Accred Qual Assur (2012) 17:129–138
DOI 10.1007/s00769-011-0827-5
GENERAL PAPER
Uncertainty of standard addition experiments:
a novel approach to include the uncertainty associated
with the standard in the model equation
Anna-Lisa Hauswaldt • Olaf Rienitz • Reinhard Jährling
Nicolas Fischer • Detlef Schiel • Guillaume Labarraque •
Bertil Magnusson
•
Received: 17 June 2011 / Accepted: 26 August 2011 / Published online: 16 September 2011
Ó Springer-Verlag 2011
Abstract A new model equation for determining the
measurement result in standard addition experiments was
derived and successfully applied to the quantitative determination of rhodium in automotive catalysts. Existing
equations for standard addition experiments with gravimetric preparation were changed in order to integrate the
novel idea of including the uncertainty associated with the
standard into the model equation. Using this novel equation
combined with the ordinary least squares algorithm for the
regression line also yielded a new formula for the associated measurement uncertainty. This uncertainty accounts
for the first time for the uncertainty associated with the
standard. The derivation for the model equation and the
resulting associated measurement uncertainty is shown for
gravimetric standard addition experiments both with and
without an internal standard.
Keywords Standard addition Measurement uncertainty Internal standard Gravimetric preparation Metrological traceability
Electronic supplementary material The online version of this
article (doi:10.1007/s00769-011-0827-5) contains supplementary
material, which is available to authorized users.
A.-L. Hauswaldt (&) O. Rienitz R. Jährling D. Schiel
Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100,
38116 Braunschweig, Germany
e-mail: [email protected]
N. Fischer G. Labarraque
Laboratoire National de Métrologie et d’Essais (LNE),
1 rue Gaston Boissier, 75724 Paris Cedex 15, France
B. Magnusson
SP Sveriges Tekniska Forskningsinstitut, P.O. Box 857,
501 15 Borås, Sweden
Introduction
Standard addition is applied to liquid samples or solutions
of solid samples in case of complex matrices and low
concentrations, if external calibration methods provide
wrong results (due to different sensitivities in the sample
and calibration solution) and if the method of isotope
dilution is too tedious and expensive or impossible
(monoisotopic analytes).
Standard addition can be performed based on either
volumetric or gravimetric sample preparation. As the volumetric sample preparation causes a loss of accuracy by
demanding simplifying preconditions, a model equation for
gravimetric standard addition experiments was set up [1].
To reduce the contribution to the measurement uncertainty
caused by indication drifts and fluctuation, an internal
standard can additionally be used. The accordingly modified gravimetric equation was successfully tested in an
international pilot study [2, 3].
A new approach to include the uncertainty associated
with the standard directly into the model equation was
presented in [4]. Overcoming the limitations of this latest
approach, a new model equation was derived from the
equations in [1] and [2] for gravimetric standard addition
experiments both with (section Standard addition combined with internal standard) and without (section
Gravimetric standard addition) an internal standard.
Conveniently, both versions of gravimetric standard
addition experiments yield the same model equation and
have hence the same associated measurement uncertainty,
see section Measurement uncertainty for standard addition.
An example is given for evaluating the measurement
result of standard addition experiments using an internal
standard and gravimetric preparation. The method was
applied to the quantitative determination of the mass
123
130
wz
Standard z
wx
Sample x
Solvent
y
fraction of rhodium (Rh) in automotive catalysts, which
was the measurand of an international pilot study (CCQMP63, see [2, 3]).
The experiments were performed on a high-resolution
inductively coupled plasma mass spectrometer (HR-ICPMS) and a multicollector inductively coupled plasma mass
spectrometer (MC-ICP-MS), respectively, acquiring the
indications of 103Rh and 115In.
It was shown that the method of gravimetric standard
addition with an internal standard (indium) yields better
results than the gravimetric standard addition without an
internal standard and much better results than the method
of external one-point calibration for determining the mass
fraction of Rh in automotive catalysts.
For the calculation of the measurement uncertainty, the
uncertainties of the following influences are considered:
The uncertainty of the x-intercept, the mass fraction of the
Rh standard, the weighing (including the air buoyancy
correction), the dry mass correction, the homogeneity of
the sample and the sample preparation (digestion). While
the last four contributions have to be introduced by
extending the model equation of the standard addition, the
uncertainty of the x-intercept and—for the first time—the
uncertainty associated with the standard are accounted for
directly in the equation of the standard addition.
