CH217
Fundamentals of Analytical
Chemistry
Module Leader: Dr. Alison Willows
Created with MindGenius Business 2005®
Created with MindGenius Business 2005®
Assessment

Practicals 60%




Practical 1: online quiz during lab session
Practicals 2 & 3: electronic reports, see lab
scripts
End of module examination 40%
In addition you are also required to:



Complete the guided study (not assessed)
Attend all the labs
Attend at least 80% lectures/workshops
Created with MindGenius Business 2005®
Studentcentral




Module content and assignments are available
through studentcentral
You will be required to submit your coursework
electronically via studentcentral
The guided study will be an electronic test on
studentcentral
Feedback on assessments will also be electronic
Please familiarise yourself with studentcentral!
Created with MindGenius Business 2005®
Recommended reading




The module descriptor tells you what you should
know by the end of this module
The information given in lectures and on
studentcentral is only a guideline to aid your study
Please refer to the module learning handbook and
studentcentral for a list of recommended books and
other useful resources.
You will not achieve a good grade in this module
without doing additional reading outside of the
lectures
Created with MindGenius Business 2005®
Principles of Analytical design
DTI's Valid Analytical Measurement programme
The six principles of good analytical practice
 Analytical measurements should be made to satisfy an agreed
requirement.
 Analytical measurements should be made using methods and
equipment which have been tested to ensure they are fit for
purpose.
 Staff making analytical measurements should be both qualified and
competent to undertake the task.
 There should be a regular independent assessment of the technical
performance of a laboratory
 Analytical measurements made in one location should be consistent
with those elsewhere.
 Organisations making analytical measurements should have well
defined quality control and quality assurance procedures.
Created with MindGenius Business 2005®
Role of analytical chemistry in
science
Do I need analytical chemistry?
Analytical chemistry might:
 enable you to pass your course
 help you to understand other modules
 be useful in your career
 be interesting
 help with your final year project
 change your life!
Created with MindGenius Business 2005®
What is analytical chemistry?
Dictionary definitions


Analytical (adj) examining or tending to examine things very carefully
Chemistry(noun) 1.(the part of science which studies) the basic
characteristics of substances and the different ways in which they
react or combine with other substances. 2. INFORMAL understanding
and attraction between two people
Cambridge Advanced Learner's dictionary


Analytical chemistry encompasses any
type of test that provides information
on the amount or identification of the
chemical composition of a sample.
This breaks down into two main
areas of analysis:
qualitative and quantitative
Created with MindGenius Business 2005®
Qualitative vs.. Quantitative


Qualitative analyses give a
positive/negative or yes/no answer. This
tells us whether a substance (the analyte)
is present but doesn't tell us how much is
there. A qualitative analysis may also
identify substances in a sample
Quantitative analyses tell us how much of
a substance is in the sample.
Created with MindGenius Business 2005®
When and where is analytical
chemistry used?








Food industry - wine production; contaminants;
process lines
Medical - blood analysis; imaging;
Pharmaceutical - drug analysis
Environmental - water, gas & soil analysis
Engineering - materials characterisation
Crime - forensics (CSI)
Sport & leisure - pool chlorination; drugs tests
Research
Created with MindGenius Business 2005®
Analytical Process



Formulating the question
Selecting analytical procedures
Conducting the analysis





Sampling
Sample preparation
calibration of method
Sample analysis
Collection and processing of data and
calculation of errors
Created with MindGenius Business 2005®
Analytical Process, cont.



Method validation
Reporting and interpretation (results &
discussion)
Drawing conclusions (answering the
question!)
Created with MindGenius Business 2005®
Method selection
Valid Analytical Measurement (VAM)
A result is fit for purpose when its uncertainty
maximises its expected utility (cost, usually)




reducing uncertainty generally increases the cost of
analysis
most users have tight budgets
uncertainty in measurement should be as large as can
be tolerated to keep costs down
other factors can affect fitness for purpose




sensitivity of technique
sample throughput
accuracy and precision that is obtainable
sample type and preparation
Created with MindGenius Business 2005®
VAM, cont

