NOTES
588 agree more close$ with the corresponding measured T1,,traH than for CE1J a t the same (r value as spinrotation effects upon T1 lntraH should be reduced relative to the more efficient homonuclear-dipolar process; the results in Table I1 for u = 20 support this position. While a complete analysis of molecular motion in liquids requires the relaxation times and q values of more than one m a gnetic nuclei in a molecule, the above study does illustrate the facility with which the dipolar contribution bo 'C relaxation times can be evaluated from TI and ?lonochromatic radiation of intensity, Io(in einstein/cm2 sec), at the surface of the layer is passed through the sample. After time, t, at a distance, X , across the layer, the intensity is I. The dependent variables, I and C, are to be related to the independent variables, x and t , Only two equations relating the variables are known, the bert equation which is valid in the differential form dI/dx
=
-&I
(1)
-$uCI
(2)
and the local rate equation dC/dt
=
where a is the absorption coefficient and 4 the quantum yield. Solutions which satisfy eq 1 and 2 are6 I=IoX
The Photo8cbemisitry o f Solid Layers. eaetion Rates
exp ( -U~C _o X_) _ _ _ _ _ _ _ _ _ ~exp ( -cpaI,t) [ 1 - exp ( - ~ C o x] ) exp ( -~ C O X ) (3)
+
and by E. L. Simmons Department of Chemistry, University of Houston, Houston, Texa:i W O O 4 (Receiued August 31, 1970) Publication costs bwne cowwletely by The Journal of Physical Chemistry
Recently I h c x has been some interest shown in the photochemical reactions of solids in the form of thin films or layers,'-5 As yet, however, the mathematical description of i-he rate of such a process has not been completely developed, Such a description is complicated by the fact bat the reactant molecules cannot readily diffuse in a d i d sample and a concentration gradient is therefore created by the photochemical reaction. The problem of the variation of the reactant concen*ration vith the distance across a solid layer during a reaction in which there is no radiation intensity gradient has been theoretically treated for special cases in which the diffusion of gaseous reactants 01" products in t h e layer is important.' Approximate solulions have a190 b w n obtained for the case in which both the reactant cowentration and the radiation intensity vary across the ,sample l a ~ e r . ~ - j I n the trentment reported here an exact solution is obtained which describes the rate of the photochemical reaction of a solid lstyr for the case in which the photoproducts are traysparent. An approximate solution is obtained for the case i n which the photoproducts absorb radiation, and 21 metliod for determining the quantum yield is described, T h e Journal of Physical Chrmistvy, Vol. 75, ATo. 4, 1971
c=cox exp ( -4aIot) exp ( -4uIot)[ 1 - exp ( - aCox) ]
4- exp( -aCox) (4)
These solutions are obtained by combining eq 1 and 2 to obtain -4dI = -(dC/dt) dz
(5)
Integration of eq 5 over x making use of the condition that I = I, when x = 0 gives
An expression for I is obtained by integrating ey 1
(7)
s,"
An expression for the parameter C' d x is obtained by substituting eq 7 into eq 6 and integrating making use of the condition that .f$C dx = Cox when t = 0 (1) J. E. Wilson, J . Chem. Phys., 22, 334 (1954). (2) P. G. Barker, M. P. Halstead, and J. H. Purnell, Trans. Faraday SOC.,6 5 , 2389 (1969). (3) T. R. Sliker, J . O p t . Soc. Amer., L3, 454 (1963). (4) P. G. Barker, >P. !II-Ialstead, . and J. H. Purtiell, Trans. Faraday Soe., 65, 2404 (1969). (5) H. E. Spencer and M. W. Schmidt, J . Phys. Chem., 74, 3472 (1970). (6) NOTEADDEDIN PROOF.It has come t o the attention of the author that Mauser [H. Mauser, Z. Naturforsch., 22b, 569 (1967) I previously obtained similar equations by a different method for
viscous samples.