Accred Qual Assur (2012) 17:129–138
0
a0
x0 0
tan
= a1
x
Fig. 1 Standard addition: Preparation of the measurement solutions
consisting of sample x, standard z (with mass fractions wx and
wz, respectively) and a solvent; applying the measurement results on
the calculation of the calibration line with y-intercept a0 and slope a1.
x, y variables of abscissa and ordinate, respectively. x-intercept x0
equals the ratio wx/wz as will be shown below
sample preparation is completely conducted on a balance
and all individual masses are arbitrary but well known.
Gravimetric standard addition
Theory
Standard addition
In case of matrices being difficult to analyze or monoisotopic analytes, the method of standard addition can be used
instead of isotope dilution (IDMS) or other techniques, see
[1, 4].
Figure 1 depicts the idea of standard addition which is
an internal calibration method. At least three different
blends are measured consisting of an aliquot of sample x,
different amounts (starting with zero) of a standard z of the
analyte concerned and a solvent. The analyte is measured
in these different solutions, and the indications, together
with their respective analyte concentrations, define an
ascending line. If linearity is given, the absolute value of
the x-intercept equals the ratio of the analyte mass fraction
wx in the original sample to the mass fraction of the analyte
in the standard wz.
In practical analytical work, it is common to do the
preparation volumetrically, see for example [5, 6]. Here,
the different blends having exactly equal volumes contain
exactly the same amount of sample. Unfortunately, this
condition is difficult or even impossible to achieve in
practical laboratory work. Hence, the gravimetric approach
was added to the method of standard addition, meaning the
123
Gravimetric standard addition is explained in Fig. 2 with
the symbols listed in Table 1. There are k solutions i with a
mass mi, containing the masses mx,i of sample x and different masses mz,i of standard z. Measuring the ith solution
yields an indication Ai.
For evaluating the standard addition, the gravimetric
approach results in a linear equation, which is based
exclusively on mass and mass fraction and accounts, in its
shown form, for the individuality of each solution prepared. Therefore, deviations from the ideal approach,
which are unavoidable in practice, do not become a source
of error.
The evaluation in the gravimetric standard addition is
done as follows. The sensitivities of the analyte from
sample x and reference z have to be equal in the final
solutions. This precondition is crucial and usually met after
mixing and equilibrating with the matrix or after a digestion. Starting from the relation between the concentration ci
of the analyte in the ith solution, the sensitivity a01 and the
indication Ai (minus the blank value and background), see
Eq. 1, and using the densities qi, all concentrations ci are
substituted by mass fractions wi. The mass fraction wi of
the analyte in the ith solution is—as shown in Fig. 2—the
sum of the analyte in the mass mx,i of sample x with the
mass fraction wx of the analyte in sample x, which is
the measurand, and the added mass mz,i of standard z with
Accred Qual Assur (2012) 17:129–138
131
Solvent
i-th
Solution
Standard z
m z,i
Sample x
m x,i
mi
i
Vessel
Fig. 2 Gravimetric standard addition: solution and indication Ai. The ith solution has a density qi and a mass mi which contains the masses of
sample x and of added standard z (mx,i and mz,i, respectively)
Table 1 Symbols used in gravimetric standard addition
Hence, we have the linear equation
Symbol
Quantity
a0, a1
y-Intercept and slope of the linear fit function
a01
Sensitivity
y i ¼ a1 x i þ a0
mi 1
with yi ¼ Ai mx;i qi
Ai
Indication of the ith solution
and a1 ¼ a01 wz ;
mi
Mass of the ith solution
mx,i
Mass of sample x in the ith solution
mz,i
Mass of added standard z in the ith solution
wx
Mass fraction of the analyte in sample x
wz
Mass fraction of the analyte in standard z
wi
Mass fraction of the analyte in the ith solution
ci
Mass concentration of the analyte in the ith solution
qi
x, y
Density of the ith solution
Variables of abscissa and ordinate
ð1Þ
where
ci ¼ qi wi
and
wi ¼
mx;i wx þ mz;i wz
:
mi
Hence,
mx;i wx þ mz;i wz
Ai ¼ qi wi ¼ qi mi
mi 1
m
w
þ
m
x;i
x
z;i wz
Ai ¼ a01 mx;i qi
mx;i
mz;i
0
¼ a1 wz þ a01 wx :
mx;i
a01
xi ¼
ð3Þ
It follows for the mass fraction of the analyte in the sample x:
a0
wx ¼ wz
ð4Þ
a1
the mass fraction of the analyte in standard z related to the
total mass mi of the ith solution. This can be written as a
linear equation 2, whose x-intercept yields the measurand,
see Eqs. 3 and 4.
