Understanding the Theory and Practice of Molecular Spectroscopy

In the Classroom

Understanding the Theory and Practice of Molecular Spectroscopy: The Effects of Spectral Bandwidth Satoshi Hirayama Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606, Japan Ronald P. Steer* Department of Chemistry, University of Saskatchewan, Saskatoon SK, Canada S7N 5C9 *[email protected]

The Beer-Lambert law, A(λ) = ε(λ)Cl , is among the most widely used relationships in analytical chemical spectroscopy and now appears as one of the “standard tools” in a Web-based calculator (1). A typical quantitative spectrophotometric or colorimetric analysis involves measuring the absorbance, A(λ), often at the absorption band maximum, A(λmax), of a solution containing a photon absorber of unknown molar concentration, C, in a cell of known light path length, l . The value of C is then determined using a calibration curve prepared by measuring A(λ) for solutions of the substance at several known concentrations in the same solvent (and the same conditions otherwise). In the simplest cases, such a calibration curve can be well represented by a linear regression from the origin, and its slope yields the value of the substance's molar absorptivity (molar extinction coefficient), ε(λ). Spectral Bandwidth and Molar Absorptivity Despite excellent references in undergraduate textbooks (2) that emphasize the importance of instrumental measurement parameters on the accuracy of such spectrophotometric determinations, the effect of spectral bandwidth (SBW) is almost always understated. Moreover, the importance of the value of the molar absorptivity, particularly at the band maximum, ε(λmax) or just εmax, is often overemphasized at the expense of the more fundamental relationships, the transition oscillator strength, f, or the Einstein coefficient. Here we discuss the pedagogical value of using quantities such as the oscillator strength that are based on integrated absorptivities. We present a simple demonstration, readily used in the classroom or as an adjunct to a laboratory exercise, that illustrates the effect of spectral bandwidth on measured values of A(λmax) or ε(λmax). We show that, unlike f, ε(λmax) can vary significantly with spectral bandwidth (when the bandwidth of the spectral transition is comparable to or less than that of the measuring instrument) and is therefore not a fundamental property of the system under study. The Importance of the Integrated Absorptivity and Oscillator Strength The oscillator strength, f, is a quantitative measure of the “degree of allowedness” of a one-photon-induced radiative transition between two quantized stationary electronic states of an atom or molecule (3). This quantity, in essence, is the ratio 1344

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of the integrated molar absorptivity (defined below) observed for a real atomic or molecular system to that of a hypothetical theoretically fully allowed electronic transition. The latter is modeled by the electric dipole allowed transition of a single electron bound to a nucleus with a simple harmonic (Hooke's law) potential. The oscillator strength is dimensionless and has a maximum value of 1 for a transition between nondegenerate states. The oscillator strength is also directly proportional to both the Einstein coefficient for spontaneous radiative decay and the radiative rate constant of the excited state, important parameters needed to describe postabsorption relaxation processes (4). A numerical value for f can be determined from the compound's absorption spectrum provided the spectrum has been recorded using samples in which the concentration of the absorber in solution (or the partial pressure of an absorbing gas) is accurately known Z 2303mc2 v2 εðvÞ dv f ¼ πe2 NA n v1 Z 4:39 10 - 9 v2 ¼ εðvÞ dv ð1Þ n v1 where n is the refractive index of the medium, m and e are the mass and charge of the electron, NA is Avogadro's constant, and c is the speed of light in a vacuum. (Digital data such as these are available through the PhotochemCAD Web site (5) and can be used directly by students in assignments.) Measurement of such an absorption spectrum then permits calculation of the molar absorptivity, ε, as a function of either wavelength, λ, or wavenumber, v. Integrating ε(v) or ε(λ) between the red and blue limits of the full electronic absorption band system then gives the integrated molar absorptivity, Z v2 Z λ2 1 εðvÞ dv ¼ ð2Þ 2 εðλÞ dλ ¼ Constant λ v1 λ1 The latter value is then substituted into eq 1 to give the oscillator strength of the transition. The only quantity in eq 1 that is not a fixed constant of nature is the refractive index of the medium, n. Note that no approximation is made in evaluating the integral, which, unlike the molar absorptivity, has a value that is independent of the spectral bandwidth of the instrument used to obtain the spectrum. The values of the definite integrals of eq 2

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Vol. 87 No. 12 December 2010 pubs.acs.org/jchemeduc r 2010 American Chemical Society and Division of Chemical Education, Inc. 10.1021/ed1005272 Published on Web 09/17/2010

In the Classroom

Figure 1. Absorption spectra of benzene vapor measured at two different spectral resolutions. (a) Bandwidth 0.1 nm and (b) bandwidth 1.0 nm. The most prominent peak of each band group is labeled as described in refs 7-9), which provide the interpretation of the spectrum. Band peak positions shift a little to the red as the spectral resolution is lowered.

