The purpose of this simulation is to compare three methods of least-squares curve fitting
to absorption spectrophotometry calibration curves with varying
degrees of non-linearity due to polychromaticity and unabsorbed
stray light (see Background information,
below). The operation is similar to "Instrumental Deviation from Beer's Law";
it computes the measured absorbance and plots the analytical curve
(absorbance vs concentration) for a simulated series of absorber
concentrations measured in a simulated absorption spectrophotometer
with variable spectral bandpass and unabsorbed stray light. You
control the maximum absorptivity, path length, half-width of the
absorber, the slit width of the monochromator, and the percent stray
light. (Number wheels below each of these parameters allow you to
change the values quickly without typing. The analytical curves
change dynamically as the variables are changed). The calibration
curve is fit using three different least-squares methods, shown from
left to right:
A first-order (straight line)
fit of measured absorbance A (y-axis) vs concentration
C (x-axis). The model equation is A =slope
* C + intercept. The
is the most common and straightforward method, but it obviously
can not compensate for non-linearity. This fit is performed
using the SLOPE and INTERCEPT functions (on Sheet2 of the
spreadsheet, in cells G51, G52, and G53). The concentration
of unknown samples is given by
(A - intercept) / slope where A is the measured absorbance and slope and intercept from the
first-order fit. If you would like to use this method of
calibration for your own data, download a
fill-in-the-blanks worksheet, in Excel or OpenOffice Calc format.
A quadratic fit of
measured absorbance A (y-axis) vs concentration C (x-axis). The
model equation is A = a*C2 + b*C + c.This fit
is performed using the LINEST
function (on Sheet2 in cells E57-G57). The concentration of
unknown samples is calculated by solving this equation for C
using the classical "quadratic formula", namely C = (-b+SQRT(b2-4*a*(c-A)))/(2*a), in cells E58:N58, whereA = measured absorbance,
and a, b, and c are the three
coefficients from the quadratic fit. If you would like to
use this method of calibration for your own data, download
in Excel or
OpenOffice Calc
format.
A reversed cubic fit
of concentration C (y-axis) vs measured absorbance A
(x-axis). The model equation is C = a*A3 + b*A2 + c*A + d. This
method reverses the usual order of axes in order to avoid the
need to solve a
cubic equation when the calibration equation is solved for
C and used to convert the measured signals of the unknowns into
concentration. (This sort of short-cut is commonly done in
least-squares curve fitting, at least by non-statisticians,
to avoid mathematical messiness). This fit is performed
using the LINEST
function (on Sheet2 in cells E73-H73). The concentration of
unknown samples is calculated directly by a*A3+b*A2+c*A+d,
where A is the measured
absorbance, and a, b, c, and d are the four coefficients
from the cubic fit. If you would like to use this method of
calibration for your own data, download in Excel or OpenOffice Calc format.
Below each calibration curve is a plot of the concentration prediction error,
the percent difference between the actual concentration of each
standard in the simulation and the concentration predicted from its
measured absorbance according to the curve fit equation, expressed
as a percentage of the highest standard concentration. For
ease in comparison, the standard deviation of all the concentration
prediction errors is computed and displayed to the left as the "σ of
errors". Version 2 allows the user to select
the quantity to plot vs concentration: either absorbance (log(Io/I))
or absorption, showing that it is better to compute absorbance
rather than absorption, even if you use a cubic least-squares fit to
the calibration curve.
Assumptions of this simulation: The
true monochromatic absorbance follows the Beer-Lambert Law; the
absorber has a single Gaussian or Lorentzian absorption band
(selectable by user); the spectral width of the light source is much
greater than the monochromator spectral bandpass; the monochromator
has a triangular slit function (i.e. entrance and exit slits are
equal); the absorption path length and absorber concentration are
both uniform across the light beam; the spectral response of the
detector is much wider that the spectral bandpass of the
monochromator. The current versions include the optional addition of
photon and detector noise to both the incident and transmitted beam
intensities; it is assumed that both beams are subject to random and
uncorrelated noise.
