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Steven D. Gammon
Beer’s Law Measurements Using Non-monochromatic Light Sources—A Computer Simulation
University of Idaho Moscow, ID 83844
W
George C.-Y. Chan and WingTat Chan* Department of Chemistry, The University of Hong Kong, Pokfulam Road, Hong Kong; *
[email protected] Background Beer’s law quantitatively describes the absorption of radiant energy by absorbing species (1–10). The derivation of Beer’s law is given in many instrumental analysis (1, pp 33–39; 2, pp 123–135; 3), analytical chemistry (5; 7, p 515), and physical chemistry (8, 9) textbooks and in this Journal (11–19), using different approaches. Application of Beer’s law in chemical analysis is an essential part of the undergraduate chemistry curriculum. We cover this topic in a second-year (junior year in the U.S. system) instrumental analytical chemistry course. A well-known requirement of Beer’s law is that the light source must be monochromatic because absorptivity and absorbance change with wavelengths (1, pp 33–39; 2, pp 123–135). Most textbooks (1, pp 113–115; 3; 7, p 629) use diagrams of Gaussian absorption and light source profiles to illustrate this criterion. If the bandwidth of the light source is larger than that of the absorption peak, intensity of the transmitted light beam is only slightly diminished by the narrow absorption peak (Fig. 1) and the apparent absorbance is much smaller than the peak absorbance. This approach enables students to understand intuitively that the bandwidth of a light source must be smaller than the absorption profile and is especially useful to illustrate the logic of selecting element-specific lamps of narrow bandwidth (e.g., hollow cathode lamps and electrodeless discharge lamps) in atomic absorption spectrometry. However, it is not obvious from Figure 1 what the value of the measured (apparent) absorbance may be and how the resultant analytical calibration curve may look like if the bandwidth of the light source is larger. In addition, the upper limit of the bandwidth of the light source that yields a linear Beer’s law plot is difficult to deduce from Figure 1. For quantitative
analysis using Beer’s law, the effects of the bandwidths and spectral profiles on the sensitivity (i.e., slope) and linearity of the analytical calibration curve can be significant but are usually not discussed in textbooks. In this paper, we describe a simple program for numerical calculation of the apparent absorbance value for arbitrary combination of absorption and light source spectral profiles using Simpson’s rule (20). Students use the program as a problem-based learning tool to study the effects of bandwidth on apparent absorbance value and the slope and linearity of an analytical calibration curve. Simulations and Data Evaluation Any modern spreadsheet program is capable of the calculation described below. In this paper, a LabTalk script (Microcal Origin Version 5.0)1 is written to simulate the spectral profile of the transmitted radiation and the apparent absorbance for different combinations of bandwidth and peak profiles. Students can run the script in the spreadsheet program and appreciate the results without tedious work on the spreadsheet program itself. Visual evaluation of data requires a lower degree of mathematical sophistication than direct evaluation of the numerical data and setting up the model, and is sufficient for our purpose to demonstrate the effects of bandwidth and spectral profile on Beer’s law measurements. The written script provides a user interface to define the peak profiles and present the results graphically. The values of the peak wavelengths (λpeak) and bandwidth (full width at half maximum, FWHM) and the profiles of both the light source and the absorption peak can be varied independently. Four predefined profiles of the light source and the absorption peak are available for simulation: rectangular, triangular, Gaussian, and Lorentzian (Table 1). Table 1. The Spectral Profiles
Figure 1. Incident and transmitted spectral profiles of a Gaussian light source. Peak absorbance of the Gaussian absorption profile is 0.5. FWHM of light source and absorption profiles are 5 nm and 1 nm, respectively.
