A set of spreadsheets that perform simulations of widely-used
analytical calibration methods. Each spreadsheet simulates an
"unknown" sample solution whose concentration is to
be measured by an
analytical instrument that generates a signal related to
concentration.
You can specify the "true" concentration of the unknown in the
sample solution,
then the spreadsheet simulates a measurement of it, using the
specified
calibration method
to establish a calibration based on one or more standard solutions
and to
convert instrument readings into concentration (including
typical
experimental errors), reports the calculated result,
and compares it to the true value that you specified. To be more
realistic, these simulations include typical systematic
and random errors in both signal and in volumetric measurements.
They demonstrate how non-linearity, interferences, and random
errors combine to influence the final result and and they allow
you to
optimize precision and
accuracy of the measurement.
Note: to run these spreadsheets,
you have to first download the OpenOffice installer (download from OpenOffice),
then install it (by double-clicking on the installer file that you
just
downloaded), and then download my spreadsheets from this page.
Once OpenOffice is installed, you can run my spreadsheets
just by
double-clicking on them. Note
1:Don't use version
3.1.
There is a bug in OpenOffice 3.1 that causes bad x-axis scaling on
some of my graphs. The problem does not occur in the more
recent versions. Note 2:
Downloading these files with some versions of Interent Explorer
will change the file types from ".ods" to ".zip"; you will have to
edit
the file names and change the extensions back to ".ods" for them
to
work properly. This problem does not occur in Firefox or in Chrome.
OpenOffice program is a free download
from OpenOffice.org
for either PCs or Macs.
Background.
In
analytical chemistry, the accurate quantitative measurement of the
composition of samples, for example by various types of
spectroscopy, usually requires that the method be calibrated
using standard samples of known composition. This is most
commonly, but
not necessarily, done with solution samples and standards
dissolved in
a suitable solvent, because of the ease of preparing and diluting
accurate and homogeneous mixtures of samples and standards in
solution
form. (Note: calibration methods may be contrasted to "absolute
analytical methods", in which the concentrations of samples are
calculated with the aid of previously-measured fundamental data
rather
than from standards that are measured along with the samples.
Absolute
methods are occasionally performed when preparing standard samples
is difficult or impossible, especially in the measurement of
atmospheric gases in situ using laser spectroscopy).
Calibration errors. Calibration
procedures are subject to error
caused by several complications:
a.
Analytical curve non-linearity. The analytical
curve is a plot of the signal from the instrument vs the
concentration of
the analyte (the chemical species whose concentration is
sought). This is closely related to the calibration curve,
which is a plot of the signal from the instrument vs the
concentration
of
the standard solutions. In
the absence of interferences, the points of the calibration curve
should fall along the analytical curve. If the
analytical curve is linear, calibration procedures are much
simpler, both mathematically and procedurally. If the relationship
in
non-linear,
a series of standard solution must be prepared and measured to
establish the shape of the curve, which is time-consuming as well
as
requiring the use of larger amounts of standard materials (which
can be
expensive
and will eventually require safe disposal).
Commonly, calibration
curves are observed to be approximately
linear over a certain concentration range, but non-linear above
that
range. In some well-defined cases, the shape of the analytical
curve can be predicted, for example in absorption
and in fluorescence
spectrophotometry.
b.
Additive interferences. Ideally, samples
and
standards should give
a zero reading when the analyte concentration is zero. Commonly,
the instrument readout is zeroed when a "blank" is measured (a "blank" is a solution
containing
zero concentration of analyte in the same solvent and containing
vessel). But in many cases this is not enough,
because some other unknown chemical components that are present
in the samples
(but not in the standards) are contributing their own signals to
the
total signal measured. Unless it it possible to resolve
(separate) the
signal
generated
by these components from that of the analyte, the signal
measured in that case will
be higher than it should be, leading to an error in the
analysis. This
is called an "additive
interference", because the signal from the interfering
components adds
to that from the analyte. So even if the analyte's concentration is zero,
you still get a signal from the sample. In spectroscopy, this is often
called
a "spectral interference".
c. Multiplicative interferences.
Ideally, a given concentration of analyte will give the same
signal reading in the
sample as in the standards (in other words, the slope of the
analytical curve is the same in the samples and standards).
But sometimes
there are conditions
or components
present in the samples (but not in the standards) that make the
analyte's signal stronger or weaker that it is in the standards;
it
might be a difference in temperature, pH, ionic strength,
density,
viscosity, surface tension, or a specific chemical component
that
reacts with or binds with the analyte. This is called an "multiplicative interference",
because the analyte's
signal is in effect multiplied
by some unknown factor.
This is distinct from an additive interference, because with a
multiplicative interference, you still get a zero signal when
the
analyte's concentration is zero.
d. Random errors (uncertainty)
and the propagation of random errors. In
addition to the systematic
errors considered above, analytical methods
are also subject to random errors
(imperfect repeatability) due to
several sources. For example, the preparation of analyte and
standard
solutions is subject to random errors in volumetric measurements,
and
the instrument's reading itself is subject to random variability
due to
electronic
noise generated by electronic amplifiers and detectors,
instability of light sources, and photon noise,
especially when the concentration of the analyte is very low.
Proper
instrument design and careful technique can reduce, but never
completely
eliminate, such errors.
The size of random errors are usually
described by the standard
deviation, s,
defined as
where N is the number of data points, x_{i} are the individual points, and x
is the mean (average) of all the x's. Spreadsheets and most programming languages
have a built-in
function for standard deviation. The relative standard deviation,
given by s/x, is
often also reported, as a percentage.
e.
Calculating the propagation of random errors. The way
that random errors combine and
influence the final precision of the measurement is called propagation
of
error.
When you compute some quantity that is based on two
or more measurements, you need to be able to figure out how
reproducible the calculated quantity will be when the input
variables
are subject to random variability. If you know the standard
deviation
of each
of those input measurements, you can calculate the expected
standard
deviation of the calculated quantity in two different ways:
1. Do the math. By using the rules for mathematical error
propagation. In principle the propagation of errors of
the entire calibration method can be
described by closed-form
algebraic formalism
by breaking down the equation into a series of
simple differences, sums, products, and ratios, and applying
the rules for error
propagation to each step. However, there are two problems
with this approach. If the
calculation is complicated, the error propagation can become
really
complicated and difficult. Secondly, the usual rules for
mathematical
error propagation assume that the random errors of the various
terms of
the calculation are not correlated:
if they are correlated, the calculations become even more
complicated.
Correlation between terms occurs in the
prediction of error
propagation of the bracket and standard addition methods. The
bottom
line is that it is often difficult to predict the propagation of
errors
by doing the math. 2.
Crunch the numbers. By repeating all the calculations over and
over again (obviously using
a computer) with random number generators employed to add
realistic
amounts of random variability ("noise") to the input variables.
