# A Study of the Critical Uncertainty Contributions in  the Analysis of PCBs in Ambient Air

Authors: Andrew S. Brown; Richard J. C. Brown
Journal: Journal of Automated Methods and Management in Chemistry
 (2008)

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## Abstract

The measurement of polychlorinated biphenyls (PCBs) in ambient air requires
a complex, multistep sample preparation procedure prior to analysis by gas
chromatography—mass spectrometry (GC-MS). Although routine analytical
laboratories regularly carry out these measurements, they are often undertaken
with little regard to the accurate calculation of measurement uncertainty, or
appreciation of the sensitivity of the accuracy of the measurement to each step of the
analysis. A measurement equation is developed for this analysis, and the contributory
sources to the overall uncertainty when preparing calibration standards and other
solutions by gravimetric and volumetric approaches are discussed and compared. For
the example analysis presented, it is found that the uncertainty of the measurement is
dominated by the repeatability of the GC-MS analysis and suggested that volumetric
(as opposed to gravimetric) preparation of solutions does not adversely affect the
overall uncertainty. The methodology presented in this work can also be applied to
analogous methods for similar analytes, for example, those used to measure polycyclic
aromatic hydrocarbons (PAHs), pesticides, dioxins, or furans in ambient air.


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## Body

## 1. Introduction

Polychlorinated
biphenyls (PCBs) (Figure1) are a group of highly toxic persistent organic
pollutants (POPs) which exist in 209 congener forms defined by the number and
location of chlorine atoms substituted onto the phenyl rings. In UK, PCBs were first produced commercially in the
1930s and by the middle of the twentieth century their production was
widespread (66500 tonnes per annum in UK
alone) [1]. Industrial
applications included the manufacture of electronic devices (as transformers
and capacitors), heat-exchange fluids, and as additives in paints, sealants,
and plastics. Their use was phased out voluntarily throughout the 1970s and a
total ban on their use in new plants and equipment was introduced in 1986 [2].
Despite this ban, concerns over the exposure of humans to PCBs remain due to
their long-term stability in the environment—the dangers to health have been identified in
a number of reports [3, 4].Figure 1
Structure of PCBs.In UK
in 2004, the
annual release of PCBs into the air was estimated to be 1330 kg, 68% of which
originated from electronic devices [5]. Other sources were identified as waste
burning and energy production. Humans are additionally exposed to PCBs through
contaminated soil [6], water [7], and food [8–10]. The presence of PCBs in
breast milk [9, 11, 12] is detrimental to the health of newborn babies, for
example, affecting their immune system [13].To limit the
release of PCBs (and other POPs) into the environment, the European Union’s
Waste Electrical and Electronic Equipment (WEEE) Directive [14, 15] sets limits
on the levels on electronic waste that may be sent to landfill. One consequence
of this is that an increasing quantity of unwanted electronic devices is being
disassembled at specialist plants, a number of which are located in East Asia. Recent measurements undertaken at these sites
show that workers are exposed to levels of POPs orders of magnitude greater
than the general population [16].The concern over
the long-term accumulation of these species in the environment leads to PCBs
being covered by the Stockholm Convention on persistent organic pollutants [17]—a treaty signed by 151 states with the aim of
protecting human health and the environment by reducing and eliminating the
release of these toxic species. The UK Government has recently developed a plan
on how to implement the findings of the treaty [1].Accurate
measurement of PCBs in ambient air (and other environmental matrices) is
therefore of great importance to legislators, industry, and the general public,
and a number of documentary standard methods are available for their analysis
[18–20]. The most widely used analytical method involves solvent extraction
followed by sample clean up and analysis by gas chromatography—mass spectrometry (GC-MS) or high-resolution
GC-MS.Although the
measurement of PCBs is now treated as routine by many laboratories (a number of who areaccredited to carry out the analysis by national accreditationorganisations), the
complex, multistage nature of the analysis means that it is likely that most
measurements of PCBs are carried out without a rigorous assessment of the
uncertainty inherent in the final result. With the increasing legislative
importance of the accurate measurement of PCBs (and similar species), there is
a pressing need to determine these uncertainties correctly in order to provide
all interested parties with the confidence in the measurements that they
require. This is exacerbated by the automation of many of the preparative and
analytical processes involved in the measurement. The absence of intervention
by skilled operators can also result in large analytical errors, especially
when the sensitivities of the result to aspects of the method are not properly
understood.This paper first presents a
measurement equation for the analysis of PCBs, using the analysis of an urban
dust certified reference material as an example. This measurement equation is
then used to develop a full uncertainty budget for the analysis, highlighting
the main contributory factors to the overall uncertainty of the final result.
The advantages and disadvantages of using gravimetric or volumetric techniques
for the preparation of calibration standards and other solutions are
quantified, and it is discussed whether calculations should be carried out in
the mass fraction or mass concentration domain in order to minimise
uncertainty. Although the measurement equation and uncertainty budget presented
here are specific to this method, it is hoped that the reader will find it easy to adapt for use in other laboratories. It
may also be used as a basis to determine the uncertainty in the measurement of
similar analytes in ambient dust, for example, polycyclic aromatic hydrocarbons
(PAHs), dioxins, furans, and chlorinated pesticides.


## 2. Experimental

### 2.1. Materials and Reagents

Calibration solutions were
prepared from a certified multicomponent PCB stock solution of nominal mass
concen-tration 2μg⋅mL−1 (LGC Promochem, Teddington, UK), di-luted as required with hexane (PAH analysis grade, Acros, Geel, Belgium, UK). Solutions of the internal standard
(d14-p-terphenyl) and one injection standard (d10-acenapthene)
were prepared from stock solutions of nominal mass concentrations 500 μg⋅mL−1 and 200 μg⋅mL−1, respectively (LGC
Promochem); solutions of the second internal standard (d12-perylene)
were prepared from the pure material (LGC Promochem). The internal standard was
used to define the efficiency of the extraction and workup process, the
injection standard to correct for the temporal drift in sensitivity of the
GC-MS.The certified reference material
(CRM) analysed was NIST SRM 1649a ‘‘urban dust’’ (National Institute ofStandards
and Technology, Gaithersburg, MD, USA). Hexane
and diethyl ether (analytical reagent grade, Fisher,Loughborough, UK) were used as extraction solvents, and
nonane (≥99% grade, Sigma-Aldrich, Gillingham, UK) was added as a ‘‘keeper’’ solvent (to prevent the solution being reduced to dryness).
Extracts were cleaned up using silica (60 to  200μm, Acros), potassium hydroxide (AnalaR grade, VWR, Lutterworth, UK), sulphuric acid (AnalaR grade, BDH), silver nitrate (reagent grade, Fisher), sodium
sulphate (reagent grade, Acros), activated aluminium oxide (50 to 200 μm, Acros), hexane, toluene (GLC analysis
grade, Fisher, UK), and dichloromethane (Baker analysed grade, J. T. Baker, Phillipsburg, NJ, USA). Helium
was used as the GC-MS carrier gas (BIP grade, Air Products, Crewe, UK).


### 2.2. Overview of Analysis Method

The extraction,
clean-up, and GC-MS procedures used were based on methods in the literature
[21] and in documentary standards covering the measurement of PCBs in ambient
air and other matrices [18–20]. An overview of the analytical procedure,
showing the solvent(s) used at each stage is given in the flowchart in Figure 2. Note that the details of the method, particularly the types and volume of
solvents used, may be unique to this analysis, but the same principles can be
easily applied to cover the methods used in other laboratories.Figure 2
Flowchart outlining the analytical method employed.


### 2.3. Extraction

An appropriate
portion (typically 0.3 g–0.5 g) of the predried CRM was Soxhlet
extracted for 24 hours at 4 cycles per hour. Prior to extraction, all samples were spiked with a predetermined quantity of the internal standard (d14-p-terphenyl)
solution added such that the target mass fraction of the internal standard in
the final solution would be approximately 50 ng⋅g−1, if complete
recovery was to be achieved. The resulting extract had 100 μL of nonane added as a ‘‘keeper’’ solvent and was
condensed using a rotary evaporator (Büchi, Flawil, Switzerland) to a volume of less
than 1 mL.


### 2.4. Clean-up

Two successive
column clean-up procedures were used—the first as a
general clean-up of the sample and the second to separate PCBs from other similar
species, for example, polychlorinated dibenzo-p-dioxins (PCDDs) and
polychlorinated dibenzofurans (PCDFs).Column one (300 mm long × 35 mm internal diameter) consisted of successive layers of silica,
potassium hydroxide-coated silica, silica, sulphuric acid coated silica, silica,
silver nitrate-coated silica, and sodium sulphate. The column was prewashed
with 120 mL of hexane and the sample then added, followed by 250 mL of hexane.
The resulting eluant contained the PCBs (and other similar species, e.g., PCDDs
and PCDFs were retained for separation by the second column).Column two (250 mm long × 22 mm internal diameter) consisted of a bottom layer of 25 g of
aluminium oxide under a layer of 5 g of sodium sulphate. The column was
pre-washed with 60 mL of hexane and the sample added. Successive elutions were then carried out using 60 mL hexane (discarded), 90 mL
toluene (eluant contained PCBs—retained for GC-MS analysis), and finally 200 mL hexane and dichloromethane (1 : 1 v/v) (eluant contained PCDDs and PCDFs—retained for
separate analysis if required). The retained eluant was condensed to a volume
of approximately 1 mL.Finally, a
quantity of the injection standard solution (containing d10-acenapthene
and d12-perylene) was added volumetrically so that the mass fraction
of injection standards in the final solution would be approximately 15 ng⋅g−1. (The mass of the solution was recorded before and after each of the above
additions.) The samples were then transferred to the GC-MS for analysis. If
this analysis was not to take place immediately, the samples were stored in
sealed amber-coloured vessels in a refrigerator in order to prevent thermal or
photodegradation.


### 2.5. Preparation of Calibration Solutions

All calibration
solutions were prepared gravimetrically in hexane using a model LA230S balance (Sartorius, Goettingen, Germany). (This is discussed fully in a subsequent section,
where the approaches of preparing solutions volumetrically or gravimetrically
are compared). A range of standards was prepared to cover the expected mass
fraction range of the extracted samples, each of which also contained a known
mass fraction of approximately 50 ng⋅g−1 of the internal standard
and approximately 15 ng⋅g−1 of the injection standard. All solutions
were stored in amber flasks in a refrigerator.


