J . Phys. Chem. 1992, 96, 10308-10315
10308
Photoinduced Oxygen Transfer from NO2 to Ethylene in the Vkkrlty of the NO2 Dissociation Threshold. A Laser Photochemical Study on Reactant Pairs Isolated in Solid Argon Donald J. Fitzmauricet and Heinz Frei* Laboratory of Chemical Biodynamics, Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (Received: June 1 1 , 1992; In Final Form: September 15, 1992)
The wavelength dependence of the photochemistry of C 2 Q N 0 2and C2D,-N02 pairs, isolated in d i d Ar,in the range 555-355 nm is reported. Continuous-wave and pulsed dye lasers were used to excite the reactants, and the products were monitored by FT infrared spectroscopy. Acetaldehyde, ethylene oxide, NO, and ethyl nitrite radical were the products observed at wavelengths longer than the 398-nm dissociation threshold of NO2. D isotope effects on the branching among these species derived from kinetic analysis of growth curves support our conclusion in previous work that the mechanism involves large-amplitude 0 transfer from NO2 to the C=C bond to yield a transient oxirane biradical. The latter acts as a common precursor of the observed products. A new species, ketene, appears at excitation energies just above the NO2 dissociation limit (385 and 355 nm). This product is assigned to a new path consisting of NO2 OcP) + NO dissociation followed by O(3P) + ethylene addition. Comparison of product growth upon 385- and 420-nm excitation indicates that there is no abrupt change in total product yield, combined aldehyde and epoxide yield, or epoxide/aldehyde branching ratio as the photolysis photon energy is tuned through the 398-nm dissociation threshold. This suggests that large-amplitude 0 transfer continues to play a dominant role upon excitation of ethylenaN02collisional pairs even at wavelengths shorter than the NO2 O(3P) dissociation limit.
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I. Introduction
SCHEME I
We have recently found that the chemical reaction of ethylene-N02 collisional pairs isolated in a solid Ar matrix can be initiated by (single) visible photons.' The photolysis wavelength range investigated was between 575 (threshold) and 488 nm, corresponding to energies between 22 and 13 kcal mol-' below the 398-nm dissociation limit of N02.2 The products identified by infrared spectroscopy were ethylene oxide, acetaldehyde, NO, and ethyl nitrite radical (cH2CH20NO). A main result of our previous studies of the ethylene + NO2 is that the mechanism and the butene NO2 photo~hemistry~-~ involves formation of a transient oxirane biradical (Scheme I). A key finding that led to this conclusion was the strict correlation between the stereochemistry of the carbon skeleton of alkyl nitrite radical and epoxide in the case of cis- and trans-2-butene + N02.kb This implies that the nitrite radical and the epoxide share a common transient precursor. An oxirane biradical is the only conceivable common precursor which is consistent with the obsewed photolysis wavelength dependence of the branching between the two products and with the observed high degree of stereochemical retention upon photolysis of alkene-N02 pair^^-^ (including cis- and trans-CHI)3CHD*NO2).' Hence, the primary step of the visiblalight-induced reaction of ethyleneN02 collisional pairs is large-amplitude 0 transfer from NO2 to the C - C bond to generate an oxirane biradical. Such largeamplitude 0 transfer from NO2 to alkene bonds, established for the first time by our previous ~ o r k , l *contrasts ~-~ with oxygen transfer involving free OCP) atoms. The latter is only accessible energetically at photon energies above the 398-nm dissociation limit of NO2. The 2B2excited state, which gives nitrogen dioxide its oscillator strength in the visible and near-infrared spectral range, is heavily mixed with vibrational levels of the 2Alground electronic state.ZH As a consequence, reaction is initiated by highly vibrationally excited NO2 with predominantly electronic ground-state character. Since two out of three NO2 vibrational modes are stretching modes, such high vibrational overtone excitation by visible light results in motion of the N 0 2 0 = C system that has a substantial component along the asymmetric N O 4 large-amplitude stretchig oootdinate, comsponding approximately to the reaction coordinate for 0 transfer. As illustrated in Scheme I, stabilization
+NO
Precrent address: Department of Chemistry, University College, Dublin
0022-3654/92/2096-10308$03.00/0
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H'
H' L
+
4, Ireland.
