Measurement of Total Reactive Phosphorus in Natural Water by Molecular Spectrophotometry (SMEWW 4500-P D)

Abstract

Phosphorus occurs in natural waters almost solely as phosphates.

Appendices

Exercise 1: Establishing Traceability in Analytical Chemistry

1. :

Specifying the analyte and measurand

Analyte	Reactive phosphorus
Measurand	Mass concentration of reactive phosphorus, estimated by measurement procedure SMEWW 4500-P D [1], in Tagus river GPS coordinates N39º 4′ 0.26, W8º 45′ 44.44 at 0.2 m depth and on 12 August 2018 at 8h20
Units	mg L−1 of P2O5

2. :

How would you demonstrate traceability of your result?

1	By analysing an adequate Certified Reference Material (CRM) and correcting measurement result of unknown samples for observed analyte recovery. In this case, measurement result would be traceable to the value embodied in the CRM
2	Through the accurate application of the measurement procedure including the use of calibrated equipment and chemical references traceable to adequate references. In this case, measurement result would be traceable to the value defined by the operationally defined measurement procedure SMEWW 4500-P D
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3. :

Any other comments, questions …

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Exercise 2: The Customer’s Requirements Concerning Quality of the Measurement Result

1. :

Specify the scope

Matrix	Natural fresh waters
Measuring range	0.02 mg L−1 to 1 mg L−1 of P2O5

2. :

Does the analytical procedure fulfil the requirement(s) for the intended use?

Intended use of the results		To check compliance of water with limits set by Directive 75/440/EEC for water intended for the abstraction of drinking water. The measurement performance requirements are laid down in Directive 79/869/EEC
Parameters to be evaluate		Value requested by the customer	Value obtained during validation
	LOD	LOD ≤ 0.02 mg L−1 of P2O5	LOD = (3sy/x/b)/V = 0.0032 mg L−1 of P; 0.0032 × 2.29 = 0.0073 mg L−1 of P2O5 (where sy/x, b and V are the residual standard deviation and the slope of a least squares calibration, and sample aliquot volume respectively; 2.29 = M(P2O5)/(2 × M(P))
	LOQ	Not specified(a)
	Repeatability	2 × standard deviation ≤ 0.04 mg L−1 of P2O (b)5	Instrumental signal repeatability: 2 × (sy/x/y) × (m/V) × 2.29 = = 2 × (0.00110/0.103) × (10/100) × 2.29 =  = 0.0049 mg L−1 of P2O5; (where y is the signal estimated for the largest mass of the calibration interval, m, 10 µg of P which corresponds to 0.23 mg L−1 of P2O5) or Measurement repeatability estimated from the standard deviation of the difference, sd, of duplicate sample results: 2 × (0.00134 mg L−1 of P/sqrt(2)) × 2.29 = = 0.0043 mg L−1 of P2O5 (Table 5 of yellow pages)
	Within-lab reproducibility	2 × standard deviation ≤ 0.04 mg L−1 of P2O (b)5	2 × 0.001725 × 2.29 = 0.0079 mg L−1 of P2O5 (estimated for a mass concentration of 0.046 mg L−1 of P2O (c)5 ; Yellow pages: Table 2)

1. ^(a)Since LOQ = LOD/0.3, acceptable LOD would result in an acceptable LOQ
2. ^(b)According to Directive 79/869/EEC “the range within which 95% of the results of measurements made on a single sample, using the same method, are located”, defined as precision, shouldn’t be larger than 10% of the limit of P₂O₅ in the water set in Directive 75/440/EEC: 0.4 mg L⁻¹ (10% × 0.4 mg L⁻¹ = 0.04 mg L⁻¹ of P₂O₅). Assuming the normal distribution of results, the half range of results specified in the Directive is estimated by two times the standard deviation of measurements precision. The Directive does not specify precision conditions. Therefore, this limit is used for estimated repeatability and within-lab reproducibility (i.e. intermediate precision) standard deviations
3. ^(c)Ideally this information should be estimated for 0.4 mg L⁻¹ of P₂O₅
4. (…)

Parameters to be evaluate		Value requested by the customer	Value obtained during validation
	Trueness	Absolute mean error ≤ 0.08 mg L−1 of P2O (d)5	The mean analyte recovery is 102.23% (Table 4 of Yellow pages) which corresponds to a mean error of 0.0089 mg L−1 at 0.4 mg L−1 of P2O5 (|102.23–100|/100 × 0.4 = 0.0089 mg L−1 of P2O5)
	Measurement uncertainty	Expanded uncertainty, U ≤ 0.10 mg L−1(e)	U = 0.025 mg L−1 for 0.40 mg L−1 of P2O5
	Other-state
The analytical procedure is fit for the intended use: Yes No