In absorption
spectroscopy, the intensity I of light passing through an
absorbing sample is given by the Beer-Lambert Law:
I = Io*10-(alpha*L*c)
where “Io” is the intensity of the light incident on
the sample, “alpha” is
the absorption coefficient of the absorber, “L” is the distance
that the light travels through the material (the path length), and
“c” is the concentration of absorber in the sample. The variables
I, Io, and alpha
are all functions of wavelength; L and c are scalar. In
conventional applications, measured values of I and Io
are used to compute the absorbance,
defined as
A = log(Io/I)
Ideally, absorbance defined in this way is proportional to
concentration, which simplifies analytical calibration. A plot of A
vs C is called the analytical
curve or the calibration
curve.
It's important to understand that the "deviations" from the Beer-Lambert
Law discussed here are not actually failures of this law but
rather apparent deviations caused by failures of the measuring
instrument to adhere to the conditions under which the law is
derived. The fundamental requirement under which then
Beer-Lambert Law is derived is that every photon of light striking the detector must have an
equal chance of absorption. Thus, every photon must have
the same absorption coefficient alpha,
must pass through the same absorption path length, L, and must
experience the same absorber concentration, c. Anything that
violates these conditions will lead to an apparent deviation from
the law.
For example, any real spectrometer has a finite spectral resolution,
meaning that the intensity reading at one wavelength setting is
actually an average over a small spectral interval. Specifically,
what is actually measured is a convolution of
the true spectrum of the absorber and the instrument function (or
"slit function"). If the absorption coefficient alpha varies over that
interval, then the calculated absorbance will no longer be linearly
proportional to concentration. This is called the “polychromicity”
error and it results in a gradual concave-down curvature of the
analytical curve.
Another source of instrumental non-ideality is stray light, which is any light
striking the detector whose wavelength is outside the spectral
bandpass of the monochromator or which has not passed through the
sample. Since in most cases the wavelength setting of the
monochromator is the peak absorption wavelength of the analyte, it
therefore follows that any light outside this spectral range is less
absorbed. The most serious effect is caused by stray light that is
not absorbed by the analyte at all; this is called unabsorbed stray light. This
effect also leads to a concave-down curvature of the analytical
curve, but the effect is relatively minor at low absorbances and
increases quickly at high absorbances. Ultimately, unabsorbed stray
light results in a flat plateau in the analytical curve at an
absorbance of -log(fsl),
where fsl is the
fractional stray light.
There are two other potential sources of deviation that are actually
not so serious in laboratory applications of absorption
spectrophotometry, because they are relatively easily avoided: those
are unequal light path lengths and unequal absorber concentration
across the light beam. In most laboratory applications, the
samples are measured in square cuvettes to insure a constant path
length for all photons. (When round test-tube sample cells are
used, the light beam passing through the sample is restricted to the
central region of the sample tube in order to minimize this effect).
Solution samples are carefully mixed before measurement to
insure homogeneity.
The simulation here includes the polychromaticity and unabsorbed
stray light errors (but not the path length and sample inhomogeneity errors, because they are seldom
important in laboratory practice). The simulation operates like any
numerical integration, by slicing up the spectral range viewed by
the detector into a large number of small slices and assuming that
the Beer-Lambert Law applies over each small slice separately. The
sample absorption is represented in this simulation by a single
absorption band of either Gaussian
or Lorentzian
shape (selectable by the user) and adjustable width. The
spectral bandpass of the monochromator is represented by a
triangular function of adjustable width. Then all the separate
slices are summed up to represent the incident and transmitted light
signal measured by the detector. As it turns out, one does not
need to use very many slices to obtain a good model of the operation
of a typical absorption spectrophotometric measurement. Student handout. Note: Instructors
are encouraged to copy, paste, and edit this material as needed to
customize for their own terminology and instructional aims.
Comparison of
Calibration Curve Fitting Methods in Absorption Spectroscopy
Open either the Excel xls or the Calc ods version of
BeersLawCurveFit. This spreadsheetsimulates a visible absorption
spectroscopy measurement, including the instrumental deviations
from the Beer-Lambert Law (a.k.a. Beer's Law) caused by polychromatic light and unabsorbed stray light. This spreadsheet is similar to
Instrumental
Deviation from Beer's Law; the
controls are similar to that simulation; the main difference is
that, instead of just a linear least-squares fit to the
calibration curve, this simulation compares the linear fit with
two non-linear (so-called "curvelinear") fitting methods, the
"quadratic" and the "reversed cubic" fits.