(This is
sometimes called a "Monte
Carlo"
approach, a reference to the famous gambling casinos in that small
country). This is relatively easily set up using
spreadsheets,
which are well suited to performing laborious repetitive
calculations
and even have built-in random number and statistical functions.
The advantage of this
approach over closed-form algebraic formalism is that it can be
applied
to essentially any arbitrarily-complicated procedure and it
automatically takes into account any correlation between
variables. The
disadvantage is that it is less "elegant" and can not be expressed
in a
neat formula.
These spreadsheets perform both of these type of
calculations, so you can compare them. It's
important to understand that even a perfectly accurate calculation
of
error propagation predicts
only the expected standard deviation "on average", for a very
large
number of repeats. If you were to repeat an actual experiment a
few
times and compute the standard deviation, you'll often get only a
very rough approximation
to the theoretical result, perhaps off by 2 or 3-fold. This is a
basic
problem of statistics in analytical chemistry; the theoretical
predictions work well for very large number of repeats, but in
analytical chemistry the cost and time of doing even a few
repeats is
often prohibitive. For this reason it is not worth
obsessing about
small differences in precision; the statistical uncertainty in
measuring the
precision of any one method is likely to be greater than the
differences
between the different methods.
Calibration methods.
The methods described below are the most commonly-used
analytical calibration methods. Each
of these methods, from the simplest to the more complex, is
modeled by a separate simulation spreadsheet, which
includes all of the above-mentioned systematic
errors, plus random errors due to both volumetric measurement
and signal measurement. These simulations allow you to
investigate
how all of these errors combine an propagate to the final
result. All of the simulations have a very similar structure and
layout, so once you learn how to work the first one, using the
others
will be relatively straightforward. The concentration of the
unknown sample, Cx, and its predicted
standard deviation, is calculated in a different way in each of
these
calibration methods.
This is the
simplest calibration method,
in which the sample and a single
separate
standard solution are measured. This method assumes that the
calibration errors a,b, and c, listed above, are absent.
The
concentration of the sample Cx is given by Cs*Sx/Ss,
where Cs is the
concentration of the standard solution, Ss is the signal given
by that
standard solution, and Sx is the signal given by the sample
solution. The predicted relative standard deviation of Cx is
easy to
compute in this particular case, if you know the standard
deviations of Cs,
Sx, and Ss: there are just three variables, all multiplied or
divided, so
according to the rules for error
propagation,
the relative standard deviation of Cx is the quadratic sum
(square root
of the sum of the squares) of the relative standard deviations
of Cs,
Sx, and Ss, which in this simulation are Ev, Es, and Es,
respectively (see cell C68).
In this calibration method, the sample is measured along with
two
standard solutions that are close in concentration to the sample
(typically one lower than and one higher than the sample
concentration). This method
has the advantage of
approximately
compensating for non-linearity in the analytical curve, if the
two
standards are close in concentration to the sample.
It's a useful method when you have many samples to analyze that
have about the same analyte concentration. However, this
method still assumes that calibration error conditions (b) and
(c) are absent. A
disadvantage of this method is that it requires more time and
uses twice the amount of standard material as the
single-standard
method.
The concentration of the sample Cx is calculated by
linear
interpolation between two standard solutions and is given
by
C1s+(C2s-C1s)*(Sx-S1s)/(S2s-S1s), where C1s and C2s are the
concentrations of the two standard solutions, S1x and S2s are
the
signal readings given by the two standards, and Sx is the signal
given
by the sample solution. The
predicted standard deviation of Cx is more complex to
compute in this case, but
it can be done by breaking down the equation into a series of
differences, sums, products, and ratios, and applying the rules for error propagation to
each step. (In the spreadsheet "BracketOO.ods", these error propagation
calculations are performed in cells C98:F103).
A
series of external standard solutions of different
concentrations is prepared and
measured. A
first-order least-squares
fit of the data is computed and the resulting
equation is used to convert readings of the unknown samples into
concentration. An
advantage of this method are that the random errors in
preparing and reading the standard solutions are averaged
over
several standards. Moreover, non-linearity in the
calibration curve can
be detected and avoided (by diluting into the linear range) or
compensated (by using non-linear curve fitting methods). An
obvious disadvantage of this method is that it requires much
more time
and
uses more standard material than other methods. The calibration
data (Cs vs Ss) are fitted with a first-order
least-squares fit. (The fit is shown as the straight red line in
the
graph).
The concentration of the sample Cx is calculated by
Cx = (Sx-intercept)/slope, where Sx
is the signal given by the sample solution, and "slope" and "intercept"
are the results of the least-squares fit. (In this
simulation, rather than choosing each standard solution
separately, you
choose the number of standards (from 2 to 18) and the
concentration of
the
highest one; the other standards are automatically evenly
distributed
between zero and the specified highest). The predicted standard
deviation of Cx is computed from the standard deviations of the
slope
and intercept given by the curve fitting procedure and
the standard deviations of Sx, as shown in cells D111:F119.
A
series of external standard solutions is prepared and
measured. A non-linear model function is selected that is
expected to
be a good fit to the calibration data (e.g. a quadratic or cubic
function), a least-squares fit of that model to the data is
computed,
and the resulting non-linear equation is solved for
concentration
and used to convert readings
of the unknown samples into
concentration. An
advantage of this method is that non-linearity in the
calibration
curve is
compensated at least approximately, depending on the
nature and
severity of the non-linearity of the calibration curve and the
choice
of non-linear fitting model equation. The
calibration data (Cs vs Ss) are fitted with a second-order
(quadratic) least-squares fit. (The fit is shown as the curved
red line
in the
graph).
The concentration of the sample Cx is calculated by
the "Quadratic equation": Cx = (-b+SQRT(b^2-4*a*(c-Sx)))/(2*a)
where Sx
is the signal given by the sample solution, and a, b, and c are
the
coefficients from the quadratic least-square fit (quadratic,
linear,
and intercept, respectively). (In this simulation, the number of
standards is fixed at 18; you
can choose the concentration of the
highest one and the other 17 standards are automatically evenly
distributed
between zero and the specified highest concentration).
In this
method, the sample is divided into two portions: one is measured
unmodified and the other is "doped" with the addition of a small
amount
of pure standard and then measured. This method has the
advantage of compensating for multiplicative interferences
(c, above),
but it still assumes that the analytical curve is linear and
that additive interferences are absent.
The
concentration of the sample is given by
(Sx*Vss*Cs)/(Ss*(Vxx+Vss)-Sx*Vxx), where Cs is the
concentration of the standard solution, Sx is the signal given
by that
sample solution by itself, and Ss is the signal given by the
sample
solution after the addition of standard, and Vxx and Vss are the
volumes of the samples and standard solution. The predicted
standard
deviation of Cx is computed by breaking down the equation
into a
series of
differences, sums, products, and ratios, and applying the rules
for error propagation to each step. These error
propagation calculations are performed in cells D111:F118.
