In the Classroom
A Unifying Approach to Absorption Spectroscopy at the Undergraduate Level Roger S. Macomber Department of Chemistry, University of Cincinnati, Cincinnati, OH 45221-0172
One of the most tantalizing, and at the same time most complex, wonders of everyday experience is the interaction of light (electromagnetic radiation) with matter. As teachers of chemistry, we usually discuss such phenomena with wave equations, quantum mechanical probabilities, selection rules, and the like, often before most students have gained even a qualitative understanding of the fundamental principles involved. It is the purpose of this brief discussion to suggest the utility of one classical aspect of the subject, in the hope that it may provide a useful springboard to the introduction of absorption spectroscopy. We teach our students that light and matter are not fundamentally different from one another, and that both forms of “stuff” from which the universe is made can exhibit the properties of either particles or electromagnetic waves, depending on the type of experiments performed. We employ the mathematical approach of wave mechanics to help compensate for the lack of precision with which we can measure certain properties of extremely small bits of “stuff.” It is within the field of spectroscopy that we quantitatively measure the consequences of the interaction of light with matter. Specifically we are concerned with energy-bartering: matter receiving energy from light (absorption) and matter yielding energy in the form of light (emission). What are the necessary conditions for such exchanges to occur? Light, with its oscillating electric and magnetic fields (Fig. 1), interacts most strongly with charged particles of matter such as the electrons and protons that make up atoms. The point of this discussion is that virtually all types of absorption spectroscopy (e.g., IR, UV-vis, NMR) share a simple but critical requirement: the frequency of absorbed radiation must exactly match the frequency of some inherent or induced periodic motion of the particles. This is not a miraculous coincidence of nature. It is a straightforward consequence of the necessity for the electric or magnetic oscillations of the incident radiation to match in both frequency and orientation (polarization) the electric or magnetic oscillations of the particles absorbing the radiation. Who has never been sitting in a quiet room when suddenly an external noise (sound waves) causes a nearby object to vibrate in sympathy? This is an example of resonance, where the frequency of the sound waves exactly matches the frequency with which the object naturally vibrates. If the pitch of the sound changes, the resonance ceases. This is because only when the two frequencies match can the sound waves constructively interfere with (reinforce) the motion of the object. At any other frequency the interaction will be one of destructive interference and the net result will be no vibration
by the object. (A short but intense sound such as a sonic boom can set many different objects vibrating because the brief sound wave acts as though it is a mixture of all audible frequencies.) There is an analogous phenomenon on the atomic scale. All the atoms within molecules, as well as all the particles that make up the atoms, are in constant periodic motion of various types, each with a characteristic frequency. Whenever the frequency of radiation incident on the collection of molecules matches one of these inherent frequencies, absorption is possible. (Pulsed mode spectroscopic techniques are analogous to the sonic boom mentioned above.) Let’s apply this idea of frequency matching to several of the common types of spectroscopy to which undergraduates are routinely introduced. Our first example is infrared (IR) absorption (vibrational) spectroscopy (1, 2), which involves the vibrational motion of atoms (nuclei) within molecules, motion that takes place even at absolute zero.1 Such vibrations are not unlike the classical situation of two masses attached by a spring, and are governed by similar mathematical equations. The important fact is that the frequency of vibration of any pair of bonded atoms is determined solely by their masses and the bond strength (the force constant of the “spring”). Only electromagnetic radiation of this exact frequency (which happens to occur in the infrared region of the electromagnetic spectrum) can be absorbed by this vibrating system, because only then will the oscillations of the electric vector of the radiation will match the vibrations of the electric dipole vector (E′) of the bond (Fig. 2).
Figure 1. A plane-polarized electromagnetic wave with wavelength λ (and frequency ν = c/λ), propagating along the x axis. E represents the electric vector oscillating in the y direction; B represents the magnetic vector oscillating in the z direction.
