Determination of Cadmium Release from Ceramic Ware by Atomic Absorption Spectrometry
This is the example A5 of the EURACHEM / CITAC Guide "Quantifying Uncertainty in Analytical Measurement", Second Edition.
The amount of released cadmium from ceramic ware is determined using atomic absorption spectrometry. The procedure employed is the empirical method BS 6748.
The item to be tested is filled with a 4 % v/v acetid acid solution for a given lenght of time, the amount of cadmium released from the item is then calculated from the measured cadmium concentration in the leach solution and the volume of the leach solution. Parameters such as leaching time, temperature, acid concentration etc. are specified in the empirical method.
Model Equation:
{calculation of the uncertainty of volume V_{L}}
V_{L} = V_{L}_{ }_{nominal} * f_{VL}_{}_{filling} * f_{VL}_{}_{temperature} * f_{VL}_{}_{reading} * f_{VL}_{}_{calibration};
{calculation of the uncertainty of the surfache area}
a_{V} = a_{V}_{ }_{nominal} * f_{aV}_{}_{length1} *f_{aV}_{}_{lenght2} * f_{aV}_{}_{area};
{calculation of the mass of cadmium leached}
r = c_{0} * V_{L} / a_{V} * d * f_{acid} * f_{time} * f_{temperature};

List of Quantities:
Quantity  Unit  Definition 

V_{L}  L  Volume of the leachate 
V_{L}_{ }_{nominal}  L  Nominal volume of the leachate 
f_{VL}_{}_{filling}  Uncertainty contribution of V_{L} due to filling of the vessel  
f_{VL}_{}_{temperature}  Uncertainty contribution of V_{L} due to temperature variation  
f_{VL}_{}_{reading}  Uncertainty contribution of V_{L} due to reading of the measuring cylinder  
f_{VL}_{}_{calibration}  Uncertainty contribution of V_{L} due to calibration of the measuring cylinder  
a_{V}  dm^{2}  Surface area of the vessel 
a_{V}_{ }_{nominal}  dm^{2}  Nominal surface area of the vessel 
f_{aV}_{}_{length1}  Uncertainty contribution to a_{V} of first length measurement (i.e. height)  
f_{aV}_{}_{lenght2}  Uncertainty contribution to a_{V} of second length measurement (i.e. lenght)  
f_{aV}_{}_{area}  Uncertainty contribution to a_{V} due to imperfect geometry  
r  mg/dm^{2}  Mass of cadmium leached per unit area 
c_{0}  mg/L  Content of cadmium in the extraction solution 
d  Dilution factor (if used)  
f_{acid}  Influence of the acid concentration  
f_{time}  Influence of the duration  
f_{temperature}  Influence of the temperature 
V_{L}_{ }_{nominal}: 
Constant Value: 0.332 L 
The nominal volume is not associated with any uncertainties. Four different factors contribute to the uncertainty of the real volume, filling, temperature, reading and calibration. These are introduced in the uncertainy budget through the factors f_{VL}_{}_{filling,} f_{VL}_{}_{temperature,} f_{VL}_{}_{reading} and f_{VL}_{}_{calibration.}
f_{VL}_{}_{filling}: 
Type B triangular distribution Value: 0.995 Halfwidth of Limits: 0.005 
The method requires the vessel to be filled 'to within 1mm from the brim'. For a typical drinking or kitchen utensil, this represents about 1% of the total height. The vessel will therefore be 99.5% ± 0.5% filled.
f_{VL}_{}_{temperature}: 
Type B rectangular distribution Value: 1 Halfwidth of Limits: =2.1e4*2 
The temperature of the acetic acid has to be 22 ±2°C, according to the method. This range leads to an uncertainty in the measured volume, due to a considerably larger volume expansion of the liquid compared to the vessel. The coefficient of volume expansion for water is 2.1·10^{4}°C^{1}. This leads to a possible volume variation of ±(332 · 2 · 2.1·10^{4}) mL. A rectangular distribution is assumed for the temperature variation of the volume. Since f_{VL}_{}_{temperature} is a multiplicative factor to the nominal volume, which is only used to introduce the temperature uncertainty, it has the value 1. Its uncertainty is calculated as the possible volume variation divided by the volume.
f_{VL}_{}_{reading}: 
Type B triangular distribution Value: 1 Halfwidth of Limits: 0.01 
f_{VL}_{}_{calibration}: 
Type B triangular distribution Value: 1 Halfwidth of Limits: =2.5/332 
The volume is calibrated within ±2.5 mL for a 500 mL measuring cylinder. No further statement is made about the level of confidence or the underlying distribution. An assumption is necessary to work with this uncertainty statement. In this case a triangular distribution is assumed. Since f_{VL}_{}_{calibration} is a multiplicative factor to the nominal volume, which is only used to introduce the calibration uncertainty, it has the value 1. The halfwidth of limits corresponds to the relative uncertainty as stated by the manufacturer (i.e. 2.5 mL / 332 mL).
a_{V}_{ }_{nominal}: 
Constant Value: 2.37 dm^{2} 
