Download 4. ANALYTICAL PROPERTIES III

ANALYTICAL
PROPERTIES
PART III
ERT 207 ANALYTICAL CHEMISTRY
SEMESTER 1, ACADEMIC SESSION 2015/16
OVERVIEW
2
ANALYTICAL SAMPLES AND METHODS
 STANDARDIZATION AND CALIBRATION
 EXTERNAL STANDARD CALIBRATION
 INTERPRETATION OF LEAST-SQUARE
RESULTS
 TRANSFORMED VARIABLES
 QUALITY ASSURANCE OF ANALYTICAL
RESULTS

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ANALYTICAL SAMPLES AND METHODS
3
Macro analysis:
 Is used for samples whose masses are > 0.1 g.
 Semimicro analysis:
 Is performed on samples in the range of 0.01 to
0.1g.
 Micro analysis:
 Is used for samples whose mass is 10-4 – 10-2g.
 Ultramicro analysis:
 Is used for samples whose mass is > 10-4g.

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ANALYTICAL SAMPLES AND METHODS
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Major constituents:
 Those present in 1 – 100% by mass.
 Minor constituents:
 Species present in 0.01-1% by mass.
 Trace constituents:
 Those present in amount between 100 ppm
(0.01%) and 1 ppb.
 Ultratrace constituents:
 Those present in amounts < 1 ppb.

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ANALYTICAL SAMPLES AND METHODS
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Fig 1:
Classification
of analytes by
sample size
Fig 2: Classification
of constituent types
by analyte level
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ANALYTICAL SAMPLES AND METHODS
6
↓
↓

Fig 3: Inter-laboratory error as a function of
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analyte concentration.
ANALYTICAL SAMPLES AND METHODS
7

Note: Samples are analysed, but species or
concentrations are determined.
 Determination of glucose in blood serum
analysis or
 the analysis of blood serum for glucose
concentration determination
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ANALYTICAL SAMPLES AND METHODS
8
The composition of the gross sample and the
laboratory sample must closely resemble the
average composition of the total mass of material
to be analyzed.
 The items chosen for analysis are often called as
sampling units or sampling increments.
 In sampling, a sample population is reduced in size
to an amount of homogeneous material that can
be conveniently handled in the laboratory and
whose composition is representative of the
population.

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ANALYTICAL SAMPLES AND METHODS

9
Statistically, the goals of the sampling process are:
1. To obtain a mean analyte concentration that is
an unbiased estimate of the population mean.
 It can be realized if all members of the
population have an equal probability of
being included in the sample.
2. To obtain variance in the measured analyte
concentration that is an unbiased estimate of
the population variance so that valid
confidence limits can be found for the mean,
and various hypothesis tests can be applied.
 It can be realized if every possible sample is
equally likely to be drawn.
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ANALYTICAL SAMPLES AND METHODS
The number of particles required in a gross
sample ranges from a few particles to 1012
particles.
 Based on the Bernoulli equation:
 The standard deviation of the number of A
particles drawn: σ  N 1  p 
Probability of

10
A

p
randomly
drawing an A
type particles
The relative standard deviation of drawing A
type particles:
σA
1 p
σr 

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Np
Np
ANALYTICAL SAMPLES AND METHODS
11

The number of particles needed to achieve a
given relative standard deviation:
1 p
N

pσ
2
t
If 80% of the particles are type A (p=0.8) and
the desired relative standard deviation is 1%
(σr=0.01), the number of particle making up the
gross sample should be:
1  0.8
N

2500
2
0.80.01
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ANALYTICAL SAMPLES AND METHODS
12
In reality:
 The type A particles contain a higher percentage
of analyte, PA and the type B particles a lesser
amount, PB.
 The average density d of the particles differs
from the densities dA and dB of these components.

 d Ad B 
N  p1  p  2 
 d 
2
 p A  pB 


 σr P 
2
Overall average percent of active ingredient (%)
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ANALYTICAL SAMPLES AND METHODS
Rearrange the equation:
 The relative standard deviation:
PA  PB d Ad B p1  p 
σr 
x 2
P
d
N
 If we make the assumption that the sample mass
m is proportional to the number of particles and
the other quantities in the equation are constant,
the product of m and σ, should be a constant.