OH
H
H,
'
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H
H '
H
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H
H +NO
H
+NO
of the transient cH2CH26biradical so formed involves competition between three ultrafast (presumably picosecond) processes, namely, ring closure to give epoxide, 1,2-H migration to yield acetaldehyde, and trapping by the NO cage neighbor to produce ethyl nitrite radical.' Interestingly, in all alkene*N02systems studied thus far, the biradical is trapped by NO in its nascent conformation. Furthermore, comparison of very recent ab initio work by Fueno et a1.9 and MeliusloJ1with the energetics of the NO2 C2H4system' shows that the 'D(uu) electronic ground state of tH2CH26(with both C and 0 unpaired p-electronorbitals in the CCO plane) is the only biradial state that is energetically accessible at the 575-nm photolysis threshold. A question of fundamental interest in bimolecular photochemistry is the behavior in the vicinity of a reactant dissociation threshold How does the branching evolve between the reaction path(s) accessible at photon energies below the dissociation limit and new one(s) opened up by reactant fragmentation as the photon excitation energy is i n d above the threshold? Our method of laser photochemistry of N02*hydrocarbonpairs in a rare gas matrix is especially suited for addressing this problem in the case of a solid matrix cage reaction. Nitrogen dioxide is an ideal reactant to study this question because optical absorption of NO2 in the entire range 350-1000 nm is dominated by the same 2B2 R2AIcomponent of the heavily mixed 2B2-%2AIstata. Hence,
+
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8 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No.25, 1992 10309
Oxygen Transfer from NO2 to Ethylene
the only change at 398 nm is switching from a region of 2Bzmixing with discrete ZAlstates to a region in which the 2B2levels couple into the R2AIcontin~um.'~-'~ The reaction with ethylene offers specialadvantages over that of higher alkenes. Fmt,interpretation of the results is aided by recent ab initio calculations on eH2CH26.9J0Js Second, the chemistry of singlet and triplet oxirane biradical eHzCH26is known in detail from extensive bulk gas and molecular beam studies on the O(3P) + C2H4 reaction,11~629 While 0 transfer upon excitation of NO2at energies below the dipsociation threshold can only proceed by large amplitude NO-C motion, two competing paths become accessible in principle when the reactant is excited above the dissociation threshold: The N02-ethylene system could react along the large-amplitude 0 transfer coordinate to yield oxirane biradical + NO, Le., follow the same reaction path already observed upon excitation to nonpredissociative levels of NO2 by visible light.' The newly accessible path is dissociation of NO2 to O(3P) + Ground-state 0 atoms could subsequently react thermally with an ethylene neighbor, producing exclusively triplet oxirane biradical provided that spin interaction with 2 N 0 coproduct plays no role. This outcome would be different from the visiblelight-induced NO2 ethylene (or butene) photochemistry where the evidence obtained thus far suggests that oxirane biradicals are formed in the sin et ground state.'*F5 Hence, if the products of singlet and triplet H2CH26were different, as they are in the gas phase,"J6 the branching between the two reaction paths accessible above the NO2dissociation limit could be established by product analysis. In fact, in a recent comparison between the photochemistry of cycloalkene;N02 pairs (cyclohexene, cyclopentene) in solid Ar induced at 355 and at A L 488 nm, we found a product, cycloalkenol, that appeared only upon 355-nm photolysis. We attributed it to the triplet oxirane biradical path opened up by NO2 O(3P) NO dissociation.M However, since no data were available on the energetics and chemistry of cycloalkene oxirane biradicals, interpretation had to be based on the analogy with ethylene oxirane biradical. Therefore, with the com hensive experimenta111J629 and ab initio data9J0Jsavailable on E 2 C H 2 6 ,much more detailed insight into the O-transfer chemistry in the vicinity of the NOz dissociation threshold could be expected from a study of the ethylene-NOz system. Comparison of bimolecular photochemistry above and below reactant dissociation thresholds is particularly interesting in light of recent results by Wittig and co-workers on molecule-molecule van der Waals complexes in molecular When photoinitiating H transfer from HX, excited above the dissociation limit, to various small polyatomics, it was found that for certain precursor geometries, product branchings and product state distributions were different from those expected for free H atom reactions. Above-the-dissociation-limit0 atom transfer in molecular bans has very recently been performed on the C2H4-NOZ van der Waals complex by Jouvet and co-workers (355-nm photolysis), which is especially relevant to the work re rted here. Reaction products characteristic for triplet cH2CHz chemistry with nonstatktical rotational and vibrational state distribution were 0bse~ed.33 In this paper, we report our results on the wavelength dependence of the photochemistry of C2H4.NO2and C2D4*N02pairs, isolated in solid Ar, in the range 555-355 nm. Dye and NdYAG lasers were used to excite the reactants, and the products were monitored by FT infrared spectroscopy.
t!
