• Table of Content
• Foreword
• Scope
• Uncertainty
• Analytical Measurement
• The Process
• Step 1: Specification
• Step 2: Identification
• Step 3: Quantification
• Step 4: Calculation
• Reporting Uncertainty
• Appendix A: Examples
  • Example 1: Calibration Standard
  • Example 2: Standardising NaOH
  • Example 3: An Acid/Base Titration
  • Example 4: In-house Validation Studies
  • Example 5: Determ. Cadmium Release
  • Example 6: Crude Fibre in Animal Feed
  • Example 7: Lead in Water - double IDMS
• Appendix B: Definitions
• Appendix C: Analytical Proc.
• Appendix D: Analyse U Sources
• Appendix E: Statistics
• Appendix F: MU at the Limit
• Appendix G: Common Sources
• Appendix H: Bibliography

Appendix - Example A5: Determination of Cadmium Release from Ceramic Ware by Atomic Absorption Spectrometry

Summary

Goal
The amount of released cadmium from ceramic ware is determined using atomic absorption spectrometry. The employed procedure is the empirical method BS 6748.

Measurement Procedure
The different stages in determining the amount of cadmium released from ceramic are given in the flow chart (figure A5.1).

Measurand:

The variables are described in table A5.1.

	Description	Value x	Standard uncertainty u(x)	Relative standard uncertainty u(x)/x
Co	Content of cadmium in the extraction solution	0.26 mg l-1	0.018 mg l-1	0.069
d	Dilution factor (if used)	1.0 Note 1	0 Note 1	0 Note 1
VL	Volume of the leachate	0.332 l	0.0018 l	0.0054
v	Surface area of the vessel	2.37 dm2	0.06 dm2	0.025
facid	Influence of the acid concentration	1.0	0.0008	0.0008
ftime	Influence of the duration	1.0	0.001	0.001
ftemp	Influence of temperature	1.0	0.06	0.06
r	Mass of cadmium leached per unit area	0.036 mg dm-2	0.0033 mg dm-2	0.09

Note 1: No dilution was applied in the present example; d is accordingly exactly 1.0

Identification of the Uncertainty Sources:
The relevant uncertainty sources are shown in the cause and effect diagram at figure A5.2.

Quantification of the Uncertainty Sources:
The sizes of the different contributions are given in table A5.1 and shown diagrammatically in figure A5.3

Figure A5.3: Uncertainties in leachable Cd Determination

The values of u(y,x_i)=(y/x_i).u(x_i) are taken from table A5.4

Example A5: Determination of Cadmium Release from Ceramic Ware by Atomic Absorption Spectrometry: Detailed Discussion

A5.1 Introduction

This example demonstrates the uncertainty evaluation of an empirical method; in this case (BS 6748), the determination of metal release from ceramic ware, glassware, glass-ceramic ware and vitreous enamel ware. The test is used to determine by atomic absorption spectroscopy (AAS) the amount of lead or cadmium leached from the surface of ceramic ware by a 4% (v/v) aqueous solution of acetic acid. The results obtained with this analytical method are only expected to be comparable with other results obtained by the same method.

A5.2 Step 1: Specification

The complete procedure is given in British Standard BS 6748:1986 "Limits of metal release from ceramic ware, glass ware, glass ceramic ware and vitreous enamel ware" and this forms the specification for the measurand. Only a general description is given here (right).

A5.2.1 Apparatus and Reagent Specifications

The reagent specifications affecting the uncertainty study are:

• A freshly prepared solution of 4% v/v glacial acetic acid in water, made up by dilution of 40 ml glacial acetic to 1 l.
• A (1000 ±1) mg l^-1 standard lead solution in 4% (v/v) acetic acid.
• A (500 ±0.5) mg l^-1 standard cadmium solution in 4% (v/v) acetic acid.

Laboratory glassware is required to be of at least class B and incapable of releasing detectable levels of lead or cadmium in 4% acetic acid during the test procedure. The atomic absorption spectrophotometer is required to have detection limits of at most 0.2 mg l^-1 for lead and 0.02 mg l^-1 for cadmium.

