Research: Science and Education edited by
Advanced Chemistry Classroom and Laboratory
Joseph J. BelBruno Dartmouth College Hanover, NH 03755
Photochemical Kinetics: Reaction Orders and Analogies with Molecular Beam Scattering and Cavity Ring-Down Experiments Michael Hippler Physical Chemistry, ETH Zürich, Zürich CH-8093, Switzerland;
[email protected] Photochemical reactions are ubiquitous and relevant; for example, chemical reactions initiated by absorption of light are involved as the primary step in the vision process (absorption of light by a pigment in the eye’s receptor cells), in photosynthesis, and as the driving force in atmospheric chemistry. Despite its fundamental importance, photochemical kinetics is not often treated in much detail in physical chemistry courses and textbooks and, as consequence, concepts are often not well-understood and remain unclear. Reaction orders of photochemical reactions remain a subject of debate. In a recent article in this Journal (1), photochemical reactions were even denied a reaction order. In this article, we want to emphasize that a photochemical reaction system is composed of several elementary steps, each of which has a defined molecularity and reaction order. The elementary, primary absorption step can be considered a bimolecular reaction. Depending on the experimental conditions, the apparent total reaction order of the mechanism may have different values, but can still be defined in most cases. The photoabsorption step should be considered a kinetic elementary reaction. A standard photochemical experiment with irradiation of a sample with constant light intensity, however, does not correspond to a “conventional” kinetic experiment with homogeneous reagent concentrations that react over time without external perturbations. Possible conceptual difficulties may be avoided by realizing two analogies between experiments involving light and kinetic experiments not involving light: A standard absorption measurement has an analogy with a molecular beam scattering experiment, and
cavity ring-down spectroscopy has an analogy with a conventional static reactor experiment. Photochemical Reaction Systems Many elementary steps are relevant for a typical photochemical reaction mechanism (Table 1). The reaction sequence is initiated by absorption of photon γ by a molecule A (Stark–Einstein law), eq 1. The excited molecule A* may then undergo a chemical transformation in a unimolecular elementary reaction, eq 2; for example by dissociation. In competition, the excited molecule may be quenched by collisions (A* + M → A + M) or by spontaneous emission of a photon γ′ (A* → A + γ′ ). γ′ is in general different from the initially absorbed photon γ (different wavelength, different propagation direction). Quenching reactions are summarized symbolically in eq 3. A* may also react with a photon and then emit both photons (stimulated emission–saturation) or become further excited (two-photon absorption), but these processes can usually be neglected at moderate photon densities (light intensity). Additional competing elementary reactions are collision-induced emission, intersystem crossing, phosphorescence, internal conversion, or triplet–triplet absorption (2, 3). The simplified reaction scheme, eqs 1–3, is a representative example to discuss important aspects of photochemical kinetics. The net reaction of the photochemical mechanism is A + γ = P, eq 4, but in the discussion one has to bear in mind that different elementary steps are involved, eqs 1–3.
Table 1. Simplified Photochemical Reaction Mechanism Equation (1)
A + γ
(2)
A*
(3) (4)
Reaction Rate Lawa
Reaction
k1
k2
A*
P
k3 A* A __________________ A + γ =
1 1 d [γ ] d [ A ]( ) d [ A* ]( ) vC(1) = − = − = = k1 [ A ][ γ ] dt dt dt 2 d [ A* ]( ) d [P ] vC(2) = − = = k2 [ A*] dt dt 3 3 d [ A*]( ) d [ A ]( ) vC(3) = − = = k 3 [ A* ] dt dt
P
aBrackets denote the number concentration of a species, v are reaction rates (based on number concentration), and k are rate c constants. Indices on differentials indicate a contribution to the total change of number concentration from one reaction, in addition to contributions from other reactions.