1. ^(d)According to Directive 79/869/EEC “the difference between the true value of the parameter examined and the average experimental value obtained”, defined as accuracy, shouldn’t be larger than 20% of the limit of P₂O₅ in the water set in Directive 75/440/EEC: 0.4 mg L⁻¹ (20% × 0.4 mg L⁻¹ = 0.08 mg L⁻¹ of P₂O₅). In this Directive accuracy term is used for the absolute mean error
2. ^(e)Assuming random and systematic effects affecting measurement results are adequately quantified by the intermediate precision and mean analyte recovery respectively, the target (maximum admissible) uncertainty is 0.10 mg L⁻¹ of P₂O₅. The following equations are used to combine the target values for precision and trueness to estimate the target expanded uncertainty, U^tg [2]
3. $U^{tg} = ku^{tg} = 2u^{tg} = 2\sqrt {\left( {\frac{0.04}{2}} \right)^{2} + \left( {\frac{0.08}{\sqrt 3 }} \right)^{2} } = 0.10{\text{ mg L}}^{ - 1} {\text{ of P}}_{ 2} {\text{O}}_{ 5}$
4. where u^tg and k are the target standard uncertainty and coverage factor respectively. The first term within the square root is the square of the intermediate precision standard deviation and the second term is the square of the bias standard uncertainty assuming mean error has a rectangular distribution. This equation assumes that the reference value of recovery tests is affected by a negligible uncertainty

Exercise 3: Enlarge the Analytical Method Scope

1. :

Specify the additional validation work needed to enlarge the scope of analysis to the analysis of wastewaters

Intended use of the results: to estimate eutrophication risk of a river by wastewater discharge. The reactive phosphorus provides an estimation of the most readily bioavailable phosphorus
Parameters to be evaluted		Value requested by the customer (relative to the request presented on Exercise 2) (“=” – Same; “<” – Smaller value; “>” – Larger value)
	LOD	Maximum LOD “>” (0.02 mg L−1 of P2O5)
	LOQ	See LOD
	Selectivity	Check if known interferences can be present in analysed wastewater
	Linearity	“=” (the same for wastewaters analysed in the same calibration range)
	Homogeneity of variances of instrumental response	“=” (the same for wastewaters analysed in the same calibration range)
	Repeatability	Maximum repeatability standard deviation “>” 0.02 mg L−1 of P2O5 (0.04/2)
	Within-lab reproducibility	Maximum within-lab reproducibility standard deviation “>” 0.02 mg L−1 of P2O5 (0.04/2)
	Reproducibility (between Lab)	Not to be included in in-house validation study
	Trueness	Maximum absolute mean error “>” 0.08 mg L−1 of P2O5
	Robustness	Not particularly relevant if within-lab reproducibility (i.e. intermediate precision) is studied in significantly different experimental and operational conditions and measurement procedure is not to be transferred to another location
	Participation in PT schemes	The z-score should be estimated with a reference standard deviation (“>”) 0.05 mg L−1 of P2O5 (half of the target expanded uncertainty). Satisfactory z-score are within the interval [−2, 2]
	Measurement uncertainty	Target expanded uncertainty “>” 0.10 mg L−1 of P2O5
	Other-state

2. :

State how should be validated the selectivity

Exercise 4: Building an Uncertainty Budget

1. :

Specify the measurand and units

Measurand	Mass concentration of reactive phosphorus, estimated by measurement procedure SMEWW 4500-P D, in Tagus river GPS coordinates N39º4′0.26, W8º45′44.44 at 0.2 m depth and on 12 August 2018 at 8h20.
Unit	mg L−1 of P2O5

2. :

Provide the model equation used to evaluate the measurement uncertainty

Measurement model:

$$\gamma ( {\text{mg L}}^{ - 1} )= \frac{{m{ (}\upmu{\text{g P)}}}}{{V ( {\text{mL)}}}} \times \frac{{M({\text{P}}_{ 2} {\text{O}}_{ 5} )}}{{2M({\text{P}})}}$$

• γ(mg L⁻¹) Reactive phosphorus (RP) mass concentration in the sample expressed as P₂O₅;

• m(µg P) RP mass interpolated in the calibration curve expressed as P;

• V(mL) Sample volumetric aliquot;

• M(P₂O₅) Molar mass of phosphorus pentoxide;

• M(P) Molar mass of phosphorus.