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+
8"
U. Experimentalsection Matrix suspensions of ethylene and NOz in solid Ar were prepared by slow continuous deposition of gas mixtures onto a 12 K cooled CsI window as described in detail in a previous report." The reactant concentration ratios used were ethylene /N02/Ar = 2.5/1/200 and 2.5/ 1/500 and C2H,/CzD4/N02/Ar = 1.25/1.25/1/200. The photocbemistry was followed by infrared spectro~copyUsing an IBM-Bruker Model 97 FT-IR spectrometer at OS-cm-' resolution, For visible/W spectroscopy, a Shimadzu spectrometer Model UV2100 was used, and the spectra were
typically recorded at 0.1-nm resolution. For laser irradiation, the 12 K cooled CsI target was rotated by 90' to expose the matrix to photolysis Light entering the vacuum shroud of the cryostat (Air Products Displex Model CSA 202) through a quartz window. Photolysis at 620,555, and 545 nm was performed with the output of an Ar-ion-pumped CW dye laser (Coherent Models Innova 90.6 and 599-01, respectively). Irradiation at 532 and 355 nm was conducted with the doubled (tripled) emission of a Quanta-Ray Nd:YAG laser Model DCR-2A ( 5 ns, 10 Hz). For photolysis at 420 and 385 nm, the 355-nm pulses of the NdYAG laser were used to pump the dyes stilbene 420 and LD 390 (Exciton), respectively, in a Quanta Ray Model PDL-1 dye laser. The power of the pulsed laser sources was measured with a Scientech power meter. Fluctuations in the laser output power at 420 and 385 nm were up to 30%. Ethylene (Matheson, 99.98%), ethylene-d4 (MSD Isotopes, 98%), and Ar (Matheson 99.998%) were used without further purification. Nitrogen dioxide (Matheson, 99.5%) was purified as described earlier.3a Acetaldehyde (Aldrich), acetaldehyde-d4 (Cambridge Isotopes, 99%), and ethylene-d4oxide (Cambridge Isotopes, 99%) were used as supplied. Ethylene oxide was obtained from Pennsylvania Engineering as a 1:8 mixture with CClZF2.The latter was removed by repeated vacuum distillation from a -40 OC bath. Removal of the Freon gas was checked by infrared spectroscopy.
III. Results We will fmt present the spectral data upon which the photolysis products were identified. Then, the photolysis wavelength dependence of the product distribution and growth kinetics will be discussed. 1. Infrared Spectra. 1.1. C2H, + NO2. Infrared data in the 2200-400-cm-' region are sufficient to identify the reaction products; hence, presentation of spectra will be limited to this region. Spectra of the reactants C2H,and NOz have been rtported in a previous paper,' as have those of N204 isomers and traces of N203 inevitably present in the matrix (Table I of ref 3a, which also shows the photoisomerization behavior of these species under visible light irradiation). No infrared absorptions of ethyleneN02 cage pairs separate from those of the isolated reactants were observed. As discussed in our earlier report on C2H4 NO2 photochemistry in solid Ar, the threshold to reaction lies at 575 nm.' In order to minimize interference by infrared spectral changes originating from N,Oy photoisomerization, ethylene/ N02/Ar matrices were irradiated at red wavelengths for 1-2 h immediately following deposition of the matrix. Such preirradiation resulted in only small intensity changes of N,Oy species when the photolysis laser was subsequently tuned to wavelengths beyond the 575-nm ethylene + NOz reaction threshold. The products observed are ethyl nitrite radical (1665,780 cm-I), NO (1872 cm-l), acetaldehyde (1749, 1726 (both overlapped by sym-N204bands), 1431,1427,1399,1349,1113,508 cm-l), and ethyleneoxide (1466,1274,1150,1126,876,818,814cm~'). The infrared spectra of