The nominal surface are is not associated with any uncertainties. Three different factors contribute to the uncertainty of the real surface area, that are the two length measurements required to calculate the surface area, and an areafactor, covering the imperfect geometry of any real vessel.
f_{aV}_{}_{length1}: 
Type B normal distribution Value: 1 Expanded Uncertainty: =0.01/1.45 Coverage Factor: 1 
Typically, two length measurements are required to calculate the surface area of a vessel. In this case, the item was approximated by a cylindrical geometry. Typical dimensions are between 1.0 and 2.0 dm, leading to an estimated uncertainty of 1 mm. The two length measurements required for this vessel were 1.45 and 1.64 dm.
f_{aV}_{}_{lenght2}: 
Type B normal distribution Value: 1 Expanded Uncertainty: =0.01/1.64 Coverage Factor: 1 
Typically, two length measurements are required to calculate the surface area of a vessel. In this case, the item was approximated by a cylindrical geometry. Typical dimensions are between 1.0 and 2.0 dm, leading to an estimated uncertainty of 1 mm. The two length measurements required for this vessel were 1.45 and 1.64 dm.
f_{aV}_{}_{area}: 
Type B normal distribution Value: 1 Expanded Uncertainty: =0.05/1.96 Coverage Factor: 1 
The item is not a perfect geometric shape (cylinder in this case). Therefore the real surface area may deviate from the caluclated one. This deviation was estimated to be 5% at 95% confidence level. To obtain the standard uncertainty the possible deviation is divided by 1.96.
c_{0}: 
Type B normal distribution Value: 0.26 mg/L Expanded Uncertainty: 0.018 mg/L Coverage Factor: 1 
The content of cadmium in the extraction solution is calculated using a calibration curve. For the calibration curve five calibration standards were prepared and measured 3 times each. Using a linear least square fit, the slope and intercept of the calibration curve have been calculated. Using this data, the concentration c_{0} was calculated from a duplicate measurement of the actual leach solution. The calculation of the uncertianty of the least square fit is described in Appendix E3 of the EURACHEM / CITAC Guide. Only the final result of this calculation is used here.
d: 
Type B normal distribution Value: 1 Expanded Uncertainty: 0 Coverage Factor: 1 
For this sample, no dilution of the leach solution was necessary, therefore no uncertainty needs to be introduced here.
f_{acid}: 
Type B normal distribution Value: 1 Expanded Uncertainty: =0.008*0.1 Coverage Factor: 1 
Data from two study on the effect of acid concentration on lead release was used to estimate this factor. One study showed that lead release was increased by approximately 0.1 when the acid concentration was increased from 4 to 5% v/v. Another study reported a 50% increase of the lead release for a change of acid concentration from 2 to 6% v/v. Assuming a liner effect, on can estimate a change of f_{acid} of 0.1 per 1% v/v change of acid concentration. In another experiment the concentration of the acetic acid and its standard uncertainty have been established using titration with standardised NaOH solution, resulting in an acetic acid concentration of 3.996 % v/v with a standard uncertainty of 0.008% v/v. The uncertainty of f_{acid} can then be calculated as 0.008 · 0.1.
f_{time}: 
Type B rectangular distribution Value: 1 Halfwidth of Limits: =0.5*0.003 
For a relatively slow process such as leaching, the amount leached will be approximately proportional to the leaching time for small changes in that time. In a study, a mean change of concentration over the last six hours of leaching of approximately 1.8 mg · L^{1} (=0.3% / h) was found. The leaching time is specified in the method as 24 ±0.5 h, the content of Cd in the extraction solution therefore needs to be corrected by a factor of 1 ±(0.5 · 0.003). A rectangular distribution is assumed for this factor.
f_{temperature}: 
Type B rectangular distribution Value: 1 Halfwidth of Limits: 0.1 
A number if studies on the effect of temperature on metal release from ceramic ware have been undertaken. The temperature effect is substantial, and a nearexponential increase in metal release with temperature is observed until limiting values are reached. Nevertheless, only one study gives information for the temperature range 2025°C. The metal release approximately linear wit temperature in this temperature range, with a gradient of approximately 5% °C^{1}. For the ±2°C range allowed by the empirical method, this leads to a factor f_{temperature} of 1 ± 0.1. A rectangular distribution is assumed for this factor.
Interim Results:
Quantity  Value 
Standard Uncertainty 