A
series of aliquots of the sample solution are taken, increasing
amounts
of standard material are added to each one, and the signals from
the
resulting mixtures are measured and plotted against the
concentration
of added standard. If the resulting calibration curve is
sufficiently
linear, a
first-order least-squares fit of the data is computed. The
sample
concentration is given by the negative of the x-axis intercept
(and to
the ratio of the y-intercept to the slope). This method has the
advantage of compensating for multiplicative interferences.
Compared to
the single addition method (below), this method reduces the
random
errors in
preparing and reading the standard solutions. Moreover,
non-linearity
in the calibration curve can
be detected and avoided (by diluting into the linear range) or
compensated (by using non-linear curve fitting methods). An
obvious disadvantage of this method is that it requires much
more time
and
uses more standard material than most other methods.
The concentration of the sample is given by intercept/slope, where "slope" and "intercept"
are the results of the first-order least-squares fit of the
standard
addition calibration
curve, shown as the straight red line in the graph. The
predicted
relative standard deviation of Cx is the quadratic sum (square
root of
the sum of the squares) of the relative standard deviations of
the
slope and intercept computed by the curve fitting procedure.
These
error propagation calculations are performed in cells B82:F87.
When you are using these spreadsheets, you can inspect the equations
that
perform these calculations by clicking on a calculated cell and
looking
for the equation that calculates that cell in the
rectangular box at the top of the screen. You'll notice that some
cells
in these spreadsheets have a tiny red squares in their upper right
corners; that means they have an attached note, which you can read
just
by mousing over the cell without clicking.
Brief
operating Instructions.
The screen display of each of the simulations have five
similar areas:
The
yellow table on the top left of the screen are independent
variables that
you can change. Click on the dark
blue boldface numbers, type a new
value and press the enter key. Some of these variables can be
controlled continuously by the sliders on the top right. (The units
of
concentration in these simulations are normalized to
the range of 0 - 10 for convenience in entering and plotting; you
can
think of them as mmoles/liter, µmoles/liter, grams/liter or any
other
convenient unit (1 mmole = 0.001 moles; 1 µmole = 10^{-6}
moles). Similarly, the signal units are arbitrary
for
similar reasons).
The graph
on the lower right shows the actual analytical curve
(blue
line)
over the concentration range from 0 to 10 (arbitrary units), with
the
actual concentration of the unknown sample marked as a yellow
triangle.
The red triangles are the standards. The green triangle is the
calculated concentration of the unknown sample according to that
calibration method. In the linear calibration curve and multiple
standard addition methods, the red
line represents the
linear least-squares fit to the calibration curve. The graph
responds
dynamically as you adjust the variables (e.g. with
the sliders).
The table
in the middle left of the screen, labeled "Computed results", are
dependent variables that are automatically calculated from
the
independent variables (don't type in those cells or you will
delete the
formulae). The most important dependent variable is "result",
which is
a single simulated experimental measurement of the analyte
concentration Cx
based on that calibration method.
The "Statistics"
section at the lower left of the screen shows
the mean, standard deviation, and % relative standard deviation
(%RSD) of 20
simulated repeat calibrations and measurements of the unknown
concentration Cx. Here, a repeat calibration means that a complete set of new standards
are prepared for each repeat. This gives an idea of the
reproducibility
if the entire procedure is repeated. The "% RSD" in the table is
the
relative standard
deviation of 20 repeated simulated calibrations; it can be
compared to the "Est. RSD" in the Computed results table above,
which
is the predicted relative standard. based onpropagation
or error rules.
Both should ideally be the same, but for a variety of
reasons will usually not agree exactly. The statistics are
re-calculated each time an input variable is changed or a slider
is
moved.
In
addition to these user-interface areas, there are "off-screen"
areas,
below and to the right, that are used by the spreadsheet for
graphing,
statistics, and error propagation calculations. You don't need to
change anything there, but you may inspect those areas if you are
interested in the technical aspects of how the simulation
spreadsheets
work internally.
Assumptions:
1. The only sources of random error are random errors in volume
and
signal measurement. They apply equally to all solutions and
readings of
the samples
and of the standards. Errors due to interference and blank
correction
errors apply only to the sample readings and are systematic
(constant
between measurements).
2. Random errors are expressed as a percentage of the quantity
measured
(relative error rather than absolute error).
3. Non-linearity of the analytical curve is introduced by a
quadratic
term whose coefficient is the variable "n" (controlled by the
first
slider). (This is not rigorously realistic in the case of
the
non-linearity in absorption spectroscopy caused by
polychromaticity and
unabsorbed stray light. See Instrumental
Deviation
from Beer's Law for a treatment of non-linearity in that
particular case).
Cell definitions and equations (for Bracket method,
OpenOffice
version):
Inputs: mo : Analytical curve slope without interference z : Interference factor (zero -> no interference) n : Analytical curve non-linearly (0 = linear) Ev : Random volumetric error (% RSD ) Es : Signal measurement error (% RSD) Cx : True analyte concentration in sample C1s : Concentration of standard solution 1 C2s : Concentration of standard solution 2 blank : (Uncorrected) blank signal
Outputs:
Analytical curve slope in actual sample m = mo+z
Signal given by standard 1 S1s =(mo*C1s-mo*C1s^2*n)*(1+0.01*2.5*Ev*(RAND()-RAND()))*(1+0.01*2.5*Es*(RAND()-RAND()))
Signal given by standard 2 S2s =(mo*C2s-mo*C2s^2*n)*(1+0.01*2.5*Ev*(RAND()-RAND()))*(1+0.01*2.5*Es*(RAND()-RAND()))
Signal given by unknown sample Sx =(blank+m*Cx-m*Cx^2*n)*(1+0.01*2.5*Ev*(RAND()-RAND()))*(1+0.01*2.5*Es*(RAND()-RAND()))
Measured analyte concentration in sample result = C1s+(C2s-C1s)*(Sx-S1s)/(S2s-S1s)
Relative % effect of interference on signal recovery = m/mo
Array calculations for statistics (performed off-screen): Average: mean = AVERAGE(I99:I118) Standard deviation: s = STDEV(I99:I118) Relative standard deviation: RSD = s/mean Accuracy = (mean-Cx)/Cx
Note: The formulation 2.5*(RAND()-RAND()) seen in the above equations is simply a way of generating random numbers with a "haystack" distribution, a mean of zero and a standard deviation of 1.0, using the RAND() function that by itself gives a uniform distribution between 0 and 1.
Suggested activity:
OpenOffice versions.
Error
propagation in analytical calibration methods: Step by Step
1. OpenSingleStandardOO.ods (view Screen
Shot).
This is the simplest calibration method, in which the only two
things
measured are the unknown sample and a single separate
standard solution of known concentration. The table in the
upper
left lists the variables that you can change in this simulation.