Ai ¼ a01 ci
mz;i
mx;i
a1
0
a0 ¼ a1 wx ¼
wx :
wz
and
Note, that Eq. 1 contains the concentration ci and not the mass
fraction wx of the analyte in sample x. Nevertheless, the Eqs.
2–4 can be expressed in terms of mass fraction wx by using the
different densities qi of the solutions. Since almost all analytical instruments generate indications proportional to the
amount of analyte in a certain volume, due to sample loops or
pumps used to introduce the sample, Eq. 1 as the basis has to be
written in terms of a concentration. The density qi directly
connects the concentration ci and the mass fraction wi and has
therefore to be considered in the model equation after substituting the concentration with the mass fraction. In practice, the
densities qi are nearly equal for all solutions i and usually equal
to the density of the solvent used to prepare the solutions.
As all quantities are known, this approach does not contain
any approximations or preconditions (with the exception of
linearity), and it is independent of temperature [1]. Even
though the densities qi are temperature dependent in theory,
the final result is nevertheless independent of the temperature
since simulations have shown, that, as long as the densities qi
are equal, the value of qi does not change the result at all.
Standard addition combined with internal standard
a01
ð2Þ
In the standard addition with an internal standard, the
solutions consist of sample x, different amounts of a
standard z, a solvent and additionally an internal standard
y, which is contained in neither the sample x, the standard z
nor the solvent, see Fig. 3.
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132
Accred Qual Assur (2012) 17:129–138
With
wy
Internal standard y
wz
Standard z
a01 ¼
b0
:
b00 wy
Equation 5 yields after rearranging
Ri my;i ¼ a01 mx;i wx þ mz;i wz
Ri wx
Sample x
my;i
mx;i
¼ a01 mx;i wx þ a01 mz;i wz
mz;i
¼ a01 wz þ a01 wx :
mx;i
Hence, we have again the linear equation
Solvent
Fig. 3 Gravimetric standard addition with an internal standard:
Preparation of the measurement solutions consisting of sample x,
standard z, internal standard y (with the mass fractions wx, wz and wy,
respectively) and a solvent
Then, not one indication Ai but the ratio Ri of the indications of the analyte and the internal standard is considered as
shown in Fig. 4. The resulting linear equation yields again
the measurand via its x-intercept, see Eq. 6.
As in section Gravimetric standard addition the linear
relation of the indication A and the respective concentration
c is the base for deducing the equation of the standard
addition with an internal standard. From Eq. 1 and the
equations
my;i wy
mx;i wx þ mz;i wz
wi ðXÞ ¼
; wi ðYÞ ¼
mi
mi
y i ¼ a1 x i þ a 0
my;i
Ai ðXÞ
mz;i
; xi ¼
with yi ¼ Ri ; Ri ¼
Ai ðYÞ
mx;i
mx;i
a1
and a1 ¼ a01 wz ; a0 ¼ a01 wx ¼
wx :
wz
It follows that
a0
wx ¼ wz
a1
with the symbols explained in Table 2.
The resulting linear equation can be applied to the
evaluation of gravimetric standard addition experiments,
which additionally use an internal standard. In addition to
the advantages of the gravimetric approach, the standard
addition with an internal standard has the following
advantages:
1.
it follows that
Ai ðXÞ b0 ci ðXÞ
b0 q wi ðXÞ
¼ 00
¼ 00 i
Ai ðYÞ b ci ðYÞ b qi wi ðYÞ
b0 wi ðXÞ b0 mx;i wx þ mz;i wz =mi
¼
¼ 00
b w i ðY Þ
b00 my;i wy =mi
b0 mx;i wx þ mz;i wz
¼
b00 my;i wy
0
b
mx;i wx þ mz;i wz
¼ 00
:
b wy
my;i
No exact knowledge about the mass fraction wy of the
internal standard is needed.
The result is not changed by dilution or storage as the
mass of the ith solution mi is not contained in the
equation.
The result is independent of temperature because the
density qi was canceled from the equation.
The ratio of indications is evaluated; hence, possible
changes caused by the measuring device are canceled
to a certain degree.
2.
Ri ¼
ð6Þ
3.
4.