the latter two products were identical with those of authentic samples obtained in our laboratory (see below). Figure l a shows product growth upon 5-h irradiation of a matrix C2H4/NOz/Ar= 2.5/1/200 at 555 nm (400mW No other product bands due to reaction of C2H4 with NOz were observed at any of the visible photolysis wavelengths used (555, 545,532, and 420 nm). In particular, independent of whether the continuous output of the CW dye laser or the 532-nm (420-nm) beam of the Nd:YAG (pulsed dye laser) source was used, the products were the same in each case. By contrast, photolysis at wavelengths shorter than the 398-nm dissociation threshold of NO2? namely, at 380 and 355 nm, resulted in observation of a new C2H4 + NO2 product absorption at 2143 cm-' (with a shoulder, presumably a second site, at 2147 cm-l). Figure 2a shows the growth of this band upon prolonged 355-nm photolysis of a matrix C2H4/ N02/Ar = 2.5/1/200. Due to its characteristic frequency and D isotope shift (see below), the band can be unambiguously assigned to the v(C-0) mode of ketene.34v35Aside from CH2= C - 0 + NO, the final oxidation products, acetaldehyde and
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10310 The Journal of Physical Chemistry, Vol. 96,No. 25, I992
I
C2tl4/N02/Ar-
Fitzmaurice and Frei C2H41N021Ar = 2.5111200
0.5
2.5111200
a Q
P
0
2000
le00
moo
MOO
1203
io00
e00
600cm-l
0 300
400
500
600
700nm
300
400 500
600
700nm
0.005 T
2000
1800
1600
1400
1200
1000
800
ewcm-1
Figure 1. Infrared difference spectrum obtained upon 555-nm photolysis for 5 h. (a) C2H4/N02/Ar = 2.5/1/200. CH3CH==0 at 400 mW bands at 1749 and 1726 cm-' are overlapped by sym-NzO4 absorptions. (b) CZD4/NO2/Ar= 2.5/1/200. Dashed bands are due to ethylene, NOz, N203, and N204. The latter two species isomerize under laser irradiation.
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C ~ U ~ I N O I I A ~ 2 5111200 355 nm
C204lN021A.r
-
355nm
25111200
Figwe 3. (a) UV-vis spectrum of matrix CzH4/NO2/Ar = 2.5/1/200. Fine structure originates from vibronic transitions. (b) Difference of spectra taken before and after 10-min photolysis of matrix C2H4/ NOZ/Ar = 2.5/1/200 by 488/514 nm Ar ion laser emission at 200 mW This was preceded by accumulation of ethyl nitrite radical by 7-h irradiation of the matrix at 555 nm. Aside from the continuous a b sorption tail, no features were reproducible (noise = 0.001 au).
at 21 13 cm-l (with a second site at 2118 cm-l), shown in Figure 2b. The 3O-m-' shift relative to the corresponding C2H4 NO2 product absorption at 2143 cm-l allows identification as CD2=
+
c4.34.35
2. Electronic Spectra. Electronic absorption of a matrix C2H4/N02/Ar = 2.5/1/200 in the 300-700-nm region is displayed in Figure 3a. Although the infrared spectra show that 0.01 I 0 I 2160 2150 2140 2130cm-' 2125 2120 2115 2110 2105cm-' N204species (both symmetric and asymmetric are present in the matrix even at this relatively low NO2 concentration, a b comparison of Figure 3a with the gas-phase spectra of NO2 and Figure 2. (a) Growth of CHz==C=O upon 13-h irradiation of matrix N2042'10.41indicates that the latter does not contribute detectably CZH,/NO2/Ar = 2.5/1/200 at 355 nm (10 mW (b) Growth of to the optical absorption at wavelengths longer than 300 nm. For CD,-C--O upon 8-h irradiation of matrix C2D4/N02/Ar = 2.5/1/200 example, the ratio of the absorbance at 340 nm (where N204 at 355 nm (10 m W would have its maximum) and at 400 nm (NO2 peak) of the spectrum in Figure 3a is the same (0.69) as in a NOz reference ethylene oxide,already observed upon X 2 420-nm photolysis were spectrum free of dimers. Moreover, no charge-transfer absorption