Ultimately, the results are fit for purpose if
they meet the specific needs of the
customer, the customer is confident in the
results and they represent value for
money.
Created with MindGenius Business 2005®
Valid Analytical Measurement
(VAM)
Goldmine
A sampling and analysis game for Minitab
can be found here

http://www.rsc.org/Membership/Networking/InterestGroups/Analytical/
AMC/Software/goldmine.asp
Created with MindGenius Business 2005®
Comparing techniques statistically
The F test and Student's t test





F test -Is there a significant difference between the
precision of two methods? i.e. are the standard
deviations of the two methods significantly different?
Student’s t test - used to decide if two sets of results are
"the same" or to compare a set of results with a known
value.
You will have learnt these tests in your QS modules, please refresh
your memory if you are unsure how to perform it.
You will be expected to be able to compare a set of results with a
known value, compare two sets of matched results and compare
two sets of unmatched results, please see me if you can not do this
Further information and worked examples are available on the
CH217 studentcentral website
Created with MindGenius Business 2005®
Samples - sampling strategy


Probably the most important stage in any
analysis.
If the sample taken is not representative
of the original material everything you do
next is worthless.
Created with MindGenius Business 2005®
Sample nomenclature







lot - quantity of material which is assumed to represent
a single population for sampling purposes
batch - quantity of material known (or assumed) to have
been produced under uniform conditions
increments - portions of material obtained using a
sampling device from lot/batch
primary/gross sample - combination of increments
composite/aggregate sample - combination of primary
samples
laboratory sample - portion of material delivered to lab
for analysis
test (analytical) portion - material actually submitted for
analysis
Created with MindGenius Business 2005®
Sampling - stages
Horwitz. Pure and Applied Chemistry, 1990, 62, 1193-1208.
Created with MindGenius Business 2005®
Obtaining a representative sample




Usually the lot is not homogeneous but may be
randomly heterogeneous (different compositions
occur on a small scale and randomly) or
segregated heterogeneous (large patches of different
compositions)
A representative sample will not reflect the composition
of the target exactly but will be adequate enough to be
'fit for purpose'. There will always be a degree of
uncertainty from sampling.
Created with MindGenius Business 2005®
Sampling - n numbers
How many replicate samples do we need to analyse?
 Often in biology you will come across n=6 for all
analyses. so where does this come from?
Confidence limits - met in QS modules
  x
ts
n


Rearrange to make n the subject
2
n
t s
2
  x 
2
Use the acceptable error    x  and confidence level
(to find t) to calculate n.
Created with MindGenius Business 2005®
Sampling - n numbers
Worked Example


The concentration of lead in the bloodstream was measured for a
sample of children from a large school near a busy main road. A
preliminary sampling of 50 children gave a mean concentration of
10.12 ng ml-1 and standard deviation of 0.64 ng ml-1. How big does
the sample need to be to give an error of less than ±0.1 ng ml-1
with 95% confidence?
For 95% confidence t = 1.96 (n = ∞)
2
n

t s
2
   x 2

1 . 96 0 . 64 

2
0 . 1 
2
2
 157 . 4  160
So 160 children would need to be tested
Created with MindGenius Business 2005®
sample preparation
Preparing samples for analysis
Depends on the form required for analysis
Samples may require





Moisture control
Grinding
Dissolving
Ashing




Fusion
Extraction
Preconcentration/dilution
Derivatisation
or a combination of several of these
 Instruments such as microwave ovens, sonicating baths, pressure
vessels (digestion bombs) and extraction cartridges may also be
used.

Please see recommended reading for further details on these preparation
techniques (ch28 Harris)
Created with MindGenius Business 2005®
solid phase extraction



Analyte is removed from sample by
passing a solution over a solid.
Analyte is adsorbed, or absorbed by the
solid and the remaining liquid can be
discarded
Analyte is eluted by use of a stronger
solvent
Created with MindGenius Business 2005®
solid phase extraction
Created with MindGenius Business 2005®
Sample storage
To keep samples reflective we must prevent contamination
& decomposition
Problems & Solutions
1. Dirty containers - ensure adequate washing; use disposable
containers
2.
Type of Container - Avoid “ion-exchange” and adsorption of
analyte
3.
4.
5.
6.
7.
Light - use brown/foil-covered bottles
Air may oxidise sample - store under vacuum, or in a
protective atmosphere
Moisture - keep tightly sealed
Evaporation - keep tightly sealed
Heat/cold - store in fridge/temperature controlled room
The measures chosen will depend on the analyte and its sample matrix
Created with MindGenius Business 2005®
Calibration
Analytical methods, particularly those using instruments, frequently
require calibration procedures
These are to establish:
 the response to known quantities of analyte (standards) within the
range used
 the reliability/drift of the method
 limits beyond which detection/quantitation is unreliable
Calibration normally involves:
 measurement of samples of known concentrations
 measurement of a relevant range of concentrations
 a range in which the response is linear
 graphical treatment of results
 modified calculation of errors
Created with MindGenius Business 2005®
External Standard