ð5Þ
Ri
Solvent
i-th
Solution
Standard z
mz,i
Int. std. y
my,i
Sample x
mx,i
mi
i
Vessel
Fig. 4 Standard addition with internal standard: solution and measurement of the ratio Ri of the indications Ai(X) and Ai(Y). The ith solution has
a density qi and a mass mi which contains the masses of sample x, of internal standard y and of standard z (mx,i, my,i and mz,i, respectively)
123
Accred Qual Assur (2012) 17:129–138
133
Table 2 Symbols used in standard addition with internal standard
(IS)
Symbol
Quantity
a0, a1
y-Intercept and slope of the linear fit function
b
0
Sensitivity of analyte X
b00
Sensitivity of IS Y
a01
Ratio of the sensitivities of analyte X and IS Y
divided by the mass fraction of IS solution
Ri
Ratio of the indications of analyte X
and IS Y in the ith solution
mx,i
Mass of sample x in the ith solution
Experimental
my,i
Mass of IS solution y in the ith solution
mz,i
Mass of added standard z in the ith solution
wx
Mass fraction of the analyte in sample x
wy
Mass fraction of the IS
wz
Mass fraction of the analyte in standard z
x, y
Variables of abscissa and ordinate
Measurement uncertainty for standard addition
From the model equations 4 and 6, respectively, the measurement uncertainty for the standard addition can be
derived as follows (for details see Appendix: Derivation of
the measurement uncertainty) using the standard approach
of propagation of variances [7]:
a0
wx ¼ wz yields
a1
owx 2 2
owx 2 2
2
u ðwx Þ ¼
u ð a0 Þ þ
u ð a1 Þ
oa0
oa1
owx 2 2
u ðwz Þ
þ
owz
owx
owx
þ2
uð a0 ; a1 Þ
oa0
oa1
owx
owx
þ2
uð a0 ; w z Þ
oa0
owz
owx
owx
þ2
uða1 ; wz Þ:
oa1
owz
Hence,
2
2 3
wx
2 2 2
þ
x
wz
uð w x Þ
uð w z Þ
S 61
7
¼
þ 2 4 þ Pn
5
2
wx
wz
n
a0
Þ
i¼1 ðxi x
ð7Þ
with
2
S ¼
Basic for Applications (VBA), which is available as
Electronic Supplementary Material from the journal’s
website along with a corresponding Excel file containing
an example data set.
Pn
i¼1
½ y i ð a0 þ a1 x i Þ 2
;
n2
Pn
x ¼
i¼1 xi
n
:
To calculate the uncertainty associated with wx
according to Eqs. 17 and 7, respectively, a Microsoft
Excel macro (MU_StdAdd_wz()) was written in Visual
The sample preparation and measurements described were
part of the participation in the international pilot study
CCQM-P63. The automotive catalyst sample was provided
by LGC (UK), the co-ordinating laboratory of this study.
From the powdered, homogenized and dried catalyst
material five aliquots i ði ¼ 1; . . .; 5Þ with a mass mx,i were
sampled and weighed directly into PTFE-TFM microwave
vessels. After adding an appropriate mass of internal
standard and standard solution my,i and mz,i, respectively,
the samples were digested microwave assisted using an
ETHOS-1600 (MLS, Germany) in two steps using nitric,
hydrofluoric and hydrochloric acid. Three blanks were
prepared in the same way without adding sample, standard
or internal standard.
Additional aliquots were used to determine the water
content of the catalyst sample in order to calculate the dry
mass correction factor wdry.
All weighings were corrected for the air buoyancy. The
density of HCl with w(HCl) = 0.07 g/g (q(HCl) = 1033 kg/m3
according to [8]) was used as an appropriate approximation
of the density of the solutions qi. The Rh standard solution
was prepared from Rh powder PMR-461 (Alfa Aesar,
Germany). The powder was digested within 5 h at 290 °C
(High Pressure Asher, Anton Paar, Austria) using HCl with
w(HCl) = 0.3 g/g (Merck, suprapur) and chlorine (generated
in situ from KClO3, Merck, for analysis). The resulting solution was diluted accordingly.
The masses of 103Rh and 115In were measured using an
HR-ICP-MS (Element2, Thermo Scientific, Germany). All
parameters of the ICP-MS were optimized for signal
intensity. The standard quartz sample introduction system
was used. Ten series of measurements with increasing
concentrations of Rh were evaluated using both standard
addition and standard addition with an internal standard.
These measurements were repeated with an MC-ICP-MS
(Neptune, Thermo Scientific, Germany).