A5.2.2 Procedure

The general procedure is illustrated schematically in figure A5.4. The specifications affecting the uncertainty estimation are:

1. The sample is conditioned to (22±2) °C. Where appropriate ("category 1" articles), the surface area of the article is determined. For this example, a surface area of 2.37 dm² was obtained (table a5.1 and table a5.3 include the experimental values for the example).
2. The conditioned sample is filled with 4% v/v acid solution at (22±2) °C to within 1 mm from the overflow point, measured from the upper rim of the sample, or to within 6 mm from the extreme edge of a sample with a flat or sloping rim.
3. The quantity of 4% v/v acetic acid required or used is recorded to an accuracy of ±2% (in this example, 332 ml acetic acid was used).
4. The sample is allowed to stand at (22 ±2) °C for 24 hours (in darkness if cadmium is determined) with due precaution to prevent evaporation loss.
5. After standing, the solution is stirred sufficiently for homogenisation, and a test portion removed, diluted by a factor d if necessary, and analysed by AA, using appropriate wavelengths and, in this example, a least squares calibration curve.
6. The result is calculated (see below) and reported as the amount of lead and/or cadmium in the total volume of the extracting solution, expressed in milligrams of lead or cadmium per square decimetre of surface area for category 1 articles or milligrams of lead or cadmium per litre of the volume for category 2 and 3 articles.

NOTE: Complete copies of BS 6748:1986 can be obtained by post from BSI customer services, 389 Chiswick High Road, London W4 4AL England +44 (0) 208 996 9001.

A5.3 Step 2: Identity and Analysing Uncertainty Sources

Step 1 describes an "empirical method". If such a method is used within its defined field of application, the bias of the method is defined as zero. Therefore bias estimation relates to the laboratory performance and not to the bias intrinsic to the method. Because no reference material certified for this standardised method is available, overall control of bias is related to the control of method parameters influencing the result. Such influence quantities are time, temperature, mass and volumes, etc.

The concentration c₀ of lead or cadmium in the acetic acid after dilution is determined by atomic absorption spectrometry and calculated using

For vessels that can be filled, the result r' is then

where d is the dilution factor employed. Otherwise, the empirical method calls for the result to be expressed as mass r of lead or cadmium leached per unit area. r is given by

The first part of the above equation of the measurand is used to draft the basic cause and effect diagram (figure A5.5).

There is no reference material certified for this empirical method with which to assess the laboratory performance. All the feasible influence quantities, such as temperature, time of the leaching process and acid concentration therefore have to be considered. To accommodate the additional influence quantities the equation is expanded by the respective correction factors leading to

These additional factors are also included in the revised cause and effect diagram (figure A5.6). They are shown there as effects on c₀.

NOTE: The latitude in temperature permitted by the standard is a case of an uncertainty arising as a result of incomplete specification of the measurand. Taking the effect of temperature into account allows estimation of the range of results which could be reported whilst complying with the empirical method as well as is practically possible. Note particularly that variations in the result caused by different operating temperatures within the range cannot reasonably described as bias as they represent results obtained in accordance with the specification.

A5.4 Step 3: Quantifying Uncertainty Sources

The aim of this step is to quantify the uncertainty arising from each of the previously identified sources. This can be done either by using experimental data or from well based assumptions.

Dilution Factor d:
For the current example, no dilution of the leaching solution is necessary, therefore no uncertainty contribution has to be accounted for.

Volume V_L:
Filling: The empirical method requires the vessel to be filled "to within 1 mm from the brim". For a typical drinking or kitchen utensil, 1 mm will represent about 1% of the height of the vessel. The vessel will therefore be 99.5 ±0.5% filled (i.e. V_L will be approximately 0.995 ±0.005 of the vessel's volume).

Temperature: The temperature of the acetic acid has to be 22 ±2ºC. This temperature range leads to an uncertainty in the determined volume, due to a considerable larger volume expansion of the liquid compared with the vessel. The standard uncertainty of a volume of 332 ml, assuming a rectangular temperature distribution, is

Reading: The volume V_L used is to be recorded to within 2%, in practice, use of a measuring cylinder allows an inaccuracy of about 1% (i.e. 0.01V_L). The standard uncertainty is calculated assuming a triangular distribution.

Calibration: The volume is calibrated according to the manufacturer's specification within the range of ±2.5 ml for a 500 ml measuring cylinder. The standard uncertainty is obtained assuming a triangular distribution.