also formed. Control experiments with N02/Ar matria showed of ethyleneN02pairs is observed at wavelengths longer than 300 that all other new infrared absorptions that grew upon prolonged nm. We have recently found charge-transfer (CT) bands of 380- or 355-nm irradiation of C2H4/N02/Armatrices are due 2-buteneN02 and cycloalkeneN02 pairs in solid Ar with maxima to photodissociation of NO2 and subsequent chemistry of the around 340 nm.4* The fact that ethyleneN02 pairs do not exhibit fragments: NO, 1872 cm-I; ~ i s - ( N 0 ) ~1864 , and 1776 cm-I; a CT absorption at X >300 nm is consistent with the 1 . k V higher tr~nr-(NO)~, 1758 cm-I; 03,1035 cm-'.The (NO), and O3bands ionization potential of ethylene compared with 2-buten~.'&'~ were weak and appeared only at photolysis times well beyond thw Hence, at the matrix concentrations used in this work, the sole used to determine product growth kinetics (see section 3). species responsible for the optical absorption in the 300-700-nm 1.2. C2D, NO2. Aside from absorptions of NO2 and N,O, range prior to photolysis is spectrally unperturbed NO2. species, matrices C2D4/N02/Ar showed ethylene-d4 reactant The difference spectra of the visible spectral region recorded bands at 1495,1075,720,715, and 596 The products of upon photoinitiation of the ethylene + NOz reaction showed an the visible-light-induced reaction (555, 545, 532, and 420 nm) absorbance demasc, indicating loss of NO,. However, the product compondcdto thw observed upon CzH4 + NO2 photochemistry cH2CH20N0must poaoess an absorption in the visible range as at these wavelengths: cD2CD20N0 (1664, 915, 767 cm-I), well because PhotodiPsociation of the radical occurs at wavelengths as long as 575 nm aocording to the infrared study.' By analogy C D 3 C M (1726,1150,1044,1023,940 cm-I), and CD2CD20 with our previous work with 2-butene + NO2 and cycloalkene + (1311,1014,964,898,808,804,757,755,591 cm-I). The acNOz,'z the best method of recording the visible spectrum of etaldehyde-d, and ~thy1ene-d~ oxide spectra were identical with c H 2 C H 2 0 N 0is by first accumulating the nitrite radical by those of authentic samples of the two molecules measured in a irradiation of a matrix ethylene/N02/Ar = 2.5/1/200 for a separate series of experiments (see below) and agreed with corFigure 1b shows the responding literature gas-phase ~pectra.~'J* prolonged period of time at 555 nm, followed by brief (10 min) spectra of these products as monitored upon prolonged photolysis photolysis of the radical at 488/514 nm. Previous infrared work has shown that such brief irradiation periods with bluegreen light of a matrix C2DI/N02/Ar = 2.5/1/200 at 555 nm (400 mW results in complete depletion of ethyl nitrite radical without sigcm-2). Photolysis at 380 and 355 nm resulted in observation of a single new C2D4 + NO2 product band aside from those of nificant reaction of ethyleneN02pairs. Therefore, the difference perdeuteratcd acetaldehyde and ethylene oxide, namely, a feature of the spectra taken before and after the blue-green irradiation
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Oxygen Transfer from NOz to Ethylene
The Journal of Physical Chemistry, Vol. 94, No. 25, 1992 10311
C2H4 + No2
C2D4 t NO2
555 nm
555 nm
a06
SCHEME 11
0.06
8
6 n
18/"', O
2
4
6
8
h
+
O
2
4
6
8
h
b
a
F w 4. Absorbance growth upon 555-nm irradiation at 400 mW c d .
.
(a) CzH4/NO2/Ar= 2.5/1/200. I: c z H 4 0 N 0 , 1665 cm-I. A: CH3CHO, 1349 cm-'. E: CH2CH20, 876 cm-'. (b) C2D4/N02/Ar = 2.5/1/200. Ld4: c z D 4 0 N 0 , 1664 cm-l. A-d4: CD,CDO, 1150 cm-I. E-d4: CD2CD20, 964 cm-I. Absorbances of A-d4 and E-d4 are normalized as described in the text.