Additionally, a Rh calibration solution with a Rh mass
fraction of wz,cal was prepared and integrated in the measurement sequence. The Rh indication for the solution
(i = 1) with no standard added (mz,1 = 0) was evaluated
against the indication Az observed for the external calibration standard in order to compare the external one-point
calibration to the standard addition. The one-point calibration results were calculated according to Eq. 8. Symbols
not mentioned above are listed in Table 1.
123
134
Accred Qual Assur (2012) 17:129–138
wx ¼
A1
m1
wz;cal :
Az
mx;1
ð8Þ
The actual standard addition experiment consisted of the
following steps:
1.
Weighing:
Five sample aliquots i ði ¼ 1; . . .; 5Þ of mx;i 100 mg:
(b) Internal standard (In-solution with wy & 40 lg/g)
with my,i & 0.9 g.
(c) Standard (Rh-solution with wz & 40 lg/g) with
mz;i 0; . . .; 1:2 g.
(a)
2.
Sample preparation:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
3.
Adding of 3 mL HNO3 with w(HNO3) = 0.65 g/g
(Merck, p.a., subboiled), 9 mL HCl with
w(HCl) = 0.3 g/g (Merck, suprapur).
First microwave digestion, 1 h, 210 °C.
Hot plate evaporation (220 °C) down to 1 mL.
Adding of 6 mL HNO3 with w(HNO3) = 0.65 g/g,
3 mL HCl with w(HCl) = 0.3 g/g, 3 mL HF with
w(HF) = 0.49 g/g (Fluka, trace select ultra).
Second microwave digestion, 1 h, 210 °C.
Hot plate evaporation (210 °C) down to 0.5 mL.
Transferring to LD-PE bottle and filling up to
solution mass mi & 120 g with HCl with
w(HCl) = 0.07 g/g.
Measurement with ICP-MS:
(a) Plasma power 1200 W.
(b) Mass resolution 300.
(c) Counting mode monitoring the indications
A(103Rh) and A(115In).
4.
Evaluation (according to Table 5):
(a) Calculating xi.
(b) Subtracting the blank indications.
(c) Calculating yi from the 103Rh indication (standard
addition).
(d) Calculating yi from the ratios Ri of the indications
of 103Rh and 115In (standard addition with an
internal standard).
(e) Calculating a0 and a1 from the ordinary least
squares algorithm.
(f) Calculating wx and u(wx).
Results and discussion
Four different scenarios (A–D) were compared (see Table 4;
Fig. 5): scenario A: external one-point calibration with
measurements on an HR-ICP-MS, scenario B: standard
123
Fig. 5 Left-hand side: Intermediate result wx without dry mass
correction calculated according to Eqs. 8, 6 and 17 yielded from four
different scenarios A–D; right-hand side: final result w with (among
others) dry mass correction applied according to Eq. 9, also yielded
from the four different scenarios A–D. Error bars denote the standard
uncertainty (left-hand side) and expanded uncertainties (right-hand
side), respectively. The consensus value and its expanded measurement uncertainty from the pilot study CCQM-P63 (dotted and dashed
lines, respectively) plotted as the reference. See Table 4 for details
addition with measurements on an HR-ICP-MS, scenario C:
standard addition combined with an internal standard measured on an HR-ICP-MS, scenario D: like C but with
measurements on an MC-ICP-MS. The pilot study consensus
value [3] from CCQM-P63 served as the reference.
Scenario A was evaluated according to Eq. 8. The
method of external one-point calibration resulted in a
rhodium mass fraction, which differed from the consensus
value by almost 100%. Evaluating the associated measurement uncertainty as well as applying the dry mass
correction was, thus, superfluous in this case. Contrarily,
with the gravimetric standard addition scenario B (presented in section Gravimetric standard addition), the
deviation from the consensus value was covered by the
measurement uncertainty associated with the rhodium mass
fraction. However, this uncertainty was considerably large
(13%). Scenario C, the additional use of an internal standard, (presented in section Standard addition combined
with internal standard) resulted in a dramatically reduced
uncertainty compared to scenario B (2.1%). The deviation
from the consensus value was already covered by the
measurement uncertainty associated with the consensus
value. Obviously, the indication ratios—accomplished by
the addition of the internal standard—rather than absolute
indications improved the precision, even on a single-collector and therefore sequentially measuring ICP-MS. Since
the analyte was not separated from the very complex
matrix—except for major parts of the silicon—by means of
chromatography (or other techniques), the absolute indications showed large standard deviations (drift and noise).