13
K s  m x σ r x100
2
σr x 100% = % relative standard deviationbblee@unimap
ANALYTICAL SAMPLES AND METHODS
14

When σr = 0.01, σr x 100% = 1%, Ks = m.
 Ks is the minimum sample mass required to
reduce the sampling uncertainty to 1%.
bblee@unimap
ANALYTICAL SAMPLES AND METHODS
15
EXAMPLE 1
 A column packing material for chromatography
consists of a mixture of two types of particles.
 Assume that the average particle in the batch
being sampled is approximately spherical with a
radius of about 0.5 mm.
 Roughly 20% of the particles appear to be pink in
color and are know to have about 30% by mass of
a polymeric stationary phase attached (analyte).
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ANALYTICAL SAMPLES AND METHODS
16
The pink particles have a density of 0.48 g/cm3.
 The remaining particles have a density of about
0.24 g/cm3 and contain little or no polymeric
stationary phase.
 What mass of the material should the gross sample
contain if the sampling uncertainty is to be kept
below 0.5% relative?

bblee@unimap
ANALYTICAL SAMPLES AND METHODS
17

Number of laboratory samples:
 How many samples should be taken for the
analysis?
2 2
t
ss
 The number of samples N,
N
2
xσ
2
r
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ANALYTICAL SAMPLES AND METHODS
18
EXAMPLE 2.
 The determination of copper in a seawater sample
gave a mean value of 77.81 μg/L and a standard
deviation ss of 1.74 μ g/L.
 How many samples must be analysed to obtain a
relative standard deviation of 1.7% in the results
at the 95% confidence level?
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STANDARDIZATION AND CALIBRATION
19

Calibration:
 It determines the relationship between the
analytical response and the analyte
concentration.
 The relationship is usually determined by the use
of chemical standards.
 Interference could be reduced from other
constituents in the sample matrix, called
concomitants by using standards added to the
analyte solution or by matrix matching or
modifications.
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STANDARDIZATION AND CALIBRATION
The absolute method (e.g. gravimetric method) do
not rely on calibration with chemical standards.
 Comparison with standards:
 2 types of comparison methods:
i. Direct comparison techniques
ii. Titration procedures
 Null camparison or isomation methods:
 Comparison a property of the analyte with
standards such that the property being tested
matches or nearly matches that of the
standard.
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
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STANDARDIZATION AND CALIBRATION

21
With some modern instruments, a variation of this
procedure is used to determine if an analytes
concentration exceeds or is less than some
threshold level.
 Comparator can be used to indicate that the
threshold has been exceeded.
 Titration is a type of chemical comparison.
 The amount of the standardized reagent needed
to achieve chemical equivalence can be related
to the amount of analyte present by
stoichiometry.
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EXTERNAL STANDARD CALIBRATION
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A series of standard solutions is prepared
separately from the sample.
 The standards are used to establish the instrument
calibration function.
 It is obtained from analysis of the instrument
response as a function of the known analyte
concentration.
 Ideally, 3 or more standard solutions are used
in the calibration process, although in some
routine determinations, 2 point calibration can
be reliable.
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
EXTERNAL STANDARD CALIBRATION
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
The calibration function can be obtained
graphically or in mathematical form.
 A plot of instrument response versus known
analyte concentrations is used to produce a
calibration curve, sometimes called a working
curve.
 It is often desirable that the calibration curve be
linear in at least the range of the analyte
concentrations.
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EXTERNAL STANDARD CALIBRATION
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Fig. 1:
Calibration curve
of absorbance
versus analyte
concentration for
a series of
standards.
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EXTERNAL STANDARD CALIBRATION
25
The linear relationship is then used to predict the
concentration of an unknown analyte solution.
(1) The Least-squares Method
 The investigator must try to draw the ‘best’
straight line among the data points.
 Regression analysis provides the means for
objectively obtaining such a line and also for
specifying the uncertainties associated with its
subsequent use.

bblee@unimap
EXTERNAL STANDARD CALIBRATION

26
Assumptions of the leastsquare method:
1)There is actually a linear
relationship between the
measured response y
and the standard
analyte concentration x.
The mathematical
relationship that
describes this assumption
is called the regression
model (y = mx + c).
Fig. 2: The slopeintercept form of
a straight line.
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EXTERNAL STANDARD CALIBRATION
27
2) We assume the y deviation from the individual
points from the straight line arises from error in
the measurement.
 There is no error in x values of the points
(concentrations).
 When the uncertainties in the y values vary
significantly with x, basic least-squares
analysis may not give the best stright line.
 Instead, a correlation analysis (or weighted least
squares analysis) should be used.
bblee@unimap
INTERPRETATION OF LEAST-SQUARE
RESULTS
28
The closer the data points are to the line predicted
by a least-square analysis, the smaller are the
residuals.
 The sum of the squares of the residuals, SSresid,
measures the variation in the observed values of
the dependent variables (y values) that are not
explained by the presumed linear relationship
2
between x and y:
N

SS resid    yi  c  mxi 
i 1
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INTERPRETATION OF LEAST-SQUARE RESULTS

A total sum of the squares,

y

  y  y    y 
N
2
29
SStot  S yy
2
i
2
i
i
SStot is a measure of the total variation in the
observed values of y since the deviations are
measured from the mean value of y.
 The coefficient of determination (R2) measures the
fraction of the observed variation in y that is
explained by the linear relationship:

SS resid
R  1
SStot
2
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INTERPRETATION OF LEAST-SQUARE
RESULTS
30
The closer R2 is to unity, the better the linear
model explains the y variations.
 The difference between SStot and SSresid is the sum
of the squares due to regression, SSreg.
 In contrast to SSresid, SSreg is a measure of the
explained variation.