NOTES
589
PZ
J, c dz
=
eq 1 must be modified by adding to the right-hand side the term - xa,gjC, where a3 is the absorption coef-
(a-1)$6
exp ( 4alot) [ B -- exp ( - aCoz) 1 exp ( -aCox)
+ exp ( -aCox)
--_l__-____l_l_
(8)
3
ficient of the j t h product, g3 the stoichiometry coefficient of the 2th product, and Cj the concentration of the j t h product. Since
+
co = c (gj)-lC? (9) Differentiation of eq 8 with respect to 5 along with the proper algebraic rearrangements gives eq 4. Subeq 1 may be written as stitution of eq 8 into eq 7 gives eq 3. dl/dz = -(a - b)CI - bCoI Figure I illustrates the intensity and reactant con(10) centration profiles across a layer. These curves were uhere b = z a j g l 2 . By carrying out the same proobtained using eq 3 and 4 and by arbitrarily choosing 3 a = lo4, C0 = l P 3 , I, = lo-*, and 4 = 1. Curves cedures using eq 2 and 10 as for case 1 given in the are given for variou,s reaction times. The concentraprevious section (eq 5) the follo\\ing equation is obtained tion gradient created by the photochemical reaction is evident. The gradient causes deviations from the exponential deciease of the radiation intensity across tlie layer predicted by the Beer-Lambert equation as can toe seen in the lower part, of Figure 1. Case 2. ~ ~ ~Absoyb~ Radiation. ~ ~ Although ~ ~ o ~ ~ c t ~ an exact solulion for the case in which the photoFor small t when C/Co does not differ greatly from one, products also absorb radiation was not obtained, an the logarithmic term in eq I1 may be approximated by approximate solution which is valid for short reaction the first term of the series expansion times was ctervved. For this case, eq 2 is valid, but In (C/Co) C/Co - 1 (12) m
On substituting eq 12 into eq 11, eq 10 and 11 may be treated in a manner similar to that for eq I and G in the previous section to obtain the following solutions
f=O
0.8
IZIOX
t 0.6 3
exu( - aCoz) exp[-(a - b)tplot][l - exp(-uCoz)]
0.4
4- exp(-uaC~.c) (13)
.2
and t
1
exp [ - (a - b)410tJ exp( - aCoz) ____ .- exp( - aC@) ____--______ exp[-(a - b)410t][1- exp(-aCox)] -b exp(-aCoz)
t
(14)
x, cm. Figure 1. Radiation intensit,y and reactant concentration a t various distances across the layer and various reaction times. The curves were obtained using eq 3 and 4 and arbitrarily choosing the following values: a = 104 cmz mol-1, Co = 10-3 molic~n-3,I , = 10-4 ainstein ern-2 sec-1, and = 1 mol einsiein-1. Reti,ction t,imes given are seconds.
+
As expected, eq 13 and 14 reduce to ey 3 and 4 when b = 0. On varying b , there is a discontinuity in eq 14 when b = a; thus, for this cas?, eq I4 cannot be used to calculate the reactant concent,ration. Applications. The most apparent application for the equations obtained in the previous sections is connected with the quantum yield determination of the photochemical reaction of a solid layer. To determine the quantum yield it is convenient t o have some easily measured parameter related t o the reaction time. For solid layers the most easily measured parameter is usually the transmittance, T = 1 / 1 0 , of the layer. Equation 3 may be written in icrms of the transmittance as The Journal of Phvasical Chemistrg, Val. 76, N o . 4, 1971
NOTES
590
T
+
To'D[(1-- To) @ ~ p ( - # d o t ) To]-'
(18)
where To = expj -aCod) (z is replaced by the total layer thickness, cl). By rearranging eq 15 and converting it to the logarithmic form, eq 16 may be obtained In [(I -- 'T)To/'(l. - To)T] = - & d o t
(16)
Thus, a p!ot of In [(I - T ) / T ] us. t is a straight line with a slope related to the quantum yield. If a and 6, are known, the quantum yield can be determined from such R plot. For the C A W in ahich the photoproducts absorb radiation, eq 14 may be treated in the same fashion to obtain In [(I - T)To/(l - To)T]% --$(a - b)Iot (17) which i s valid for small reaction times. It i s therefore apparently possible to determine the quantum yield of the photochemical reaction of a solid layer by meawring She transmittance of the layer as a function of time Proof of the applicability of the method must, howver, await experimental evidence. Ack~iotdtdgme~ai.The author wishes to thank the Science Research cIounci1, London, England, and the Sandia Corporation, Albuquerque, New Mexico, for financial support during a large portion of this work.