The most important one is
Cx, which is the true concentration of
the sample solution. (Of course, in the real world, you
wouldn't know
this beforehand, but in these simulations you can set the true
sample
concentration as you wish. The simulation "pretends not to
know" the
true value and computes the measured sample concentration from the
sample and standard signals, just as you would in the real world,
then
compares that calculated value to the true value to determine the
accuracy of the simulated measurement). The other important
variable
is Cs, the concentration of the standard solution that you prepare
to
calibrate the system. The variable mo controls the slope of the
analytical curve, that is, the magnitude of the simulated signals.
The other variables
control simulated imperfections and sources of error: z controls multiplicative
interferences, blank controls additive interference, n controls the
non-linearity of the analytical curve, and Ev and Es
control the random errors in volume measurement and signal
measures,
respectively. If these variable are set to zero, the
simulated
measurement should be perfect.
Some of these variables
can be varied continuously by means of the sliders on the top
right.
The table in the center
left lists the quantities that are computed by the simulation.
The most important of
these is the measured concentration of the sample, "result".
In this calibration method, it is given by Cs*Sx/Ss),
where Cs is the
concentration of the standard solution, Ss is the signal given by
that
standard solution, and Sx is the signal given by the sample
solution.
You can click on the numbers in this table and look at the
input
line at the top to see the equations that the simulation uses to
calculate that number. The measured signals, Sx and Ss, take
into
account all the sources of error due interferences, non-linearity,
and
random errors. The "Est. RSD" is the estimated relative
standard deviation of the result, computed as described
above for that calibration method.
The
graph on the lower right shows the actual analytical curve
(blue
line)
over the concentration range from 0 to 10 (arbitrary units), with
the
actual concentration of the unknown sample marked as a yellow
triangle.
The red triangle represents the standard solution. The green
triangle
is the
calculated concentration of the unknown sample (which should
ideally
overlay exactly the yellow triangle representing the true sample
concentration). The entire graph responds
dynamically as you adjust the variables (e.g. with
the sliders).
The "Statistics" section at the lower left of
the screen shows
the
result of 20 simulated repeat measurements. The statistics are
re-calculated each time an input variable is changed or a slider
is
moved.
Start the
experiment with mo=2, blank=0, Ev and Es=0 and use the
sliders to set
z=0, n=0, Cx=5, and Cs=10. This represents an ideal case - a
perfect
world with no interferences, no random errors and a perfectly
linear
analytical curve. Under these conditions, the sample gives a
reading of Sx=10.000 units and the standard gives a reading of
Ss=20.000 units. So of course the calibration works perfectly
and the "result" equals the true Cx = Cs*Sx/Ss = 10*10/20 = 5.000.
The error, percent difference between the true and measured
concentrations,
is zero. Now use
the Cx slider to vary over its whole range and
you'll see that the results remain perfect for any value of
Cx.
Now use the Cs slider to vary the concentration of the standard
over
its whole range and you'll see that it also have no effect, as
long as
it is not zero. Even "mo",
the slope of the analytical curve, also has no effect as long as
it is
not zero, because it effect the signals of samples and standards
equally.
Now let's make the
simulation a little more realistic by
introducing some random variability. There are two variables
here, the random volumetric error Ev, and the random signal error
Es.
The random volumetric error refers to the random error in
measuring
volume or weights when preparing the sample and standard
solutions. The
magnitude of these errors depends on the technique used and on the
solution volumes involved. Then using accurate quantitative
glassware (volumetric flasks and pipettes) for volumes in the 10
mL - 1
L range, a volumetric precision of 0.1% is achievable, but a very
small
volumes below 1 mL a volumetric precision of 1%
is more likely. The signal measurement error refers to the
reproducibility of the signal output of the instrument, that is,
the
signal-to-noise ratio. This varies greatly with the analytical
instrument type and the concentration level of the analyte. It may
be
as good as 0.1% under optimum conditions, but is more likely to be
in
the 1-10% range, especially at lower concentrations. The signal
measurement error, like the random volumetric error, applies to
both
the sample and the standard. Both errors are expressed in
terms
of the relative standard deviation (ratio of the standard
deviation to
the mean).
For
starters, set Ev = 1% and Es = 1%. Now you'll see that the
instrument readings Ss and Sx are no longer exactly 20.000 and 10.000
as before; they are a little off because of the effect
of random
signal measurement error. If you click on the numbers for Ss
and Sx, and look at the entry bar at the top, you'll see the
equations for this numbers. Note that they both involve Es
because the same instrument is used to measure both Ss and
Sx. This causes the calculated sample concentration "result"
to be a little off as well. In fact, if you press the f9 function
key at the top of your keyboard, it will cause the spreadsheet to
recalculate with different random errors. You can see the Ss
and Sx and result jumping around slightly each time you press f9. (Also, the little triangles
on the graph move ever so slightly).
But actually the spreadsheet does this automatically, in the Statistics
table.
The Statistics
table, in the bottom left, computes the mean, the standard
deviation, and the percent relative standard deviation (% RSD) of
20
repeat measurements (including both signal measurement and
volumetric
error). Notice that the predicted % RSD of result (in cell C68) is actually larger
that the 1% RSD that you set for the random signal measurement
error Es.
Why? That's because Cs, Ss and Sx are subject to random errors: Cs is subject to random error
Ev and Ss and Sx are subject to random error Es. But the errors do not simply add up
linearly.
Theoretically,
according to the rules for
mathematical error propagation, the % RSD of Cx is predicted
to be =SQRT((Es)^2+
(Es)^2+(Ev)^2)/100,
if the errors are independent and uncorrelated. If Es = Ev = 1.00,
as
in this illustration, this works out to about 1.7%. This is
reported
as the "Est. RSD". But this
is the predicted standard
deviation "on
average", for a very large number
of repeats. Cell C72 gives the actual % RSD of 20 simulated repeat
experiments, which should turn out to be somewhere around the Est.
RSD, but not exactly because 20 repeats is not really a sufficiently "large"
number from a statistical point of view. But
from an analytical laboratory of view, doing 20 repeats of an
analytical calibration is a lot of work, time, and expense.
Sometimes
you can only afford to do 3 or 4 repeats, in which case you'll get
an experimental RSD even more approximate, possibly differing from
the
predicted by a factor of 2 or 3.
This is a basic problem of statistics in analytical
chemistry; the theoretical predictions work well for very large
number
of repeats, but in analytical chemistry the cost and time of doing
even
a few repeats is often prohibitive).
Now let's introduce a
larger random error. Set Es = Ev = 5% and look at
the % RSD of the result. It's
much larger than
before - theoretically
8.7% - because of the extra effect of Ev. But again the actual
results
bounce around quite a bit as you press f9, in this case mostly between 7 and 11%. What
if you use
accurate volumetric glassware (which can have an RSD of 0.1%) and
a
highly precise measurement technique that also gives a 0.1% RSD
signal
measurement precision)? Set Es = Ev = 0.1% and see what you get for the %
RSD of result.