Accred Qual Assur (2012) 17:129–138
Table 3 Symbols used in Eq. 9
135
Symbol
Unit
Quantity
w
lg/g
Mass fraction of the analyte Rh in the automotive catalyst sample—measurand
wdry
g/g
Dry mass correction—result of repeated measurements
fexp
1
Sampling, sample preparation and inhomogeneity
wx
lg/g
Result of the standard addition model equation (mass fraction of the analyte
in the sample without moisture correction)
dmx
1
Uncertainty contribution from the sample mass
dmz
1
Uncertainty contribution from the mass of standard added
dmi
1
Uncertainty contribution from the mass of solutions measured
Table 4 Rhodium mass fractions without (intermediate result) and
with dry mass correction applied (final result) along with their
associated uncertainties (standard uncertainties u in case of the
Scenario
Method
Device
intermediate result, expanded uncertainties U with the coverage factor
k (p = 0.95) in case of the final result)
Intermediate result
wx
(lg/g)
u
(lg/g)
A
External one-point calibration
HR-ICP-MS
450
B
Gravimetric standard addition
HR-ICP-MS
215.6
C
Std. add. with internal standard
HR-ICP-MS
234.6
4.8
D
Std. add. with internal standard
MC-ICP-MS
232.9
2.1
Final result
urel
(%)
w
(lg/g)
U
(lg/g)
Urel
(%)
k
(1)
–
–
450
–
–
28
13
218
58
27
2.1
237.0
19
8.0
2.2
0.9
235.4
15
6.3
2.0
1.8
2.0
Pilot study (consensus value)
234.0
4.2
–
2.0
Four scenarios A–D represent combinations of different methods and mass spectrometers (devices)
Using a simultaneously measuring MC-ICP-MS to analyze
the solutions (scenario D) took full advantage of the
combination of standard addition and internal standard
resulting in a further reduced uncertainty (0.9%).
The rhodium mass fraction wx calculated according to
Eq. 6 without dry mass correction is only an intermediate
result and becomes therefore one input quantity of the final
rhodium mass fraction w described with Eq. 9. The
meaning of the symbols used are compiled in Table 3.
w¼
1
fexp wx dmx dmz dmi :
wdry
ð9Þ
The uncertainties associated with these final (dry mass
corrected) results w were determined using the GUMWorkbenchTM [9] and Eq. 9 as the model equation. Table 4
and Figure 5 summarize the rhodium mass fractions without and with the dry mass correction applied in case of the
four scenarios A–D. The dry mass correction and sample
preparation increased the measurement uncertainty considerably. This way the advantage of the MC-ICP-MS
(scenario D) became less obvious. Fig. 6 shows the relative
contribution of all relevant input quantities to the measurement uncertainty associated with the dry mass corrected
rhodium mass fraction w. In case of scenario D the
Fig. 6 The measurement uncertainty associated with the dry mass
corrected Rh mass fraction w according to Eq. 9 is dominated by
different input quantities in case of the different scenarios B–D.
Relative contributions (calculated as the fraction of the squared
product of the sensitivity coefficient and the standard uncertainty of
each input quantity from u2c (w), [7]) to the uncertainty associated with
w arising from the following input quantities (compare Eq. 9): light
gray: intermediate result/measurement wx, white: sample preparation/
dry mass correction fexp, dark gray: sample mass mx, black: masses of
standard added and masses of final solutions (mz,i and mi, respectively). See Table 4 for details about the different scenarios
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Accred Qual Assur (2012) 17:129–138
Table 5 Summary of equations in standard addition using a linear model yi ¼ a1 xi þ a0
m
z;i
xi ¼ mx;i
OLS algorithm
Model equation
wx ¼ aa01 wz
Gravimetric preparation
plus internal standard
yi ¼ Ai mmx;ii q1
yi ¼ Ri my;i
x;i
m
i
a0 ¼
Pn
Pn 2 Pn
Pn
xi x y i¼1 xi
i¼1 i i
Pi¼1
P
¼ y a1 x
2
n
n
n
x2 ð
xÞ
i¼1 i
i¼1 i
y
i¼1 i
a1 ¼
Measurement uncertainty
2
2
2
ðwx þxÞ
u2 ðwx Þ ¼ w2x uðwwzz Þ þw2z Sa2 1n þ Pnwz
2
1
i¼1
n
and
Þ
Ri ¼ AAii ððX
YÞ
Pn
Pn Pn
Pn
xy
y i¼1 xi
ðyi yÞðxi xÞ
i¼1 i i
i¼1 i
i¼1
P
P
P
¼
2
n
2
n
n
n
x2 ð
xÞ
ðxi xÞ
i¼1
i¼1 i
i¼1 i
where
Pn
Pn
2
xi
½yi ða0 þa xi Þ
; x ¼ i¼1
S2 ¼ i¼1 n2 1
n
ðxi xÞ
All indications Ai must be blank corrected prior to the calculations
measurement itself did not contribute significantly to the
combined uncertainty. As already mentioned, it looks as if
the use of MC-ICP-MS seems to be an inappropriate effort.
