Instrumental Deviation from Beer's Law

Instrumental Deviation from Beer's Law

Version 1.1: May, 2000. Click to see larger view.

[Operating instructions] [Cell definitions and equations] [Student assignment handout]

A numerical simulation of absorption spectroscopy. Students visualize graphically how the spectral distribution of the light source and the absorbers influence the transmitted light profile and the calculated absorbance, discover under what conditions Beer's law is obeyed, observe and measure the Beer's law deviation caused by polychromatic radiation and unabsorbed stray light. The plot shows the transmission (T=I/Io) of the absorbing sample vs wavelength in nm as a blue line, the spectral profile of the light (slit function) as the grey line, and the spectral profile of the light transmitted by the sample (and measured by the detector) as a red line. Version 1.1 incorporates some minor re-wording and layout changes.

Download links: Beer.WKZ.
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Operating Instructions

In this simulation (based on the equations in Ingle and Crouch, "Spectrochemical Analysis", Chapter 13) the absorption spectrum of a imaginary absorbing sample is represented by a single Gaussian absorption band. "Peak Wavelength" and "Absorption width" are the wavelength and half-width (in nm) of the absorption band. "Source width" is the width of the spectral profile of the light source passing through the sample, represented as a Gaussian band centered at 300 nm (in uv-vis. spectrophotometry this would be the spectral bandpass; in atomic absorption this would be the width of the hollow cathode lamp line); "Absorber conc" is the relative number of absorbers in the light beam (arbitrary units); "% stray light" is the unabsorbed stray light expressed as a percent of the total light intensity. To change any of these, click on the number, type a new value, and press the enter key. Calculated values are shown just above the graph. The "Width Ratio" is the ratio of the source width to the absorber width, "Peak Absorbance" is the true value of the sample absorbance that would be measured with an instrument of infinite resolution and zero stary light, "Measured I-zero" is the total intensity of the light source incident on the sample as viewed by the detector in the 200-400 nm region, "Measured I" is the total intensity of the light source transmitted by the sample as viewed by the detector, "Meas absorbance" is the absorbance as measured by the spectrophotometer (log of the ratio of measured I-zero to measured I). "Detector SNR" is the the signal-to-noise ratio in the detector noise limit (the typical limit in IR spectroscopy or in uv-visible spectroscopy with solid-state array detectors). "Photon SNR" is the the signal-to-noise ratio in the photon noise limit (the typical limit in uv-visible spectroscopy with photomultiplier detectors).

Clicking the "Analyt. Curve" button creates a table of measured absorbances for absorber concentrations from 4 to 80 units and puts it in columns J and K. Clicking the "Plot" button creates a plot of the analytical curve, computes the linear least-squares best-fit line and an ANOVA table, and display it on a new sheet.

Spreadsheet cell definitions and equations:

Inputs:

PeakWavelength, nm	= B4	
AbsorptionWidth,nm 	= C4
SourceWidth,nm		= D4
AbsorberConc (arbitrary units)	= E4
% stray light = StrayLight = F4


Array calculations:

A37..A139: wavelength = 200 - 400 nm in 2 nm steps
B37..B139: Absorbance	=PeakAbs*exp(-((wavelength-PeakWavelength)/AbsorptionWidth)^2)	
C37..C139: Transmission	=10^(-Absorbance)	
D37..D139: SourceIntensity	=exp(-((wavelength-300)/SourceWidth)^2)	
E37..E139: TransmittedIntensity = Transmission*SourceIntensity

Graph:
X-axis: wavelength = 200 - 400 nm 
Y-axis: Blue = Transmission
        Gray = SourceIntensity
        Red = TransmittedIntensity 

Display calculations:

Width Ratio = D4/C4	
Peak Absorbance	= PeakAbs = AbsorberConc/(1.7725*AbsorptionWidth)	
Meas. Absorbance = MeasAbs = log(MeasIo/MeasI)	
Measured I-zero	= MeasIo = sum(SourceIntensity)+StrayLight*sum(SourceIntensity)/100	
Measured I = MeasI = sum(TransmittedIntensity )+StrayLight*sum(SourceIntensity)/100	
Detector SNR = DeltaI = 100*MeasAbs*MeasI	
Photon SNR  =1000*MeasAbs*sqrt(MeasI)


"Analyt. Curve" button

Creates a table of measured absorbances
for absorber concentrations from 4 to 80 units and puts it
in columns J and K.

Button script:

column letters
define standard
for standard=2 to 40
    put standard*2 into E4
    put B7 into "J"&standard
    put C7 into "K"&standard
end for

"Plot" button

Creates a plot of the analytical curve, computes 
the linear least-squares best-fit line and an ANOVA table,
and display it on a new sheet.

