l[NTERillfOLECULklt Il:N Elt(;Y TK,ANSIJE:Ii IN (+*ISItEACTIOSS] RY KARL Criow A N I ) UAVIU J, WILWN Dcpurlmcr~lfdChemistry, U>izvvr\il?/ oJ I I ‘ ~h u x l v i , 1Corhrdcr $0, iVno York Ri’cPivd O C ~ O 8. ~ Ji901 ~T
Photochcmical and unirnoleculur reactions in whicli a rwwtive moleculc may rcnct by tvo diffcrcnt path8 to yield diutinnuishublo roducls are audyzed theoretically In roritrad to reactions which tlnlreplacc by only onc path, reactionr of I h i s ;type muy e o expected to vield con4drrst)lr qii:di(:itiw infox ination nhoiit, thc naturc of oollkionul cnergy transfer pro‘ esst+ involving highly energetic niolecules
I. Introduction A 1:irgo riiimbcr of papcrs have appeared in the last few years on the subject of internal encrgy transfer by molecular collisions in gases.* 1 1 difficulty noticed prcvioasly by scrcral authors is that of distinguishing, by conventional photochemical3& or unimolecular thcrma13b reaction rate studies, between the various types of plausiblc collisional energy tra,nsfer mechanisms. For unimolecular thermal reactions the situation appears hopeless; theoretical curves of rate constant us. pressure are essentially indistinguishable for strong and stepwise collision mechanisms. (A strong collisioii mechanism is one in which the average amount of energy transferred per effective colIision is large compared to k T ; a stepwise collision mechanism is one in which this quant,ity is comparable to 1 0 . ) Porter and Connolly showed that the situation was almost as bad for photochemical reactions; that very precise measurements of quantum yield as rz fiinctkm of prcssurc at rclat>ivelylow pressures would bc necessary to distinguish between strong arid stepwise collision mechanisms. Cnrrington2 has indicntcd thc difficulties of extracting prcniso qiianti1;:ilivo iiiforrnution even from transiciit or steady-state spectroscopic mcasuremnnt,s. (Ione f i s interested in only n qurtlitativc discrimination hcl;wcnn strong uiid stepwise collision rriocliani~ms,however, we iinil t.liat, the situation is iiut so black for spectmsnopin st,iidin,q.2.4. Rcccnt,ly, howevw, ~ O I K K rcmlts ? oil 3 particular 1,yptt of innimolcculw rewcl,ioii were piihliahcd by C l i e ~ i c k lLaLiiiovitchIR ,~ Wfilt,crs,’ l?rcy,* and I M r co-tvorkcrfi. Tn t,hcsr, rcac:t ions :L rcac tant riiolecule (mi rtxot, unimolccularly by two (or morr) different niodes 1 o yield chemically disidnguishable procluct,s. Wc felt, that thr: dopandeuce of the rntio of t,hc rate constants of the dil‘ferent mntlrs on pressure might yield informfition ahout, t,ha naturc of collisionnl cncrgy tmnfifcr in gas rcactious not obtainahlo from convcritioiiui .rate studies in which there is only one mode of :rcact:ion. It also was (1) This work was supportcd by grants from the Research Corporation a n d the National Science Foundation. (2) See, for cxampln, T. Carrington, J . Chem. Phys., 35, 807 (1961); D. J. Wilson, ibid., to be published (this paper contains several recent references); K. E, Shiiler, ibid., 32, 1692 (1960); E. W. Montroll a n d K. E. Shiiler, “Advances in Physics,” Interscience Publishers, Inc., New York, 1958, Vol. I (this article list,s many of the earlier papers). (3) (a) G . B. Porter and I3. T. Connelly, J . Chem. I‘hya., 33 81(1060); (b) F. P. Buff and D. J. Wilson, ibid., 32, 677 (1960). (4) 1). J. Wilson, B. Xohlo and I3. 1,ce. ibid., Sa, 1392 (1961). (-5) .I. 1’. Chcsick, J . Ana. Chem. Soc.. 82, 3277 (1960). ( 6 ) 13. S. Rabinovitch, E. W. Schlag and K. R. Wiberg, J . Chem. I’h?/%, 28, 504 (1958). (7) 11. R, Grrbcrich, 1’h.D. ?‘hr&. IJnircrsity of Rochrstrr, 1959. ( 8 ) A I . C. I’linvers and I f . M. Frry. J . Chem. Soc., 3953 (1959).