In thequadratic
method, the
measured absorbance A (y-axis) vs concentration C (x-axis) is fit
to a quadratic model, A =aC2 + bC + c, using standard least-squares
techniques, yielding the three coefficientsa,
b, and c. Thereafter, the concentration of unknown
samples is calculated by solving this equation for C using the
classical "quadratic formula" using those same
coefficients, namely C = (-b2+SQRT(b-4*a*(c-A)))/(2*a), where A = measured absorbance, and a, b, and c are the three coefficients
from the quadratic fit.
In the "reversed cubic" fit, the measured absorbances
of the standard solutions are treated as the independent variable
and the concentrations of the standard solutions are treated as the dependent
variable. Then
the calibration data set is fitted to a cubic equation, C =aA3 + bA2 + cA + d, where C is concentration, A
is absorbance, and a, b, c, and d
are the coefficients from the least-squares fit. Thereafter, the concentrations
of unknown samples are calculated from their measured absorbances
by using those
same coefficients: C =aA3 + bA2 + cA + d. Clearly, both curvilinear
methods are more complicated computationally than a simple straight-line fit, but they can handle curvature in the
calibration curve. The cubic version is capable of handling more
complex curve shapes.
The purpose of this simulation is to discover which of these
methods is best able to fit the kind of non-linearity that results
from the polychromatic and stray light errors in absorption
spectroscopy. Moreover, the simulation can demonstrate how
the use of a curvilinear fitting methods can influence the optimum
choice of slit width and spectral bandpass.
1. Start the experiment with a nearly ideal case. Set wavelength = 500
nm, spectral bandpass = 10 nm, absorber width = 200
nm, maximum absorptivity = 4, path length = 1 cm, unabsorbed
stray light = 0; maximum concentration = 10, no photon or detector
noise (both boxes unchecked) and Gaussian peak shape. You
can see that the ideal absorbances (red line), the measured
absorbances (blue dots), and the least-squares fit (blue line) are
essentially identical, even at the highest concentrations (where
the absorbance is 4), showing that the instrument readings follow Beer's Law.
The concentration prediction error plots below each
calibration curve show almost no error due to curve-fitting and
the "σ of errors" is
zero in all cases. In this case the linear curve-fit methods
works perfectly, so you really don't need to use more complicated
methods.
2. Unabsorbed stray light limit only. Leave the settings as they
were, except increase the unabsorbed stray light to 0.1%. Now
you can see some serious non-linearity at high absorbances, caused
by the stray light, as the absorbance approaches the plateau at
3.0 absorbance. Right away, you can see that the straight-line fit fails badly,
and that the quadratic and cubic fits are a little better but not
perfect. The "σ of
errors" number at the bottom provides a quick single-number
comparison: the curvilinear methods do a little better than the
straight-line fit, but even the cubic fit does not do a very good job if fitting
the curvature in this case.
3. Effect of spectral bandpass.
Leave the
settings as they were in #2, except reduce the absorber width to 100 nm. Then
increase the spectral bandpass step-by-step and watch what
happens. You can see the slope of the analytical curve
decreases slightly as spectral bandpass increases. Why? This is caused by the polychromatic
light effect, as the increasingly wide spectral bandpass includes
wavelengths where the analyte's absorptivity is lower. But
the reduced slope also means that the absorbances at high
concentrations are lower, and therefore stray light effect is
less serious. You can also see that the shape of the calibration
curve changes as the spectral bandpass increases,
exhibiting a more gradual concave-down curvature. As the result of
this, the quadratic and cubic fits are better able to model the
curvature, and the σ
of errors actually decreases
as the spectral bandpass increases, even though the increasingly
serious polychromatic effect itself is expected to increase
non-linearity. At a spectral bandpass of 90 nm, the σ of errors drops to 1.1 for the quadratic fit and
only 0.08% for the cubic fit!
So the bottom line is that you should not automatically assume
that the smallest slit width will always give you the best
calibration linearity or the smallest calibration errors, at least
when stray light is a factor.