3. :

Identify (all possible) sources of uncertainty

	RP mass interpolated in the calibration curve, m(µg P) (statistical interpolation and calibrator mass uncertainty components)
	sample volumetric aliquot, V(mL)
	Recovery (mean analyte recovery, $\bar{R}$, and CRM certified value uncertainty components)
	Dilution of sample volumetric aliquot by acid addition, fdil (negligible component for most samples)
	Molar mass of phosphorus pentoxide (g mol−1) (negligible component)
	Molar mass of phosphorus (g mol−1) (negligible component)

3.1. :

Build cause/effect diagram

Updated measurement model:

$$\gamma ( {\text{mg L}}^{ - 1} )= \frac{{m{ (}\upmu{\text{g P)}} \times f_{\text{std}} }}{{V ( {\text{mL)}}}} \times \frac{{M({\text{P}}_{ 2} {\text{O}}_{ 5} )}}{{2M({\text{P}})}} \times \frac{1}{{\bar{R}}}$$

• $\bar{R}$ mean analyte recovery;

• f_std unitary factor for accounting for calibrator mass uncertainty.

4. :

Evaluate values of each input quantity

Input quantity	Value	Unit	Remark
m	6.39	µg	(0.067 a.u. − 0.00206 a.u.)/(0.0102 a.u. μg−1)
V	100	mL	–
$\bar{R}$	1	–	$\bar{R}$ is set equal to 1 since analyte recovery is metrologically equivalent to 100%(a)
f std	1	–	–
M(P2O5)	141.94	g mol−1	negligible uncertainty component
M(P)	30.97	g mol−1	negligible uncertainty component

1. ^(a)Significance test: $\left( {\left| {\bar{R} - 1} \right|} \right) /u\left( {\bar{R}} \right) \le {\text{t(}}N{ - 1; 0} . 0 5 )= {\text{t(}}1 8 ; { 0} . 0 5 )\Leftrightarrow 1.63 \le 2.10$ (see following section)

5. :

Evaluate the standard uncertainty of each input quantity

Input quantity	Standard uncertainty	Unit	Remark
m	0.117	µg	This uncertainty component represents the statistical interpolation of sample signal in the calibration curve(a)
V	0.0684	mL	See following calculations
$\bar{R}$	0.0135	–	See following calculations
f std	0.0166	–	The standard uncertainty associated with fstd, u(fstd), is estimated from the RP mass relative standard uncertainty of the calibrator with lowest quantity except the blank, uStd2/2 (i.e. u(fstd) = fstd × uStd2/2) [3]
M(P2O5)	negligible	g mol−1
M(P)	negligible	g mol−1

1. ^(a)The estimated mass of RP in the diluted sample is affected by both the statistical interpolation of sample signal in the calibration curve and by calibrator mass uncertainty. The ratio of the masses of RP of any pair of calibrators should have a negligible uncertainty given the instrumental signal precision to allow the reliable estimation of the interpolation uncertainty from least squares regression model [2]

Calculations:

f_std:

$\frac{{u\left( {f_{\text{Std}} } \right)}}{{f_{\text{Std}} }} = \frac{{u_{\text{Std2}} }}{2} = \frac{{ 0. 0 3 3 2 { }\upmu{\text{g P}}}}{{ 2 { }\upmu{\text{g P}}}} = 0.0166$ and, since f_std = 1, u(f_std) = 0.0166.

$\bar{R}$:

$$\begin{aligned} u\left( {\bar{R}} \right) & = \bar{R}\sqrt {\left( {\frac{{s_{R} }}{{\bar{R}\sqrt N }}} \right)^{2} + \left( {\frac{{u\left( {\gamma_{CRM} } \right)}}{{\gamma_{CRM} }}} \right)^{2} } \\ & = 102.2\sqrt {\left( {\frac{5.43}{{102.2\sqrt {19} }}} \right)^{2} + \left( {\frac{{\frac{0.0076}{2}}}{0.7410}} \right)^{2} } = 1.35\% { [0} . 0 1 3 5 ]\\ \end{aligned}$$

V:

$$\begin{aligned} u\left( V \right) & = \sqrt {\left( {\frac{tol}{\sqrt 3 }} \right)^{2} + \left( {s_{r} } \right)^{2} + \left( {\frac{{V \times 4 \times \alpha_{{{\text{H}}_{ 2} {\text{O}}}} }}{\sqrt 3 }} \right)^{2} } = \sqrt {\left( {\frac{0.08}{\sqrt 3 }} \right)^{2} + \left( {0.014} \right)^{2} + \left( {\frac{100 \times 4 \times 0.00021}{\sqrt 3 }} \right)^{2} } \\ & = 0.0684 \\ \end{aligned}$$

• tol Tolerance associated with the pipette nominal volume (0.08 mL);

• s_r repeatability of pipette manipulation (0.014 mL);

• α_H20 volume expansion coefficient for water (2.1 × 10⁻⁴ °C⁻¹).

m:

$$u_{Int} = \frac{{s_{y/x} }}{b}\sqrt {\frac{1}{n} + 1} = \frac{ 0. 0 0 1 1 0}{ 0. 0 1 0 2}\sqrt {\frac{1}{6} + 1} = 0. 1 1 7 { }\upmu{\text{g P}}$$