But the MC-ICP-MS opens up the possibility to monitor the
influence of the dry mass correction and different sample
preparation procedures, which is nearly impossible using
the HR-ICP-MS, because the information is hidden by the
uncertainty associated with the measurement itself.
The gravimetric standard addition yielded a Rh mass
fraction with the relative expanded (p = 0.95) measurement uncertainty Urel = 27%. Applying the method of
gravimetric standard addition with an internal standard (see
section Standard addition combined with internal standard) decreased considerably the measurement uncertainty
(Urel = 8%). Using the simultaneously measuring MCICP-MS causes further improvements (Urel = 6.3%). Even
though these uncertainties seem to be rather large compared to IDMS, the small sample sizes (50–100 mg) have
to be considered. Most of the other participants of CCQMP63 used sample sizes of 1–12 g. The relative expanded
uncertainty of 0.9% associated with the uncorrected Rh
mass fraction calculated from scenario D demonstrates the
potential of the gravimetric standard addition in combination with an internal standard to compete with IDMS.
Table 5 summarizes all equations needed to perform the
data evaluation of standard addition experiments without
and with the additional use of an internal standard. Note
that merely the definition of yi changes in case an internal
standard is used.
The determination of Rh in an automotive catalyst
sample in the framework of an international pilot study
underpinned the potential of the gravimetric standard
addition especially in combination with an internal standard to yield results with a performance close to that of
IDMS even in demanding matrices.
The method described can therefore easily be adopted in
clinical chemistry where samples usually feature complex
matrices and the applicability of IDMS is restricted to a
certain number of analytes. Gravimetric standard addition
will therefore help to set up reference methods without the
efforts of IDMS. This way it could help to save money in
the public health sector.
Acknowledgments The authors wish to thank Carola Pape who
tackled the tricky sample preparation. The research within this
EURAMET joint research project receives funding from the European
Community’s Seventh Framework Programme, ERA-NET Plus,
under Grant Agreement No. 217257.
Appendix: Derivation of the measurement uncertainty
Conclusion
A gravimetric standard addition can be modeled mathematically without using simplifying preconditions, which
are difficult or even impossible to fulfill. It offers an
alternative approach to the more sophisticated IDMS in
case of complex matrices, monoisotopic analytes or in case
no mass spectrometer is available in the laboratory. In the
form presented here, even the uncertainty of the added
reference is included in a closed form—to our knowledge—for the first time. This way it becomes easier to
claim metrological traceability of measurement results
determined with standard addition.
123
From the model equations 4 and 6, respectively, the measurement uncertainty for standard addition can be derived
as follows following the guidelines in [7]. With
wx ¼
a0
wz
a1
the partial derivatives are:
owx wz
¼ ;
oa0
a1
owx
wx
¼ ;
oa1
a1
owx wx
¼ :
owz wz
Using the approach of propagation of variances and
inserting the above partial derivatives yields:
Accred Qual Assur (2012) 17:129–138
137
owx 2 2
owx 2 2
u ðwx Þ ¼
u ð a0 Þ þ
u ð a1 Þ
oa0
oa1
owx 2 2
þ
u ðw z Þ
owz
owx
owx
þ2
uða0 ; a1 Þ
oa0
oa1
owx
owx
þ2
uð a0 ; w z Þ
oa0
owz |fflfflfflfflffl{zfflfflfflfflffl}
¼0
owx
owx
þ2
uð a1 ; w z Þ
oa1
owz |fflfflfflfflffl{zfflfflfflfflffl}
uð a0 ; a1 Þ ¼ Pn
2
i¼1
and
1 Xn
n
ð10Þ
The uncertainty u2(wz) was estimated with an additional
uncertainty budget, which includes all preparation steps
along with the air buoyancy correction and concentration
changes due to storage losses of the solvent.