Figure 2. Absorption spectra of benzene measured in n-hexane with bandwidths of (a) 0.1 nm and (b) 1.0 nm. The spectra are intrinsically broad and the spectral shapes or peak positions are almost independent of the spectral bandwidth, but a significant decrease in the absorbance is observed at the lower spectral resolution.

are therefore constant for a given electronic transition of a given molecule and, for a given solvent, the corresponding value of f is therefore also constant, irrespective of the SBW of the measuring system. We use the near-UV absorption spectrum of benzene to illustrate these points. Readers may choose to present this material as a visual illustration in the classroom, as a dry lab exercise, or as part of an experimental laboratory in which the spectra are acquired and analyzed, as described in the supporting information. The Near-UV Spectrum of Benzene: The Effect of Phase and SBW Measurement and analysis of the near-UV spectrum of benzene is a common upper-level undergraduate laboratory experiment in spectroscopy, one that has been described in detail in this Journal by Campbell (6). Figures 1 and 2 show the electronic spectrum of pure benzene taken with a common laboratory UV-visible double-beam spectrophotometer in the

r 2010 American Chemical Society and Division of Chemical Education, Inc.

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gas phase at spectral bandwidths of 0.1 and 1.0 nm and in n-hexane solution under the same conditions of spectral bandwidth. The details of how these measurements are made, together with Excel files of the digitized spectra, are provided in the supporting information. The analysis of the spectrum may be made with reference to the original measurements of Callomon, Dunn, and Mills (7), Atkinson and Parmenter (8), and Stephenson, Radloff, and Rice (9), as well as the article by Campbell (6). We focus here on the effects of spectral bandwidth on both the gas-phase and solution-phase spectra. At the highest spectral resolution employed in this experiment, some of the vibrational structure in the gas-phase spectrum is well resolved. However, the observed resolution is sensitive to the SBW of the spectrophotometer. The observed spectrum broadens with increasing SBW, bands coalesce, and the observed band intensities decrease dramatically. This simple observation shows that, for gas-phase samples under these conditions, the measured value of the molar absorptivity at a given wavelength, ε(λ), is not constant but, instead, is strongly dependent on the spectral resolution employed. In solution, the fine vibrational structure seen in the gasphase spectra disappears and the absorption bands are intrinsically broad even when measured at the higher resolution. The band shapes and the wavelengths of the band maxima do not depend appreciably on the spectral resolution, but the measured values of absorbance, A(λ), and of ε(λ) are still weak functions of the SBW of the spectrophotometer. The value of ε(λ) is often regarded as constant and characteristic of the molecule and this assumption underlies the use of the Beer-Lambert law in analytical spectroscopy. However, only when absorbances are measured under the same conditions (solvent, temperature, SBW, geometry) for both calibration curves and unknowns is this assumption not a limitation on the accuracy of the analytical measurement. The values of εmax are not fundamental properties of the absorber; literature values cannot be used in accurate measurements of C unless the same SBW is employed or unless the spectrum is so broad that εmax is constant over the entire instrumental SBW. Simulations Once the digitized data of a more highly resolved absorption spectrum such as the one shown in Figure 1a are obtained, then it is feasible to generate a more poorly resolved spectrum by convolution of the observed spectrum with a slit function of finite width. We describe the details of how this is done in the supporting information and provide only an illustration here. The SBW of the spectrophotometer is obtained from the monochromator's reciprocal dispersion in nm/mm and a specified slit width in mm, and its functional form can be most simply represented by a triangular function in an Excel format, as depicted in Figure 3a. Here the horizontal axis represents cell numbers, each of which corresponds to a certain central wavelength and a given width, for example, 0.05 nm, and the vertical axis represents the normalized (transmitted) light intensity at the corresponding wavelength. For the symmetrical triangular function shown in Figure 3a, the nominal spectral bandwidth corresponds to 10 cells, which is the number of cells used to represent the full width of the function at half-maximum (fwhm). Thus, the spectral bandwidth for this function will be 0.5 nm when the cell width is 0.05 nm. Choosing a larger number of cells

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In the Classroom

Figure 3. SBW functions used in Excel to mimic finite spectral resolution: (a) triangular function and (b) Gaussian function; exp[-(x - 10)2/20] where x is the cell number. The cell number corresponds to the central wavelength of the increment in the digital spectrum and the y axis gives the light intensity normalized to the central wavelength. The intensities at cells number 0 and 20 are zero.