### 2.6. GC-MS Analysis

GC-MS analysis was performed using a model 5890 gas chromatograph with a model 6973 electron
ionisation mass spectrometer detection system (Agilent, Wokingham, UK).
An autosampler was used to allow automated injection of a large number of
samples and calibration standards. Full GC-MS method and optimisation
parameters are given in Table1. The mass spectrometer was operated in selected ion monitoring (SIM) mode with the m/z ratio(s) monitored for
each analyte shown in Table 2. Each sample and calibration solution was
analysed at least three times. To correct for the drift of the instrument, the
ratio of analyte peak area to the injection standard peak area was calculated
for each run—the mean and standard deviation of these
ratios were then determined. The generalised least squares calculations were
carried out using XLGENLINE (National Physical Laboratory, UK)
[22].Table 1
GC-MS method and optimisation parameters.
Gas chromatograph parametersInjection modeSplitlessInjection port temperature275°CInjection volume2μLGC to MS transfer line temperature290°CCarrier gas and flow rateHelium; 1.8 mL⋅min−1ColumnHP-5MS [(5% phenyl-) methyl-polysiloxane],30 m × 0.25 mm × 0.25μm (film thickness)Temperature program(1) 70°C-hold for 2 minutes(2) ramp to 150°C at 15°C⋅min−1-no hold(3) ramp to 300°C at 10°C⋅min−1-hold for 5 minutes[Total time of program = 27.3 minutes]Mass spectrometer parametersIon source temperature250°CQuadrupole temperature150°CElectron multiplier voltage2800 VOther tuning parametersOptimised automaticallyTable 2
Ions monitored in GC-MS single ion monitoring mode.
GroupPCB congener(s) measuredm/z (1)m/z (2)Mono (Cl1) PCBs1, 3188.1152.1Di (Cl2) PCBs4, 15222.0152.1Tri (Cl3) PCBs19, 37255.9186.0Tetra (Cl4) PCBs54, 77, 81291.9220.0Penta (Cl5) PCBs104, 105, 114, 118, 123, 126325.9254.0Hexa (Cl6) PCBs155, 156, 157, 167, 169359.8289.9Hepta (Cl7) PCBs188, 189393.9323.9Octa (Cl8) PCBs202, 205429.8359.9Nona (Cl9) PCBs206, 208463.8393.8Deca (Cl10) PCB209499.6429.8AnalyteUsem/z (1)m/z (2)d14-p-terphenylInternal standard164.1162.1d10-acenaptheneInjection standard 1224.2—d12-peryleneInjection standard 2264.1—


## 2.1. Materials and Reagents

Calibration solutions were
prepared from a certified multicomponent PCB stock solution of nominal mass
concen-tration 2μg⋅mL−1 (LGC Promochem, Teddington, UK), di-luted as required with hexane (PAH analysis grade, Acros, Geel, Belgium, UK). Solutions of the internal standard
(d14-p-terphenyl) and one injection standard (d10-acenapthene)
were prepared from stock solutions of nominal mass concentrations 500 μg⋅mL−1 and 200 μg⋅mL−1, respectively (LGC
Promochem); solutions of the second internal standard (d12-perylene)
were prepared from the pure material (LGC Promochem). The internal standard was
used to define the efficiency of the extraction and workup process, the
injection standard to correct for the temporal drift in sensitivity of the
GC-MS.The certified reference material
(CRM) analysed was NIST SRM 1649a ‘‘urban dust’’ (National Institute ofStandards
and Technology, Gaithersburg, MD, USA). Hexane
and diethyl ether (analytical reagent grade, Fisher,Loughborough, UK) were used as extraction solvents, and
nonane (≥99% grade, Sigma-Aldrich, Gillingham, UK) was added as a ‘‘keeper’’ solvent (to prevent the solution being reduced to dryness).
Extracts were cleaned up using silica (60 to  200μm, Acros), potassium hydroxide (AnalaR grade, VWR, Lutterworth, UK), sulphuric acid (AnalaR grade, BDH), silver nitrate (reagent grade, Fisher), sodium
sulphate (reagent grade, Acros), activated aluminium oxide (50 to 200 μm, Acros), hexane, toluene (GLC analysis
grade, Fisher, UK), and dichloromethane (Baker analysed grade, J. T. Baker, Phillipsburg, NJ, USA). Helium
was used as the GC-MS carrier gas (BIP grade, Air Products, Crewe, UK).


## 2.2. Overview of Analysis Method

The extraction,
clean-up, and GC-MS procedures used were based on methods in the literature
[21] and in documentary standards covering the measurement of PCBs in ambient
air and other matrices [18–20]. An overview of the analytical procedure,
showing the solvent(s) used at each stage is given in the flowchart in Figure 2. Note that the details of the method, particularly the types and volume of
solvents used, may be unique to this analysis, but the same principles can be
easily applied to cover the methods used in other laboratories.Figure 2
Flowchart outlining the analytical method employed.


## 2.3. Extraction

An appropriate
portion (typically 0.3 g–0.5 g) of the predried CRM was Soxhlet
extracted for 24 hours at 4 cycles per hour. Prior to extraction, all samples were spiked with a predetermined quantity of the internal standard (d14-p-terphenyl)
solution added such that the target mass fraction of the internal standard in
the final solution would be approximately 50 ng⋅g−1, if complete
recovery was to be achieved. The resulting extract had 100 μL of nonane added as a ‘‘keeper’’ solvent and was
condensed using a rotary evaporator (Büchi, Flawil, Switzerland) to a volume of less
than 1 mL.


## 2.4. Clean-up

Two successive
column clean-up procedures were used—the first as a
general clean-up of the sample and the second to separate PCBs from other similar
species, for example, polychlorinated dibenzo-p-dioxins (PCDDs) and
polychlorinated dibenzofurans (PCDFs).Column one (300 mm long × 35 mm internal diameter) consisted of successive layers of silica,
potassium hydroxide-coated silica, silica, sulphuric acid coated silica, silica,
silver nitrate-coated silica, and sodium sulphate. The column was prewashed
with 120 mL of hexane and the sample then added, followed by 250 mL of hexane.
The resulting eluant contained the PCBs (and other similar species, e.g., PCDDs
and PCDFs were retained for separation by the second column).Column two (250 mm long × 22 mm internal diameter) consisted of a bottom layer of 25 g of
aluminium oxide under a layer of 5 g of sodium sulphate. The column was
pre-washed with 60 mL of hexane and the sample added. Successive elutions were then carried out using 60 mL hexane (discarded), 90 mL
toluene (eluant contained PCBs—retained for GC-MS analysis), and finally 200 mL hexane and dichloromethane (1 : 1 v/v) (eluant contained PCDDs and PCDFs—retained for
separate analysis if required). The retained eluant was condensed to a volume
of approximately 1 mL.Finally, a
quantity of the injection standard solution (containing d10-acenapthene
and d12-perylene) was added volumetrically so that the mass fraction
of injection standards in the final solution would be approximately 15 ng⋅g−1. (The mass of the solution was recorded before and after each of the above
additions.) The samples were then transferred to the GC-MS for analysis. If
this analysis was not to take place immediately, the samples were stored in
sealed amber-coloured vessels in a refrigerator in order to prevent thermal or
photodegradation.


## 2.5. Preparation of Calibration Solutions

All calibration
solutions were prepared gravimetrically in hexane using a model LA230S balance (Sartorius, Goettingen, Germany). (This is discussed fully in a subsequent section,
where the approaches of preparing solutions volumetrically or gravimetrically
are compared). A range of standards was prepared to cover the expected mass
fraction range of the extracted samples, each of which also contained a known
mass fraction of approximately 50 ng⋅g−1 of the internal standard
and approximately 15 ng⋅g−1 of the injection standard. All solutions
were stored in amber flasks in a refrigerator.


## 2.6. GC-MS Analysis

GC-MS analysis was performed using a model 5890 gas chromatograph with a model 6973 electron
ionisation mass spectrometer detection system (Agilent, Wokingham, UK).
An autosampler was used to allow automated injection of a large number of
samples and calibration standards. Full GC-MS method and optimisation
parameters are given in Table1. The mass spectrometer was operated in selected ion monitoring (SIM) mode with the m/z ratio(s) monitored for
each analyte shown in Table 2. Each sample and calibration solution was
analysed at least three times. To correct for the drift of the instrument, the
ratio of analyte peak area to the injection standard peak area was calculated
for each run—the mean and standard deviation of these
ratios were then determined. The generalised least squares calculations were
carried out using XLGENLINE (National Physical Laboratory, UK)
[22].Table 1
GC-MS method and optimisation parameters.
Gas chromatograph parametersInjection modeSplitlessInjection port temperature275°CInjection volume2μLGC to MS transfer line temperature290°CCarrier gas and flow rateHelium; 1.8 mL⋅min−1ColumnHP-5MS [(5% phenyl-) methyl-polysiloxane],30 m × 0.25 mm × 0.25μm (film thickness)Temperature program(1) 70°C-hold for 2 minutes(2) ramp to 150°C at 15°C⋅min−1-no hold(3) ramp to 300°C at 10°C⋅min−1-hold for 5 minutes[Total time of program = 27.3 minutes]Mass spectrometer parametersIon source temperature250°CQuadrupole temperature150°CElectron multiplier voltage2800 VOther tuning parametersOptimised automaticallyTable 2
Ions monitored in GC-MS single ion monitoring mode.
GroupPCB congener(s) measuredm/z (1)m/z (2)Mono (Cl1) PCBs1, 3188.1152.1Di (Cl2) PCBs4, 15222.0152.1Tri (Cl3) PCBs19, 37255.9186.0Tetra (Cl4) PCBs54, 77, 81291.9220.0Penta (Cl5) PCBs104, 105, 114, 118, 123, 126325.9254.0Hexa (Cl6) PCBs155, 156, 157, 167, 169359.8289.9Hepta (Cl7) PCBs188, 189393.9323.9Octa (Cl8) PCBs202, 205429.8359.9Nona (Cl9) PCBs206, 208463.8393.8Deca (Cl10) PCB209499.6429.8AnalyteUsem/z (1)m/z (2)d14-p-terphenylInternal standard164.1162.1d10-acenaptheneInjection standard 1224.2—d12-peryleneInjection standard 2264.1—


## 3. Results and Discussion

As discussed in
the introduction, there is a pressing need to determine the uncertainty in the
measurement of PCBs rigorously and in a robust manner in order to provide
legislators, government, industry, and the general public withsufficient
confidence in the measurement. This paper determines the uncertainty of the
measurement by developing an uncertainty budget for three different approaches
in turn:(1)
gravimetric preparation of solutions: calculations carried out in the mass fraction domain;(2)
volumetric preparation of solutions: calculations carried out in the mass concentration domain;(3)
an intermediate
approach where the solutions are prepared gravimetrically, but calculations and the labelling of solutions are carried out in the mass fraction domain.