Button script:

{ INSTRUCTIONS: Put data to be fit into two adjacent columns, the x-axis data   }
{ (i.e. independent variable) in the first column and the y-axis data (i.e.     }
{ (the dependent variable) in the second column.  The first row of each column  }
{ should contain column labels.  Select the data range and run this script.     }
{ The script creates a new sheet containing a copy of the data values, the best }
{ fit data, an ANOVA table, and an XY plot of the data and best fit lines.      }
{ Print page 2 of this new sheet to get report contain ANOVA table and plot.    }
repaint off
column numbers
Define datarange,numrows
select range R1C10..R20C11
Copy
unselect
New Worksheet ""
window size (12780, 7540)
paste values
Select last cell
numrows=row()
select range "R1C1..R"&numrows&"C1"
Copy
select range R1C3
Paste
datarange="$R1$C2..$R"&numrows&"$C3"
Select range datarange
select more range R1C5
select more range R1C6
column letters
Regress
column numbers
select range R1C4
Put "best fit line" Into R1C4
select range R2C4
Put "=R2C3*$R2$C5+$R1$C5" Into R2C4
select range "R2C4..R"&numrows&"C4"
Copy Down
select range "R1C1..R"&numrows&"C4"
Unselect Add Chart Range frac(R25C6..R51C11,0,62,149,252) Using "R1C1..R"&numrows&"C4"
select object 1
XY
Legend Bottom
select chart 1 series 1
symbol color 0
symbol width 20
symbol type 3
symbol size 7 points
line fg 0
line bg 16777215
line pattern 0
line width 40
select chart 1 series 2
line fg 0
line bg 16777215
line pattern 1
line width 20
select range R1C2
select more chart 1 axis 3
axis title range
select range R1C1
select more chart 1 axis 1
axis title range
column letters
column width range E1 to 2780
select range F8..J22
precision 4
select range A1
hide headings
hide tool box
hide entry bar
hide cell grid
repaint on
select more chart 1 axis 1
select range L50

Student assignment:

1. Nearly ideal case (source width << absorption width; no stray light). Set wavelength = 300 nm, abs width = 500 nm, source width = 5 nm, % stray light = 0. Vary absorber conc from 1, 10, 100, 1000, 2000, 4000. Note that the peak absorbance and measured absorbance are essentially identical, even at the highest concentrations, proving that Beer's law holds in this ideal case.

2. Effect of absorbance on signal to noise ratio. Using the same settings above, vary absorber conc until the photon SNR is maximum and record the absorbance at that point. Repeat for the detector SNR. Compare your results with Ingle and Crouch, chapter 13, page 370, figure 13-11a, line 1 for detector noise and line 3 for photon noise? (The book plots relative standard deviation, the reciprocal of SNR).

3. Unabsorbed stray light limit only. Leave abs width = 500, source width = 5, Set % stray light = 0.1 and vary absorber conc from 1000 to 10000. Notice that measured absorbance approaches a plateau as absorber concentration increases. With absorber conc = 10000, try % stray light = 0.01 and 0.001 and observe the measured absorbance. Devise a simple rule that will predict the plateau absorbance for a given stray light percent.

4. Realistic case for condensed-phase molecular absorption. Set abs width = 50, source width = 5, % stray light = 0.01. Set absorber conc to 9, 90, and 900 and record peak absorbance and measured absorbance. Note that the linearity is very good between 9 and 90 (measured absorbance increases by factor of 10), but not between 90 and 900. Why not?

5. Measuring higher concentrations on the side of the absorption band. Leave abs width = 50, source width = 5, % stray light = 0.01 and set wavelength to 240. This reduces the absorptivity (sensitivity) by about 4-fold. Now set absorber conc to 9, 90, and 900 as before and record measured absorbance. Now you get useful results (approximate linearity) between 90 and 900 (measured absorbance increases by approximately factor of 10), but the linearity between 9 and 90 is not so good as it was at 300 nm (in part 4). Why not?

6. Narrow absorption band. Set abs width = 10, source width = 5, % stray light = 0.01. This is similar to atomic absorption. Click on the auto button, sit back and watch while the computer runs a "calibration series" for you automatically and records the peak and measured absorbances in the table on the top right of the screen. When it finishes, click on the Plot button. This creates a new sheet, plots peak vs. measured absorbance, and does a least-squares fit. Print out page 2 of the new sheet (Select Print from the File menu, set print range to page 2 to 2). Close the new sheet but don't save it. Note that the plot is almost perfectly linear (correlation coefficient is close to 1.0) yet the slope is quite a bit less than unity. In other words, just because the the analytical curve seems to be linear does not mean that the measured absorbance equals the true peak absorbance.

7. Continuum-source AA. Repeat part 6 with abs width = 5, source width = 50. This is basically the situation in continuum-source AA with a medium-resolution spectrometer. Note the very poor linearity and low sensitivity (measured absorbance << peak absorbance).

8. Signal to noise optimization of source width. Set abs width = 10, absorber conc = 1, % stray light = 0, vary source width from 1 to 100 and look for optimum photon SNR. Do the results surprise you? Why does the SNR decrease at small source width, even though the absorbance is a maximum at small source widths? Hint: look at the measured I-zero as source width approaches zero.