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In the Classroom
Note that this classical model also accounts for the polarization of the absorption, the necessity for the radiation to propagate in a direction perpendicular to the bond so that vectors E (Fig. 1) and E′ (Fig. 2) are parallel. But where does the vanishing photon’s energy go? Because the frequency of bond vibration does not change as a result of energy absorption (assuming harmonic oscillations), it is the amplitude of the vibration that increases when energy is absorbed. This is analogous to pushing someone on a swing: you must push “in phase” with the oscillations of the swing, and your energy goes to increasing the amplitude, not the frequency, of the swinging motion. Thus, when a vibrational excited state is generated, the “quantum” of electromagnetic energy is converted to vibrational kinetic energy, one manifestation of heat. But only this discrete (specific) amount of energy could be absorbed, because only a photon with this exact energy has the appropriate frequency to match the vibrations! Next, let’s look at visible–ultraviolet (UV-vis, or electronic) spectroscopy (1, 3), where photon energies are sufficient to interact directly with valence electrons, most commonly π electrons. Here, it becomes a little less intuitive to identify the periodic electronic oscillations, and this is one reason that quantum mechanics treats electrons as waves. The larger the region of space in which a given electron is confined, the lower will be the frequency of its oscillation through that space. Consider a classical picture of ethylene, where the two π electrons are free to oscillate in a cloud above and below the molecular plane (Fig. 3). Ethylene exhibits an intense π absorption at 165 nm, equivalent to a frequency of 1.82 × 1015 s{1 (in the far UV). As the IR absorption discussed above, this UV absorption is polarized along the C=C bond axis so that E′′ (Fig. 3) is parallel to E (Fig. 1). We can crudely equate the 1.82 × 1015 s{1 to the frequency of π electronic oscillations in a “box” of length 133 pm (1.33 × 10{8 cm). From this we can estimate the ground state velocity of the π electrons: v = (2 × 1.33 × 10{8 cm) (1.82 × 1015 s{1) = 4.84 × 107 cm s{1 or about 0.15% of the speed of light in a vacuum.2 If the π electron ground state velocity remains reasonably constant from one molecule to the next, we might expect larger molecules to exhibit lower frequency (longer wavelength) absorptions. This is exactly what is observed. Some molecules, notably dyes, have very extended π systems which permit oscillation of the π electrons throughout the entire molecule. Therefore, the electronic oscillation frequency for this type of system is quite low, and consequently the absorbed radiation falls in the visible region, hence their color. How does the absorbed radiant energy manifest itself in the case of electronic absorptions? Short of invoking wave functions or molecular orbitals, a classical view is that the excited electron expands both the region through which it oscillates and its velocity, so that its oscillation frequency remains unchanged. It is possible, of course, to irradiate molecules with such high energy radiation that absorption causes the molecule to ionize or fragment, processes known respectively as photoionization and photodissociation. Finally, let’s consider nuclear magnetic resonance (NMR) spectroscopy (1, 4). When placed in a strong external magnetic field (Bo in Fig. 4), nuclei of certain isotopes (most notably 1H and 13C) act like tiny magnets and align themselves parallel or antiparallel to the applied field, with a very slight excess parallel (the more
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Figure 2. Vibration of the A–B bond, with its electric dipole vector E′, along the y axis.
Figure 3. The molecular structure of ethylene, with π electron vector E′′ along the y axis.
Figure 4. (a) Precession of magnetic moment M around external magnetic field Bo (aligned along the z axis). (b) Circularly polarized electromagnetic field, with magnetic vector B1 oscillating in the xy plane. (c) Orientation of M after a photon is absorbed. Note the change in axis designations from Figures 1–3.