### 3.1. Gravimetric Preparation of Solutions  (Mass Fraction Domain)

When following the procedure outlined in the Guide to Uncertainty in
Measurement (GUM) [23], the first step to the development of an uncertainty
budget is to produce a measurement equation. Assuming that all calibration
standards and other solutions are prepared gravimetrically, the measurement
equation for the analysis method described above is shown below. The equation
presented here is of course specific to this experimental method, but it is
intended that it is presented in such a manner that it should be able to be
easily adapted by the reader for use with other similar analyses.The measurement equation is(1)xPCB,SRM=xPCB,ext⋅mext⋅δηe⋅mSRM,(2)xPCB,ext=A¯PCB,extV•PCB,(3)ηe=A¯int,ext⋅xint,cal⋅δA¯int,cal⋅xint,ext′,(4)δ=ρcalρext.
In (1), xPCB,SRM is the calculated mass fraction of the PCB
analyte in the SRM, xPCB,ext is the measured mass fraction of
the PCB analyte in the extract, mext is the mass of the
extract, δ is a bias correction to account for
differences between the densities of the analysed extract and calibration
solutions, ηe is the extraction efficiency, and mSRM is the mass of SRM extracted.In (2), A¯PCB,ext is the average, injection standard-corrected, peak area of the PCB analyte in the extract, and V•PCB is the gradient of the calibration curve for
the PCB analyte.In (3), A¯int,ext and A¯int,cal are the average, injection
standard-corrected, peak areas of the internal standard in the extract, and a
calibration solution, respectively, xint,cal is the mass
fraction of the internal standard in this calibration solution, xint,ext′ is the theoretical mass
fraction of the internal standard in the extract (assuming complete recovery),
and δ is the bias correction which also appears in
(1).In (4), ρext is the density of extract
and ρcal is the density of calibration solution.Equation (1) is
the top-level measurement equation and is input into directly by each of  (2), (3), and (4). Equation (4) also inputs
into (3). It is important to note that when (3) is substituted into (1), the δ terms cancel—this is the
subject of later discussion. Each of these equations is now discussed in turn
and the uncertainty of each component in each equation estimated.Equation (4)
describes a correction required because the density of the calibration
solutions is very likely to be different than the density of the extract [24].
When, as in the case described here, calculations are carried out in terms of
mass fraction, this correction is required to account for differences in the
mass injected into the GC-MS using a fixed-volume injection. This difference
arises because the density of the calibration solutions and the extract is
likely to vary for one of two reasons. Firstly, following the multistage sample
workup process (outlined in Figure 2), the resulting extract solution is likely
to comprise a mixture of solvents and will inevitably have a different density
to that of the calibration solutions, which are prepared in hexane. Secondly,
the extract contains a complex mixture of species extracted from the CRM. It is
obvious from simple observation of the condensed extract that its properties
differ significantly from the calibration solutions—the extracts are
often yellow-brown in colour and viscous in nature. In this example, it is
assumed that the density of the calibration solution is the same as hexane, that
is, ρcal=(0.65936±0.00137) g⋅mL−1 (from above), and the density of the
extract, that is, ρext is (0.85±0.07) g⋅mL−1. This gives a value for δ of
0.775. (In this example, this value for ρext has been approximated and a
large standard uncertainty (8% relative) with a rectangular uncertainty
function has therefore been applied. ρext could also be determined
experimentally by drawing the extract into a suitable syringe, recording its
volume and using this value along with the mass of the extract. This mass may
be measured by weighing the syringe with and without extract or by transferral
to an alternative vessel.)As described
above, the twoδ terms in the measurement equation cancel. This
is because each injection into the GC-MS is used for two purposes:
determination of the amount of PCB present in the sample (by (1)), and
calculation of the extraction efficiency (by (3)). As both of thesedeterminations suffer from the same bias, δ, associated with the difference in
density, this means that the uncertainty in δ can
be assigned to be zero. Despite the δ terms cancelling, the presence of (3)
increases the overall uncertainty through the introduction of two further
instrument repeatability terms and two further standard preparation terms. This
would not be required if a method was used, where ηe is not
part of the measurement equation but is simply calculated as part of a quality
assurance procedure, that is, (3) is not part of the measurement equation. In
these cases, which are discussed further below, the uncertainty contribution
from δ in
(1) must be considered fully.Equation (3)
calculates the extraction efficiency of the analysis by comparing the measured
mass fraction of the internal standard in the analysed extract to its
theoretical mass fraction calculated from the quantity spiked prior to
extraction. This theoretical mass fraction, xint,ext′ in (3), can be
determined from the mass fraction and volume of the internal standard solution
used to carry out the spiking. In this study, an internal standard (d14-p-terphenyl) solution of mass fraction (1284.6±4.5) ng⋅g−1 was used. (The
calculations of the uncertainty in the mass fraction of this and all other
solutions used have been carried out assuming that the uncertainty in a balance
reading is 0.5 mg). A volume of 50 μL of this solution was used to spike the sample
prior to extract. If it is assumed that the standard uncertainty in the volume
spiked was 0.25 μL and the density of the solution was the same
as that of hexane (i.e., (0.65936±0.00137) g⋅mL−1), this gives the total mass of solution
spiked as (32.97±0.18) mg and therefore a d14-p-terphenyl mass of (42.35±0.27) ng. (The value used for the density
of hexane is the figure at 20°C given by [25] and the uncertainty in this
figure covers the given densities for temperatures between 18.5°C and 21.5°C—a reasonable
assumption in a controlled laboratory environment.)  Using data from a typical analysis as example,
the mass of the total extract after condensation to just over 1 mL in volume
was (1032.2±0.6) mg, giving xint,ext′ in this instance to be (41.03±0.27) ng⋅g−1. (The uncertainty in each of the above
quantities has been calculated by combination of the constituent uncertainties
in  quadrature.)The above
calculation demonstrates the degree that knowledge of the density of the
organic solvent(s) used to prepare the solutions has on the overall
measurement. As stock solutions purchased from suppliers are in the vast
majority of cases certified in mass concentrations units (e.g., ng⋅mL−1),
the use of a gravimetric preparation approach, such as that described here,
necessitates that the density is first used to convert these into mass
fractions before being used again in the above calculation.The other mass
fraction term in (3), xint,cal, is the mass fraction of the
internal standard in one of calibration solutions used. As described in the
above experimental section, the target mass fraction of internal standard in
each calibration solution was 50 ng⋅g−1, but the actual value and
uncertainty can be determined accurately from the gravimetric data, ensuring
that the uncertainties are propagated correctly. Taking one of the midrange calibration solutions as an example,the mass fraction and standard uncertainty
of the internal standard in this solution were determined to be (58.90±0.45) ng⋅g−1, with the uncertainty here being calculated by propagation of the
uncertainties of each individual weighing (0.5 mg), and the uncertainties in
the density and mass concentration of the stock solution.The bias correction term,δ, is discussed above. The two remaining
terms in the equation, A¯int,ext and A¯int,cal,
are the mean, injection standard-corrected responses for the internal standard in the extract and a calibration solution,
respectively. An injection standard correction is used to improve the
repeatability of a measurement by reducing effects caused by instrumental drift
and differences in the volume of sample injected into the GC-MS. In the
method used here, all solutions analysed contained two injection standards—it was the injection standard with the
retention time nearest to the analyte of interest that was used to provide the
correction. The uncertainties in A¯int,ext and A¯int,cal can be estimated by simply calculating the
standard deviation of the ratios Aint,ext,i/Ainj,ext,i and Aint,cal,i/Ainj,cal,i, respectively,
where Ainj,ext,i is the peak area recorded for the injection
standards from i repeat analyses of the extract and Ainj,cal,i the analogous term for the calibration solution. Exemplar standard deviations
for both of the above ratios are 2.5%, however this standard deviation is
likely to vary with mass fraction [26, 27].
It should also be noted that the peak areas discussed throughout this
paper are blank subtracted peak areas, that is, peak areas after subtraction
of a solvent blank (which for these analytes are often found to be zero).Returning to the
discussion of extraction efficiency,ηe, if all of the
internal standard spiked prior to extraction is recovered in the final GC-MS
analysis, then ηe is equal to unity. It is more than likely
that ηe<1, and in this case two different strategies can
be applied. The first is to use the calculated value of ηe,
no matter how large or small. The alternative approach is to set minimum and/or
maximum allowable values for ηe, such as
0.50<ηe<1.2. In this approach, if the value of ηe falls within this range, then the extraction is said to be valid and a value of ηe=1 is used in (1). On the other hand, if the value of ηe is below this minimum, the extraction is said to be not fit for purpose and the
results discarded. Although both of these approaches have their advantages and
disadvantages, here the former is chosen, that is, the calculated values of ηe are input into (1). (Some documentary standards for similar analyses (e.g.,
[28])  use the latter approach, however, this is not strictly valid without the use of an uncertainty contribution for the term
(1−ηe), a fact ignored by most analysts.)Equation (2)
calculates the mass fraction of PCB congener in the extract by the use of
generalised least-squares (GLSs) [29, 30]. GLS is a fitting procedure that takes
account of the uncertainties inherent in both the x-axis and y-axis data and
performs a fit weighted to these uncertainties. The term on the denominator of
(2), V•PCB,
is the gradient of the GLS fitted curve of the mass fraction of PCB congener in
each calibration solution, xPCB,cal (x-axis), against the average
injection standard corrected peak area from the PCB congener in the calibration
solution, A¯PCB,cal (y-axis). V•PCB and its uncertainty are determined
automatically in a GUM-compliant manner by the software program used [22]  and
the uncertainty includes contributions from the uncertainties in the data on
the x-axis and y-axis, u(xPCB,cal) and u(A¯PCB,cal),
respectively. Note that (2) assumes that the intercept of the calibration curve
is zero—this is a valid assumption that can be tested
by analysing solvent blank samples for the PCB congeners of interest—their levels are typically found to be below
the instrumental of detection. The GLS fit can be constrained to give a zero
intercept by including a point at (0, 0) with uncertainties on both axes which
are very small compared to those of the other data points into the calibration
curve. A typical value for V•PCB and its uncertainty obtained here is (2.58×10−3±0.08×10−3) g⋅ng−1.The remaining
term in (2) is analogous to A¯int,cal in (3), but in this instance A¯PCB,ext relates to the corrected intensity of the
PCB congener in the extract. u(A¯PCB,ext) can be estimated by calculating the standard
deviation of  the ratios APCB,ext,i/Ainj,ext,i obtained from each repeat injection. Due to the complex nature of the
sample and the large number of potentially interfering species in the sample
(even following clean-up), separation of the species by GC retention time
and/or MS target ion(s) is more difficult  than for the calibration standards.
Typical values of u(A¯PCB,ext) are between approximately 5% and 20%
relative depending on the quantity of PCB present in the sample: in this
example, a value of u(A¯PCB,ext) of 10% relative is used.Finally, (1) calculates the mass fraction of the PCB in the SRM and
contains the terms δ, ηe, and xPCB,ext described in (4), (3), and (2), respectively. The values and uncertainties of
the remaining two terms, the mass of the extract analysed by GC-MS (mext),
and the mass of the SRM used (mSRM) can be determined
straightforwardly by gravimetry. Typical values from this work are mext=(1032±1) mg and  mSRM=(397.1±0.5) mg.Tables3(a) to 3(d) show the full uncertainty budget for the
analysis determined from  (1) to (4)
using the values and uncertainties discussed above. Each table represents one
of these equations and contains the following columns (from left to right):
(i)
the quantity contained within the equation;(ii)
the symbol used to represent the quantity;(iii)
the estimated value of the quantity;(iv)
the sensitivity coefficient (the partial derivative of the output quantity with respect to the quantity in question);(v)
the estimated uncertainty of the quantity;(vi)
the shape of the probability distribution of the uncertainty;(vii)
the divisor which this distribution confers on this uncertainty;(viii)
the contribution to the standard uncertainty.Table 3(a)
Table calculating the uncertainty in the mass fraction of PCB in the SRM using (1) and the gravimetric approach (mass fraction domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertMass fraction of PCB in the extract (ng⋅g−1)xPCB,ext8.2442.970.857Normal12.545Total mass of extract (g)mext1.032223.70.0005Normal10.010Bias correction (no units)δ0.77531.60Normal10.000Extraction efficiency (no units)ηe0.679−36.10.029Normal1−1.030Mass of SRM extracted (g)mSRM0.397−61.60.001Normal1−0.031Mass fraction of PCB in the SRM (ng⋅g−1)xPCB,SRM24.475——Normal12.746(b)
Table calculating the uncertainty in the measured mass fraction of PCB in the extract using (2) and the gravimetric approach (mass fraction domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage injection standard corrected peak area of PCB in the extract (no units)A¯PCB,ext0.02133870.0021Normal10.824Gradient of the calibration curve for the PCB (g⋅ng−1)V•PCB0.00258−31910.00007Normal1−0.236Measured mass fraction of PCB in the extract (ng⋅g−1)xPCB,ext8.244——Normal10.857(c)
Table calculating the uncertainty in extraction efficiency using (3) and the gravimetric approach (mass fraction domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage, injection standard-corrected, peak area of the internal standard in the extract (no units)A¯int,ext0.2982.280.007Normal10.017Mass fraction of the internal standard in a calibration solution (ng⋅g−1)xint,cal58.900.0120.45Normal10.005Bias correction (no units)δ0.7750.8760Normal10.000Average, injection standard-corrected, peak area of the internal standard in a calibration solution (no units)A¯int,cal0.489−1.790.012Normal1−0.022Theoretical mass fraction of the internal standard in the extract (assuming complete recovery) (ng⋅g−1)xint,ext′41.03−0.0210.27Normal1−0.006Extraction efficiency (no units)ηe0.679——Normal10.029(d)
Table calculating the uncertainty in the bias correction to account for differences between the densities of the analysed extracts and calibration solutions using (4) and the gravimetric approach (mass fraction domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertDensity of calibration solution (g⋅mL−1)ρcal0.6591.180.001Normal10.002Density of extract (g⋅mL−1)ρext0.850−0.9120.070Rectangular√3−0.064Bias correction (no units)δ0.775————0**As discussed in the main text, the standard uncertainty in δ has been assigned to be zero.Note that some values
in the table contain a large number of significant figures—these are used to avoid introducing rounding
errors into the calculations. Fewer significant figures (appropriate to the
overall relative uncertainty) are used when reporting the final results and,
even then, the nonlinearity of the GUM for large relative uncertainties [31]
such as those here means that the final significant figure should be used with
caution.The calculated output quantity for each table is given at the bottom
of the third column. The uncertainty in this value, which is calculated by
combining the individual contributions in quadrature, is given in the bottom
right-hand cell of each table. In the example shown here, the measured mass
fraction of the PCB analyte in the SRM,xPCB,SRM, is therefore calculated from
Table 3(a) to be 24.5 ng⋅g−1 with a
standard uncertainty of  2.7 ng⋅g−1.The relative contribution to the overall
uncertainty of each of the quantities in (1) taken from Table 3(a) is shown in
Figure 3. This shows that by far the largest contributor is xPCB,ext (as calculated by (2)) and then ηe (as calculated by (3)). The contribution from
the two gravimetric quantities, mext and mSRM, are relatively very small and, as discussed above, the uncertainty in δ has been
assigned to be zero.Figure 3
Graph showing the contribution of each of the five quantities in (1)
to the uncertainty in xPCB,SRM for the gravimetric approach
(mass fraction domain).Finally, the expanded uncertaintyU(xPCB,SRM) can
be calculated by multiplying the standard uncertainty by a coverage factor:
(5)U(xPCB,SRM)=k⋅u(xPCB,SRM),
where k is the coverage factor and u(xPCB,SRM) is the standard uncertainty in xPCB,SRM. For this
application, it is assumed that the effective degrees of freedom of the
measurement are sufficient to assign a value of k=2 in order to
calculate the expanded uncertainty with a level of confidence of approximately
95%.From Table3(a), it can be determined that the overall expanded uncertainty for this exemplar analysis is 22.4% relative. Although this initially seems large in comparison to some analytical chemical measurements, it is likely to be more than fit for purpose for these challenging analyses at low levels and is of the order of magnitude generally expected for measurements of the composition of ambient material. As a comparison, the target uncertainties for similar trace analytes in ambient air are significantly
larger than this. For example, for benzo(a)pyrene and other PAHs in ambient
air, the legislative requirement is an expanded uncertainty of 50% for the
particulate phase and 70% for total deposition at the target mass
concentration of 1 ng⋅m−3 [32].