Simplest and most common form of calibration.
Prepare samples containing known quantities of analyte
over a relevant range including blanks
Controls for sample preparation/matrix should be used,
matched to the unknown samples
Carry out and record measurements
Plot quantity/concentration of analyte vs. response
Linear regression with least squares analysis is used to
determine response (expressed as y = bx+a)
Repeat as and when appropriate (when it is likely that
an unacceptable drift will have occurred)
Created with MindGenius Business 2005®
External Standard
Advantages
 May only need one calibration plot (of 5-10 samples) for
10’s to 100’s of unknown samples
 Can be easily automated
 Simple statistics will provide estimates of uncertainty for
the method
Disadvantages
 Requires care to match conditions and matrix to that of
the unknown samples
 Does not control for sudden changes in method
performance
Created with MindGenius Business 2005®
External standard
You will have done this in more detail in BY131
You should be able to use linear regression to
calculate the line of best fit and the errors in the
calibration line to calculate the concentration of
the analyte and its error from this information
(see sec 5.4, 5.5, 5.6 in Miller & Miller)
The ability to do this is assumed in this module.
Created with MindGenius Business 2005®
Internal Standard





Useful for methods which are not very reproducible; e.g.
Gas chromatography uses very small volumes (<1 ml) difficult to measure accurately
The instrument responses to mixtures of known amounts
of analyte and of a different compound (internal
standard) are measured, and response factor
determined
A known amount of internal standard is added to the
unknown sample.
Signals from the analyte and from the internal standard
are measured
Response factor allows determination of analyte
concentration
Created with MindGenius Business 2005®
Internal Standard
Advantages
 Can control for loss during sample preparation
 Controls for unexpected changes in method
performance
Disadvantages
 Requires suitable reference standard
 The two compounds (standard and analyte)
must be quantifiable independently and have
linear responses over a range of concentrations
 Must account for dilution steps in calculations
Created with MindGenius Business 2005®
Internal Standard-Worked Example
Measurement of caffeine concentration by HPLC, using theophyline as
an internal standard. Standard solutions containing a range of known
amounts of both caffeine and theophyline are prepared. These are
subjected to HPLC and the relative instrument response (area under
each peak) is determined, and response factor determined.
Absorbance

Caffeine
Theophyline
solution caffeine
Conc./mg.l-1
Theophyline
Peak area Conc./mg.l-1
Peak area
A
1
20000
1
50000
B
2
38400
1
48000
c
4
89600
1
56000
Created with MindGenius Business 2005®
Internal Standard-Worked Example

Response Factor
Analyte signal
 F
Analyte conc .
a)
20000
 F
1
b)
c)
38400

signal
standard
conc.
 F  0 .4
1
 F
48000
2
1
89600
56000
4

50000
Standard
 F
 F  0 .4
 F  0 .4
1
In reality there would be some variation and multiple calibration
samples would be used to determine precision of response factor
A 10ml of a 1mg.L-1 internal standard is added to 10ml of an
unknown sample . Instrument signals measured: Analyte: 30,000,
Internal Standard: 27,000
Created with MindGenius Business 2005®
Internal Standard-Worked Example

Response factor allows determination of analyte concentration in
sample:
Analyte signal
 F
Standard
signal
standard
conc.
Analyte conc .
30000
x
 0 .4
27000
0.5
x  1 . 39 mg . L

1
 Original concentration = 1.39 x 20/10
= 2.78mg.L-1
Created with MindGenius Business 2005®
Standard addition
Frequently used where matrix effects and interferents
are prevalent e.g. atomic absorption/emission