.
period, displayed in Figure 3b, is attributed to CH2CH20N0. Although the signal-to-noise ratio is rather small due to the low steady-state buildup of ethyl nitrite radical that can be achieved in the matrix, an absorption tail extending to approximately 580 nm can clearly be seen. By analogy to the spectra of the higher alkyl nitrite radicals studied previously, the absorption is assigned to excitation of a CT state involving transfer of the electron on the C radical center to the strongly attracting nitrite 3. Photolysis Wavelength Dependem of Product Bmochhgs. Infrared absorbance growth kinetics was measured upon photolysis at 555,420,380, and 355 nm for CzH4 + NO2 and CzD4 NO2 in order to obtain insight into the change of product branchings as the excitation photon energy is tuned through the 398-nm dissociation threshold of NO2. Since measured widths of the infrared bands were practically constant during photolysis, peak heights were used for analysis. Figure 4a shows absorbance growth curves for cH2CHzON0 (I, 1665 cm-I), acetaldehyde (A, 1349 cm-I), and ethylene oxide (E, 876 cm-I) upon prolonged 555-nm photolysis (400 mW cm-2) of a matrix C2H4/N02/Ar= 2.5/1/200. Growth curves of species I-d4 (1664 cm-I), A-d4 (1 150 cm-I), and E d 4 (964 cm-I) in an identical experiment with a matrix C2D4/N02/Ar = 2.5/1/200 are depicted in Figure 4b. In the latter plot, CD2CDz0 and C D $ M absorbances were normalized by extinction coefficient ratios E-h4 (876 cm-I)/E-d4(964 cm-I) = 3.82 and A-h4 (1349 cm-')/A-d4 (1 150 cm-')= 2.60. The extinction coefficient ratios were determined by preparation of matrices CH3CHO/ CD3CDO/Ar = 1/ 1/ 1OOO and CHzCH20/CDzCD20/Ar = 1/ 1/ 1OOO. Hence, comparison of parts a and b of Figure 4 shows directly the D isotope effect on the change of the acetaldehyde/ethylme oxide product branching. The partial induction period of the E h 4 and -d4growth curves indicates that sbcondary photolysis of trapped ethyl nitrite radical, h4 and d4 forms, contributes significantly to ethylene oxide absorbance growth. The acetaldehyde-h4growth also exhibits slight induction behavior. On the other hand, the non-zero slope of all curves at the start of photolysis implies that all products are formed by direct single-photon photolysis of ethylene-N02pairs as well. Approximating the growth arising from reaction of ethylene*N02pairs by the product increase over the fist hour of photolysis, we find absorbance growth ratio M(E-h,, 876 cm-l)/M(A-h4, 1349 cm-') = 0.93 and M(E-d4, 964 cm-')/M(A-d4, 1150 cm-I) = 3.17 (in the latter case ratio of normalized absorbances is given, as described above). Therefore, the epoxide/aldehyde branching ratio increases by a factor of 3.4 as C2H4 is replaced by CzD4. Using the measured extinction coefficient ratio CH,CHO( 1349 cm-')/CH2CHz0(876 cm-I) = 0.89 and assuming identical extinction coefficients for u ( N 4 ) of I-h4 and Id4, we find that
+
the branching ratio I/(A E) increases by a factor of 1.7 when C2H4 is replaced by C2D4. These D isotope effects were calculated more accurately by determining the rate constants k associated with the reaction steps shown in Scheme 11. According to the partial induction shape of the ethylene oxide and acetaldehyde curves and the non-zero slopes at the start of photolysis, both aldehyde and epoxide are produced by direct,one-photon photolysis of ethyleneNOZpairs as well as by secondary photolysis of ethyl nitrite radical at this irradiation wavelength. The corresponding differential equations are d[RI /dt = -(h + k3a + k3e) [RI
(1)
d[Il/dt = h [ R I - (k2a + kze)[Il
(2)
+ k2AIl d[Al/dt = h [ R I + kzJI1
(3)
d[El/dt = M R I
(4) where R stands for the fixed reservoir of ethylene*N02reactant pairs and kl= e!FdlZIur, &k = &4hIl kh = e&Zhr, kk = c ~ F ~ and ~~ k3eZ = evil I ~l4)gbr ~, and 4" are decadic extinction coefficients of NOz and ethyl mtnte radical, respectively, at 555 nm; I$ is the reaction quantum efficiency;Z,.- is the laser photon intensity). Application of the weak absorption limit a p proximation is justified by observation of weak absorption in the 550-nm region at these NO2 concentrations (see Figure 3a). Integration of the differential equations (1)-(4) using the integrating factor method in the case of eq 244gives (expressed in absorbances)
($!
AR
AgRe