### 3.2. Volumetric Preparation of Solutions  (Volume Fraction Domain)

The implications
on the uncertainty of the measurement if the solutions
used in the procedure are prepared volumetrically rather than gravimetrically are now
considered. Volumetric production of solutions as used by the majority of
analytical laboratories generally results in a higher uncertainty than
gravimetry but has the advantage of being a more rapid process, thus saving
time and costs.  For the analysis of PCBs
and similar species, the issue likely to work in the favour of volumetric
preparation is that the density of the extract solution, which is required for
the solely gravimetric approach, is difficult to determine accurately.The measurement
equation for the volumetric preparation approach is very similar to that for
the gravimetric case, but as no correction is necessary to account for the
volumetric injection into the GC-MS, the bias correction termδ is not
required. The measurement equation is outlined below:
(6)xPCB,SRM=cPCB,ext⋅vextηe⋅mSRM,(7)cPCB,ext=A¯PCBV•PCB,(8)ηe=A¯int,ext⋅cint,calA¯int,cal⋅cint,ext′.
In the above equations, c represents mass concentration and
these symbols replace the corresponding symbols x (mass fraction) in
(1), (2), and (3), for example, cPCB,ext is the measured mass concentration of the PCB
analyte in the extract. The only ‘‘new’’ term in the above expression is the
total volume of extract vext. Note that (7) and (8) both
input directly into (6), and, importantly, the above expressions do not require
the determination of the density of any solution because injection into the
GC-MS is also carried out in a volumetric manner.As before, each equation is now studied in turn, but this
time the discussion is limited to the parameters that are different to those in
the gravimetric approach. Note that this assumes that similar solutions to
those used in the gravimetric case had been prepared volumetrically by usual
laboratory methods. The calculations of the uncertainty in the mass
concentrations of the solutions used here have been carried out assuming that
the standard uncertainty in a single volumetric addition is as stated on a
pipette (e.g., 0.05 mL for a 10 mL pipette and 0.1 mL for a 25 mL
pipette) or equal to half the graduation of a syringe (e.g., 2.5μL for a 250 μL syringe with 5 μL graduations).In (8), cint,ext′ is the theoretical mass
concentration of internal standard in the extract, which is calculated to be
(33.21±0.54) ng⋅mL−1 for the example presented here.
This value is determined from the mass concentration of the spiking solution
((826.3±12.7) ng⋅mL−1), the volume spiked ((50.0±0.25) μL), and the
measured volume of the extract ((1.244±0.006) mL) assuming that the temperature of the
laboratory was (20 ± 2)°C). The mass concentration of the internal
standard, cint,cal, in one of calibration solutions is
calculated to be (33.21±0.54) ng⋅mL−1, and the remaining two
terms in the expression, A¯int,ext and A¯int,cal are assumed to be the same as for the gravimetric approach. This is an
approximation as due to small differences in the preparation methods, the mass
concentrations of the solutions used in the approach are not exactly equivalent
to those in the gravimetric approach (if the latter were calculated by
converting from mass fraction). However, this approximation is valid for this
example considering the relatively large uncertainty of the overall method.Note that the value ofηe calculated by this approach is slightly different than from the
gravimetric approach (although the two values do actually agree within their
expanded uncertainties). This is a result of two factors:the assumption
regarding the use of the same values for A¯int,ext and A¯int,cal (discussed above) and the fact that, due to
the complex preparation route of the calibration standards which involves a
number of stock solutions in different solvents, the ratio of mass fraction to
mass concentration is similar to, but not exactly equal to, the density of
hexane—the solvent in which the calibration solutions
are prepared.Equation (7)
calculates the mass concentration of PCB congener in the extract, again by the
use of GLS. If a similar solution was analysed as in the gravimetric approach,
the value of A¯PCB,cal would not change, but the gradient V•PCB would report a different value and
uncertainty, in this case (3.78 × 10−3 ± 0.12 × 10−3) mL⋅ng−1.Finally, (6)
calculates the mass concentration of the PCB in the SRM. vSRM can be determined by measuring the volume of the extract in a suitable syringe
and in this example was (1.244±0.006) mL.
All the other parameters in the equation either come directly from previous
equations or remain unchanged from the gravimetric example.Tables4(a) to 4(c) show the full uncertainty
budget for the analysis for the volumetric preparation approach. The format of
the table is identical to that of Table 3. In the example shown here, the
measured mass fraction of the PCB analyte in the SRM, cPCB,SRM,
is calculated to be 25.1 ng⋅g−1 with a standard uncertainty of 2.9 ng⋅g−1. This result differs slightly from that from the gravimetric
approach due to the differences in extraction efficiency discussed above, but
it should be noted that this difference is more than covered by the
uncertainty. The expanded uncertainty, U(cPCB,SRM),
calculated by the use of (5) is 23.3% relative, slightly larger than that for
the gravimetric approach.Table 4(a)
Table calculating the uncertainty in the mass concentration of PCB in the SRM using (6) and the volumetric approach (mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertMass fraction of PCB in the extract (ng⋅mL−1)cPCB,ext5.6404.450.593Normal12.638Total volume of extract (mL)vext1.24420.20.006Normal10.113Extraction efficiency (no units)ηe0.704−35.70.035Normal1−1.266Mass of SRM extracted (g)mSRM0.397−63.20.001Normal1−0.032Mass fraction of PCB in the SRM (ng⋅g−1)xPCB,SRM25.108——Normal12.929(b)
Table calculating the uncertainty in the measured mass concentration of PCB in the extract using (7) and the volumetric approach (mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage injection standard corrected peak area of PCB in the extract (no units)A¯PCB,ext0.02132640.0021Normal10.564Gradient of the calibration curve for the PCB (mL⋅ng−1)V•PCB0.00378−14940.00012Normal1−0.182Measured mass concentration of PCB in the extract (ng⋅mL−1)cPCB,ext5.640——Normal10.593(c)
Table calculating the uncertainty in extraction efficiency using (8) and the volumetric approach (mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage, injection standard-corrected, peak area of the internal standard in the extract (no units)A¯int,ext0.2982.360.007Normal10.017Mass concentration of the internal standard in a calibration solution (ng⋅mL−1)cint,cal38.320.0181.24Normal10.023Average, injection standard-corrected, peak area of the internal standard in a calibration solution (no units)A¯int,cal0.489−1.440.012Normal1−0.018Theoretical mass concentration of the internal standard in the extract (assuming complete recovery) (ng⋅mL−1)cint,ext′33.21−0.0210.54Normal1−0.012Extraction efficiency (no units)ηe0.704——Normal10.035The relative contribution to the overall
uncertainty of each of the quantities in (6) as taken from Table 4(a) is shown
in Figure 4. As for the gravimetric approach, the largest contributors are cPCB,ext (as calculated by (7)) followed by ηe (as calculated by (8)). The contributions from vext and mSRM are much smaller.Figure 4
Graph showing the contribution of each of the four quantities in (6)
to the uncertainty in xPCB,SRM for the volumetric approach
(mass concentration domain).