Prepare samples containing equal volumes of unknown
analyte concentration
“Spike” each sample with known, different amounts of
standard (same analyte, including a range from 0 to ~5x
expected unknown concentration)
Dilute all samples to the same volume
Carry out and record measurements
Plot quantity/concentration of known analyte added vs..
response
Linear regression with least squares analysis is used to
determine response (expressed as y = bx+a)
Concentration of unknown = - (x-intercept) = a/b
Repeat for each unknown sample
Created with MindGenius Business 2005®
Standard addition
Advantages
 Controls for matrix effects
 Controls for unexpected changes in method
performance
Disadvantages
 Requires several measurements for each unknown
 May use more unknown sample than other methods
 Must be careful to account for dilution steps in
calculations
Created with MindGenius Business 2005®
Standard addition - Worked Example
Measurement of Copper concentration by atomic
absorption spectrometry

Five 10ml solutions of unknown (approx. 2mg.L-1) copper
concentration were prepared and to these was added: 0, 2, 4, 6 and
8 cm3 of 10mg.L-1 standard analyte solution in water (one volume to
each flask). All samples diluted to 25cm3 with water and mixed well.
The solutions were then measured using AAS and the results
recorded
Solution Added volume/
cm3
Absorbance
1
0
0.150
2
2
0.312
3
4
0.446
4
6
0.580
5
8
0.762
Created with MindGenius Business 2005®


Calculate concentration of copper added to solution, using c1V1 =
c2 V 2
i.e. 2 cm3 added: 10 x 2/1000 = c2 x 25/1000
c2 = 0.8 mg.L-1 etc
Solution Added volume/
cm3

Absorbance
Added concentration/
mg.l-1
1
0
0.150
0
2
2
0.312
0.8
3
4
0.446
1.6
4
6
0.580
2.4
5
8
0.762
3.2
Plot quantity/concentration of known analyte added vs. response,
and plot line using linear regression with least square analysis
(expressed as y = bx+a)
Created with MindGenius Business 2005®
0.1865 ± 0.006028
Y-intercept
0.1516 ± 0.01181
X-intercept
-0.8129
S ta n d a r d a d d itio n p lo t
1 .0
x- interc ept = -ve c onc entration
of unknown
A bso rbance
Slope
0 .8
0 .6
0 .4
0 .2
-1
0
1
2
3
4
C o n cen tr atio n ad d ed to sam p le
(m g .L

Conc. of unknown in samples = - (x-intercept) = a/b
= 0.813mg.L-1


-1 )
NB: 10cm3 aliquots of the original solution were diluted to 25cm3 in
the samples, so concentration of original solution = 0.813 x 25/10 =
2.0325 ~ 2.03mg.L-1
Created with MindGenius Business 2005®
validation




Standards
Performance parameters
Errors in Analysis
Record Keeping
Created with MindGenius Business 2005®
“How long is a piece of string?”


The results from any analytical measurement
depends upon and is traceable to the
measurement standards used in the process.
These include standards for mass, volume and
amount of a chemical species.
Equipment is usually periodically calibrated using
standards that can be traced back to an
International Primary Standard.
Created with MindGenius Business 2005®
Example
1.
2.
3.
4.

An analytical balance will be calibrated periodically using calibrated
weights.
These weights are regularly checked against a set of weights held
at a reference laboratory.
The reference laboratory's weights will be checked periodically
against the national standard kilogram (held at the National
Physical Laboratory, NPL).
This national standard kilogram is occasionally compared to the
international standard kilogram.
Each stage introduces a measurement uncertainty which has to be
taken into account. This means that the standards used in a
laboratory will always have a greater uncertainty associated with
them than those from the reference laboratories.
Created with MindGenius Business 2005®
Standard solutions


Standard solutions can be used to help with
calibration and to compare results against to
establish the accuracy of a technique.
The two main grades of standard are:



Primary
Secondary
Certified Reference Materials (CRM) - specially
prepared samples containing an analyte at a
pre-determined concentration .
Created with MindGenius Business 2005®
Primary standards


Primary standards are highly purified compounds that
are used, directly or indirectly, to establish the
concentration of standard solutions.
Primary standards should meet the following
requirements:






High purity
Stability toward air
Absence of hydrate water so composition does not change with
variations in humidity
Ready availability at reasonable cost
Reasonable solubility in titration medium
Reasonably large molar mass so that relative error associated
with weighing the standard is minimised
Created with MindGenius Business 2005®
Secondary standards

There are few compounds that meet these criteria. So
often a less pure compound has to be used:
secondary standard