stable orientation). In addition, the spinning of these nuclei induces the magnetic moment (M, Fig. 4) of each nucleus to precess around Bo (the z axis). This periodic motion describes a cone, and is analogous to the wobbling of a top spinning on its axis. The nuclear precessional frequency is determined solely by the type of nucleus and the strength of Bo. If circularly polarized radiation of this same frequency (in this case, radio frequencies), propagating along the z axis, is passed through the sample, it can be absorbed because the frequencymatching resonance condition is fulfilled. The principal difference is that in this case the magnetic field (B1, Fig. 4) oscillations of the circularly polarized radiation match the magnetic precessional frequency of Mxy, the component of M in the xy plane. Where does the absorbed energy go in the case of NMR? It causes a flip in the orientation of the nucleus from parallel to antiparallel with the magnetic field (Fig. 4c), an orientation of higher energy (and less stability). Yet the magnetic moment (M) of the nucleus continues to precess at the original frequency. We have hedged a little in this discussion by avoiding the details of selection rules and transition probabili-
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In the Classroom
ties. The probability of any absorption or stimulated emission depends, in part, upon the ratio of particles in the higher energy (excited) state to those in the lower (ground) state. If this ratio is less than one, absorption is favored; if it is greater than one, stimulated emission is favored. If the ratio is unity, both processes are equally likely, so no net absorption can occur; this state is called saturation. This constraint is quite critical in NMR because the ratio is about 0.9999. Saturation is of much less importance in IR and UV-vis spectroscopy, where the ratio is generally much less than 10{2. A type of spectroscopy very similar to NMR is known as electron spin resonance (ESR), the only difference being the particle investigated. Here it is an unpaired electron that is precessing in an external magnetic field. But just as before, the electron precesses with a characteristic frequency, and whenever this frequency is matched by that of incident radiation, absorption (resonance) can take place. What other parts of atoms can absorb light? Although it is usually not discussed at any length at the undergraduate level, nucleons themselves exist in various nuclear (as opposed to magnetic) energy states, just as do electrons. Each nucleus has associated with it (even in the absence of a magnetic field) certain intranuclear energy levels and periodic motion, which can be excited with very high frequency radiation (X-rays and γ-rays). This is the basis of Mössbauer spectroscopy. Students sometimes wonder if it is possible for a particle to absorb two or more photons simultaneously, or for a particle already excited to absorb another photon. The answer under normal conditions is no, primarily because the probability of two photons simultaneously passing close enough to the absorbing unit (chromophore) is vanishingly small, and because the lifetime of species in excited states is usually very short. However, the extremely high intensities available with pulsed lasers is making multiphoton processes a field of active
investigation. Finally, it is interesting to use this classical model to speculate about how long it takes for a photon to be absorbed. If we regard the photon as defining a point, that point would pass through a sphere of diameter 133 pm (the C=C bond length in ethylene) in 4.4 × 10{19 s! This is about 0.1% of one π electron oscillation. Is this enough time for absorption to occur? Alternatively, what are the dimensions of a photon wavelet, and how near to the chromophore must it pass to be absorbed? Maybe it is fortunate that we have the Uncertainty Principle to fall back on. The principle of frequency-matching resonance is thus a valuable and unifying heuristic approach when introducing the phenomena of light’s interaction with matter, and can be extended to fluorescence and other emission processes. One can even introduce the various types of scattering phenomena in this context. Notes 1. As one referee pointed out, if bonds did not possess this “zero-point” vibration, they would be unable to absorb in the IR! 2. The free electron (particle in the box) quantum mechanical model predicts a ground state frequency of 5.13 × 10 15 s {1 , corresponding to a classical electron velocity of 2.7 × 10 8 cm s{1.
Literature Cited 1. Two very useful texts for further reading on the theories and applications of these and other spectroscopic techniques are (a) Orchin, M.; Jaffe, H. H. Symmetry, Orbitals, and Spectra; Wiley Interscience: New York, 1971; (b) Silverstein, R. M.; Bassler, G. C.; Morrill, T. C. Spectrometric Identification of Organic Compounds, 5th ed.; John Wiley and Sons: New York, 1991. 2. Colthup, N. B.; Daly, L. H.; Wiberly, S. E. Introduction to Infrared and Raman Spectroscopy, 2nd ed.; Academic: New York, 1975. 3. Jaffe, H. H.; Orchin, M. Theory and Applications of Ultraviolet Spectroscopy, John Wiley and Sons: New York, 1962. 4. Macomber, R. S. NMR Spectroscopy, Essential Theory and Practice; Harcourt Brace Jovanovich: San Diego, 1988.
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