### 3.3. Gravimetric Preparation of Solutions: Conversion to Mass Concentration Domain

A third approach may also be taken—preparation of the solutions gravimetrically
but then converting to the mass concentration domain for analysis. This approach benefits from the small uncertainties produced by the gravimetric process, although this
is to some extent compromised by the need to convert each weighing into a
volume by use of the density of the solvent, which also has an inherent
uncertainty. An example of the respective uncertainties from each approach can
be obtained by comparing the standard uncertainty of the PCB content in
nominally the same calibration standard prepared by each approach (before the
addition of the injection standards)—0.60% relative for the gravimetric approach
(mass fraction domain), 1.58% relative for the volumetric approach (mass
concentration domain), and an intermediate 0.76% relative for this third
approach—gravimetry followed by conversion to the mass
concentration domain.Tables5(a)-5(c) show the full uncertainty
budget for the analysis for this third approach; the format of the table is
identical to that of Tables 3 and 4. Because the solutions used are identical
to those in the discussion of the gravimetric approach, the values and
uncertainties of the three average, internal standard corrected peak areas, A¯PCB,ext,A¯int,ext, and A¯int,cal are the same as
in Table 3.Table 5(a)
Table calculating the uncertainty in the mass concentration of PCB in the SRM using (6) and the third approach (gravimetric preparation: calculations in the mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertMass fraction of PCB in the extract (ng⋅mL−1)cPCB,ext5.6404.760.593Normal12.821Total volume of extract (mL)vext1.24421.60.006Normal10.121Extraction efficiency (no units)ηe0.658−40.80.028Normal1−1.139Mass of SRM extracted (g)mSRM0.397−67.60.001Normal1−0.034Mass fraction of PCB in the SRM (ng⋅g−1)xPCB,SRM26.853——Normal13.045(b)
Table calculating the uncertainty in the measured mass concentration of PCB in the extract using (7) and the third approach (gravimetric preparation: calculations in the mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage injection standard corrected peak area of PCB in the extract (no units)A¯PCB,ext0.02132650.0021Normal10.564Gradient of the calibration curve for the PCB (mL⋅ng−1)V•PCB0.00378−14940.00012Normal1−0.182Measured mass concentration of PCB in the extract (ng⋅mL−1)cPCB,ext5.640——Normal10.593(c)
Table calculating the uncertainty in extraction efficiency using (8) and the third approach (gravimetric preparation: calculations in the mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage, injection standard-corrected, peak area of the internal standard in the extract (no units)A¯int,ext0.2982.210.007Normal10.016Mass concentration of the internal standard in a calibration solution (ng⋅mL−1)cint,cal36.790.0180.83Normal10.015Average, injection standard-corrected, peak area of the internal standard in a calibration solution (no units)A¯int,cal0.489−1.350.012Normal1−0.016Theoretical mass concentration of the internal standard in the extract (assuming complete recovery) (ng⋅mL−1)cint,ext′34.10−0.0190.28Normal1−0.005Extraction efficiency (no units)ηe0.658——Normal10.028In the example shown here, the measured mass
fraction of the PCB analyte in the SRM,xPCB,SRM is
calculated to be 26.9 ng⋅g−1 with a standard uncertainty of 3.0 ng⋅g−1—the expanded
uncertainty U(cPCB,SRM) is therefore 22.7% relative. Again,
this value differs from that calculated by the gravimetric (mass fraction
domain) approach, but the two values are clearly within the uncertainty of the
measurement. The relative contribution to the overall
uncertainty of this third approach (taken from Table 5)(a) is shown in Figure 5, which shows a very similar pattern to Figure 4.Figure 5
Graph showing the relative contribution of each of the four
quantities in (3) to the uncertainty in xPCB,SRM for the
third approach (gravimetric preparation: calculations in the mass concentration
domain).


### 3.4. Comparison of Approaches

The values ofxPCB,SRM calculated by each of the three approaches are compared in Table 6. The three relative uncertainties are very similar (between 22.4% and 23.3%) and, as discussed earlier in this paper, are all fit forpurpose for the
analysis as they exceed the confidence levels provided by legislators. The
similarity between these values indicates that the vast majority of uncertainty
in the example presented here comes from the GC-MS analysis—the contributions from the gravimetric and
volumetric preparation methods are relatively small. This can be seen in
Figures 3–5 and is
demonstrated further by Figure 6, which shows the relative contributions to the overall uncertainty of ηe as calculated by (3) for the
gravimetric approach (mass fraction domain). The dominant factors are A¯int,ext and A¯int,cal,
the uncertainties of which are determined by the repeatability of the GC-MS
analysis. The contributions of these factors to the overall uncertainty in ηe are approximately three
to five times those from xint,cal and xint,ext′. The bias correction δ has been assigned an uncertainty of zero. A
further indication of the dominance of the uncertainty contributions from the
GC-MS analysis is obtained if the overall uncertainty of the process is
calculated assuming the uncertainty in the GC-MS analysis is zero, that is,  if u(A¯int,ext), u(A¯int,cal),
and u(A¯PCB,ext) are all set to zero. In this case, the
expanded relative uncertainties of each approach are 2.5% for the gravimetric
approach (mass fraction domain), 8.2% for the volumetric approach (mass
concentration domain) and 7.6% for the third approach—gravimetry followed by conversion to the mass
concentration domain.Table 6
Comparison of the values ofxPCB,SRM and its expanded uncertainty calculated by each of the three approaches discussed in the main text.
ApproachxPCB,SRMU(xPCB,SRM)U(xPCB,SRM)/xPCB,SRM(1) Gravimetric (mass fraction domain)24.55.522.4%(2) Volumetric (mass concentration domain)25.15.923.3%(3) Gravimetric: calculations in the mass concentration domain26.96.122.7%Figure 6
Graph showing the contribution of each of the five quantities in (3)
to the uncertainty ηe (normalised to the largest
contribution) for the gravimetric approach (mass fraction domain).The dominance of the uncertainty of the GC-MS analysis over that
from the preparation of the solutions used in the analysis means that in
addition to being more efficient, the volumetric approach does not compromise
the uncertainty of the overall measurement. We therefore propose that,for the example
presented here, the preferred approach for the analysis of PCBs in ambient air
is to remain in themass concentration domain at all times, that is, to preparesolutions volumetrically.