SS regr  SStot  SS resid & R 
2
SS regr
SStot
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INTERPRETATION OF LEAST-SQUARE RESULTS
EXAMPLE 3
 Find the coefficient of determination for the
chromatographic data below.
31
Mole percent isooctane, xi
0.352
0.803
Peak area, yi
1.09
1.78
1.08
1.38
1.75
2.60
3.03
4.01
The least-square line:
 SStot = 5.07748.

y  2.09 x  0.26
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TRANSFORMED VARIABLES
32


Transformations to linearize function:
Linear least squares gives best estimates of the
transformed variables, but these may not be
optimal when transformed back to obtain estimates
of the original parameters.
 Nonlinear regression methods may give better
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estimates.
QUALITY ASSURANCE OF
ANALYTICAL RESULTS
33
Control chart:
 A sequential plot of some characteristic that is a
criterion of quality (quality assurance).
 It shows the statistical limits of variation that are
permissible for the characteristic being
measured.
3σ μ = population
UCL

μ

mean
 Upper control limit:
N

3σ
 Lower control limit: UCL  μ 
N
σ = population
standard
deviation
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QUALITY ASSURANCE OF ANALYTICAL RESULTS
34

Fig 3: A
control
chart for a
modern
analytical
balance.
For example, from independent experiments,
estimates of the population mean and standard
deviation were found to be μ = 20.000 g and σ =
0.00012 g, respectively.
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QUALITY ASSURANCE OF ANALYTICAL RESULTS
35

The mean of 5 measurement,
0.00012 = 0.00016
3x
5
UCL = 20.00016 g & LCL = 19.99984 g. (see Fig
3).
 As long as the mean mass remains between the
LCL and the UCL, the process is said to be in
statistical control.

bblee@unimap
QUALITY ASSURANCE OF ANALYTICAL RESULTS
Fig. 4 shows the results of 89 production runs of a
cream containing a nominal 10% benzoyl
peroxide measured on consecutive days.
 Each sample is represented by the mean
percent benzoyl peroxide determined from the
results of five titrations of different analytical
samples of the cream.
 The chart shows that, until day 83, the
manufacturing process was in statistical control
with normal random fluctuations in the amount of
benzoyl peroxide.

36
bblee@unimap
QUALITY ASSURANCE OF ANALYTICAL RESULTS
Fig 4: A control
37
chart for
monitoring the
concentration of
benzoyl
peroxide in a
commercial
acne
preparation

On day 83, the system went out of control with a
dramatic systematic increase above the UCL.
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QUALITY ASSURANCE OF ANALYTICAL RESULTS
38
This increase caused considerable concern at the
manufacturing facility until its source was
discovered and corrected.
 These examples show how control charts are
effective for presenting quality control data in real
world problems.

bblee@unimap
EXAMPLE 1
 The
values for the average density and
percent polymer:
d = 0.20 x 0.48 + 0.80 x 0.24
= 0.288 g/cm3

0.20 x0.48 x0.30g polymer / cm
P
0.288 g sample / cm
= 0.10%
 Then,
3
3
x100%
2
 0.48 x0.24   30  0 
N  0.201  0.20

2  
 0.288   0.005 x0.10 
= 1.11 x 105 particles required
39
bblee@unimap
2
EXAMPLE 1
 Mass
of sample =
3
4
cm
0.288g
3
1.11x10 particles x π 0.05
x
3
3
particle
cm
5
= 16.7 g
40
bblee@unimap
EXAMPLE 2
 Assume
we have infinite number of
samples, t = 1.96 at 95% confidence
level.
 Since σr = 0.017, ss = 1.74 and
Ẋ=77.81, 2
2

1.96 x1.74
N
2
2
0.017  x77.81
= 6.65
 We
round the result to 7 samples, the
value for t for 6 degrees of freedom is
2.45.
 Using the t value (2.45), we calculate N
= 10.38.
41
bblee@unimap
EXAMPLE 2
 Now
we use 9 degree of freedom, t =
2.26, N = approximately 9.
 Note that it would be good strategy to
reduce the sampling uncertainty so that
fewer samples would be needed.
42
bblee@unimap
EXAMPLE 3
 Finding
the sum of the squares of the
residuals:
xi
yi
ẏi
yi-ẏi
(yi-ẏi)2
0.352
1.09
0.99326
0.09674
0.00936
0.803
1.78
1.93698
-0.15698
0.02464
1.08
2.60
2.51660
0.08340
0.00696
1.38
3.03
3.14435
-0.11435
0.01308
1.75
4.01
3.91857
0.09143
0.00836
5.365
12.51
 SStot
0.06240
= 5.07748,
SS resid
0.0624
R  1
 1
= 0.9877
SStot
5.07748
2
43
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