to Eq. (16):

$$ u_{53rel} = \frac{1}{{\overline{m}}}\sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {(m_{i} - \overline{m})^{2} } }}{n - 1}} = 0.581\% $$

(16)

Overall, the synthesis expression of the uncertainty of sample volume u_5rel is shown in Eq. (17).

$$ u_{5rel} = \sqrt {u_{51rel}^{2} + u_{52rel}^{2} + u_{53rel}^{2} } = 0.681\% $$

(17)

3.6 Uncertainty of Weighing Samples is (u_6rel)

The uncertainty of weighing sample size u_6rel is composed of the uncertainty of weighing accuracy u_61rel and uncertainty of quality weighing repeatability of the balance u_62rel.

3.6.1 Uncertainty of Weighing Accuracy u_61rel

The weighing accuracy of the electronic balance is 0.1 mg. Assuming rectangular distribution, take the factor $k = \sqrt 3$ and weigh the balance with 0.05 g sample’s weighing accuracy uncertainty u_61rel according to the class B uncertainty evaluation method, and as shown in Eq. (18):

$$ u_{61rel} = \frac{{0.1\,{\text{mg}}}}{{0.05\,{\text{g}} \times \sqrt 3 }} = 0.11{\text{\% }} $$

(18)

3.6.2 Uncertainty of Quality Weighing Repeatability of the Balance u_62rel

According to the method of class A uncertainty, 0.05 g biological samples were found in measuring bottles and weighed every 10 min, n = 10. Results of weighing biological samples are shown in Table 6. After calculation, u_62rel equal to 0.09%.

Table 6. Reproducibility of mass weighing of biological samples

Therefore, the synthesis expression of uncertainty of weighing accuracy u_6rel is shown in Eq. (19).

$$ u_{6rel} = \sqrt {u_{61rel}^{2} + u_{62rel}^{2} } = 0.11\% $$

(19)

3.7 Uncertainty of Total Chemical Recovery(u_7rel)

Two 0.05 g of the same biological ash samples were weighed, one was added with 0.5 μg/mL uranium standard solution, the other was not added with uranium standard solution, and the control experiment was conducted. Uranium concentration was measured after pretreatment of the two samples according to HJ840-2017, and the chemical recovery Y was calculated according to Eq. (20).

$$ Y = \frac{{C_{2} - C_{1} }}{{C_{0} }} \times 100\% $$

(20)

where C₂ represent Determination of uranium content in a standard sample (ng), C₁ represent Determination of uranium content in blank samples (ng) and C₀ represent Quantity of uranium standard sample (ng).

10 groups of control experiments were repeated (n = 10), and the total chemical recovery Y of the same biological ash sample was obtained for 10 times. The detailed data were shown in Table 7, so the total chemical recovery uncertainty u_7rel was calculated to be 0.58%.

Table 7. Overall chemical recovery of biological samples

3.8 Uncertainty of the Ash/fresh Ratio (u_8rel)

The uncertainty of the ash/fresh ratio is mainly caused by mass weighing, which is composed of the uncertainty of fresh weighing u_81rel and ash weighing u_82rel. The uncertainty of fresh weighing u_81rel consists of the uncertainty of fresh weighing accuracy u_811rel and the uncertainty of fresh weighing repeatability u_812rel. The uncertainty of ash weighing u_82rel, consists of the uncertainty of ash sample weighing accuracy u_821rel and the uncertainty of ash sample weighing repeatability u_822rel.

3.8.1 Uncertainty of Fresh Weighing u_81rel

• (1) Uncertainty of fresh weighing accuracy u_811rel

  According to the test certificate of platform scale, the accuracy of platform scale can be obtained as s = 50 g, according to rectangular distribution, including factor $k = \sqrt 3$. The mass of typical fresh sample measured is 4.10 kg, and the mass of empty bag is 0.10 kg. The calculation of the uncertainty of weighing accuracy of fresh sample, u_811rel, is shown in Eq. (21):

  $$ u_{811rel} = \frac{{\sqrt {2\left( \frac{s}{k} \right)^{2} } }}{{m - m_{0} }} = 1.02\% $$

  (21)

• (2) Uncertainty of fresh weighing repeatability u_812rel

  The evaluation method of class A method was adopted, and fresh samples were weighed for 10 times. The data were shown in Table 8. According to Bessel formula, the weighing repeatability of fresh samples u_812rel was 0.41%.