## 3.1. Gravimetric Preparation of Solutions  (Mass Fraction Domain)

When following the procedure outlined in the Guide to Uncertainty in
Measurement (GUM) [23], the first step to the development of an uncertainty
budget is to produce a measurement equation. Assuming that all calibration
standards and other solutions are prepared gravimetrically, the measurement
equation for the analysis method described above is shown below. The equation
presented here is of course specific to this experimental method, but it is
intended that it is presented in such a manner that it should be able to be
easily adapted by the reader for use with other similar analyses.The measurement equation is(1)xPCB,SRM=xPCB,ext⋅mext⋅δηe⋅mSRM,(2)xPCB,ext=A¯PCB,extV•PCB,(3)ηe=A¯int,ext⋅xint,cal⋅δA¯int,cal⋅xint,ext′,(4)δ=ρcalρext.
In (1), xPCB,SRM is the calculated mass fraction of the PCB
analyte in the SRM, xPCB,ext is the measured mass fraction of
the PCB analyte in the extract, mext is the mass of the
extract, δ is a bias correction to account for
differences between the densities of the analysed extract and calibration
solutions, ηe is the extraction efficiency, and mSRM is the mass of SRM extracted.In (2), A¯PCB,ext is the average, injection standard-corrected, peak area of the PCB analyte in the extract, and V•PCB is the gradient of the calibration curve for
the PCB analyte.In (3), A¯int,ext and A¯int,cal are the average, injection
standard-corrected, peak areas of the internal standard in the extract, and a
calibration solution, respectively, xint,cal is the mass
fraction of the internal standard in this calibration solution, xint,ext′ is the theoretical mass
fraction of the internal standard in the extract (assuming complete recovery),
and δ is the bias correction which also appears in
(1).In (4), ρext is the density of extract
and ρcal is the density of calibration solution.Equation (1) is
the top-level measurement equation and is input into directly by each of  (2), (3), and (4). Equation (4) also inputs
into (3). It is important to note that when (3) is substituted into (1), the δ terms cancel—this is the
subject of later discussion. Each of these equations is now discussed in turn
and the uncertainty of each component in each equation estimated.Equation (4)
describes a correction required because the density of the calibration
solutions is very likely to be different than the density of the extract [24].
When, as in the case described here, calculations are carried out in terms of
mass fraction, this correction is required to account for differences in the
mass injected into the GC-MS using a fixed-volume injection. This difference
arises because the density of the calibration solutions and the extract is
likely to vary for one of two reasons. Firstly, following the multistage sample
workup process (outlined in Figure 2), the resulting extract solution is likely
to comprise a mixture of solvents and will inevitably have a different density
to that of the calibration solutions, which are prepared in hexane. Secondly,
the extract contains a complex mixture of species extracted from the CRM. It is
obvious from simple observation of the condensed extract that its properties
differ significantly from the calibration solutions—the extracts are
often yellow-brown in colour and viscous in nature. In this example, it is
assumed that the density of the calibration solution is the same as hexane, that
is, ρcal=(0.65936±0.00137) g⋅mL−1 (from above), and the density of the
extract, that is, ρext is (0.85±0.07) g⋅mL−1. This gives a value for δ of
0.775. (In this example, this value for ρext has been approximated and a
large standard uncertainty (8% relative) with a rectangular uncertainty
function has therefore been applied. ρext could also be determined
experimentally by drawing the extract into a suitable syringe, recording its
volume and using this value along with the mass of the extract. This mass may
be measured by weighing the syringe with and without extract or by transferral
to an alternative vessel.)As described
above, the twoδ terms in the measurement equation cancel. This
is because each injection into the GC-MS is used for two purposes:
determination of the amount of PCB present in the sample (by (1)), and
calculation of the extraction efficiency (by (3)). As both of thesedeterminations suffer from the same bias, δ, associated with the difference in
density, this means that the uncertainty in δ can
be assigned to be zero. Despite the δ terms cancelling, the presence of (3)
increases the overall uncertainty through the introduction of two further
instrument repeatability terms and two further standard preparation terms. This
would not be required if a method was used, where ηe is not
part of the measurement equation but is simply calculated as part of a quality
assurance procedure, that is, (3) is not part of the measurement equation. In
these cases, which are discussed further below, the uncertainty contribution
from δ in
(1) must be considered fully.Equation (3)
calculates the extraction efficiency of the analysis by comparing the measured
mass fraction of the internal standard in the analysed extract to its
theoretical mass fraction calculated from the quantity spiked prior to
extraction. This theoretical mass fraction, xint,ext′ in (3), can be
determined from the mass fraction and volume of the internal standard solution
used to carry out the spiking. In this study, an internal standard (d14-p-terphenyl) solution of mass fraction (1284.6±4.5) ng⋅g−1 was used. (The
calculations of the uncertainty in the mass fraction of this and all other
solutions used have been carried out assuming that the uncertainty in a balance
reading is 0.5 mg). A volume of 50 μL of this solution was used to spike the sample
prior to extract. If it is assumed that the standard uncertainty in the volume
spiked was 0.25 μL and the density of the solution was the same
as that of hexane (i.e., (0.65936±0.00137) g⋅mL−1), this gives the total mass of solution
spiked as (32.97±0.18) mg and therefore a d14-p-terphenyl mass of (42.35±0.27) ng. (The value used for the density
of hexane is the figure at 20°C given by [25] and the uncertainty in this
figure covers the given densities for temperatures between 18.5°C and 21.5°C—a reasonable
assumption in a controlled laboratory environment.)  Using data from a typical analysis as example,
the mass of the total extract after condensation to just over 1 mL in volume
was (1032.2±0.6) mg, giving xint,ext′ in this instance to be (41.03±0.27) ng⋅g−1. (The uncertainty in each of the above
quantities has been calculated by combination of the constituent uncertainties
in  quadrature.)The above
calculation demonstrates the degree that knowledge of the density of the
organic solvent(s) used to prepare the solutions has on the overall
measurement. As stock solutions purchased from suppliers are in the vast
majority of cases certified in mass concentrations units (e.g., ng⋅mL−1),
the use of a gravimetric preparation approach, such as that described here,
necessitates that the density is first used to convert these into mass
fractions before being used again in the above calculation.The other mass
fraction term in (3), xint,cal, is the mass fraction of the
internal standard in one of calibration solutions used. As described in the
above experimental section, the target mass fraction of internal standard in
each calibration solution was 50 ng⋅g−1, but the actual value and
uncertainty can be determined accurately from the gravimetric data, ensuring
that the uncertainties are propagated correctly. Taking one of the midrange calibration solutions as an example,the mass fraction and standard uncertainty
of the internal standard in this solution were determined to be (58.90±0.45) ng⋅g−1, with the uncertainty here being calculated by propagation of the
uncertainties of each individual weighing (0.5 mg), and the uncertainties in
the density and mass concentration of the stock solution.The bias correction term,δ, is discussed above. The two remaining
terms in the equation, A¯int,ext and A¯int,cal,
are the mean, injection standard-corrected responses for the internal standard in the extract and a calibration solution,
respectively. An injection standard correction is used to improve the
repeatability of a measurement by reducing effects caused by instrumental drift
and differences in the volume of sample injected into the GC-MS. In the
method used here, all solutions analysed contained two injection standards—it was the injection standard with the
retention time nearest to the analyte of interest that was used to provide the
correction. The uncertainties in A¯int,ext and A¯int,cal can be estimated by simply calculating the
standard deviation of the ratios Aint,ext,i/Ainj,ext,i and Aint,cal,i/Ainj,cal,i, respectively,
where Ainj,ext,i is the peak area recorded for the injection
standards from i repeat analyses of the extract and Ainj,cal,i the analogous term for the calibration solution. Exemplar standard deviations
for both of the above ratios are 2.5%, however this standard deviation is
likely to vary with mass fraction [26, 27].
It should also be noted that the peak areas discussed throughout this
paper are blank subtracted peak areas, that is, peak areas after subtraction
of a solvent blank (which for these analytes are often found to be zero).Returning to the
discussion of extraction efficiency,ηe, if all of the
internal standard spiked prior to extraction is recovered in the final GC-MS
analysis, then ηe is equal to unity. It is more than likely
that ηe<1, and in this case two different strategies can
be applied. The first is to use the calculated value of ηe,
no matter how large or small. The alternative approach is to set minimum and/or
maximum allowable values for ηe, such as
0.50<ηe<1.2. In this approach, if the value of ηe falls within this range, then the extraction is said to be valid and a value of ηe=1 is used in (1). On the other hand, if the value of ηe is below this minimum, the extraction is said to be not fit for purpose and the
results discarded. Although both of these approaches have their advantages and
disadvantages, here the former is chosen, that is, the calculated values of ηe are input into (1). (Some documentary standards for similar analyses (e.g.,
[28])  use the latter approach, however, this is not strictly valid without the use of an uncertainty contribution for the term
(1−ηe), a fact ignored by most analysts.)Equation (2)
calculates the mass fraction of PCB congener in the extract by the use of
generalised least-squares (GLSs) [29, 30]. GLS is a fitting procedure that takes
account of the uncertainties inherent in both the x-axis and y-axis data and
performs a fit weighted to these uncertainties. The term on the denominator of
(2), V•PCB,
is the gradient of the GLS fitted curve of the mass fraction of PCB congener in
each calibration solution, xPCB,cal (x-axis), against the average
injection standard corrected peak area from the PCB congener in the calibration
solution, A¯PCB,cal (y-axis). V•PCB and its uncertainty are determined
automatically in a GUM-compliant manner by the software program used [22]  and
the uncertainty includes contributions from the uncertainties in the data on
the x-axis and y-axis, u(xPCB,cal) and u(A¯PCB,cal),
respectively. Note that (2) assumes that the intercept of the calibration curve
is zero—this is a valid assumption that can be tested
by analysing solvent blank samples for the PCB congeners of interest—their levels are typically found to be below
the instrumental of detection. The GLS fit can be constrained to give a zero
intercept by including a point at (0, 0) with uncertainties on both axes which
are very small compared to those of the other data points into the calibration
curve. A typical value for V•PCB and its uncertainty obtained here is (2.58×10−3±0.08×10−3) g⋅ng−1.The remaining
term in (2) is analogous to A¯int,cal in (3), but in this instance A¯PCB,ext relates to the corrected intensity of the
PCB congener in the extract. u(A¯PCB,ext) can be estimated by calculating the standard
deviation of  the ratios APCB,ext,i/Ainj,ext,i obtained from each repeat injection. Due to the complex nature of the
sample and the large number of potentially interfering species in the sample
(even following clean-up), separation of the species by GC retention time
and/or MS target ion(s) is more difficult  than for the calibration standards.
Typical values of u(A¯PCB,ext) are between approximately 5% and 20%
relative depending on the quantity of PCB present in the sample: in this
example, a value of u(A¯PCB,ext) of 10% relative is used.Finally, (1) calculates the mass fraction of the PCB in the SRM and
contains the terms δ, ηe, and xPCB,ext described in (4), (3), and (2), respectively. The values and uncertainties of
the remaining two terms, the mass of the extract analysed by GC-MS (mext),
and the mass of the SRM used (mSRM) can be determined
straightforwardly by gravimetry. Typical values from this work are mext=(1032±1) mg and  mSRM=(397.1±0.5) mg.Tables3(a) to 3(d) show the full uncertainty budget for the
analysis determined from  (1) to (4)
using the values and uncertainties discussed above. Each table represents one
of these equations and contains the following columns (from left to right):
(i)
the quantity contained within the equation;(ii)
the symbol used to represent the quantity;(iii)
the estimated value of the quantity;(iv)
the sensitivity coefficient (the partial derivative of the output quantity with respect to the quantity in question);(v)
the estimated uncertainty of the quantity;(vi)
the shape of the probability distribution of the uncertainty;(vii)
the divisor which this distribution confers on this uncertainty;(viii)
the contribution to the standard uncertainty.Table 3(a)
Table calculating the uncertainty in the mass fraction of PCB in the SRM using (1) and the gravimetric approach (mass fraction domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertMass fraction of PCB in the extract (ng⋅g−1)xPCB,ext8.2442.970.857Normal12.545Total mass of extract (g)mext1.032223.70.0005Normal10.010Bias correction (no units)δ0.77531.60Normal10.000Extraction efficiency (no units)ηe0.679−36.10.029Normal1−1.030Mass of SRM extracted (g)mSRM0.397−61.60.001Normal1−0.031Mass fraction of PCB in the SRM (ng⋅g−1)xPCB,SRM24.475——Normal12.746(b)
Table calculating the uncertainty in the measured mass fraction of PCB in the extract using (2) and the gravimetric approach (mass fraction domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage injection standard corrected peak area of PCB in the extract (no units)A¯PCB,ext0.02133870.0021Normal10.824Gradient of the calibration curve for the PCB (g⋅ng−1)V•PCB0.00258−31910.00007Normal1−0.236Measured mass fraction of PCB in the extract (ng⋅g−1)xPCB,ext8.244——Normal10.857(c)
Table calculating the uncertainty in extraction efficiency using (3) and the gravimetric approach (mass fraction domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage, injection standard-corrected, peak area of the internal standard in the extract (no units)A¯int,ext0.2982.280.007Normal10.017Mass fraction of the internal standard in a calibration solution (ng⋅g−1)xint,cal58.900.0120.45Normal10.005Bias correction (no units)δ0.7750.8760Normal10.000Average, injection standard-corrected, peak area of the internal standard in a calibration solution (no units)A¯int,cal0.489−1.790.012Normal1−0.022Theoretical mass fraction of the internal standard in the extract (assuming complete recovery) (ng⋅g−1)xint,ext′41.03−0.0210.27Normal1−0.006Extraction efficiency (no units)ηe0.679——Normal10.029(d)
Table calculating the uncertainty in the bias correction to account for differences between the densities of the analysed extracts and calibration solutions using (4) and the gravimetric approach (mass fraction domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertDensity of calibration solution (g⋅mL−1)ρcal0.6591.180.001Normal10.002Density of extract (g⋅mL−1)ρext0.850−0.9120.070Rectangular√3−0.064Bias correction (no units)δ0.775————0**As discussed in the main text, the standard uncertainty in δ has been assigned to be zero.Note that some values
in the table contain a large number of significant figures—these are used to avoid introducing rounding
errors into the calculations. Fewer significant figures (appropriate to the
overall relative uncertainty) are used when reporting the final results and,
even then, the nonlinearity of the GUM for large relative uncertainties [31]
such as those here means that the final significant figure should be used with
caution.The calculated output quantity for each table is given at the bottom
of the third column. The uncertainty in this value, which is calculated by
combining the individual contributions in quadrature, is given in the bottom
right-hand cell of each table. In the example shown here, the measured mass
fraction of the PCB analyte in the SRM,xPCB,SRM, is therefore calculated from
Table 3(a) to be 24.5 ng⋅g−1 with a
standard uncertainty of  2.7 ng⋅g−1.The relative contribution to the overall
uncertainty of each of the quantities in (1) taken from Table 3(a) is shown in
Figure 3. This shows that by far the largest contributor is xPCB,ext (as calculated by (2)) and then ηe (as calculated by (3)). The contribution from
the two gravimetric quantities, mext and mSRM, are relatively very small and, as discussed above, the uncertainty in δ has been
assigned to be zero.Figure 3
Graph showing the contribution of each of the five quantities in (1)
to the uncertainty in xPCB,SRM for the gravimetric approach
(mass fraction domain).Finally, the expanded uncertaintyU(xPCB,SRM) can
be calculated by multiplying the standard uncertainty by a coverage factor:
(5)U(xPCB,SRM)=k⋅u(xPCB,SRM),
where k is the coverage factor and u(xPCB,SRM) is the standard uncertainty in xPCB,SRM. For this
application, it is assumed that the effective degrees of freedom of the
measurement are sufficient to assign a value of k=2 in order to
calculate the expanded uncertainty with a level of confidence of approximately
95%.From Table3(a), it can be determined that the overall expanded uncertainty for this exemplar analysis is 22.4% relative. Although this initially seems large in comparison to some analytical chemical measurements, it is likely to be more than fit for purpose for these challenging analyses at low levels and is of the order of magnitude generally expected for measurements of the composition of ambient material. As a comparison, the target uncertainties for similar trace analytes in ambient air are significantly
larger than this. For example, for benzo(a)pyrene and other PAHs in ambient
air, the legislative requirement is an expanded uncertainty of 50% for the
particulate phase and 70% for total deposition at the target mass
concentration of 1 ng⋅m−3 [32].