Name
Definition
Rectangular
Triangular
f λ = 1 for λpeak – 1 λFWHM ≤ λ ≤ λpeak + 1 λFWHM 2 2 f λ = 0 elsewhere λpeak – λ f λ =1– for λpeak –λFWHM ≤ λ ≤ λpeak λFWHM λ – λpeak f λ =1– for λpeak < λ ≤ λpeak +λFWHM λFWHM f λ = 0 elsewhere
Gaussian
f λ = exp
Lorentzian
f λ =
ln 16 × λ – λpeak λFWHM
2
2
λFWHM 2 4 λ – λpeak 2 + λFWHM 2
for ∞ < λ < ∞
for ∞ < λ < ∞
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In this simulation, the light-source profile is divided into 2000 equal intervals. The absorptivity value and the power of the transmitted light at each interval are calculated to obtain the transmitted spectral profile. The overall transmitted power is then obtained by summing the transmitted power of the 2000 intervals using Simpson’s rule (20). Finally, the apparent absorbance is calculated and the calibration curve is calculated and presented graphically. A detailed description of the model can be found in the supplemental material.W Computers with the software installed are set up in the chemistry library. Students run the script according to the procedures given in their assignment. To familiarize students with the program, their first task is to obtain the transmitted spectral profiles of a Gaussian light source for Gaussian absorption profiles of different bandwidths. They are asked to observe the shape of the transmitted spectral profiles and note the resultant absorbance. They are then asked to tabulate the appearance absorbance values for a wide range of bandwidth ratios between the light source and the absorbance profile and summarize the effect of bandwidth ratio on the shape and slope of the calibration curve. Finally, they compare the relative sensitivity and linearity of a calibration curve using different bandwidth ratios of light source to absorption profile.
Figure 2. Incident and transmitted spectral profiles of a Gaussian light source. Parameters are the same as in Figure 1, except that the FWHM of light source and absorption profiles are 0.5 nm and 1 nm, respectively.
Results and Discussion
Effects of Relative Bandwidths of Absorption Profile and Light Source Students are first asked to plot the incident and transmitted spectral profiles of Gaussian light sources and absorption profiles (Figs. 1 and 2). The peak wavelength of the absorption profile and the light source profile coincide exactly. The ratios of the bandwidths of the absorption peak to the light source are 1:5 and 1:0.5, respectively. (The ratio of the bandwidth of the absorption profile to the light source profile is referred to as “bandwidth ratio” in the remainder of this paper.) The profiles in Figures 1 and 2 show that the fraction of the attenuated light intensity increases as the bandwidth ratio increases. However, the apparent absorbance value of 0.4420 for a bandwidth ratio of 2 (Fig. 2) is still smaller than the peak absorbance value of 0.5. The lower apparent absorbance value is due to smaller absorptivity away from the absorption peak maximum. An important conclusion is that the sensitivity of the absorbance measurement is reduced when a non-monochromatic light source is used. Apparent Absorbance Value versus Bandwidth Ratios Students are then asked to tabulate the apparent absorbance values for a wide range of bandwidth ratios (0.01 to 100) of Gaussian peaks and make a plot similar to Figure 3. Figure 3 plots the value of apparent absorbance versus bandwidth ratio for different absorption profiles. The light source is Gaussian and the peak absorbance value is 0.5 in all cases. Again, the peak wavelengths coincide exactly. A general trend observed is that the apparent absorbance approaches the peak absorbance when the bandwidth of the light source is smaller than the absorption profile (bandwidth ratio > 1). The apparent absorbance decreases quickly with decreasing bandwidth ratio once the bandwidth of absorption profile is equal to or smaller than the light source. From this plot, students will notice that the bandwidth of the light source must be smaller than that 1286
Figure 3 Apparent absorbance value versus bandwidth ratio for a Gaussian light source and four absorption profiles: rectangular, triangular, Gaussian and Lorentzian. Peak absorbance value is 0.5.
Figure 4. Calibration curves (apparent absorbance versus peak absorbance) for a Gaussian light source and a Gaussian absorption profile with bandwidth ratios of 2 and 0.5. The dotted line represents the ideal situation that peak absorbance equals to measured absorbance.
Journal of Chemical Education • Vol. 78 No. 9 September 2001 • JChemEd.chem.wisc.edu
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Figure 5. Linearity plot for Gaussian light source and absorption profile. Bandwidth ratio is 1.
Figure 6. Relative sensitivity versus bandwidth ratio for a Gaussian light source and different absorption profiles
of the absorption profile in order to obtain an apparent (measured) absorbance value that is close to the peak absorbance value. For example, to obtain an apparent absorbance value ≥ 95% of the peak absorbance value, a bandwidth ratio of 3 or above must be used if the absorption profiles are of Gaussian or Lorentzian shape.
