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, click here to download a fill-in-the-blank worksheet).
A quadratic fit of measured absorbance
A (y-axis) vs concentration C (x-axis). The model equation is A = aC2 + bC + 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, click here to download a fill-in-the-blank worksheet).
A reversed cubic fit
of concentration C (y-axis) vs measured absorbance A (x-axis). The model equation is C = aA3 + bA2 + cA + d. This method reverses the usual order of axes in order to avoid the need
to solve a cubic equation. (This coordinate transformation is
not strictly speaking proper, because the presumed errors are in the
absorbances and not in the concentrations. Nevertheless, 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 aA3+bA2+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, click here to download a fill-in-the-blank worksheet).
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". Newly added as of Sept 16, 2008: version 2,
which allows the user to select the quantity to plot vs concentration:
either absorbance (log(Io/I)) or absorption (1-I/Io). (This version
shows that it is better to compute absorbance 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
August 7 and later 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 http://www.wam.umd.edu/~toh/SimpleModels/BeersLawCurveFit.ods
in OpenOffice Calc (August 11, 2008 version or later). 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%.