## 3.2. Volumetric Preparation of Solutions  (Volume Fraction Domain)

The implications
on the uncertainty of the measurement if the solutions
used in the procedure are prepared volumetrically rather than gravimetrically are now
considered. Volumetric production of solutions as used by the majority of
analytical laboratories generally results in a higher uncertainty than
gravimetry but has the advantage of being a more rapid process, thus saving
time and costs.  For the analysis of PCBs
and similar species, the issue likely to work in the favour of volumetric
preparation is that the density of the extract solution, which is required for
the solely gravimetric approach, is difficult to determine accurately.The measurement
equation for the volumetric preparation approach is very similar to that for
the gravimetric case, but as no correction is necessary to account for the
volumetric injection into the GC-MS, the bias correction termδ is not
required. The measurement equation is outlined below:
(6)xPCB,SRM=cPCB,ext⋅vextηe⋅mSRM,(7)cPCB,ext=A¯PCBV•PCB,(8)ηe=A¯int,ext⋅cint,calA¯int,cal⋅cint,ext′.
In the above equations, c represents mass concentration and
these symbols replace the corresponding symbols x (mass fraction) in
(1), (2), and (3), for example, cPCB,ext is the measured mass concentration of the PCB
analyte in the extract. The only ‘‘new’’ term in the above expression is the
total volume of extract vext. Note that (7) and (8) both
input directly into (6), and, importantly, the above expressions do not require
the determination of the density of any solution because injection into the
GC-MS is also carried out in a volumetric manner.As before, each equation is now studied in turn, but this
time the discussion is limited to the parameters that are different to those in
the gravimetric approach. Note that this assumes that similar solutions to
those used in the gravimetric case had been prepared volumetrically by usual
laboratory methods. The calculations of the uncertainty in the mass
concentrations of the solutions used here have been carried out assuming that
the standard uncertainty in a single volumetric addition is as stated on a
pipette (e.g., 0.05 mL for a 10 mL pipette and 0.1 mL for a 25 mL
pipette) or equal to half the graduation of a syringe (e.g., 2.5μL for a 250 μL syringe with 5 μL graduations).In (8), cint,ext′ is the theoretical mass
concentration of internal standard in the extract, which is calculated to be
(33.21±0.54) ng⋅mL−1 for the example presented here.
This value is determined from the mass concentration of the spiking solution
((826.3±12.7) ng⋅mL−1), the volume spiked ((50.0±0.25) μL), and the
measured volume of the extract ((1.244±0.006) mL) assuming that the temperature of the
laboratory was (20 ± 2)°C). The mass concentration of the internal
standard, cint,cal, in one of calibration solutions is
calculated to be (33.21±0.54) ng⋅mL−1, and the remaining two
terms in the expression, A¯int,ext and A¯int,cal are assumed to be the same as for the gravimetric approach. This is an
approximation as due to small differences in the preparation methods, the mass
concentrations of the solutions used in the approach are not exactly equivalent
to those in the gravimetric approach (if the latter were calculated by
converting from mass fraction). However, this approximation is valid for this
example considering the relatively large uncertainty of the overall method.Note that the value ofηe calculated by this approach is slightly different than from the
gravimetric approach (although the two values do actually agree within their
expanded uncertainties). This is a result of two factors:the assumption
regarding the use of the same values for A¯int,ext and A¯int,cal (discussed above) and the fact that, due to
the complex preparation route of the calibration standards which involves a
number of stock solutions in different solvents, the ratio of mass fraction to
mass concentration is similar to, but not exactly equal to, the density of
hexane—the solvent in which the calibration solutions
are prepared.Equation (7)
calculates the mass concentration of PCB congener in the extract, again by the
use of GLS. If a similar solution was analysed as in the gravimetric approach,
the value of A¯PCB,cal would not change, but the gradient V•PCB would report a different value and
uncertainty, in this case (3.78 × 10−3 ± 0.12 × 10−3) mL⋅ng−1.Finally, (6)
calculates the mass concentration of the PCB in the SRM. vSRM can be determined by measuring the volume of the extract in a suitable syringe
and in this example was (1.244±0.006) mL.
All the other parameters in the equation either come directly from previous
equations or remain unchanged from the gravimetric example.Tables4(a) to 4(c) show the full uncertainty
budget for the analysis for the volumetric preparation approach. The format of
the table is identical to that of Table 3. In the example shown here, the
measured mass fraction of the PCB analyte in the SRM, cPCB,SRM,
is calculated to be 25.1 ng⋅g−1 with a standard uncertainty of 2.9 ng⋅g−1. This result differs slightly from that from the gravimetric
approach due to the differences in extraction efficiency discussed above, but
it should be noted that this difference is more than covered by the
uncertainty. The expanded uncertainty, U(cPCB,SRM),
calculated by the use of (5) is 23.3% relative, slightly larger than that for
the gravimetric approach.Table 4(a)
Table calculating the uncertainty in the mass concentration of PCB in the SRM using (6) and the volumetric approach (mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertMass fraction of PCB in the extract (ng⋅mL−1)cPCB,ext5.6404.450.593Normal12.638Total volume of extract (mL)vext1.24420.20.006Normal10.113Extraction efficiency (no units)ηe0.704−35.70.035Normal1−1.266Mass of SRM extracted (g)mSRM0.397−63.20.001Normal1−0.032Mass fraction of PCB in the SRM (ng⋅g−1)xPCB,SRM25.108——Normal12.929(b)
Table calculating the uncertainty in the measured mass concentration of PCB in the extract using (7) and the volumetric approach (mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage injection standard corrected peak area of PCB in the extract (no units)A¯PCB,ext0.02132640.0021Normal10.564Gradient of the calibration curve for the PCB (mL⋅ng−1)V•PCB0.00378−14940.00012Normal1−0.182Measured mass concentration of PCB in the extract (ng⋅mL−1)cPCB,ext5.640——Normal10.593(c)
Table calculating the uncertainty in extraction efficiency using (8) and the volumetric approach (mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage, injection standard-corrected, peak area of the internal standard in the extract (no units)A¯int,ext0.2982.360.007Normal10.017Mass concentration of the internal standard in a calibration solution (ng⋅mL−1)cint,cal38.320.0181.24Normal10.023Average, injection standard-corrected, peak area of the internal standard in a calibration solution (no units)A¯int,cal0.489−1.440.012Normal1−0.018Theoretical mass concentration of the internal standard in the extract (assuming complete recovery) (ng⋅mL−1)cint,ext′33.21−0.0210.54Normal1−0.012Extraction efficiency (no units)ηe0.704——Normal10.035The relative contribution to the overall
uncertainty of each of the quantities in (6) as taken from Table 4(a) is shown
in Figure 4. As for the gravimetric approach, the largest contributors are cPCB,ext (as calculated by (7)) followed by ηe (as calculated by (8)). The contributions from vext and mSRM are much smaller.Figure 4
Graph showing the contribution of each of the four quantities in (6)
to the uncertainty in xPCB,SRM for the volumetric approach
(mass concentration domain).