Analytical Calibration Curve Using Different Bandwidth Ratios Students also explore the effects of bandwidth ratios on the linearity and slope of Beer’s law plots. Figure 4 shows the plot of apparent absorbance versus peak absorbance for Gaussian light sources and absorption peaks. (Peak absorbance is used because it is directly proportional to the concentration of absorbing species according to Beer’s law.) Bandwidth ratios of 2 and 0.5 are used (Fig. 4). Both curves show negative deviation, but the deviation is much more severe for the smaller bandwidth ratios. Relative sensitivity and linearity of a calibration curve can be used to compare the quality of the curves. Relative sensitivity, S, is defined as S=
A apparent A peak
S is equal to 1 if there is no deviation from Beer’s law. Also, S is independent of Apeak if the calibration curve is linear. A linearity plot (21) can be used to show the linear range of the Beer’s law plot. The plot (S versus Apeak) is a horizontal straight line for an ideal linear calibration curve. A linearity plot that deviates from the horizontal straight line indicates a nonlinear calibration curve. For example, for Gaussian spectral profiles and bandwidth ratio of 1, the calibration curve is linear up to ∼ 0.1 peak absorbance value (Fig. 5). In addition, the relative sensitivity is 0.7071 instead of 1. To compare the linearity of calibration curves for different bandwidth ratios, linear dynamic range (the region where the linearity plot shows a horizontal straight line) is used. The linearity plots for various combinations of bandwidth ratios and profiles have shapes similar to the curve in Figure 5; that is, a linear region at low Apeak values and negative deviation at high Apeak values. Therefore, comparing the upper linearity limit is sufficient to indicate the rank of linearity of different calibration curves. In the following discussions, a tolerance of ±5% is allowed for S; that is, the linear range is defined as
Figure 7. Upper linearity limit of the calibration curves versus bandwidth ratio for a Gaussian light source and different absorption profiles. Maximum peak absorbance is 5 in this simulation.
the range for which S is constant to within a tolerance of ±5% (21). The relative sensitivity within the linear region and the upper linearity limit for a Gaussian light source passing through different absorption profiles are plotted against bandwidth ratios in Figures 6 and 7, respectively. The Apeak values used in this simulation range from 0.0001 to 5 (i.e., the maximum theoretical upper linearity limit is 5 in Figure 7). This range of absorbance is sufficient to simulate practical situations. From Figures 6 and 7, it is noted that if the bandwidth of the light source is larger than that of the absorption profile, the linearity and sensitivity degrade significantly as the bandwidth ratio decreases. The bandwidth of the light source must be smaller than the bandwidth of the absorption profile (i.e., bandwidth ratio > 1) to obtain good sensitivity and large linear dynamic range. It is also drawn to the students’ attention that similar trends are displayed in Figures 6 and 7 for all four profiles, even though the natures of the four absorption profiles differ drastically, from idealistic rectangular and triangular shapes to more realistic Gaussian and Lorentzian shapes. Therefore, one may propose that similar effects of bandwidth should be observed for a real absorption profile. From Figures 6 and 7, one may conclude that a minimum bandwidth ratio 3 is required to obtain an analytical calibration curve with relative sensitivity > 0.9 and linearity up to a peak absorbance value of 2. Therefore, for good sensitivity and large linear dynamic range, an element-specific light source
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linearity up to peak absorbance value of 2), a large bandwidth ratio must be used. They come to appreciate the rationale of selection of light sources for different spectroscopic techniques (e.g., hollow cathode lamps for atomic absorption spectrometry and deuterium or tungsten lamps plus a monochromator for UV–vis spectrophotometry). Students had no difficulty running the program. One student even extended the virtual experiment to an assignment problem in the textbook to simulate the measurement of a high concentration of Na, which absorbs at 330.299 nm, using a Zn line at 330.294 nm (1, p 121 question 1). Supplemental Material A detailed description of the model and its theoretical background, and LabTalk program source code are available in this issue of JCE Online. W
Note Figure 8. Simulated calibration curve using a Zn line at 330.294 nm to measure Na absorption at 330.299 nm.
of narrow bandwidth (e.g., hollow cathode lamps) must be used in atomic absorption spectrometry, whereas a relatively broadband light source passing through a spectrometer is sufficient for molecular absorption spectrometry (e.g., UV–vis spectrophotometry). A typical bandwidth for an elementspecific light source is approximately 1 to 3 pm and the bandwidth for atomic absorption profile is 2 to 5 pm (2, p 211; 4, p 479; 22). Typical bandwidth of a monochromator is on the order of 0.1 nm (1, pp 113–115) and the absorption bandwidth for molecular transition is on the order of 10 nm (1, pp 60–66).