The ideal standard solution should:







Be sufficiently stable that its concentration needs to be
determined only once
React rapidly with the analyte
React more or less completely with the analyte for good end
points
Undergo selective reaction with simple balanced equation
Few reagents meet all of these requirements
Created with MindGenius Business 2005®
Performance parameters






Accuracy – measure of agreement between a single analytical
result and the true value
Precision – measure of agreement between observed values
obtained by repeated application of the same analytical procedure
Selectivity – measure of the discriminating power of an analytical
procedure in differentiating between the analyte and other
components in the test sample
Sensitivity – the change of the measured signal as a result of one
unit change in the content of the analyte (calculated from the
calibration line)
Limit of Detection – calculated amount of analyte in the sample
which corresponds to 3 times the sd of the blank sample
Limit of Quantitation – minimum content of the analyte that can
be quantitatively determined with reasonable statistical confidence.
Equivalent to 6 time the sd of the blank sample
Created with MindGenius Business 2005®



Linearity – a measure of the linearity of the calibration
Range – concentration range to which the technique is applicable
Ruggedness – insensibility of the method for variations during
execution




Standard deviation and relative standard deviation (RSD)
– measures of the spread in the observed values as a result of
random errors
Repeatability – expected maximum difference between two
results of identical test samples obtained under identical conditions
Within-lab reproducibility – expected maximum difference
between two results obtained by repeated application of the
analytical procedure to an identical test sample under different
conditions (e.g. different operator, different days) but in the same
laboratory
Between-lab reproducibility - expected maximum difference
between two results obtained by repeated application of the
analytical procedure to an identical test sample in different
laboratories (e.g. different operators, different instrumentation in
different labs on different days using same method
Created with MindGenius Business 2005®
Errors in Analysis




The key to any successful analysis is ensuring
that it will “answer the question”
No analysis can be absolutely error-free
All analyses must be designed to produce
acceptable levels of errors and uncertainty
The best way to minimise errors is by careful
experimental design
Created with MindGenius Business 2005®
types of error

Three main types of error

Gross: So serious the experiment must be abandoned. e.g.
dropping a key sample, instrumental breakdown

Random: When an experiment is repeated as exactly as
possible, the replicate results will differ due to random errors.
Estimates of random errors gives the precision or reproducibility
of the analysis.

Systematic: An experimental method gives a reproducible
under- or overestimate of the real result. Total of all systematic
errors gives the bias of an analysis.
Created with MindGenius Business 2005®
Typical sources of error
May be personal, instrumental or methodological
 Random



Systematic



Volume - not reading the burette reproducibly
Weight - sensitivity of the balance
Volume - glassware not exact; ”indicator errors”; incomplete
drainage of pipette/burette; lab temperature
Weight - vessel at different temperature to balance; air
buoyancy effect
Both



Incomplete transference between vessels
Incomplete reaction, decomposition or moisture absorbance of
sample/analyte
Interfering species
With good tools and careful measurement, traditional
methods (gravimetry, titrimetry) are generally more
accurate than instrumental method.
Created with MindGenius Business 2005®
Error Avoidance
After considering each stage of the process, employ:
 Random Errors





Systematic errors








improved technique (e.g. reading burette volumes)
a more accurate balance
sufficient repeated measurements (replicates)
a different scale (g are easier to weigh than mg)
replicates in different glassware
temperature controls
difference weighing
“reference standards” and “blank” measurements
purified reagents
a different/additional method
interlaboratory trials
Systematic errors are not always obvious - but the
methods above can often be used to detect them!
Created with MindGenius Business 2005®
Accuracy and precision
Accurate and Precise
Precise but not accurate
Accurate but not precise
Inaccurate and
Created
with MindGenius Business 2005®
Imprecise
accuracy & precision


Uncertainty - A measure of both precision and accuracy,
i.e. is an indicator of overall errors associated with the
method.
May be quoted using s, RSD or CI (should state which)



s and RSD should be quoted with the relevant n
Analytical results are quoted as a mean ± uncertainty
Size of the uncertainty dictates how many significant
figures to quote
Created with MindGenius Business 2005®
Calculating uncertainty

Bottom-up method



Combine all known errors (e.g. weighing, glassware, reagent
purity) to give an estimate of uncertainty
Problem: This can be very complex, and it is difficult to include
systematic errors
Top-down method