Now let's make the
simulation even more realistic by
introducing interferences. Use the "Interference factor" slider to set z
to about 0.5, which causes the analytical signal in the samples to
be
substantially stronger than that in the standards (this is type of
multiplicative interference). Note that this causes the
calculated Cx to increase about 25% (as indicated by the Accuracy). Clearly, the single standard
method can not compensate for this type of interference. Note:
interferences are systematic errors that effect the
accuracy but not the precision (% RSD).
Return z
to zero and set "blank" = 1. This simulates an additive
interference,
such as a spectral overlap or background interference. Note that
this causes the calculated Cx to be too high (Accuracy is about
10%).
Clearly, the single standard method can not compensate for
this
type of interference either
Now test the
effect of analytical curve linearity. Return "blank" to zero.
Drag the
"Analytical
curve non-linearity"
slider
to the right and watch the analytical curve
(blue line) in the graph. The curve becomes concave down and the
accuracy degrades as the curvature increases, as indicated by the
fact
that the green triangle on the graph (representing the calculated
concentration of the unknown sample) is no longer on top of the
yellow
triangle
(representing the true concentration). Clearly, the
single standard method
depends on having a linear analytical curve. But the problem is
that, in the real world, you
wouldn't even have a clue that the analytical curve is non-linear
if
you used only one standard. For that, you'd need to measure more
than a single standard.
The
Two-Standard
Bracket Method
OpenBracketOO.ods
(view Screen
Shot). In this method the sample is
measured along with twostandard solutions that are (ideally)
close in concentration to the sample
(typically one lower than and one higher than the expected sample
concentration). The concentration of the sample Cx is calculated
by
linear
interpolation
between the two standards (cell C65). The
bottom two sliders allow you to adjust the two standard
concentrations.
The closer the the two standards are together, the smaller the
error
due to
analytical curve non-linearity. Of course, this assumes that
know
the sample concentrations beforehand, at least approximately, in
order
to be able to make up appropriate bracketing standards. (For
this
reason, the two-standard
bracket
method is mostly used when the approximate range of unknown
concentrations is narrow and fairly well known, as in quality
control
applications. It is not well suited when there are a large
number
of samples of widely and unpredictable varying concentrations). Start the
experiment with mo=2, blank=0, Ev and Es=0 and use the
sliders to set
z=0, n=0, Cx=5, and C1s=4,3 and C2s=5.7. In this case (linear
calibration curve, zero noise) everything works perfectly.
Now
slide the non-linearity slider up gradually and watch the shape of
the analytic curve (blue) change.
Note that the error
(cell 66) stays fairly low, even as the calibration curve becomes
noticeably non-linear. Even when the
non-linearity slider is all the way up (n=0.1), the error is less
than
1%, compared to a 5% error for the single-standard method with
Cs=5.7.
So the the
two-standard
bracket method is effective in reducing, but not completely
eliminating, the non-linearity error. Try moving the
sample
concentration slider Cx just outside
the range of the two standards; it still works pretty well as long
you
don't get too far off. Cx does not actually have to fall between the two standards,
just close to them.
Now set Ev and Es=1. Note that the predicted RSD
(based on error-propagation calculations) is greater than the
measured
RSD in the
statistics section. This is caused by the correlation
between the terms in the expression for sample concentration;
simple
error propagation math won't work well in this case. Comparing the measured RSD of
this method with that of the previous (single standard) method, you can see that the
two-standard
bracket method is very slightly less precise, because of the
random
error in
preparing and measuring two standards rather than one, but this
hardly
matters if method suffers from a significant calibration curve
non-linearity that the bracket method can compensate for.
The Calibration Curve Method
with Linear Curve Fit
OpenCalCurveOO.ods
(view Screen Shot).
This simulates a calibration curve with 2 to 18 standard solutions and a
linear least-squares fit. This is probably the most
common calibration method in general use. It is laid out just like
the previous simulations, with a few additions. You can choose the
number of standards (ns)
by typing into cell C57 or by clicking on the arrowheads of the
"spin
button". Cs (controlled by the bottom slider) now controls the
concentration of the highest standard solution. The
concentrations of
the other
standards are spaced out evenly between 0 and Cs.
The slope and intercept
of the linear least-squares fit to the calibration curve (the red
line
on the graph) is shown in the computed results section, and the
equation of the fit and the R^{2}
value (the "Coefficient of Determination", sometimes called the
"Correlation Coefficient") is shown in the upper left of the
graph. The
R^{2}
value is one way to estimate the "goodness-of-fit" of the
least-squares
line to the data; it is 1.000 when the fit is perfect and less
than
1.000 when the fit is imperfect.
To start with, set mo=2,
blank=0, Ev and Es=0, and use the sliders to set
z=0, n=0, Cx=5, and Cs=10. Now move the linearity slider
(variable "n") to the right to
introduce non-linearity. As the analytic curve becomes more
curved, you can clearly see that the linear least-squares fit no
longer
describes the curve well. Also, you'll see the R^{2}, which is 1.000 for a perfect
straight-line, begin to drop gradually, but R^{2 }still
reads 0.99 when the curve is already severely non-linear and the
error has already begun to degrade seriously (see error in the
Statistics table) to about 7% error. Even an R^{2 }value
of 0.999 results in an error of 2%. Maybe 2% sounds
pretty good, and in some applications that may be adequate, but
sometimes analytical methods are called upon to make measurements
as
accurate as 0.1% or even better. So this tells us thatR^{2 }must be
expressed to several (3 or 4) decimal places for analytical calibration
purposes.
Test
this simulation also for interference (variables "z" and "blank");
you'll see that it is no better than the single standard method
that
that respect.
Set
Ev and Es=1 to introduce a small random error. You'll see
some small
random scatter in the calibration points, with some slightly above
and some slightly below the "best fit" line in red, and the R2 value
will dropslightly below 1.0. Also the
measured
Cx ("result") will no longer be exact. In the Statistics section, the entire
calibration curve and measurement procedure is repeated 20 times
(not
just 20 repeat readings of the sample). With the conditions
set
the same as before (mo=2, blank=0, Ev and Es=1,
z=0, n=0, Cx=5, and Cs=10, and ns=2)
you'll notice that the %RSD in the statistics section is slightly
higher than Es and Ev (around 1.4%). The increase is caused by the
variability of the
calibration curve.
There is really no way to prepare a
perfect calibration curve without random error. But is is possible
to
reduce the reduce the variability of the computed slope and
intercept of the calibration curve by using more standards,
thereby "averaging
out" some of the random variability. Try setting ns to 2 and
then to 18.