## 3.3. Gravimetric Preparation of Solutions: Conversion to Mass Concentration Domain

A third approach may also be taken—preparation of the solutions gravimetrically
but then converting to the mass concentration domain for analysis. This approach benefits from the small uncertainties produced by the gravimetric process, although this
is to some extent compromised by the need to convert each weighing into a
volume by use of the density of the solvent, which also has an inherent
uncertainty. An example of the respective uncertainties from each approach can
be obtained by comparing the standard uncertainty of the PCB content in
nominally the same calibration standard prepared by each approach (before the
addition of the injection standards)—0.60% relative for the gravimetric approach
(mass fraction domain), 1.58% relative for the volumetric approach (mass
concentration domain), and an intermediate 0.76% relative for this third
approach—gravimetry followed by conversion to the mass
concentration domain.Tables5(a)-5(c) show the full uncertainty
budget for the analysis for this third approach; the format of the table is
identical to that of Tables 3 and 4. Because the solutions used are identical
to those in the discussion of the gravimetric approach, the values and
uncertainties of the three average, internal standard corrected peak areas, A¯PCB,ext,A¯int,ext, and A¯int,cal are the same as
in Table 3.Table 5(a)
Table calculating the uncertainty in the mass concentration of PCB in the SRM using (6) and the third approach (gravimetric preparation: calculations in the mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertMass fraction of PCB in the extract (ng⋅mL−1)cPCB,ext5.6404.760.593Normal12.821Total volume of extract (mL)vext1.24421.60.006Normal10.121Extraction efficiency (no units)ηe0.658−40.80.028Normal1−1.139Mass of SRM extracted (g)mSRM0.397−67.60.001Normal1−0.034Mass fraction of PCB in the SRM (ng⋅g−1)xPCB,SRM26.853——Normal13.045(b)
Table calculating the uncertainty in the measured mass concentration of PCB in the extract using (7) and the third approach (gravimetric preparation: calculations in the mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage injection standard corrected peak area of PCB in the extract (no units)A¯PCB,ext0.02132650.0021Normal10.564Gradient of the calibration curve for the PCB (mL⋅ng−1)V•PCB0.00378−14940.00012Normal1−0.182Measured mass concentration of PCB in the extract (ng⋅mL−1)cPCB,ext5.640——Normal10.593(c)
Table calculating the uncertainty in extraction efficiency using (8) and the third approach (gravimetric preparation: calculations in the mass concentration domain).
QuantitySymbolValueSensitivity coefficientUncertaintyProbability distributionDivisorContribution to standard uncertAverage, injection standard-corrected, peak area of the internal standard in the extract (no units)A¯int,ext0.2982.210.007Normal10.016Mass concentration of the internal standard in a calibration solution (ng⋅mL−1)cint,cal36.790.0180.83Normal10.015Average, injection standard-corrected, peak area of the internal standard in a calibration solution (no units)A¯int,cal0.489−1.350.012Normal1−0.016Theoretical mass concentration of the internal standard in the extract (assuming complete recovery) (ng⋅mL−1)cint,ext′34.10−0.0190.28Normal1−0.005Extraction efficiency (no units)ηe0.658——Normal10.028In the example shown here, the measured mass
fraction of the PCB analyte in the SRM,xPCB,SRM is
calculated to be 26.9 ng⋅g−1 with a standard uncertainty of 3.0 ng⋅g−1—the expanded
uncertainty U(cPCB,SRM) is therefore 22.7% relative. Again,
this value differs from that calculated by the gravimetric (mass fraction
domain) approach, but the two values are clearly within the uncertainty of the
measurement. The relative contribution to the overall
uncertainty of this third approach (taken from Table 5)(a) is shown in Figure 5, which shows a very similar pattern to Figure 4.Figure 5
Graph showing the relative contribution of each of the four
quantities in (3) to the uncertainty in xPCB,SRM for the
third approach (gravimetric preparation: calculations in the mass concentration
domain).


## 3.4. Comparison of Approaches

The values ofxPCB,SRM calculated by each of the three approaches are compared in Table 6. The three relative uncertainties are very similar (between 22.4% and 23.3%) and, as discussed earlier in this paper, are all fit forpurpose for the
analysis as they exceed the confidence levels provided by legislators. The
similarity between these values indicates that the vast majority of uncertainty
in the example presented here comes from the GC-MS analysis—the contributions from the gravimetric and
volumetric preparation methods are relatively small. This can be seen in
Figures 3–5 and is
demonstrated further by Figure 6, which shows the relative contributions to the overall uncertainty of ηe as calculated by (3) for the
gravimetric approach (mass fraction domain). The dominant factors are A¯int,ext and A¯int,cal,
the uncertainties of which are determined by the repeatability of the GC-MS
analysis. The contributions of these factors to the overall uncertainty in ηe are approximately three
to five times those from xint,cal and xint,ext′. The bias correction δ has been assigned an uncertainty of zero. A
further indication of the dominance of the uncertainty contributions from the
GC-MS analysis is obtained if the overall uncertainty of the process is
calculated assuming the uncertainty in the GC-MS analysis is zero, that is,  if u(A¯int,ext), u(A¯int,cal),
and u(A¯PCB,ext) are all set to zero. In this case, the
expanded relative uncertainties of each approach are 2.5% for the gravimetric
approach (mass fraction domain), 8.2% for the volumetric approach (mass
concentration domain) and 7.6% for the third approach—gravimetry followed by conversion to the mass
concentration domain.Table 6
Comparison of the values ofxPCB,SRM and its expanded uncertainty calculated by each of the three approaches discussed in the main text.
ApproachxPCB,SRMU(xPCB,SRM)U(xPCB,SRM)/xPCB,SRM(1) Gravimetric (mass fraction domain)24.55.522.4%(2) Volumetric (mass concentration domain)25.15.923.3%(3) Gravimetric: calculations in the mass concentration domain26.96.122.7%Figure 6
Graph showing the contribution of each of the five quantities in (3)
to the uncertainty ηe (normalised to the largest
contribution) for the gravimetric approach (mass fraction domain).The dominance of the uncertainty of the GC-MS analysis over that
from the preparation of the solutions used in the analysis means that in
addition to being more efficient, the volumetric approach does not compromise
the uncertainty of the overall measurement. We therefore propose that,for the example
presented here, the preferred approach for the analysis of PCBs in ambient air
is to remain in themass concentration domain at all times, that is, to preparesolutions volumetrically.


## 4. Conclusions

This paper has
presented a detailed measurement equation and developed a full uncertainty
budget for the analysis of PCBs in an urban dust reference material. Three
approaches for preparing calibration standards and other solutions have been
compared, namely,(1)
gravimetric preparation of solutions: calculations carried out in the mass fraction domain;(2)
volumetric preparation of solutions: calculations carried out in the mass concentration domain;(3)
an intermediate approach where the solutions are prepared gravimetrically, but calculations and the labelling of solutions are carried out in the mass fraction domain.Calculation of
the overall expanded uncertainty of the measurement resulting from these three
approaches has shown that they are very similar for the example presented here,
ranging from 22.4% to 23.3%. These values are all fit for purpose for the
analysis. Examination of the uncertaintybudget has revealed that the dominant
contributory factor to the calculated uncertainty is the repeatability of the
GC-MS analysis, and that the method chosen to prepare the calibration standards
and other solution contributes relatively little. For this reason, it is
suggested that thesolutions should be prepared in the most convenient manner
possible—this is likely to
be the volumetric approach  (mass concentration domain). Of course, different laboratories us-ing different methods may obtain different conclusions, for
example, if the accuracy of the GC-MS analyses was much improved, the
volumetric approach may have a significant contribution to the overall
uncertainty and a gravimetric approach may be preferred. This can easily be
investigated by use of the measurement equation presented here.The use of a
bias correction term,δ, introduced to account for differences in
the densities of the calibration solution and analysed extract has also been
discussed. This term is only required for the gravimetric approach (mass
fraction domain) to correct for the response of the GC-MS to a fixed volume
injection. However, as δ is
used to calculate both the amount of PCB in the sample and the extraction
efficiency of the method, the two δ terms
cancel in the top-level measurement equation. Although δ can
therefore be assigned an uncertainty of zero, knowledge of the density of both
the calibration solutions and analysed extract is still required at other
stages in the measurement equation. If ηe is not included in the
measurement equation, then accurate knowledge of δ and
its uncertainty are
required.Although this
work has focussed on the analysis of PCBs, the authors hope that it can be
applied easily to analogous methods for similar analytes, for example, those
used by routine analytical laboratories to measure polycyclic aromatic
hydrocarbons (PAHs), pesticides, dioxins, or furans.



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*Source: 179498-2008-05-22.xml*