This section is based on a measurement described in Appendix 5 of the 3rd edition of the EURACHEM / CITAC Guide1.
The CITAC Guide gives a careful discussion of the uncertainty analysis leading to particular numerical values. The following shows only how the data processing could be preformed using GTC.
The experiment determines the amount of cadmium released from ceramic ware.
The measurand can be expressed as
- where
- r is the mass of cadmium leached per unit area (mg dm − 2),
- c0 cadmium content in the extraction solution (mg L − 1),
- VL is the volume of leachate (L),
- d is the dilution factor,
- av is the surface area of the liquid (dm2).
- facid is the influence of acid concentration.
- ftime is the influence of the duration,
- ftemp is the influence of temperature.
Section A5.4 of the CITAC Guide provides numerical estimates of these quantities that can be used to define uncertain numbers for the calculation.
In this example there was no dilution.
Several factors contribute to the uncertainty of VL:
- VL − fill the relative accuracy with which the vessel can be filled
- VL − temp temperature variation affects the determined volume
- VL − read the accuracy with which the volume reading can be made
- VL − cal the accuracy with which the manufacturer can calibrate a 500 mL vessel
Uncertain numbers for each contribution can be defined and combined to obtain an uncertain number for the volume. In this case, the volume of leachate is 332 mL. :
>>> v_leachate = 332 # mL
>>> a_liquid = 2.1E-4 # liquid volume expansion per degree
>>> v_fill = ureal(0.995,tb.triangular(0.005),label='v_fill')
>>> v_temp = ureal(0,tb.uniform(v_leachate*a_liquid*2),label='v_temp')
>>> v_reading = ureal(1,tb.triangular(0.01),label='v_reading')
>>> v_cal = ureal(0,tb.triangular(2.5),label='v_cal')
# Change units to liters now
>>> V_L = result(
... (v_leachate * v_fill * v_reading + v_temp + v_cal)/1000,
... label='V_L') # L
...
>>> print( "V leachate: {}".format(V_L) )
V leachate: 0.3303(18) The amount of leached cadmium is calculated using a calibration curve. A linear relationship is assumed between observed absorbance and cadmium concentration.
Ai = ci ⋅ B1 + B0 + Ei ,
where B1 and B0 are the slope and intercept, respectively, of the line, Ai is the observed absorbance, ci is the concentration of the ith calibration standard and Ei is the unknown measurement error incurred during the ith observation.
Three repeat observations are made for each of five calibration standards and the parameters of the calibration line are estimated by ordinary least-squares regression.
The GTC calculation uses the ~type_a.line_fit function :
>>> x_data = [0.1, 0.1, 0.1, 0.3, 0.3, 0.3, 0.5, 0.5, 0.5, 0.7, 0.7, 0.7, 0.9, 0.9, 0.9]
>>> y_data = [0.028, 0.029, 0.029, 0.084, 0.083, 0.081, 0.135,
... 0.131, 0.133, 0.180, 0.181, 0.183, 0.215, 0.230, 0.216]
...
>>> fit = ta.line_fit(x_data,y_data,label='regression')
>>> B_0 = fit.intercept
>>> B_1 = fit.slope
>>> print( "Intercept: {}".format(B_0) )
Intercept: 0.0087(29)
>>> print( "Slope: {}".format(B_1) )
Slope: 0.2410(50)There is correlation between these uncertain numbers (the estimates are correlated) :
>>> print( get_correlation(B_0, B_1) )
-0.87038...The object fit contains information about the regression that can be used to make predictions about cadmium concentration from subsequent observations of absorbance. In this case, two further values of absorbance are used to estimate the concentration c0.
Using the function ~type_a.LineFitOLS.x_from_y we write (the label 'absorbance' will be attached to the mean of the observations and identify this influence in the uncertainty budget below) :
>>> c_0 = fit.x_from_y( [0.0712,0.0716], x_label='absorbance',y_label='noise' )
>>> print( "absorbance: {}".format(c_0) )
absorbance: 0.260(18)There are two contributions to the uncertainty of aV:
- aV − dia uncertainty due to the diameter measurement
- aV − shape uncertainty due to imperfect shape
Uncertain numbers for each contribution can be combined to obtain an estimate of the surface area :
>>> dia = ureal(2.70,0.01,label='dia')
>>> a_dia = math.pi*(dia/2)**2
>>> a_shape = ureal(1,0.05/1.96,label='a_shape')
>>> a_V = result( a_dia * a_shape, label='a_V' )
>>> print( "a_V: {}".format(a_V) )
a_V: 5.73(15)The temperature factor is given as ftemp = 1 ± 0.1. Assuming a uniform distribution we define :
>>> f_temp = ureal(1,tb.uniform(0.1),label='f_temp')The time factor is given as ftime = 1 ± 0.0015. Assuming a uniform distribution we define :
>>> f_time = ureal(1,tb.uniform(0.0015),label='f_time')The acid concentration factor is given as facid = 1 ± 0.0008. This is already in the form of a standard uncertainty so we define :
>>> f_acid = ureal(1,0.0008,label='f_acid')To estimate r we now evaluate :
>>> r = c_0 * V_L / a_V * f_acid * f_time * f_temp
>>> print( "r: {}".format(r) )
r: 0.0150(14)The contribution from the different influences can be examined :
>>> for i in rp.budget(r,influences=[c_0,V_L,a_V,f_acid,f_time,f_temp]):
... print( " {}: {:G}".format(i.label,i.u) )
...
absorbance: 0.00102956
f_temp: 0.00086663
a_V: 0.000398736
V_L: 8.28714E-05
f_time: 1.29994E-05
f_acid: 1.20084E-05The results (which can be compared with Figure A5.8 in the Guide) show that the content of cadmium in the extraction solution is the dominant component of uncertainty.
The full uncertainty budget can be obtained by writing :
>>> for i in rp.budget(r,trim=0):
... print( " {}: {:G}".format(i.label,i.u) )
...
noise: 0.000928623
f_temp: 0.00086663
a_regression: 0.000688685
a_shape: 0.00038292
b_regression: 0.000311899
dia: 0.000111189
v_reading: 6.128E-05
v_cal: 4.63764E-05
v_fill: 3.0794E-05
f_time: 1.29994E-05
f_acid: 1.20084E-05
v_temp: 3.65814E-06This reveals that the additional observations of absorbance have contributed most to the uncertainty (so perhaps a few more observations would help)
Footnotes
On-line: http://www.citac.cc/QUAM2012_P1.pdf↩