of the same width simulates a proportionately larger SBW. The triangular function can be turned into a delta-function (δ-function) when only one cell corresponding to the chosen wavelength is used and is assigned a value of 1 for the light intensity. In this case, applying the spectral bandwidth function to the measured spectrum will just reproduce the spectrum with no change in its apparent resolution. More realistic SBW functions, such as the Gaussian function given in Figure 3b, or a slit width function sin½x - 10 2 , 1 e x e 19 ð3Þ y ¼ x - 10 can also easily be used in the convolution procedure described in the supporting information. The results of convoluting the spectrum shown in Figure 1a with the triangular bandwidth function having SBWs of 0.2, 0.5, and 1.0 nm (the values for the fwhm are then 2, 5, and 10 cells, respectively) are shown in Figure 4. As the bandwidth increases, the fine structure in the high-resolution benzene vapor spectrum merges into a single group of broad bands with an interval of ca. 6 nm in the wavelength range shown in Figure 4. This interval corresponds to the most Franck-Condon active vibrational mode in benzene's lowest excited singlet state (7), measured here as 920 ( 8 cm-1. Note that the peak intensity of any coalescing band group falls significantly with increasing SBW. Considering the fact that the triangular SBW function used in these simulations is a crude approximation to more realistic functions (e.g., a Gaussian or a [(sin2x)/x2] slit function), it is perhaps surprising to find that the simulated spectrum generated with a SBW of 1.0 nm (Figure 4d) accurately reproduces the measured one with a comparable spectral resolution (Figure 1b). Using the simulation method outlined above, it is easy to confirm that the area under the absorption spectrum associated with a given electronic transition, and hence the integrated molar absorptivity and oscillator strength, must remain constant whether the spectrum is measured under high or low resolution or in units of nm or cm-1. That is, the values of the definite integrals of eqs 1, 2, and 4 Z λ2 εðλÞ dλ ¼ Constant 0 ð4Þ λ1

hold irrespective of SBW. The validity of eq 2 or 4 can easily be tested by using the data for any of the simulated spectra (such as those 1346


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Figure 4. Simulation of the absorption spectra at lower spectral resolution. The spectra are shown only in the range of 245-265 nm for the sake of magnification. (a) The experimentally measured spectrum with a bandwidth of 0.1 nm and simulated spectra with bandwidths of (b) 0.2 nm, (c) 0.5 nm, and (d) 1.0 nm. The band maxima of spectrum (d) are shifted slightly to the red compared with those for spectrum (a). The absorbance of the highest peak of each band group drops significantly with decreasing spectral resolution.

shown in Figure 4), because the integrals of eqs 2 and 4 are proportional to the summation of the values for the absorbance over the entire set of increments of wavenumber or wavelength (Excel cells) used to record the spectrum. In practice, the areas under the absorption spectra taken under the various SBW conditions described above are constant within a few percent. Conclusions and Suggestions for Use Demonstration of the effects of SBW on vibrationally resolved spectra can be carried out with a few keystrokes in a lecture or in a prelab discussion once digital data such as that described for benzene are in hand. Provided the students have acquired a prior working knowledge of Excel or other similar programs for digital data manipulation, they can carry out any of the simulations described. An interesting follow-up exercise involves asking students to go to the literature to find a spectrophotometric method that relies on a published value of the molar absorptivity and then to see if sufficient SBW data are provided to enable them to use this value with confidence in an experiment using their own instrumentation. Curious students may ask about the red (bathochromic) spectral shift observed when benzene is taken from the gas phase to solution. Reference may be made to recent articles in this Journal (10). Acknowledgment The authors wish to acknowledge their respective universities, the Kyoto Institute of Technology (SH) and the University of Saskatchewan (RPS), for supporting this work. Literature Cited 1. Chang Bioscience. http://www.changbioscience.com/calculator/ BeerLambert.html. (accessed Sep 2010). 2. Harris, D. C. Quantitative Chemical Analysis, 7th ed.; W. H. Freeman: New York, 2007. 3. See, for example, Hollas, J. M. High Resolution Spectroscopy; Butterworths and Co.: London, 1982; p 42 and p 387 ff.


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In the Classroom

4. See for example, Turro, N. J. Modern Molecular Photochemistry; Benjamin/ Cummings Pub. Co.: Menlo Park, CA, 1978; p 87 ff. 5. (a) Du, H.; Fuh, R. A.; Li, J.; Corkan, A.; Lindsey, J. S. Photochem. Photobiol. 1998, 68, 141. (b) Dixon, J. M.; Taniguchi, M.; Lindsey, J. S. Photochem. Photobiol. 2005, 81, 212. The Web site at which PC downloadable programs may be obtained is: http://www.photochemcad.com/ (accessed Sep 2010). 6. Campbell, M. K. J. Chem. Educ. 1980, 57, 756. 7. Callomon, J. H.; Dunn, T. M.; Mills, I. M. Philos. Trans. R. Soc. London 1966, 259A, 499.


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8. Atkinson, G. H.; Parmenter, C. A. J. Mol. Spectrosc. 1978, 73, 52. 9. Stephenson, T. A.; Radloff, P. L.; Rice, S. A. J. Chem. Phys. 1984, 81, 1060. 10. Hirayama, S.; Steer, R. P. J. Chem. Educ. 2008, 85, 317 and references therein.

Supporting Information Available Methods to present this material; details of how the measurements are made; details of how the simulations are done; Excel data. This material is available via the Internet at http://pubs.acs.org.


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Understanding the Theory and Practice of Molecular Spectroscopy

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