Note that the measured Cx ("result") is more more accurate and
that the %RSD is also lower (about 1) with the higher number of
standards. This
is what you get in return
for all that extra work of preparing and running a larger number
of
standards. Whether it's worth it or not depends on the
situation.
Compared
to the single-standard method,
the calibration curve method give a slightly lower %RSD as long as
the
number of standards is greater than 2, because a calibration based
on
several standards is better than one based on a single standard.
However, the difference is not as much as you might think,
because the
reading of the unknown signal Sx has the same uncertainty as in
the
single standard method, and that uncertainty is not decreased by using more standards.
How can we predict
how much random error we can expect in the result (Cx), without
performing a series of experiments or
creating a simulation?The
standard way to do this is to perform a propagation of error calculation on
the least-squares slope and intercept and on the equation that
calculates the sample concentration:
namely Cx = (Sx-intercept)/slope.
This is done in the table D110:F119, and the result of this
calculation is shown as "Pred RSD" in
cell C74. The prediction is
based only on a single calibration
curve and is good only insofar as that calibration curve is
typical of
others that might be obtained in repeated trials.
If your random errors happen to be small when you run your calibration curve,
you'll get a deceptively good-looking
calibration curve, but then your estimates of the random error in
the slope and intercept will be too
low. If your random errors happen to be large, you'll get a
deceptively bad-looking
calibration curve, but then your estimates of the random error in
the slope and intercept will be too high. (Here are two examples taken from a
set of 20 repeats, one "good"
and one "bad", that illustrate
this point). Some days it just does not pay to be lucky.
One way to help this situation is to use more standards. Try
varying the
number of standards, ns; you will also discover that,
if the number of standards
is
very small, the agreement between the "Predicted % RSD" and
the % RSD of 20 repeat calibrations is very poor. As the number of standards increases, then
agreement improves and the actual error decreases. What's the minimum
number of standards needed? There is no hard and fast answer to
that question; it all depends on the quality of the data and the
required quality of results.
These simulated
experiments demonstrate
two things: first, the predicted
RSD (because it is based on a single
calibration curve) is extremely
unreliable when the number of standards is small, and second, the
%RSD of the result improves slightly when the number
of standards is increased greatly. We rightly expect that the
precision
of measurement of concentration should improve if more standards
are
used, but not so much as you might expect. Looking at the
expression
for the sample
concentration, Cx = (Sx-intercept)/slope, the precision of the slope and
intercept are inversely proportional to the square
root of number
of standards, ns,
but
the precision of Sx does not depend on the number of standards.
For
example, if we go from using 4 standards to using 16 standards (4
times
as many), the RSD the slope and intercept does decrease by half (the square
root of 4), but the RSD
of calculated concentrations decreases only from 1.5% to 1.2%. So,
using
a larger number of standards has some benefits, but it may or may
not
be "worth it" considering the time
and expense of preparing and running more standards.
The Calibration Curve Method
with Non-Linear Curve Fit
OpenCalCurveQuadFitOO.ods
(view Screen Shot). This simulates a
calibration curve with 10 standards solutions
and a quadratic least-squares fit. Set
the usual starting conditions: mo=2, blank=0, Ev and Es=0 and use
the
sliders to set
z=0, n=0, Cx=5, and Cs=10. Obviously in this perfect linear
case
the results are essentially perfect (zero standard deviation,
almost
perfect accuracy, and R^{2} = 1.000).
Now
increase the calibration curve non-linearity with the n slider,
about
half-way up (about n=0.05) and compare the error (in cell C66)
with
the error of the linear
method (in adjacent cell B67). You will find that this
method is effective at fitting
moderate degrees of non-linearity, and (unlike the bracket method) it does so over the entire
range
of concentrations (test this by varying the Cx slider). It fails,
however,
if the analytical curve is too
non-linear, especially if it goes to a
flat plateau or doubles back on
itself. Try increasing n all the way up to 0.1 and note the
error is not so low.
Another problem with non-linear fits occurs when there is lots of
random error (noise) in the data. Return n to 0.05 (half-way up) and set Ev and Es=1. With a
modest amount of random noise such as this, the quadratic fit
works pretty well. Compare the error (in cell C66) with the error of the
linear
method (in adjacent cell B67). In
this case, the non-linearity is the dominant source of inaccuracy.
Note that the relative standard deviation of 20 repeat
calibrations (cell C72) is about 2%, a little higher than a linear
calibration
curve with 10 standards (about 1.5%), but that's hardly a
deal-breaker if the error due to
non-linearity is greater than that due to random noise.
But now set Ev
and Es=5. Press the f9 key a few times to simulate different
calibration curves. Now the plot shows a good bit of discrepancy
between the actual analytical curve (blue) and the quadratic fit
to the
data points (red). The curve fit does its best to fit the
data
points, even if it has to weave a wavy line through and between
the
points. With more random error, you can get some truly strange
fits in some
cases.
The bottom line is that, if you know
from previous experience that the true calibration curve is
linear,
then a linear fit will be better than a non-linear fit, especially
if
the data are very noisy, because a
non-linear fit will try to "fit the noise", occasionally leading
to very great errors. If the calibration
curve is clearly non-linear, and the potential errors due to
linear
curve-fitting are greater than the random errors due to noise,
then a
non-linear fit is a good choice. On the other hand, a linear fit
may be best with
really noisy data, even if
the calibration curve is slightly non-linear, because the error
caused by a non-linear fit trying to "fit the noise" may exceed the error casued by a linear fits
inability to fit
the curve.
Reversed-axis
fits (Optional):
The
application of curve fitting to analytical calibration requires
that the fitting equation be solved for concentration as a
function of
signal in order to be applied to the measurements of unknown
samples. This is trivial in the case of a linear fit, and
not so
hard for a quadratic fit (requiring the use of the well-known
"quadratic equation" found in any algebra textbook), but it
becomes
more difficult for higher-order polynomial fits. One technique
that is
sometimes used in these cases is to reverse the x and y axes, that
is,
to plot concentration on the y axis as a function of signal on the
x
axis. This is not really justified statistically, but is is
nevertheless sometimes done in practice because it avoids the need
to
solve the fitting
equation. For example, consider the quadratic fit: in a
normal
quadratic fit (plotting concentration on the x axis and signal on
the y
axis as usual), the concentration of unknowns is calculated using
the
quadratic equation as Cx = (-b+SQRT(b^2-4*a*(c-Sx)))/(2*a), where
Sx is
the signal given by the unknown sample solution, and a, b, and c
are
the coefficients from the quadratic least-square fit (quadratic,
linear, and intercept terms, respectively). If the axes are
reversed (plotting concentration on the y axis and signal on the x
axis), the
concentration of unknowns
is calculated by the simpler expression Cx = ax^2+Bx+(c-Sx).