Possible correlations between the regression line and wz
were determined by simulating the experiment. No correlations were observed; hence, we have: uða0 ; wz Þ ¼ 0 and
uða1 ; wz Þ ¼ 0:
The ordinary least squares (OLS) algorithm for linear
regression curves yields the formulae for a0 and a1 (see
Table 5) as well as for u2(a0), u2(a1) and the correlation
u(a0, a1) [10]. The OLS was chosen because it is widely
used and suited to the described experiment with uncertainties larger in y than in x. The derivation of the
uncertainty associated with the analyte mass fraction u(wx)
can be performed analogously with any other suitable fitting algorithm.
Using Eq. 14, the uncertainties associated with a0 and
a1 (Eqs. 11 and 12) were rearranged in the following
way:
Pn
ð11Þ
n S2
S2
u2 ð a1 Þ ¼ Pn
¼
P
P
2
n
n
Þ2
n i¼1 x2i i¼1 ðxi x
i¼1 xi
ð12Þ
u ð a0 Þ ¼
S2 1
¼ Pn n
Pn
2
i¼1 xi
2
2
i¼1 xi
2
Pn 2 Pn
n i¼1 xi i¼1 xi
i¼1
ðxi xÞ
X
1
n
x2
2
i i
n X 2 X
1
2
¼ 2 n i ðxi xÞ þ
x
i i
n
Pn
ðxi xÞ2
¼ i¼1
þ x2 :
n
x2 ¼
i¼1 i
¼0
S2 ð13Þ
ðxi xÞ2
Additionally, the following identities are needed:
X 2
X
X
2
n
x
x ¼n
ðxi xÞ2
ð14Þ
i i
i i
i
2
wx
¼
u2 ðwz Þ
wz
wz
wx
þ2
uð a0 ; a1 Þ
a1
a1
2
wx 2 2
wz
þ u ða1 Þ þ
u2 ð a0 Þ
a1
a1
u2 ðwz Þ
¼ w2x w2z
"
#
w2z
a0 2 2
a0
2
þ 2
u ð a1 Þ 2 uð a0 ; a1 Þ þ u ð a0 Þ :
a1
a1
a1
2
S2 x
ð15Þ
Using the Eq. 11 for u2 ða0 Þ; Eq. 12 for u2 ða1 Þ and Eq. 13
for uða0 ; a1 Þ; as well as Eqs. 14 and 15 the expression in
square brackets from Eq. 10 can be evaluated as follows.
" #
a0 2 2
a0
u ð a1 Þ 2 uð a0 ; a1 Þ þ u2 ð a0 Þ
a1
a1
!
2
a0
S2
a0
S2 x
¼
Pn
2 Pn
a1
a1
Þ2
Þ2
i¼1 ðxi x
i¼1 ðxi x
P
S2 1 n x 2
þ Pn n i¼1 i 2
Þ
i¼1 ðxi x
" #
S2
a0 2
a0
1 Xn 2
¼ Pn
þ2 x þ
x
i¼1 i
n
a1
a1
Þ2
i¼1 ðxi x
¼ Pn
S2
ðxi xÞ2
"i¼1 !#
Pn
Þ2
a0 2
a0
2
i¼1 ðxi x
þ x
þ2 x þ
a1
a1
n
"
#
Pn
2
1
ð
x
x
Þ
i
¼ S2 Pn
i¼1
n
Þ2
i¼1 ðxi x
"
!#
2
1
a0
a0
2
2
þ S Pn
þ2 x þ x
a1
a1
Þ2
i¼1 ðxi x
2
2 3
a0
a1 þ x
61
7
¼ S2 4 þ Pn
25
n
ð
x
x
Þ
i
i¼1
2
2 3
wx
þ
x
wz
61
7
¼ S2 4 þ Pn
:
ð16Þ
25
n
Þ
i¼1 ðxi x
From Eq. 10 by using Eq. 16 it follows
2
2 3
wx
2
2
þ
x
wz
uðwz Þ
S 61
7
u2 ðwx Þ ¼ w2x þw2z 2 4 þ Pn
25
wz
a1 n
ð
x
x
Þ
i
i¼1
ð17Þ
123
138
or written in terms of the relative uncertainty
2
2 3
wx
2 2 2
wz þ x
uð w x Þ
uð w z Þ
S 61
7
:
¼
þ 2 4 þ Pn
25
wx
wz
a0 n
ð
x
x
Þ
i
i¼1
Accred Qual Assur (2012) 17:129–138
ð7Þ
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