wggcslcd l o us by Ur, It. Srinivasan (thcn at, thr Uiiiveisity of Rochester) that, photorhomica1 reactions In which a rractant molrculo in oiic excitd elcctronic siatc can decompose by two or more dif-
ferent modes to yield chemically distinct products might be of interest. Wc are indcbtcd to Dr. Srinivasan for this suggestion and for discussing with us his results on what appears to be one such reaction, the photolysis of cyclopentanonc. The remainder of this paper describes briefly and reports the results of calculations on these unimolecular and photochemical “bireactions.” 11. Unimolecular Bireactions The commonly accrpted mechanism for a unimolecular rcaction is9 A,
+ M +AI + M A, --+
(Products),
a,, ci*
(1)
A; represents a reactant molccule in its ith internal energy state, M represents an inert “heat-bath” molcculc (the inert gas is presiimcd prcscnt in large excess in comparison with rcactant, and is assiimcd to be in lhcrmal equilibrium), and a,, t~ridci are the microscopic rate constants for thc processes with which they are associated. We gcncmlina this mcchunism to include a second mode of docompositiou by including the proccss A, --+(Prodilcts), ele (2) where (Prodiicta)l tirc chcmic:dly distiuguikhslble from (PPoducts),. Kat;wl’s model of R reacting rnoIrciilo10 as u sybtem or s coupled dcgcnmhtr hnr~nonicowillatoi s \ms iiscd to d c u l a t c thc rriivrobcopic clccomposifhn rate functiuris q’and c,l c,1
=
U,UD(t
- n”l)/g,(c)
(3)
where gs(j) is thc dcgrrioracy of the jth eiiergy state of t,hc systcm (containing J quanta of vihrational energy), v1 is a frequency factor, and n * I is the minimllm number of quanta required to yicld (Products)l from a molecule of renctnnt. A similar expression was used t o compute c,?. The quantities a,,were computed according to two models: (1) the strong collision model used by Kassel,lO Slatcr,ll& and others,llb in which activated molecules are deactivated into an equilibrium distribution (cssentially) by single collisions, and (2) the stepwise collision mechanism we treated = a(n+s)e-O, where a is in which (9) Sec 11. S Johnston a n d J. R White, J Chem P h y s , 22, l W 9 (1954),for oxample (10) L. S. Kassrl, ”Kinetics of IIornogencoiis C a s Reactions ” Chemical Cntalog Co , Inc , NIV York, N Y., 1932 (11) (a) N D. Slater, “Throry of Unimolcciilar Reactions,“ Corndl University Press, Ithaca, New York, Ic159, (1)) C S t r d , -1. Chem P h y s , 31, 899 (1059), for exarnplr.