  Table 8. Results of Fresh sample weighing

  Therefore, the synthesis expression of the uncertainty of fresh weighing U_81rel is shown in Eq. (22).

  $$ u_{81rel} = \sqrt {u_{811rel}^{2} + u_{812rel}^{2} } = 1.10\% $$

  (22)

3.8.2 Uncertainty of Ash Weighing u_82rel

• (1) Uncertainty of ash sample weighing accuracy u_821rel

  According to the verification certificate, the accuracy of the balance is s = 0.4 mg ($k = \sqrt 3$), the mass of the typical gray sample measured is m = 55.7833 g, and the mass of the empty plate is m₀ = 0.3920 g. The calculation of the uncertainty of the weighing accuracy of the gray sample u_821rel, is shown in Eq. (23):

  $$ u_{812rel} = \frac{{\sqrt {2\left( \frac{s}{k} \right)^{2} } }}{{m - m_{0} }} = 0.00059\% $$

  (23)

• (2) Uncertainty of ash sample weighing repeatability u_822rel

  The evaluation method of class A method was used, and the gray samples were weighed for 10 times. The data were shown in Table 9. The weighing repeatability uncertainty u_822rel was 0.001% calculated by Bessel equation.

  Table 9. Results of Ash sample weighing

  Therefore, the synthesis expression of uncertainty of ash weighing u_82rel is shown in Eq. (24).

  $$ u_{82rel} = \sqrt {u_{821rel}^{2} + u_{822rel}^{2} } = 0.001\% $$

  (24)

  Overall, the synthesis expression of uncertainty of the ash/fresh ratio (u_8rel) is shown in Eq. (25).

  $$ u_{8rel} = \sqrt {u_{81rel}^{2} + u_{82rel}^{2} } = 1.10\% $$

  (25)

3.9 Uncertainty of Dilution Ratio (u_9rel)

When the concentration of the sample is too high or the sample precipitates after adding the fluorescence enhancer, it is often necessary to dilute the solution under test.

Sample dilution usually uses pipette to remove a quantitative amount of liquid into a volumetric flask and then determine volume. The uncertainty of dilution ratio is composed of Uncertainty of sample volume u_91rel and the uncertainty of dilute solution volume u_92rel. The uncertainty of sampling volume u_91rel was composed of the uncertainty of pipette measurement accuracy u_911rel, the uncertainty of pipette temperature correction u_912rel, and the uncertainty of removal repeatability u_913rel. The uncertainty of diluted solution volume u_92rel was composed of the uncertainty of volumetric flask standard measurement accuracy u_921rel, Uncertainty of volumetric flask temperature correction u_922rel and Uncertainty of constant volume repeatability u_923rel. Taking dilution by a factor of 10.

3.9.1 Uncertainty of Sample Volume u_91rel

• (1) Uncertainty of pipette measurement accuracy u_911rel.

  5.00 mL pipette with accuracy of 0.03 mL was used to remove 5.00 mL original solution. Assuming rectangular distribution, the inclusion factor $k = \sqrt 3$ was taken. According to class B uncertainty evaluation method, the pipette measurement accuracy uncertainty u_911rel was calculated as shown in Eq. (26).

  $$ u_{911rel} = \frac{{0.03\,{\text{mL}}}}{{5\,{\text{mL}} \times \sqrt 3 }} = 0.35\% $$

  (26)

• (2) Uncertainty of pipette temperature correction u_912rel

  The temperature of the laboratory varies at ±5 ℃, and the calculation method of the temperature correction uncertainty is consistent with the calculation method of the temperature correction uncertainty u_222rel in Sect. 3.2.2. The calculated u_912rel is 0.0606%.

• (3) Uncertainty of removal repeatability u_913rel

  Sample removal was 5.00 mL, 5.00 mL water sample was absorbed with a 5 mL pipette and weighed with a balance for 20 times. The deviation introduced by the repeatability of removal was determined experimentally, and the results were shown in Table 5. According to the evaluation method of class A uncertainty, the calculated uncertainty u_913rel of removal repeatability is 0.581%.

