KINETICS OF PHOTOCHEMICAL DECOMPOSITION OF SOLIDS (I) Introduction. The Rate Equation

Transcription

1 KINETICS OF PHOTOCHEMICL DECOMPOSITION OF SOLIDS (I) D. Ftu model whch descrbes the knetcs of photochemcal decomposton: a (S) b (S) cc (the reacton area beng located near the reactant surface) s presented. The general rate equaton s dscussed for varous ratos between the radaton absorpton coeffcents of the solds and. The curves (reacton rate, degree of converson) have been calculated usng a program wrtten n SIC language. (g) Introducton The nterest generated by the nvestgaton of photochemcal decomposton of solds are determned manly by ts applcatons n the photographc technque [1,2]. Reacton mechansms for photochemcal decomposton of alkalne haldes, azdes, perchlorates have been worked out [3]. Ths paper ams a general knetc model, whch takes nto account the mass absorpton coeffcents of X-Ray of the reactant and product. The results are valoo-wth some modfcaton for other radatons ether n statc or n knematc condtons. The Rate Equaton In order to descrbe a reacton whch occurs accordng the equaton: a S h b (S) cc the degree of converson defned by the relatonshp (1) s gong to be used: N N (g) N m m (1) m m where N and N are the number of partcles of the reactant at the ntal moment t and respectvely at the moment t; m, m and m f are the sample mass for = at the ntal moment t, for the moment t and respectvely for the total decomposton of (S). In the followng we shall express the reacton rate through d/dt. Takng nto account the relaton (1) t follows that: f Department of Physcal Chemstry, Faculty of Chemstry, Unversty of ucharest, 4-12 lvd. Elsabeta, ucharest, Romana

2 54 D. FTU d 1 N d N The photophyscal prmary processes, generate actve ntermedate speces lke exctons, postve holes whch mgrate and near rradated surface of sold reactant (S) are trapped by lattce defects (vacances, dslocatons, F centers etc.). The groups of such assocated and captured actve ntermedates decompose generatng the reacton products. The gaseous product C (g) s evolved and (S) forms a surface layer. The further consderatons are based on the followng hypotheses: 1. The lmtng step of the decomposton s the smultaneous capture of a number of ( = 1, 2, 3) actves partcles at a defect level located near the reactant surface and thus generatng one actve group. 2. Durng the rradaton a steady state s reached, thus the number N g of the actve groups s constant anher decomposton rate equals ther generaton rate. In such condtons the decomposton rate can be expressed as: d N (2) d N g (3) takng nto account that an actve group consst of reactant partcles. 3. The capture process can be characterzed by a constant K, whch s the capture probablty W of an actve ntermedate by a trap n the surface vcnty for a radaton ntensty I a (watt/m 2 or ensten*s 1 *m 2 ) absorbed by the reactant Thus, accordng to the defnton: W K (4) I a 4. The rate of actve group generaton equals the product between the smultaneous probablty of partcles captured anhe number of defects. Thus: d N dt g W n S (5) where n d and S are the surface defects concentraton and respectvely the rradated surface area. The capture of the I partcles s a compound event whose probablty s expressed by the product of the probabltes of ndvdual events. Combnng relatonshps (2) (5) we get: d KI d a nd N In order to evaluate the ntensty I a one has to notce that the energy of the ncdent radaton s partly absorbed by the product. ccordng to the hypothess no. 1 the S (6)

3 KINETICS OF PHOTOCHEMICL DECOMPOSITION OF SOLIDS (I) 55 decomposton s located at the reactant surface where the volatle component s evolved. s a consequence the product (S) appears as a layer wth growng thckness n tme. Fg. 1 shows the successon of sold phases and n the reacton system, as well as the change of the radaton ntensty wth the space coordnate. Fg. 1: The successon of sold phases and n the reacton system: M-N phases nterface (upper part). Curve I(x) (ntensty vs. poston coordnate; downsde of the fgure). The ntensty of the radaton (transmsson flux densty) whch penetrates a layer from a materal wth the lnear absorpton coeffcent (m 1 ) decreases wth the thckness x accordng to the exponental law: I x where I s the ntensty of the ncdent radaton. x The rate of absorpton R a (watt. m 3 ) s defned by: I e (7) d I x Ra I e (8) d x ecause the sold phases (s) and (s) are homogeneous, µ s a constant. For the layer of reactant (s) (fgure 1), wth thckness l and absorpton coeffcent µ, the absorbed power densty I a s obtaned by ntegraton. On the other hand ' I a a l ' x I e ' l l dx I I e (9) I s the ntensty of the beam transmttehrough the layer of the ' product wth the thckness l anhe absorpton coeffcent. Thus I could be expressehrough the ntensty I of the ncdent radaton at the external surface of the system by the relatonshp:

4 56 D. FTU ' l I I e (1) Takng nto account ths result the relatonshp (9) can be transcrbed as: a l l l I I e e (11) In order to reduce the number of varables, whch descrbe the knetc behavour of the system we must express l and l through the degree of converson: l l m 1 (12) S mbm (13) M a S where m s the ntal mass of the reactant and x the densty of phase x. The molecular weght M and M as well as the coeffcents a and b enter n relaton (13) because the mass of the product was expressehrough the mass of the transformed reactant. m From relatons (6), (11) (12) the followng rate equaton s obtaned: d 1 e m KI n S 1 S a M S d N e m b M The rato / (m 2 /kg) s the mass absorpton coeffcent of the radaton n the phases and and can evaluated usng the lterature data. If phases and are well separated, the reacton occurrence can be followed ether measurng the product layer thckness or measurng the reactant layer thckness. In such condtons: f (14) l l 1 (15) l l where superscrpts and f mean ntal and fnal thckness of the reactant and product. For a poor separaton of the phases and, the reacton s followed by recordng the mass change of the system tme. In such condtons the degree of converson s calculated by the relaton (1). Obvously f the system does not exhbt a phase stratfcaton and reacton product s dssemnated n the reactant mass, equaton (14) cannot be appled and hypothess 1) should be modfed. Relaton (14) shows that the photochemcal decomposton rate of solds depends on the radaton ntensty, on the wavelength (through the coeffcent ), on the temperature whch changes the concentraton n d accordng to an exponental law, on the rradaton surface

5 KINETICS OF PHOTOCHEMICL DECOMPOSITION OF SOLIDS (I) 57 area and obvously on the degree of converson and on the nature of the reactant anhe sold product. The mechancal, thermal and chemcal treatments, whch modfy the defects concentraton, could nfluence the knetcs of the photodecomposton. In order to express equaton (14) n a more compact form, the followng notaton s ntroduced: k KI nd N S k (16) k m (17) S m b M (18) S a M d v (19) where k s the overall rate constant anhe ratos k and k wll be called dmensonless absorpton coeffcents of the radaton. These are not materal constants as they depend not only on the ntrnsc propertes of phases and on the surface densty (m /S) too, thus on the sample thckness, whch can be changed from an experment to the other. Takng nto account ths notatons the reacton rate can be expressed by the equaton: for I = or = 1 v =. k k v ki 1 e 1e (2) Introducng n equaton (2) the condton = the ntal rate v results: k v ki 1e (21) ccordng to the presented model the reacton begns to occur wth ts maxmum value (v = v max ). The reacton rate can be evaluated from the expermental curves = f(t) anhe radaton ntensty by help of the defnton relatonshp. I P / S (watt/m2 ) (22) where P s the beam power. If the steady state settles up after an nducton perod, the ntal rate can be evaluatehrough extrapolaton. The logarthmc form of the equaton (21): lg v lg k k 1e lg I shows that the plot lgv f I s lnear wth the slope. (23)

6 58 D. FTU If the photochemcal decomposton s a sngle reacton anhe descrbed model can be appled one should obtan = 1 or = 2. Hgher values of mean smultaneous capture of three actve partcles by the same lattce defect, whch s a rare event. The coeffcent k can be evaluated by help of the relaton (17) anhe constant k from the ntercept. The shape of the rate curves The shape of the rate curves depends on the rato of the absorpton coeffcents of the reactant and product. To plot the rate curves s useful to defne a relatve rate v r accordng to the relaton: Introducng relaton (2) and (21) n (24), one obtans: v r v v r (24) v k k k 1 e e 1e 1 (25) The advantage of usng v r conssts on ts dmensonless character. Fg. 2 shows the rate curves v r f for the followng values of the rato k / k a).5; b) 1; c) 2 and for = 1. '' Fg. 2: The curve v r for three values of the rato k /k : a),5; b) 1; c) 2. For smplcty sake for all three curves k was kept at a constant values k 1. The used program wrtten n SIC language [4] allows to dsplay the numercal values of the reacton rate, the plot of the curves as well as the placng of expermental ponts on ths curves n order to compare the expermental results wth the theoretcal ones [5].