For this example a volume of 332 ml is used and the four uncertainty components are combined accordingly

Cadmium Concentration c₀:
The amount of leached cadmium is calculated using a manually prepared calibration curve. For this purpose five calibration standards, with a concentration 0.1 mg l^-1, 0.3 mg l^-1, 0.5 mg l^-1, 0.7 mg l^-1 and 0.9 mg l^-1, were prepared from a 500 ±0.5 mg l^-1 cadmium reference standard. The linear least squares fitting procedure used assumes that the uncertainties of the values of the abscissa are considerably smaller than the uncertainty on the values of the ordinate. Therefore the usual uncertainty calculation procedures for c₀ only reflect the uncertainty in the absorbance and not the uncertainty of the calibration standards, nor the inevitable correlations induced by successive dilution from the same stock. In this case, however, the uncertainty of the calibration standards is sufficiently small to be neglected.

The five calibration standards were measured three times each, providing the results in table A5.2.

Table A5.2: Calibration Results

Concentration [mg l-1]	1	2	3
0.1	0.028	0.029	0.029
0.3	0.084	0.083	0.081
0.5	0.135	0.131	0.133
0.7	0.180	0.181	0.183
0.9	0.215	0.230	0.216

The calibration curve is given by

and the results of the linear least square fit are

	Value	Standard deviation
B1	0.2410	0.0050
B0	0.0087	0.0029

with a correlation coefficient r of 0.997. The fitted line is shown in figure A5.7. The residual standard deviation S is 0.005486.

Figure A5.7: Linear Least Square Fit and Uncertainty Interval for duplicate Determinations

The actual leach solution was measured twice, leading to a concentration c₀ of 0.26 mg l^-1. The calculation of the uncertainty u(c₀) associated with the linear least square fitting procedure is described in detail in appendix E3. Therefore only a short description of the different calculation steps is given here.

u(c₀) is given by

with the residual standard deviation S given by

and

Area a_V :
Length measurement: The total surface area of the sample vessel was calculated, from measured dimensions, to be 2.37 dm². Since the item is approximately cylindrical but not perfectly regular, measurements are estimated to be within 2 mm at 95% confidence. Typical dimensions are between 1.0 dm and 2.0 dm leading to an estimated dimensional measurement uncertainty of 1 mm (after dividing the 95% figure by 1.96). Area measurements typically require two length measurements, height and width respectively (i.e. 1.45 dm and 1.64 dm)

Area: Since the item has not a perfect geometric shape, there is also an uncertainty in any area calculation; in this example, this is estimated to contribute an additional 5% at 95% confidence.

The uncertainty contribution of the length measurement and area itself are combined in the usual way.

Temperature Effect f_temp:
A number of studies of the effect of temperature on metal release from ceramic ware have been undertaken^(1-5). In general, the temperature effect is substantial and a near-exponential increase in metal release with temperature is observed until limiting values are reached. Only one study¹ has given an indication of effects in the range of 20-25°C. From the graphical information presented the change in metal release with temperature near 25°C is approximately linear, with a gradient of approximately 5% °C^-1. For the ±2°C range allowed by the empirical method this leads to a factor f_temp of 1±0.1. Converting this to a standard uncertainty gives, assuming a rectangular distribution:

u(f_temp)=

Time Effect f_time:
For a relatively slow process such as leaching, the amount leached will be approxi-mately proportional to time for small changes in the time. Krinitz and Franco¹ found a mean change in concentration over the last six hours of leaching of approximately 1.8 mg l^-1 in 86 mg l^-1, that is, about 0.3%/h. For a time of (24±0.5)h c₀ will therefore need correction by a factor f_time of 1±(0.5x0.003) =1±0.0015. This is a rectangular distribution leading to the standard uncertainty

Acid Concentration f_acid:
One study of the effect of acid concentration on lead release showed that changing concentration from 4 to 5% v/v increased the lead released from a particular ceramic batch from 92.9 to 101.9 mg l^-1, i.e. a change in f_acid of or close to 0.1. Another study, using a hot leach method, showed a comparable change (50% change in lead extracted on a change of from 2 to 6% v/v)³. Assuming this effect as approximately linear with acid concentration gives an estimated change in f_acid of approximately 0.1 per % v/v change in acid concentration. In a separate experiment the concentration and its standard uncertainty have been established using titration with a standardised NaOH titre (3.996% v/v u = 0.008% v/v). Taking the uncertainty of 0.008% v/v on the acid concentration suggests an uncertainty for f_acid of 0.008x0.1 = 0.0008. As the uncertainty on the acid concentration is already expressed as a standard uncertainty, this value can be used directly as the uncertainty associated with f_acid.