V_{L}  0.33034 L  1.82·10^{3} L 
a_{V}  2.3700 dm^{2}  0.0643 dm^{2} 
Uncertainty Budgets:
r: Mass of cadmium leached per unit area
Quantity  Value 
Standard Uncertainty 
Distribution 
Sensitivity Coefficient 
Uncertainty Contribution 
Index 

V_{L}_{ }_{nominal}  0.332 L  
f_{VL}_{}_{filling}  0.99500  2.04·10^{3}  triangular  0.036  74·10^{6} mg/dm^{2}  0.0 % 
f_{VL}_{}_{temperature}  1.000000  242·10^{6}  rectangular  0.036  8.8·10^{6} mg/dm^{2}  0.0 % 
f_{VL}_{}_{reading}  1.00000  4.08·10^{3}  triangular  0.036  150·10^{6} mg/dm^{2}  0.2 % 
f_{VL}_{}_{calibration}  1.00000  3.07·10^{3}  triangular  0.036  110·10^{6} mg/dm^{2}  0.1 % 
a_{V}_{ }_{nominal}  2.37 dm^{2}  
f_{aV}_{}_{length1}  1.00000  6.90·10^{3}  normal  0.036  250·10^{6} mg/dm^{2}  0.5 % 
f_{aV}_{}_{lenght2}  1.00000  6.10·10^{3}  normal  0.036  220·10^{6} mg/dm^{2}  0.4 % 
f_{aV}_{}_{area}  1.0000  0.0255  normal  0.036  930·10^{6} mg/dm^{2}  7.3 % 
c_{0}  0.2600 mg/L  0.0180 mg/L  normal  0.14  2.5·10^{3} mg/dm^{2}  53.9 % 
d  1.0  0.0  normal  0.0  0.0 mg/dm^{2}  0.0 % 
f_{acid}  1.000000  800·10^{6}  normal  0.036  29·10^{6} mg/dm^{2}  0.0 % 
f_{time}  1.000000  866·10^{6}  rectangular  0.036  31·10^{6} mg/dm^{2}  0.0 % 
f_{temperature}  1.0000  0.0577  rectangular  0.036  2.1·10^{3} mg/dm^{2}  37.5 % 
r  0.03624 mg/dm^{2}  3.42·10^{3} mg/dm^{2} 
Results:
Quantity  Value 
Expanded Uncertainty 
Coverage factor 
Coverage 

r  0.0362 mg/dm^{2}  6.8·10^{3} mg/dm^{2}  2.00  95% (ttable 95.45%) 