Conduct multiple replicates of the experiment, varying as many
conditions that cause bias as possible - operator, reagent
source, glassware etc. - then mathematically estimate the
uncertainty.
Created with MindGenius Business 2005®
Measures of Spread

Often quoted as a indicators of uncertainty





Range: Difference between highest and lowest values
Standard Deviation (s): A good measure of precision. A small s
means that the data is more precise than data with a large s but not necessarily more accurate
Variance (s2): The square of the standard deviation
Coefficient of Variation (CV) OR Relative Standard Deviation
(RSD): A relative error estimate expressed as a % of the mean
of the measurements. Used to compare the precision of methods
with different units/ranges.
Confidence interval (CI): A range which has a high statistical
likelihood (e.g.95%) of containing the true value
Created with MindGenius Business 2005®
significant figures




You should have covered this in more detail in BY131
(also see Harris 3.1-3.3)
Don’t just write down all the digits your calculator gives
you!
Quote the minimum number of digits needed to write a
value in scientific notation without loss of accuracy
e.g. 9.34 (±0.02) x 102 not 93400, and not 9.34567
±0.02
Created with MindGenius Business 2005®
Rules






Generally only the last digit should have uncertainty
associated with it
The last digit will always have uncertainty associated
with it (unless the data is discrete)
Zeros at the end of a number imply you know the value
ends in 0 (4.56 is not the same as 4.560)
Calculations should be carried out without rounding only round up the answer
If you are worried about loss of information you may
put an extra digit as a subscript (e.g. 4.562)
Use literature examples and common sense if you are
unsure!
Created with MindGenius Business 2005®
propagation of errors

Necessary to calculate combined errors for:


“bottom up” estimation of uncertainty
estimating uncertainty for results based on two or more values
each with its own uncertainty
e.g. For data reported as ratios
Value = (sample result (±error) : control result (±error))
- we cannot simply add the errors - sometimes they will
cancel each other out
Created with MindGenius Business 2005®
Relative and Absolute uncertainties

Uncertainties of a measurement x can be quoted as



absolute – ex (in same units as x)
Relative - %ex ( a percentage of x)
Conversion:
Relative
y
uncertaint
Absolute
uncertaint
Measured
%e
Absolute
uncertaint
y
x

ex
y
 100 %
value
 100 %
x
Relative
uncertaint
y  Measured
value
100%
ex 
ex  x
100 %
Created with MindGenius Business 2005®
Example Question

A sample weight was measured as 5.1g. The balance
used was known to be accurate to 0.02g. What is the
relative uncertainty associated with this measurement?
Relative
uncertaint
y
Absolute
uncertaint
Measured
%e
%e
%e

So,
sample weight
x

ex
y
 100 %
value
 100 %
x
0.02g
x

x
 0 .4 %
 100 %
5 . 10 g
 5 . 10 g   0 . 4 % 
Created with MindGenius Business 2005®
How do we combine uncertainties?


We can use simple formulae to combine uncertainties
Combine one stage at a time
e.g. x = (a/b) + c
1) Combine a and b uncertainty, then
2) Combine the result with c to get uncertainty in x

NOTE: The methods described here are only used for
random errors, and assume that systematic errors have
been identified and eliminated
Created with MindGenius Business 2005®
Combining uncertainties addition and subtraction




Where the calculation to find x includes addition or
subtraction e.g. x = a+b-c
We need to combine the absolute uncertainties for a, b
and c, i.e. Combine ea eb and ec
Uncertainty in x:
2
2
2
ex 
e a  eb  ec
Method:
 Calculate x
 Calculate ex
 Quote result as x ± ex
Created with MindGenius Business 2005®
Example question
Uncertainty in reading a burette:
 You measure a volume by subtracting the initial reading
from the final reading.
 Initial reading is 0.05 (0.02) ml
 Final reading is 17.88 (0.02) ml
 If the uncertainty in each reading is known to be 0.02ml
what is the volume measured and its overall uncertainty

Measured volume is 17.88 - 0.05ml = 17.83ml

Absolute uncertainty, ex =

Volume =17.83 (±0.03)ml

e f  ei 

8  10
2
2
4
0 . 02 ml  0 . 02 ml
2
2
ml
 0 . 02 8 ml  0 . 03 ml
Created with MindGenius Business 2005®
Combining uncertainties –
multiplication and division