The
practical difference between these two approaches is demonstrated
by
the spreadsheet NormalVsReversedQuadFit2.ods (Screen shot),
which applies both techniques to the same set of simulated
calibration
data. This spreadsheet shows that the normal method is in fact
slightly
better on average, although the difference is slight in most
cases, especially if
the random errors in signal reading (Es) and in concentration (Ev)
are
comparable. Clearly, the reversed-axis approach is really not
needed for the
quadratic case. It us usually reserved for cubic and higher-order
fits,
where the difficulty of solving the fitting equation is much
greater;
for example, CalCurveCubicFitOO.ods
(Screen shot) applies this technique to a
cubic (third-degree) calibration fit, and
it could be easily extended to even higher order polynomial fits,
even those for
which the solution of the fitting equation is mathematically
impossible. A related spreadsheet (ReversedQuadraticVsCubic.ods,
Screen shot)
compares the reversed quadratic and reversed cubic fits applied
to the
same calibration data, showing that there is nothing significant
to be
gained by going to a cubic fit, at least for the type of
non-linearity
simulated here. On the other hand, the cubic fit can be
useful in
some practical cases where the non-linearity of the analytical
curve is
not well matched by a quadratic fit, for example in absorption
spectroscopy (see BeersLawCurveFit.html).
Technical note: All of these these non-linear
curve fitting spreadsheets use the LINEST function (common
to Excel
and OpenOffice
Calc). For example, in cell B136 of CalCurveCubicFitOO.ods,
the syntax is LINEST(E117:E126;B117:D126;0;0),
where E117:E126 are the 10 concentrations of the
standards, D117:D126 are the measured
absorbances, C117:C126 are the absorbances
squared, and B117:B126 are
the absorbances cubed. (Important detail: Because this is an
array function, rather
than a normal function, when you enter this function into the cell
you have to press Ctrl-Shift-Enter
rather than just Enter).
The function returns the first-order
coefficient (equivalent to the slope) in cell B136 (the variable
named
"qa" in the spreadsheet), the second-order coefficient in cell
C137
(the variable "qb"), and the third-order coefficient in cell D137
(the variable "qc"). The constant term is zero. These
coefficients are then used to compute the concentrations C of
unknown
samples from their measured absorbance A: C = qa*A+qb*A^{2}+qc*A^{3}.
In the Statistics section, this entire cubic calibration
procedure is repeated 20 times, in the 20 bordered blocks of cells
that
extend to the right between rows 115 and 140 out to column DP, and
the
results for each repeat are collected in the Results table in
column
J.
The Single Standard Addition
Method
OpenSingleStandardOO.ods
(view Screen
Shot).
In this
method, the sample is divided into two portions: one is measured
unmodified and the other is "doped" with the addition of a small
amount
of pure standard and then measured. This is similar to the single
standard method, in that only the sample and a single standard are
measured, but the difference is that in this case the standard
solution
is in the same matrix as the sample,
so it is effected by the same multiplicative interference, no
matter
what the origin of that interference might be.
The downside of this method is
that
each separate sample requires the preparation of its own standard,
whereas in the other methods one standard (or one set of
standards) can
be used to analyze a whole series of different samples. Also, the
calculations must compensate for the fact that the concentration
of the
standard solution now contains an unknown contribution from the
unknown
sample, but this is easily taken care of by a little algebra. The
result is only that the equation used to calculate the unknown
concentration is little more complicated, Cx =
Sx*Vxx*Cs)/(Ss*(Vxx+Vss)-Sx*Vxx), than the equation for the single
standard method, Cx = Cs(Sx/Ss).
To test this method, set
mo=2, blank=0, Ev and Es=0 and use the sliders to set
z=0, n=0, Cx=5, and Cs=10 as before. Now move the interference
slider (variable "z") to the right to
introduce an increasingly severe multiplicative interference. You can
see the analytical curve changes slope as you do
this, but that
both the sample signal (yellow triangle) and the standard signal
(green
triangle) track this change, and so the calculated sample
concentration
(red triangle) remains accurate.
Now
try setting blank to 1 or 2, to test the affect of an additive
interference. Unfortunately, the standard addition method does
not correct for additive interferences, only for
multiplicative
interferences. (You have to rely on other methods to compensate
for additive interferences, such asmultiwavelength
methods, wavelength
modulation,
derivative
methods, peak fitting, high-resolution
spectroscopy, separation methods, etc). Also, a linear analytical
curve is a requirement.
Set
Ev and Es=1 to introduce a small random error. The
predicted standard
deviation of Cx (Cell C70) is computed by breaking down the
equation for Cx into a
series of
differences, sums, products, and ratios, and applying the rules
for error propagation
to each step. These error propagation calculations are performed
in cells D111:F118. Comparing
the measured RSD of this method with that of the single standard method,
you can see that the single standard addition method is less
precise by
about a factor of 2, which might seem surprising considering that
both
methods measure the unknown sample along with a single standard
solution. You can understand what is going on here by looking at
the
expressions for Cx for the two methods: for the single standard
method,
it is Cx=Cs*Sx/Ss; for the standard addition method, it is Cx
=
Sx*Vss*Cs)/(Ss*(Vxx+Vss)-Sx*Vxx). The extra volume terms Vss
and
Vxx, both of which are subject to random volumetric errors, do not
occur in the single standard in the single standard method.
Moreover,
the denominator is the difference between two noisy quantities, Ss*(Vxx+Vss) and Sx*Vxx, which increases the
relative standard deviation of the difference. The result is that the
precision of standard addition is noticeably poorer than the
single standard method, but this the price for correcting for multiplicative
interference.
The Multiple Standard Addition Method
The
standard addition method can also be used with multiple standards:
(StandardAdditionOO.ods , view Screen
Shot). In this
method a
series of aliquots of the sample solution are taken, increasing
amounts
of standard material are added to each one, and the signals from
the
resulting mixtures are measured and plotted against the
concentration
of added standard. If the resulting calibration curve is
sufficiently
linear, a
first-order least-squares fit of the data is computed. The sample
concentration is given by the negative of the x-axis intercept
(and to
the ratio of the y-intercept to the slope). The advantage over
the single addition method is that you can verify the linearity of
the
calibration curve.
To test
this method, keep the same conditions as before (mo=2,
blank=0, Ev and Es=0 and use the sliders to set
z=0, n=0, Cx=5, and Cs=10) and set the number of standards ("ns")
to 4.
You can see that the calibration curve is linear and that the x-axis intercept is exactly -5 (which agrees with the
negative of Cx). Now move the interference slider (variable "z")
to the right to
introduce an increasingly severe multiplicative interference. You
can
see the analytical curve changes slope as you do this, but that
the x-axis
intercept remains unchanged, proving that this method corrects
perfectly for multiplicative interferences (slope changes).
If
you move the Cx slider to change the analyte concentration, the
whole
curve slides up and down, so that the x-axis intercept tracks the
changes in Cx.