Peak Wavelength of Light Source at the Wing of an Absorption Profile Another exercise for the students is to determine the effects of wavelength mismatch on calibration. Figure 8 shows an analytical calibration curve with a Gaussian light source profile centered at 330.294 nm (FWHM = 0.002 nm) and a Gaussian absorption profile centered at 330.299 nm (FWHM = 0.004 nm). This is a simulation of using a Zn line at 330.294 nm to measure high concentration of Na that absorbs at 330.299 nm (1, p 121 question 1). Absorbance can be measured even if the center of the light source is on the wing of the absorption profile. The analytical calibration curve is surprisingly linear, as reflected by the correlation coefficient (r = .9992), although the sensitivity is low because of the small absorptivity at the wing. It is interesting that a linear analytical calibration curve can be obtained even when the spectral profile of the light source is located at the wings of the absorbance peak. Absorptivity changes slowly at the wing of a Gaussian absorption profile (i.e., the absorptivity is relatively constant at the wing). As a result, the calibration curve is linear. Summary It is reported that departures from Beer’s law are not significant if the bandwidth of the absorption peak is 10 times larger than the bandwidth of the light source (4, p 556). Such a statement can be “verified” easily using this simulation program. In addition, students better understand that to obtain a good analytical calibration curve (e.g., relative sensitivity >0.9 and 1288
1. A demonstration version of the program can be downloaded from the home page of Microcal Software (http://www. originlab.com) to run the script in this manuscript.
Literature Cited 1. Ewing, G. W. Instrumental Methods of Chemical Analysis, 5th int. ed.; McGraw-Hill: Singapore, 1985. 2. Skoog, D. A. Principles of Instrumental Analysis, 4th int. ed.; Saunders: Fort Worth, TX, 1992. 3. Willard, H. H.; Merritt, L. L. Jr.; Dean, J. A.; Settle, F. A. Jr. Instrumental Methods of Analysis, 7th ed.; Wadsworth: Belmont, CA, 1988; p 244. 4. Strobel, H. A.; Heineman, W. R. Chemical Instrumentation: A Systematic Approach, 3rd ed.; Wiley: New York, 1989. 5. Skoog, D. A.; West, D. M.; Holler, F. J. Fundamentals of Analytical Chemistry, 7th ed.; Saunders: Fort Worth, TX, 1996; pp 511–513. 6. Christian, G. D. Analytical Chemistry, 4th ed.; Wiley: New York, 1986; pp 371–373. 7. Harris, D. C. Quantitative Chemical Analysis, 5th ed.; Freeman: New York, 1999. 8. Moore, W. J. Physical Chemistry, 5th ed.; Longman: London, 1972; pp 750–751. 9. Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, 1994; p 546. 10. Metcalfe, E. Atomic Absorption and Emission Spectroscopy; Wiley: Chichester, UK, 1987; pp 61–65. 11. Daniels, R. S. J. Chem. Educ. 1999, 76, 138–141. 12. Berberan-Santos, M. N. J. Chem. Educ. 1990, 67, 757–759. 13. Pinkerton, R. C. J. Chem. Educ. 1964, 41, 366–367. 14. Liebhafsky, H. A.; Pfeiffer, H. G. J. Chem. Educ. 1953, 30, 450–452. 15. Goldstein, J. H.; Day, R. A. J. Chem. Educ. 1954, 31, 417–418. 16. Lohman, F. H. J. Chem. Educ. 1955, 32, 155. 17. Holleran, E. M. J. Chem. Educ. 1955, 32, 636–637. 18. Swinehart, D. F. J. Chem. Educ. 1962, 39, 333–335. 19. Solomons, C. C. J. Chem. Educ. 1965, 42, 226. 20. Atkinson, K. E. An Introduction to Numerical Analysis, 2nd ed.; Wiley: New York, 1989; Chapter 5. 21. Dorschel, C. A.; Ekmanis, J. L.; Oberholtzer, J. E.; Warren, F. V. Jr.; Bidlingmeyer, B. A. Anal. Chem. 1989, 61, 951A–968A. 22. Welz, B.; Sperling, M. Atomic Absorption Spectrometry, 3rd ed.; Wiley-VCH: Weinheim, 1999; p 115.
Journal of Chemical Education • Vol. 78 No. 9 September 2001 • JChemEd.chem.wisc.edu