7 KINETICS OF PHOTOCHEMICL DECOMPOSITION OF SOLIDS (I) 59 s shown n Fg. 1 for the same value of the degree of converson ( =,5 for nstance), the reacton rate ncreases wth the rato k = k. Thus, the photochemcal decomposton s favoured for hgh values of the absorpton of the reactant and low values of the absorpton coeffcent of the product. Partcular cases a) Low values of the absorpton coeffcents. The fact that k and k depend on the layer thckness s an advantage snce usng samples of low mass and hgh exposton surfaces the condtons: k k 1, k 1 1 are fulflled. s a consequence the exponentals can be expanded n seres. Keepng only the frst two terms, relatons (2)-(23) and (25) take the approprate forms: k 1 v kk I 1 (26) lgv v kk I (27) lg kk lg I (28) k k 1 vr k 1 (29) more sutable for processng the data as these equatons can be ntegrated. b) The reacton product s transparent. If the reacton product does not absorb the radaton, ts accumulaton does not nhbt the decomposton. From (25) for k one obtans: k k 1 e 1e c) The reacton product s opaque k v r 1 (3) If the sold product s photo protectve the reacton s stopped vr as soon as the surface s covered by a thn layer of t. d) The reactant s transparent k. Obvously n such condtons vr anhe reacton does not occurs as beng condtoned by the generaton of actve ntermedate by the observed radaton. e) The reactant s opaque k. In such condtons, the reacton rate: e k r does not depend on the amount of the undecomposed reactant. v 1 (31)

8 6 D. FTU Irradaton wth movng sample When usng hgh power beams (UV, X ray, or laser radaton) the sample s placed n a movng carrer wth the constant speed [6, 7]. The speed (v l ) (fg. 3) s orented normally wth respect to the beam (F) wth the wdth (L) anhckness (d). Fg. 3: Irradaton wth movng samples: F the rradaton beam, N the sample carrer, x poston coordnate. Relaton (22) should be modfehrough the substtuton of the samples area S by the secton area S = Ld. The crossng tme t 1 of a sample surface pont wth the beam can be expressed as t 1 d / vl. In order to reach a gven degree of converson, a number of n passng through the beam are necessary. Thus, total rradaton tme can be calculated accordng to the relaton: nd t (32) v l Wth these changes the derved knetc equatons can be used for movng samples. Conclusons 1. general model of the photochemcal decomposton of solds was worked out. 2. The model allows plottng the curves of the reacton rate aganst the degree of converson. 3. The theoretcal model ndcates that the photochemcal decomposton of solds s favoured by hgh values of the absorpton coeffcent of the reactant and low values of the product absorpton coeffcent. 4. Fve partcular cases are represented n the framework of the general model wth respect to the absorpton coeffcent of reactant and product. 5. The changes of the obtaned equatons for movng samples are equally consdered.

9 KINETICS OF PHOTOCHEMICL DECOMPOSITION OF SOLIDS (I) 61 REFERENCES 1. Mees, K. (1954) The Theory of the Photographc Process, New York. 2. Mtchell, J.W. (1955) The Photographc Process n Chemstry of the Sold State, W.E. Carner edtor, London. 3. Jakobs, P.W. M. and Tomkns, F.C. The cton of Lght on Solds bdem. 4. Ftu, D. (1993) n.alele Unv. ucharest, Chm. II, Ftu, D. and Dnculescu, C. (2) nalele Unv. ucharest Chm. II, Popescu, C., lexandrescu, R., Morjan, I. and Popescu, M. (1991) Thermochm cta 184, Popescu, C., Vocu, I., lexandrescu, R., Morjan, I., Dumtras, D., Popescu, M. and Fatu, D. (1999) J. Thermal. nal. Cal. 56,