Where the calculation to find x includes multiplication or
division e.g. x = (a×b)/c
We need to combine the relative uncertainties for a, b
and c, i.e. Combine %ea %eb and %ec
Uncertainty in x:
2
2
2
% e x  % e a  % eb  % ec
Method:
 Calculate x
 Convert absolute uncertainties to relative uncertainties
 Calculate %ex
 Convert %ex to ex
 Quote result as x ± ex
Created with MindGenius Business 2005®
Example question
Calculate the value and uncertainty of x where: x= (a.b)/c
and: a = 1.76 ( 0.03), b = 1.89 ( 0.02) and c = 0.59 ( 0.02)

Calculate x
x 
1 . 76
 0 . 03   1 . 89  0 . 02 
0 . 59
 0 . 02 
 5 . 64  e x

Relative uncertainties: %e
a

e a  100

a
0 . 03   100
1 . 76
 1 .7 %
% eb 
e b  100
b
 1 .1 %

0 . 02   100
1 . 89
%e c 
e c  100
c

0 . 02   100
0 . 59
 3 .4 %
Created with MindGenius Business 2005®
Example question cont.

Combine
Combined
Relative
% ex 
% e1  % e 2  % e 3
2
2
2

1 .7 %  1 .1 %  3 .4 %

15 . 66 %
2
2
2
 4 .0%

Convert to absolute uncertainty
Absolute
Uncertain
ty in x 
% ex  x
100

4 . 0  5 . 64
100
 0 .2 3

So,
x  5 . 64  0 . 2 3  5 . 6  0 . 2
Created with MindGenius Business 2005®
Combining uncertainties –
powers and roots



Where the calculation to find x includes a power or root
e.g. x = ab or x = √a
Using relative uncertainties
Uncertainty in x:
% ex  b  % e
a

Method:
 Express roots as ab e.g. √a = a½
 Calculate x
 Convert ea to %ea
 Calculate %ex by multiplying by b
 Convert %ex to ex
 Quote result as x ± ex
Created with MindGenius Business 2005®
Example question
Calculate the value and uncertainty of x where: x= a3 and
a = 1.76 ( 0.03)

Express as x = ab
x  1 . 76   0 . 03 

Calculate x
x  1 . 76   5 . 4 5

Convert absolute uncertainty to relative uncertainty
3
3
% ea 
0 . 03
 100 %
1 . 76
 1 .7 %
Created with MindGenius Business 2005®
Example question, cont.

Multiply by b
% e x  1 .7 %  3
 5 .1 %

Convert to absolute uncertainty
ex 
5 . 1 %  5 . 45
100 %
 0 .2 8

So,

(using correct s.f.)
x  5 .5  0 .3
Created with MindGenius Business 2005®
Combining uncertainties constants
Where a constant k is part of the calculation, and has no
uncertainty associated with it.
Rule of thumb: If you are uncertain about the effect of k,
include it as a term with an associated uncertainty of 0


Case 1: k is added or subtracted

Value and uncertainty of x where x = k + a or x = a - k

k does not affect the absolute uncertainty - but will affect relative uncertainty
Case 2: k is multiplied or divided

Value and uncertainty of x where x = ka or a/k

k affects the absolute uncertainty - but not the relative uncertainty
Created with MindGenius Business 2005®
Combining uncertainties –
combinations
Solve each type of combination separately, one at a time

e.g
x = (a + b)
c


•
Combine errors for a + b to get absolute ea+b
•
Convert ea+b to %ea+b using (a + b) as the measured value
•
Convert ec to %ec and combine with %ea+b to get %ex
•
Calculate x
•
Convert %ex to ex
•
Answer is expressed as x± ex
NOTE: All these examples give ex (absolute uncertainty) as an answer.
You may be asked to calculate just %ex (relative uncertainty)
Read the question carefully!
Created with MindGenius Business 2005®
Example question
A 50cm3 burette can be read to ± 0.02cm3. In a particular analysis
the result is calculated using the formula
y = xm/(Ts - Tb)
where y is the analyte concentration, in mol.dm-3, and Ts and Tb and
the sample and blank titres respectively in cm3. Calculate the
uncertainty in the final result when:
x = (0.150 ± 0.002) mol.dm-3, m=300, Ts = 15.01 cm3 and Tb =
0.04 cm3. m is known absolutely.
Created with MindGenius Business 2005®

y = xm/(Ts - Tb) Look at subtraction first

Measured volume Ts - Tb = 15.01 - 0.04ml = 14.97ml

Calculate eTs-Tb
e Ts  Tb 


Convert to %eTs-Tb
e Ts  e Tb 
2
0 . 02  0 . 02
2
8  10
2
4
 0 . 02 8  0 . 03 cm
% eT s  Tb 