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Journal of Chemical Education • Vol. 80 No. 9 September 2003 • JChemEd.chem.wisc.edu
Research: Science and Education
Reaction Orders The photoabsorption step, eq 1, is clearly a bimolecular elementary reaction with a total reaction order of two, if one considers the photon as particle and the absorption process as an inelastic collision of two particles. The reaction rate of the bimolecular reaction is vC(1) = k1[A][γ]. If [γ] remains constant (irradiation with constant light intensity), the reaction is “pseudo-first order”; that is, the apparent total reaction order is one. If, in addition, [A] is only slightly diminishing during the measurement, for example as a result of low absorption cross-sections or if [A] is very large, vC(1) may appear approximately constant. The apparent total reaction order is then zero. The “true” total reaction order of a bimolecular elementary reaction, however, is always two. The most relevant quantity in the reaction mechanism is the rate of product formation, vC = d[P]兾dt. According to eq 2, vC = k2[A*]. To evaluate this rate further, we need an expression for [A*], which can be provided by the assumption of a quasi stationary-state condition for A* (2–4). Under normal conditions, thermal population of A* is negligible and deactivation of A* by reaction or quenching is much faster than excitation of A by absorption of a photon. With [A*] 99.9%). Within the cavity, a constant light intensity I0 is rapidly attained in an equilibrium between gain by light input and loss by either imperfect mirrors or as a result of absorption by gasphase species inside the cavity. The laser source is then switched off and light intensity decays within the cavity as a result of losses on the mirrors and to absorption. A small amount of light I′ is always leaking out of the cavity. I′(t) is monitored by a photodiode or photomultiplier; it is directly proportional to the light intensity I(t) inside the cavity. Since light is typically reflected back and forth thousands of times inside the cavity during a decay event, effective absorption path lengths of several km are easily achieved, which explains the enormous sensitivity of CRD spectroscopy to the absorbing species (Figure 2; refs 5, 6 ). After one pass through the cavity of length l from one mirror to the other, light intensity is attenuated by
I = R exp [−α l ] = exp − (− ln R + α l ) I0 ≈ exp − ((1 − R ) + α l )
(9)
where ln R ≈ ᎑(1 − R) holds for R ≈ 1 (highly-reflective mirrors). In eq 9, the Beer–Lambert absorption law has been used. The absorption coefficient is α = σ[A] and [A] the number density of absorbing species A inside the cavity with absorption cross-section σ. After n passes,
I n = (R exp [−α l ]) ≈ exp − ((1 − R ) + α l ) n I0
(10)
and substituting n = (ct)兾l, where c is the speed of light, we finally arrive at,
c I (t ) = I 0 exp − (1 − R ) + α c l k =
(1 − R ) c l
t = I 0 exp [−k t ]; (11)
+ αc
where I(t) is the light intensity inside the cavity and I′(t) is
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Figure 2. Diagram of a cavity ring-down experiment. I–incoming light from a laser source, switched-off at t = 0; I’(t)–light intensity monitored outside the cavity with length l; M–two highly reflective, concave cavity mirrors.
the light intensity leaking out of the cavity, which is proportional to I(t). The ring-down constant k is obtained by a single-exponential fit of the observed decaying light intensity. If the spectral baseline (1 − R)c兾l is determined in a separate experiment (empty cavity, without absorbing species A), the absorption coefficient α can be measured, from which follows the absorption cross-section σ (if [A] is known) or the concentration [A] (if σ is known). In the derivation of eq 11 it has been assumed that [A] remains approximately constant with time, and hence α = σ[A]. This is adequate since CRD spectroscopy is usually applied to measure very weak absorptions (see below for the more general case where [A] is not constant). It has also been assumed that both mirrors have the same reflectivity R. If they are different (R1 and R2), a similar derivation shows that R in eq 11 is replaced by (R1 + R2)兾2. As an example, typical decay curves I′(t) observed by a photo diode in the near-infrared spectral region close to 1.3 µm (6) are shown in Figure 3. In the experiment, ∼1 mW infrared light from a continuous diode laser is coupled into a cavity composed of two highly-reflective mirrors (R = 99.988%, l = 29.5 cm). The cavity is filled with 0.53 mbar nitrous oxide gas (N2O) at room temperature. At t = 0, the diode is switched off, and the decay of the light intensity is observed. The curves are characterized by an exponential decay with ring-down constant k. The decay is fast at a wavelength where N2O absorbs light and much slower at a slightly different wavelength where N2O does not absorb light (Figure 3). The 1兾e decay time of the empty cavity is about 8 µs; within that time interval, light makes about 4000 round trips in the cavity and travels a distance of 2.4 km, which is a measure for the effective absorption path length. Knowing the concentration of N2O, k can be converted into the absorption cross-section σ. By measuring σ at different wavelengths, an absorption spectrum is obtained. The very weak rovibronic R(7) transition of the ν1 + 3ν3 combination band of 14N216O near 7788.5 cm-1 obtained with the described setup (6) is shown in Figure 4. Taking the noise level of the spectral baseline in Figure 4 as measure for the detection limit, the root-mean-square noise equivalent absorption coefficient α is about 2 × 10᎑8 cm᎑1 (6). This detection limit compares favorably with other extremely sensitive detection techniques, for example, frequency modulation diode laser spectroscopy. The relationship between the CRD parameters in eqs 9–11 and the rate constants of the various elementary steps involved can be derived as follows: if the absorption and pho-
Journal of Chemical Education • Vol. 80 No. 9 September 2003 • JChemEd.chem.wisc.edu
Research: Science and Education
Figure 3. Two typical decay curves in a CRD experiment after switching-off a near-infrared diode laser at t = 0. The fast decay corresponds to a wavelength where the N2O inside the cavity absorbs light (spectrum shown in Figure 4) and the slower decay to a slightly different wavelength where N2O does not absorb light.