6. :

Calculate the value of the measurand, using the model equation

$$\begin{aligned} \gamma ( {\text{mg L}}^{ - 1} )& = \frac{{m{ (}\upmu{\text{g P)}} \cdot f_{\text{std}} }}{{V ( {\text{mL)}}}} \cdot \frac{{M({\text{P}}_{ 2} {\text{O}}_{ 5} )}}{{2M({\text{P}})}} \cdot \frac{1}{{\bar{R}}} \\ & = \frac{{\left( {\frac{ 0. 0 6 7- 0. 0 0 2 0 6}{ 0. 0 1 0 2}} \right) \times 1}}{ 1 0 0} \times \frac{141.94}{2 \times 30.97} \times \frac{ 1}{ 1} = \frac{ 6. 3 9}{ 1 0 0} \times 2.291 = 0. 1 4 5 9 0 {\text{ mg L}}^{ - 1}{\text{of P}}_{ 2} {\text{O}}_{ 5} \\ \end{aligned}$$

7. :

Calculate the combined standard uncertainty ( u _c ) of the result and specify units

Using: Mathematical solution; Spreadsheet Approach; Commercial Software

Input quantity	Value	Standard uncertainty	Unit	Remark
m	6.39	0.117	µg
V	100	0.0684	mL
$\bar{R}$	1	0.0135	–
f std	1	0.0166	–
M(P2O5)	141.94	–	g mol−1
M(P)	30.97	–	g mol−1

Calculations:

$$\begin{aligned} & u(\gamma ) = \gamma \sqrt {\left( {\frac{{u_{Int} }}{m}} \right)^{2} + \left( {\frac{u\left( V \right)}{V}} \right)^{2} + \left( {\frac{{u\left( {\bar{R}} \right)}}{{\bar{R}}}} \right)^{2} + \left( {\frac{{u\left( {f_{\text{Std}} } \right)}}{{f_{\text{Std}} }}} \right)^{2} } \\ & u\left( \gamma \right) = 0. 1 4 5 9 0\sqrt {\left( {\frac{0.117}{6.39}} \right)^{2} + \left( {\frac{0.0684}{100}} \right)^{2} + \left( {\frac{0.0135}{1}} \right)^{2} + \left( {\frac{0.0166}{1}} \right)^{2} } \\ & u\left( \gamma \right) = 0.0041{\text{ mg L}}^{ - 1} {\text{ of P}}_{ 2} {\text{O}}_{ 5} \\ \end{aligned}$$

8. :

Calculate the expanded uncertainty ( U _c ) & specify the coverage factor k and the units

$$U_{c} (\gamma ) = 0.0082\,{\text{ mg L}}^{ - 1}\,{\text{of}}\,{\text{P}}_{2} {\text{O}}_{5}$$

For a confidence level of approximately 95% considering a coverage factor, k, of 2.

Result to be reported: (0.1459 ± 0.0082) mg L⁻¹ of P₂O₅ (for k = 2 and ≈95%)

9. :

Analyse the uncertainty contribution & specify the main input quantity contributing the most to U _c

1	Statistical interpolation component: 42.2%
2	Calibrator mass component: 34.7%
3	Analyte recovery component: 23.0%
4	Sample aliquot: 0.06%
5
6

Calculations:

Example: Percentage contribution, P_Int, of the statistical interpolation uncertainty to the combined standard uncertainty:

$$\begin{aligned} & P_{Int} = \frac{{\left( {\frac{{u_{Int} }}{m}} \right)^{2} }}{{\left( {\frac{{u_{Int} }}{m}} \right)^{2} + \left( {\frac{{u\left( {f_{\text{Std}} } \right)}}{{f_{\text{Std}} }}} \right)^{2} + \left( {\frac{u\left( V \right)}{V}} \right)^{2} + \left( {\frac{{u_{{\bar{R}}} }}{{\bar{R}}}} \right)^{2} }} \\ & P_{Int} = \frac{{\left( {0.0183} \right)^{2} }}{{\left( {0.0183} \right)^{2} + \left( {0.0166} \right)^{2} + \left( {0.000684} \right)^{2} + \left( {0.0135} \right)^{2} }} \\ & P_{Int} = 0. 4 2 2 { [42} . 2\% ] \\ \end{aligned}$$

Graphic representation of the percentage contribution of the uncertainty components:

10. :

Prepare your Uncertainty Budget Report

(…)

Conclusion: Measurement is fit for its intended use since the reported expanded uncertainty (0.0082 mg L⁻¹ of P₂O₅) is smaller than the target expanded uncertainty (0.1 mg L⁻¹ of P₂O₅).