  Therefore, the synthesis expression of Uncertainty of sample volume u_91rel is shown in Eq. (27).

  $$ u_{91rel} = \sqrt {u_{911rel}^{2} + u_{912rel}^{2} + u_{913rel}^{2} } = 0.681\% $$

  (27)

3.9.2 Uncertainty of Diluted Solution Volume u_92rel

• (1) Uncertainty of volumetric flask standard measurement accuracy u_921rel

  A 50 mL volumetric bottle with a precision of 0.05 mL is used from constant volume to scale line, assuming trigonometric distribution is followed, and the inclusion factor $k = \sqrt 6$ is taken. According to the evaluation method of class B uncertainty, the measurement accuracy uncertainty of the volumetric bottle u_921rel is calculated as shown in Eq. (28).

  $$ u_{921rel} = \frac{{0.05\,{\text{mL}}}}{{50\,{\text{mL}} \times \sqrt 6 }} = 0.04\% $$

  (28)

• (2) Uncertainty of volumetric flask temperature correction u_922rel

  The temperature of the laboratory varies at ±5 ℃, and the calculation method of the temperature correction uncertainty u_922rel is consistent with that of the temperature correction uncertainty u_222rel in Sect. 3.2.2, and the calculated u_922rel is 0.0606%.

• (3) Uncertainty of constant volume repeatability u_923rel

  Distilled water was used for constant volume of 50.00 mL volumetric bottle, and the net weight of distilled water was weighed by a balance, n = 20 times. The deviation introduced by constant volume repeatability was determined experimentally, and the results were shown in Table 10. According to the evaluation method of class A uncertainty, the uncertainty of constant volume reproducibility u_923rel was 0.21%.

  Table 10. 50 mL pipette absorbed 50.00 mL water sample quality change

  Therefore, the synthesis expression of the uncertainty of diluted solution volume u_92rel is shown in Eq. (29).

  $$ u_{92rel} = \sqrt {u_{921rel}^{2} + u_{922rel}^{2} + u_{923rel}^{2} } = 0.22\% $$

  (29)

  Overall, the synthesis expression of the uncertainty of dilution ratio u_9rel is shown in Eq. (30).

  $$ u_{9rel} = \sqrt {u_{91rel}^{2} + u_{92rel}^{2} } = 0.72\% $$

  (30)

3.10 Total Measurement Uncertainty

Uncertainty variables u_1rel, u_2rel, u_3rel, u_4rel, u_5rel, u_6rel, u_7rel, u_8rel and u_9rel are independent and unrelated to each other. The uncertainty information of each is shown in Table 11. When dilution multiple is not considered, the synthesized uncertainty u_Ar is shown in Eq. (31):

$$ \begin{aligned} u_{Ar} = &\, \sqrt {u_{1rel}^{2} + u_{2rel}^{2} + u_{3rel}^{2} + u_{4rel}^{2} + u_{5rel}^{2} + u_{6rel}^{2} + u_{7rel}^{2} + u_{8rel}^{2} } \\ = &\, 5.39\% \\ \end{aligned} $$

(31)

When k = 2, the extended uncertainty U_r(A) = 2 × u_Ar = 10.8%.

When there is dilution process, take dilution of 10 times as an example, and the resultant uncertainty u_Ar is shown in Eq. (32).

$$ \begin{aligned} u_{Ar} = &\, \sqrt {u_{1rel}^{2} + u_{2rel}^{2} + u_{3rel}^{2} + u_{4rel}^{2} + u_{5rel}^{2} + u_{6rel}^{2} + u_{7rel}^{2} + u_{8rel}^{2} + + u_{9rel}^{2} } \\ = & \, 5.54\% \\ \end{aligned} $$

(32)

When k = 2, the extended uncertainty U_r(A) = 2 × u_Ar = 10.9%.

Table 11. Uncertainty information table of trace uranium in organisms

4 Conclusions

Uranium content in biological samples was determined by laser fluorescence method, the sources of uncertainty of the method were analyzed, and the uncertainty components were calculated. The results showed that the uncertainty of the measurement of the total uranium content of a 0.05 g environmental biological sample is 10.8% (k = 2) under the condition of no dilution, and the dominant factor is the measurement of the sample fluorescence count.