2
eT s  Tb
T s  Tb 
0 . 028
3
 100 %
 100 %
14 . 97
 0 . 18 7 %
Created with MindGenius Business 2005®





Now look at constant
y = xm/(Ts-Tb)
m only affects absolute uncertainties for y
To calculate ey we will be using relative uncertainties (division)
Convert ex to %ex
ex
% ex 
 100 %
x

0 . 002
 100 %
0 . 150

Combine relative uncertainties
 1 .3 3 %
% ey 
% eTs  Tb  % e x

0 . 18 7  1 . 33

1 . 80
2
2
2
2
 1 .3 4%
Created with MindGenius Business 2005®

Calculate y
y 

xm
T s
 Tb 
0 . 15  300 
14 . 97 
 3 . 0 1 mol .dm

3
Convert relative uncertainty of y to absolute
ey 

%ey  y
100 %
1 . 34  3 . 01 
100 %
 0 . 04 0 mol .dm

So,
3
y  3 . 01  0 . 04 mol .dm
3
Created with MindGenius Business 2005®
Limit of Detection (LoD)

The concentration which gives an instrument signal (y)
significantly different from the blank (or background)
signal

This is generally calculated as:
Concentration x which gives rise to a signal of yB + 3sB

where yB and sB are the mean and s.d. of blank solutions


NB the method used may vary according to the purpose
of the analysis - so it should always be quoted
Created with MindGenius Business 2005®
Measuring LOD/LOQ
Practically
 Perform the analysis on matched solutions containing
no analyte
 Calculate the mean (yB) and standard deviation (sB)
of the signals/measurements obtained
BUT this can be very time- and reagent- consuming


Mathematically



Use the calculated value of the intercept (a) as an estimate of
yB
Use sy/x as an estimate of sB
This is more accurate than using the single blank value
included as part of the calibration process, and
eliminates the need for repetition
Created with MindGenius Business 2005®
Sensitivity vs. Detectivity


LOD and LOQ are measures of detectivity
and are dependent on both the slope and
the intercept of the calibration plot
Sensitivity is a measure of instrument
response to changes in concentration
across the entire linear range and is only
dependent on the slope of the plot
Created with MindGenius Business 2005®
Limit of Quantitation (LoQ)

The concentration above which precise quantitative
measurement is possible

This is generally calculated as:

Concentration x which gives rise to a signal of yB + 10sB

This calculation is often conducted in different ways again the method used should always be quoted
Created with MindGenius Business 2005®
Record Keeping




Ensure results are recorded in a laboratory notebook
even if they are available electronically.
Enough information should be included to ensure a
colleague can repeat the experiment using only your
notes.
Keep a copy of the notebook (preferably in a separate
location).
Many employers have their own methods for laboratory
record keeping and usually require that each page is
signed and dated by both the employee and their line
manager. This is useful when it comes to intellectual
property rights.
Created with MindGenius Business 2005®
Record Keeping

In general for each experiment include:
 title and date
 objectives
 reaction scheme, if applicable
 hazard assessment, if necessary
 method
 results and calculations, including any instrument
readouts and graphs
 conclusion
Created with MindGenius Business 2005®
Reporting - Analytical Documentation

Used to allow other competent analysts to reproduce the method.
Sufficient detail is required to obtain consistent results. Trained and
competent personnel are still required even when a full detailed
document is available
Please see studentcentral website for further details

Drawing conclusions






In a written report of an experiment you must come to some conclusion
about the work
Use the information from the statistical tests and performance
parameters
Pull together all the information
Keep the wording ‘analytical’ i.e. use ‘accurate’ and ‘precise’ correctly,
and don’t over-generalise
Make informed judgements about the technique and compare to other
possible techniques
Created with MindGenius Business 2005®