Spectrophotometric Errors:
Choosing the Best Concentration Range

©David L. Zellmer, Ph.D.
Department of Chemistry
California State University, Fresno

Every instrument has a useful range for a particular analyte.

As an analytical chemist, it is up to you to know what this range is. The instrument will cheerfully inform you that the concentration is "7.112 mg/L" on its digital display or in a computer-generated report, when the instrumental error is actually approaching 10%.

For spectrophotometric errors, consider the following model instrument:

The monochromator splits the white light of the source into it component wavelengths, and allows a particular band of wavelengths to pass through the sample. Light of power Po goes into the cell, and light of power P comes out. The light is converted into electrical current in the Detector, and transformed into Absorbance in the readout. Absorbance is related to concentration using Beer's Law, A=abC. The equations are:

A is the Absorbance, C is the concentration, a is the absorptivity and b is the cell path length. If C is in mg/L and b in cm, then a will have the units cm-1mg-1L. Our instrument measures P and Po with an uncertainty determined by the stability of the light source and the electronics.

As %T approaches 100% and A approaches zero, the uncertainties in measuring %T dominate the error in A. It can be shown that the absolute error in A due to %T error = Error in %T*0.434/%T. [For example, in a Spectronic 20 the instrumental noise is 0.5%T. At an Absorbance of 0.100 (%T=79.4%) the error is 0.5%T*0.434/79.4%T=0.003 A, a 3% error in A.] In simpler terms, at low concentrations the instrument can no longer "see" the color present with any accuracy.

As %T approaches zero and A approaches infinity, we run out of light to measure when the concentrations become too high. In addition to the random errors inherent in measuring very low light levels, Stray Light limits our ability to measure highly absorbing solutions. All spectrophotometric instruments allow some of the light to bypass the cell and add to the light reaching the detector. In a Spectronic 20, this amounts to 0.5%T. So even if our sample is so concentrated that it absorbs nearly all the light that is put into it, the detector will not produce a readout of 0.0 %T. The actual %T is:

This causes large errors at high Absorbance. A Spectronic 20D has a digital display that can reach 1.95 A, but values above 1.00 A will no longer follow the linear behavior of Beer's Law.

A spreadsheet analysis computes the actual errors:

Absorptivity

0.25
Path Length, b
1
Po
100
Stray Light %
0.5
Pstray
0.5
Error in %T
0.5

conc

Ideal A
Ideal %T
P
P+Pstray
%T observed w Stray Light
A observed w Stray Light
Combined %error in A
0.100
0.025
94.41
94.41
94.91
94.43
0.025
9.21
0.500
0.125
74.99
74.99
75.49
75.11
0.124
2.39
1.000
0.250
56.23
56.23
56.73
56.45
0.248
1.68
1.500
0.375
42.17
42.17
42.67
42.46
0.372
1.58
2.000
0.500
31.62
31.62
32.12
31.96
0.495
1.66
2.500
0.625
23.71
23.71
24.21
24.09
0.618
1.83
3.000
0.750
17.78
17.78
18.28
18.19
0.740
2.09
3.500
0.875
13.34
13.34
13.84
13.77
0.861
2.44
4.000
1.000
10.00
10.00
10.50
10.45
0.981
2.89
4.500
1.125
7.50
7.50
8.00
7.96
1.099
3.45
5.000
1.250
5.62
5.62
6.12
6.09
1.215
4.16
5.500
1.375
4.22
4.22
4.72
4.69
1.329
5.04
6.000
1.500
3.16
3.16
3.66
3.64
1.438
6.15
6.500
1.625
2.37
2.37
2.87
2.86
1.544
7.52
7.000
1.750
1.78
1.78
2.28
2.27
1.645
9.22

In the working curve at the left we see the departure from linearity above A=1.0. In the graph on the right we see that the "best" range in which to use a Spectronic 20 lies between A=0.1 and A=1.0, with a minimum around 0.4 Absorbance. Outside of this range the errors increase rapidly.

For questions or comments contact:

David L. Zellmer, Ph.D.
Department of Chemistry
California State University, Fresno
E-mail: david_zellmer@csufresno.edu

This page was last updated on 29 March 1997