4. Effect of random noise.
But there is another reason why the best results in absorption
spectroscopy are not always obtained at the smallest slit width:
noise. This aspect is explored in more detail in the simulations "Effect of Slit Width on Signal-to-Noise
Ratio in Absorption Spectroscopy" and "Signal-to-noise
ratio of absorption spectrophotometry". The present simulation includes two of the most
commonly-encountered types of noise: photon noise and detector
noise. Photon noise (often the limiting noise in instruments that
use photomultiplier detectors) is proportional to the square root
of light intensity. Detector noise (often the limiting noise in
instruments that use solid-state array detectors such as CCD and
CID detectors) is independent of the light intensity. These
sources of noise can be introduced into the simulation by clicking
the check boxes under "Random Noise". When either of these types
of noise is included in the simulation, these noises will be added
to the incident and transmitted intensities (Izero and I, respectively) from
which the absorbances are computed, resulting in some random
variation in the measured absorbances within one calibration curve
and between recalculations of the spreadsheet (press f9 to
recalculate). The relative effect of the noise depends
strongly on the slit width and the spectral bandpass: at small
slit widths, the values of Izero and I are both low, resulting in
poor signal-to-noise ratio and a high random variation in measured
absorbance. As the slit width increases, the intensities
increase and the signal-to-noise ratio improves, resulting in less random variation in measured
absorbance, but also the slope of the analytical curve decreases,
which reduces the effect of stray light but increases the effect
of polychromatic light. The net effect is that there is an
optimum range of slit widths that is the best trade-off between
poor signal-to-noise
ratio at the low end and unacceptable non-linearity at the high
end. This optimum slit width usually corresponds to a
spectral bandpass roughly equal to the width of the absorption
peak, depending upon the type of noise and the curve fitting
method used.
The effect of noise can be dramatically demonstrated by setting wavelength = 500
nm, spectral bandpass = 10 nm, absorber width = 80
nm, maximum absorptivity = 4, path length = 1 cm, unabsorbed
stray light = 0.1%; maximum concentration = 10, and checking the
boxes for both photon and detector noise. The calibration
curve now shows a substantial degree of random noise in the
absorbances, especially at high absorbance where the transmitted
intensity (I) is therefore the signal-to-noise ratio is very low. The "σ of errors" number is high
for all three curve fitting methods. Now, increase the spectral
bandpass and watch what happens. The random noise gradually
decreases (because the intensities I and Izero are both increased)
and the concentrationpredictionerrors
begins to drop. Eventually as the spectral bandpass approaches
the absorber width, the calibration curve becomes smoothly
non-linear, but its shape is easily modeled by the cubic curve
fit. The result is that the concentrationpredictionerror
for the straight-line fit starts to go back up at high spectral bandpass, but for the cubic fit, it
continues to drop down to near 0.1%.
Conclusion: The most
common sources of deviation from Beer's Law, stray light and
polychromatic radiation, lead to concave-down non-linearity in the
calibration curves, which can be fit approximately (but not
exactly) using quadratic or cubic least-squares fits. In
most cases, the cubic fit is better that the quadratic. In
the case of an extreme unabsorbed stray light plateau, neither
method works well.
The theoretical requirement for adherence to the Beer-Lambert Law
is that the incident light be monochromatic, which implies that
the smallest possible slit width and spectral bandpass be used.
Nevertheless, this simulation shows that, in the presence of stray
light and random photon or detector noise, a larger slit width and spectral
bandpass will give better signal-to-noise ratio and concentration
prediction error, especially if a curvilinear least-squares method
(such as the reversed cubic method) is used to fit the calibration
data.
Note: The non-linearity observed here has its origin in the
spectral domain (intensity vs wavelength), not in the calibration domain
(absorbance vs concentration). Therefore it should be no surprise
that curve fitting in the calibration domain is not a perfect
solution. An alternative and more advanced approach to
quantitative measurement in absorption spectrophotometry is to
apply curve fitting to the spectral domain where the problem
originates; this is what the "Transmission
fitting" method is all about.
(c) 1991, 2015. This page is part of Interactive
Computer Models for Analytical Chemistry Instruction, created
and maintained by 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.
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