Now
introduce some random error: set Ev
and Es = 1%. The calibration curve still looks pretty good,
but
as you change the interference slider ("z") or press f9 to
recalculate,
the x-axis
intercept changes slightly, as does the "result" in cell C65. The
predicted relative standard deviation of Cx (cell C68) is
the quadratic sum (square root of the sum of the squares) of the
relative standard deviations of the slope and intercept computed
by the
curve fitting procedure. These error propagation calculations are
performed in cells B82:F87. Note that the %RSD of 20 repeats (cell
C72)
is about 2.6%, significantly
greater than Ev
or Es, and is
only roughly predicted by the Est. RSD (cell C68). However, if you
increase the
number of standards ("ns")
to 16, the
%RSD of 20 repeats is about half that with ns=4 and is much better
predicted by the Est. RSD (both about 1.3%). As you saw
before, in the linear calibration curve method, the predicted
RSD (because it is based on a single
calibration curve) is extremely
unreliable when the number of standards is small, and secondly, the
%RSD of the result improves slightly when the number
of standards is increased greatly.
If you compare the
precision of this method to that of the linear calibration curve
method, you'll notice that the multiple
standard method is poorer, even though their expressions for Cx
are
very similar: Cx = (Sx-intercept)/slope v. Cx = intercept/slope.
Here again, correlation between terms is significant: there
is
sufficient negative correlation
between the intercept and the slope in the multiple standard method (the
intercept goes down when the slope goes up and vice versa) that the relative standard deviation
(RSD) of the ratio of the two is
poorer than the square root of the sum of the squares of the relative standard deviations
of the two terms individually (as would be the case if they were
not correlated).
The
Bottom Line
The take-home lesson here is
two-fold:
1.
Each calibration method has its own advantages and
disadvantages,
compared in Table 1 below; there is not one method that is best
in all
aspects and none that compensate or emiminate all possible
errors
(none, for example, eliminate additive interferences). As
expected, the
simplest methods do the least; the more complex methods do more,
but at
a cost.
2. The random error (relative standard deviation) of the
measured concentrations (Table 2) is typically poorer (greater)
than
that of the volumetric precision or the signal precision alone,
depending on the calibration method, but is usually no more than
twice
(except for the single-addition method).
Table 3: Effect of number of standards for Linear
Calibration Curve
(Same conditions as above)
Number of standards
RSD of
slope
SD of
intercept
Predicted RSD
Measured
RSD
4
1.4
0.1
2
2
16
0.7
0.05
1.7
1.7
Student assignment for Standard Addition Method, WingZ version:
Wingz player
application and basic set of simulation modules,
for windows PCs or Macintosh
The Single Standard
Addition Method (Old 1992 version)
Our textbook, Ingle and
Crouch, Chapter 6, page 179, says "The
standard addition procedure is a powerful technique that is often
used
improperly due to a failure to understand the assumptions
involved."
This simulation will help you appreciate the capabilities
and
limitations of the standard addition procedure.
1. Open StandardAddition.wkz. This model
is
based on the text, page 178-179 and Equation 6-16. The same
terminology
is used, with the following modifications: Ss is used for the
signal
measured after standard addition instead of Sx+s. Cx means the
true
analyte concentration (the unknown in the simulated experiment);
the
experimental quantity calculated by equation Equation 6-16, which
is
supposed to be a measure of Cx, is called "result". The volumes Vx
and Vs
mean the actual volumes (including the random volumetric errors);
nomVx and nomVs are the "nominal" volumes, that
is, the labeled volumes of the pipettes and flasks.
2. The
simulation includes the effect of a multiplicative interference
(Io =
interferent concentration) and additive interference, i.e. blank
error
(blank = uncorrected blank signal), and random errors in volume
and
signal measurement. Errors are assumed to be a fixed percentage of
the
quantity measured (fixed relative error rather than fixed absolute
error). The analytical curve is assumed to be linear.
3. The following are the
independent variable that you can change:
mo
Analytical curve slope without interference
z
Interference factor (zero => no interference)
Io
Interferent concentration in original sample
Ev
Random volumetric error (% RSD )
Es
Signal measurement error (% RSD)
Cx
Analyte concentration in original sample solution
Cs
Analyte concentration of standard solution
blank
(Uncorrected) blank signal
nomVx
Nominal volume of sample solution before addition
nomVs
Nominal volume of standard added to sample
To
change any of these, click on the number (not on the symbol) in
the
spreadsheet, type a new value, and press the enter key. The other
quantities in the spreadsheet are dependent variables that are
calculated from these independent variables. The most important of
these is result, which is the experimental estimate of Cx
calculated by
equation Equation 6-16. In this simulation we will compare result
to
the correct value Cx to see how well Equation 6-16 works.
4.
Choose any value of Cx and nomVx you like, then set Cs = ten-fold
or so
larger than Cx. Start with the ideal case of no interference
(Io =
0; blank = 0) and no random errors (Ev = 0 and Es = 0). Verify
that
result = Cx for arbitrary Cs, nomVx, and nomVs.
5. Introduce a
multiplicative interference by making Io > 0 and z > 0,
keeping
blank = 0. (The recovery expresses by what percent the analytical
signal is changed by the interference). Does result = Cx? Try
arbitrary
values of Io, z, Cx, Cs, nomVx, and nomVs and notice the effect on
result.
6. Introduce an additive
interference by making blank
> 0. Compare result and Cx. What do you conclude about the
ability
of the standard addition method to compensate for additive and
multiplicative interferences?
7.
Introduce random errors into the volumetric measurement (Ev) and
the
signal measurement (Es). To start with make both 1% RSD (Ev
= Es =1). Set Io > 0 and z > 0, keeping blank = 0 to
simulate a
multiplicative interference only. Click on the 20 repeat runs
button to
simulate 20 separate standard addition measurements. (Quick repeat
does
the same thing, only faster). The table on the right shows the
result
of each measurement, and at the bottom computes the mean, standard
deviation (s), percent relative standard deviation, and the error
(%
difference between the mean and Cx). Why is it that if you perform
several successive 20-run simulations under fixed conditions, the
standard deviation is the exactly the same each time? How could
the
simulation be designed to make the standard deviation more
reproducible?
8.
Vary Cs and nomVs and observe the effect on the percent relative
standard deviation of the 20 repeats. Is there an optimum value of
Cs
and nomVs that minimizes this error? On the basis of your
observations,
formulate a rule that allows you to predict the optimum value of
Cs and
nomVs.
9. Why is it that, even
under the best condition, the % RSD of result is greater than Es
or Ev?
(c) T.C. O'Haver, 1992 (WingZ versions), 2009 (OpenOffice versions),
Prof. Tom O'Haver ,
Professor Emeritus, The University of Maryland at College Park.
Comments, suggestions and questions should be directed to Prof.
O'Haver
at toh@umd.edu.
Last updated May, 2013.
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