NOTE: In principle, the uncertainty value would need correcting for the assumption that the single study above is sufficiently representative of all ceramics. The present value does, however, give a reasonable estimate of the magnitude of the uncertainty.

A5.5 Step 4: Calculating the Combined Standard Uncertainty

The amount of leached cadmium per unit area, assuming no dilution, is given by

The intermediate values and their standard uncertainties are collected in table a5.3. Employing those values

Table A5.3: Intermediate Values and Uncertainties for leachable Cadmium Analysis

	Description	Value x	Standard uncertainty u(x)	Relative standard uncertainty u(x)/x
co	Content of cadmium in the extraction solution	0.26 mg l-1	0.018 mg l-1	0.069
VL	Volume of the leachate	0.332 l	0.0018 l	0.0054
v	Surface area of the vessel	2.37 dm2	0.06 dm2	0.025
facid	Influence of the acid concentration	1.0	0.0008	0.0008
ftime	Influence of the duration	1.0	0.001	0.001
ftemp	Influence of temperature	1.0	0.06	0.06

In order to calculate the combined standard uncertainty of a multiplicative expression (as above) the standard uncertainties of each component are used as follows:

The simpler spreadsheet approach to calculate the combined standard uncertainty is shown in table a5.4. A description of the method is given in appendix e.

Table A5.4: Spreadsheet Calculation of Uncertainty for leachable Cadmium Analysis

"	A	B	C	D	E	F	G	H
1			cO	VL	av	facid	ftime	ftemp
2		value	0.26	0.332	2.37	1.0	1.0	1.0
3		uncertainty	0.018	0.0018	0.06	0.0008	0.001	0.06
4
5	cO	0.26	0.278	0.26	0.26	0.26	0.26	0.26
6	VL	0.332	0.332	0.3338	0.332	0.332	0.332	0.332
7	av	2.37	2.37	2.37	2.43	2.37	2.37	2.37
8	facid	1.0	1.0	1.0	1.0	1.0008	1.0	1.0
9	ftime	1.0	1.0	1.0	1.0	1.0	1.001	1.0
10	ftemp	1.0	1.0	1.0	1.0	1.0	1.0	1.06
11
12	r	0.036422	0.038943	0.036619	0.035523	0.036451	0.36458	0.038607
13	u(y,xi)		0.002521	0.000197	-0.000899	0.000029	0.000036	0.002185
14	u(y)2,u(y,xi)2	1.199 E-5	6.36 E-6	3.90 E-8	8.09 E-7	8.49 E-10	0.33E-9	4.78 E-6
15
16	uc(r)	0.0034

The values of the parameters are entered in the second row from C2 to H2, and their standard uncertainties in the row below (C3:H3). The spreadsheet copies the values from C2:H2 into the second column (B5:B10). The result (r) using these values is given in B12. C5 shows the value of c₀ from C2 plus its uncertainty given in C3. The result of the calculation using the values C5:C10 is given in C12. The columns D and H follow a similar procedure. Row 13 (C13:H13) shows the differences of the row (C12:H12) minus the value given in B12. In row 14 (C14:H14) the values of row 13 (C13:H13) are squared and summed to give the value shown in B14. B16 gives the combined standard uncertainty, which is the square root of B14.

The contributions of the different parameters and influence quantities to the measurement uncertainty are illustrated in figure a5.8, comparing the size of each of the contributions (C13:H13 in table a5.4) with the combined uncertainty (B16).

The values of u(y,x_i)=(y/x_i).u(x_i) are taken from table A5.4

The expanded uncertainty U(r) is obtained by applying a coverage factor of 2

Ur= 0.0034 x 2 = 0.007 mg dm^-2

Thus the amount of released cadmium measured according to BS 6748:1986

(0.036 ±0.007) mg dm^-2

where the stated uncertainty is calculated using a coverage factor of 2.

A5.6 References for Example 5

1. B. Krinitz, V. Franco, J. AOAC 56 869-875 (1973)
2. B. Krinitz, J. AOAC 61, 1124-1129 (1978)
3. J. H. Gould, S. W. Butler, K. W. Boyer, E. A. Stelle, J. AOAC 66, 610-619 (1983)
4. T. D. Seht, S. Sircar, M. Z. Hasan, Bull. Environ. Contam. Toxicol. 10, 51-56 (1973)
5. J. H. Gould, S. W. Butler, E. A. Steele, J. AOAC 66, 1112-1116 (1983)