Figure 4. CRD spectrum of the R(7) transition of the ν1 + 3ν3 combination band of 14N216O.
tochemical reaction is described by the simplified mechanism, eqs 1–3, and if the quasi stationary-state condition for A* is applied with the previously defined effective rate constant keff = k1k2兾(k2 + k3), we arrive at
Conclusions
d [A ] d [γ ] = −keff [ γ ][ A ] and = −k1 [ γ ][ A ] − k4 [ γ ] (12) dt dt
Equation 12 is the coupled rate law that applies to the situation inside the cavity after switching off the external light source. Mirror losses contribute to the decay of photon density via the rate constant k4 = (1 − R)c兾l. Light intensity I is related to photon density [γ] by I = hνc[γ]. If [A] remains approximately constant during a CRD measurement, k1[A] is constant and corresponds to the constant αc in eq 11. Straightforward integration of the differential eq 12 shows the equivalence between eqs 11 and 12 in this case. Photon density or light intensity within the cavity is then characterized by an exponential decay, which usually applies to CRD measurements of weak absorptions. In the more general case, however, where the decrease of [A] by absorption or photochemical reaction cannot be neglected, eq 11 is not valid and a nonexponential decay will be effective described by eq 12. Since light is trapped and confined in an optical cavity and then left to react with molecules (absorption) inside the cavity, there is a perhaps surprising analogy of a cavity ringdown measurement with a static reactor experiment in kinetics. To make this analogy more evident, we describe the concepts of a conventional kinetic experiment and compare them in parentheses to the situation applying to a CRD measurement: At t = 0, the reagents (species A and the reactive species γ) are added and mixed carefully in the reactor (the cavity), and then the reaction is observed by monitoring the time-dependent concentration of one reactant (photon density [γ] or light intensity I(t)). Losses of the reactive species on the reactor walls also have to be included in the analysis (mirror losses, k4). One difference between the two experiments is that mirror losses in a cavity are in general substantial, whereas wall losses in a reactor are often negligible.
A photochemical reaction mechanism is composed of different elementary reaction steps. The reaction sequence is initiated by the absorption of one photon by a molecule. This absorption step can be considered a bimolecular elementary reaction. By analogy with molecular beam experiments and cavity ring-down experiments, it was shown that photoabsorption is a special type of elementary reaction, but can still be described by standard kinetic concepts. Depending on the experimental conditions, the apparent total reaction order of the compound photochemical reaction mechanism may have different values, but will still be defined in most cases. A photochemical reaction does therefore have a reaction order in general. Acknowledgment I am grateful to Martin Quack for stimulating discussions. Literature Cited 1. Logan, S. R. J. Chem. Educ. 1997, 74, 1303. 2. Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry; Wiley: New York, 1980. 3. Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, 1994. 4. Luckhaus, D.; Quack, M. Gas-Phase Kinetics. In Encyclopedia of Chemical Physics and Physical Chemistry; Moore, J. H., Spencer, N. D., Eds.; IOP Publishing: Bristol, England, 2001; Chapters A3.4, B2.5. 5. Cavity-Ringdown Spectroscopy: An Ultratrace-Absorption Measurement Technique; Busch, K. W., Busch, M. A., Eds.; ACS Symposium Series 720; American Chemical Society (distributed by Oxford University Press): Washington, DC, 1999. 6. He, Y.; Hippler, M.; Quack, M. Chem. Phys. Lett. 1998, 289, 527.
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