Kinetics of Chemical Ionization.
I.
eaetion of tert- : a H g + with Benzyl. Acetate
by
S.Vredenberg, L,Wojcik, and J. H. Futrell*112
Depaitment of Chcmtstry, The Universitq of Utah, Salt L ~ k City, e Liah 84112 (Rcceived September 14, 1970) Publication costs assisted 521 the Air Force Materials Laboratory, United States Air FoTce, Wright-Putterson A i r Force Base, Ohio
Recently, Field rqmrted a study of the kinetics of reaction of t e r K lRg" generated in a high-pressure ion source with benzyl ~ t c e t a t e . ~This pioneering attempt to measure rate parameters quantitatively under the conditions of chemical ionization mass spectrometry provided evidence that the rate constant was larger by a factor of 10 imder certain conditions than could be r,ztionalized on the basis of simple models for ion-molecule reactions4 and that the rate constant exhibited a negative tempwa turc coefficient. Gsing methane reagent gas to gentmte CH5+and C2H5+,Field also found a rate about five tirnes larger than could be justified theoretically ; this system, however, exhibited no variation of rate with temperature. Since there is no ready explanation for these phenomena, we have repeated the measurements using a high-resolution chemical ionization mass spec1,romeler of rather different design.5 T h e Jouinal of Physzral Cliemast?y, Vol 76, ,To. 4, 1971
Experimental Section Our experiments were carried out using a CEC Model 21-l10B mass spectrometer which was modified for chemical ionization studies as described previously.5 The gas-liquid inlet system is further modified by the addition of a septum inlet so that samples may be introduced directly into the expansion volume with a microliter syringe, ensuring precise sample measurement and good reproducibility. The samples are injected into a calibrated 3-1. volume maintained a t 130". The sample pressure is then increased to several Torr with reagent gas; a rather dilute mixture of benzyl acetate provides better sample control. It is also necessary to use enough pressure for the gas to flow into the source which is a t 0.5 Torr pressure. Typical sample sizes are approximately 15 pl, and backing pressure is about 70 Torr. The required amount of sample is introduced into the source from the reservoir through a manual leak valve, and the pressure of sample-reagent gas mixture is measured with an M I I S Raratron capacitance manometer. Additional reagent gas is then added from the high-pressure reservoir to establish and maintain the total pressure at 0.5 Torr. Source temperature is controlled in the normal fashion.
Results and Discussion Chemical ionization experiments are carried out using a sufficiently high pressure of the reagent gas so that hundreds of ion-neutral collisions occur.6 ,4 reagent gas is chosen whose primary ions react in a series of fast ion-molecule reactions to generate one or a small number of ions which are unreactive toward the reagent gas. These reagent ions react efficiently with the additive molecules in the manner indicated schematically by the reaction
E+
+ Ad *Ad+
+ hT
(1)
A host of reactions are possible: charge transfer, proton transfer, condensation, dissociative proton transfer, etc. In the particular example under consideration, with i-C4H9+as R + and benzyl acetate as Ad, the product ionsare 91 (PhCH2+),107 (PhGH,O+), 108 (PhCHzOH+), 147 (PhCH20C4Hg+), 181 (PhCH?OhcH+), 181 (protonated dimer Iess 2C&COOH?), 241 (protonated dimer less CHXCOOH?), and 301 ([PhCHsOAc]J€+). The distribution of products is a fairly strong function of temperature and at 8 given temperature our results are in generally good agreement with (1) Alfred P. Sloan Fellow. (2) This investigation was supported in part by a Public Health Service Research Career Development Award, N o . 1 KO4 GM4239001, from the National Institutes of General -Medical
Sciences. (3) F. H. Field, J . Amer. Chem. Soc., 91, 2827 (1969). (4) S. K. Gupta, E. G. Jones, A. G. Harrison, and J. J. Myher, Can. J . Chem., 45, 3107 (1967). ( 5 ) L. H. Fojcik and J. H . Futrell, Rev. Sci. Inslrum., in press. (6) F. H. Field, Accounts Chem. Res., 1, 42 (1988).