n coIistaiil, 0 is hvLYlb/kT,:ind
V,ib ifl t,hc frcqiicncy the vscillaturs ruyroscutiiig Ohr! mo10c:ulo. (Mir,roscopic r.cvrrsihilit,y I ~ L P I I yields a i l + ~ , .l-l a(n -I- l).) l'lots of kl//;im uiid kz//in, as fuiiclhtis of IC&/ klw then were computed on the Cornell Aeroiiau tics Laboratories' IRM 1704 computer. The computation based oii the strong collision model employed the quantum mcchnnical analog of the formula mentioned by Chesick.5 The computation based on the stepwise collisioiinl model was performed by means of slight. modific:it,ionsof the procicdure used by Buff and Wilsonsb t,o handle t,he case in which only one react ion occurs. l2 We found t'hat plots of log kl//clm 1's. the log of the dimensionless pressure /cloM/lc~, were essentially indistinguishable for t he strong and stepwise mechanisms, as we had suspected from our earlier ~ o r k .(It ~ is ~ assunled here and following that nl*is less than n2*- that is, that the reaction producing (l'roduct,s)z has the higher critical energy.) This is demonstrated in E'ig. 1 and 2 ; the curves for the two collision mechanisms are essentially iridistiiig~iis~~ablc over tho erit,ire pressure range. Log-log plot,s of /cl/k2 L'S. dimensionless pressure were markedly different, for the two mechanisms a t low pressures, arid the differences were more mayked t,he greater the difference Ez - El. We therefore investigated Chesick's data on the pressure dependence of the ratio of the rate constants for the production from methylcyc1oprop:ine of the two normal butenes and of isobutene. We took Chcsick's ~ a l u eof 2.3 f 0.7 kcal. for E2 - El, increased this dilfert:ncc! to 3.7 kcal., and adjusted the rat,io of the pre-exponential factors so as to rctairi agreement with Chesick's value of lc2/kl a t high pressures. I n this way we hoped to detect, by leaning over hackwtrds, any discrepancy between t,he stepwise collision theory and Chesick's data. (Chesick had shown previously that his rcsults werc consist,eiit with the appropriate modification of Kassel theoryes) As is shown in Fig. 3, OUT attempt to use these data to eliminate the stepwise mechanism from consideration was a failure. However, there remains a definite possibility of discriminating between the strong and stepwise collision mechanisms by means of the study of unirnolccular reactions of this t~ypea t sufficiently low pressures. Figiires 4 and 5 demonstrate the feasibi1it)y of discrirniiiutirig between the two mechanisms by a study of the rate constant of the reaction having the higher activat;ion energy. R e had remarked
of
(12) The onls modifications made were (1) the inclusion of an extra term in fiuff a n d Rilson'a equationx 40 a n d 4 1 , corres~,ondinyt o t h e disappearance of reactant via the second irioilr of renrtion, itrid (2) t h e partitioning of X (their notation) i n t o ki arid kz. 'rliis last is retidily accomplished by accumulating t h o e x p r ~ s s i o n q c i l ~ i a
anti
ci2~i
i
dtlring t h e numerical solution of t h e differenco equations rrbov~ (reaction 1 is presumed to h a v e t h e 1owr:r nctivntion encrgy) and tlirii noting t h t
ii*
is just k i ; thc euprr8sinn for kz is obtninrd hy prrtnuting t h e suhsrript8 I m i l 2.
10-3 I - ._ -* IO 1 1
- -_
.
IO
-
103
102
I
104
101
104
k d f l h m.
Ihg. I - RtLte const:unt of t h r reaction having thr loa rr activation cnrrgy versnh rctlucctf pressure. Tho symbols :*IC dcscribcd in thc tcxt. Strong collision mechanism. I'rom top to bottoms = 5, 7, 9, 15, 17, 19, n2* - nl* = 5, e 0 = 0 6; Y ~ / V I = 1.13.
i i
10-3 10--1
I
-
-
--10
102 holly
--
-
104
108
L.---,
104
104
/bm.
Fig. 2.-R:itc constnnt of the reaction having thc lower activation energy versus reductd prrssurc. Stepwise rolli~iori rnechnnism. From top to bottom s = 5, 7, !I, 15; n2* 7 2 1 * = 5 ; e-a = 0.6; v,/v1 = 1.13. 100
___ -
~
l
-
o Stepwise mechanism s * 15, e - @ - 0.6, n;-n;
-
5
,
3(J+p1=13
--1
,
,
, I
o C h e s i c k ' r exptl. d a t a , 4 4 6 9 ' C
I IO"
10-2
IO
I
P (cm Hg).
1
,L--0
Cheslck's exptl. data, 4 9 0 . 6 - C . ..
10-3
I
o s above
-Some
1
-_
10-2
l
--
-
- A
10-1
I
---A 10
P (cm Ha).
Fix. S.-Compurison
of the stcpwisc theory with Chrsick's results.
above that the larger the difference Ez - El, the better the discrimination between the two mechanisms; in these figures we see that the simplm t,he molecule, the better the discrimination, also. I'igure 6 shows the ratio kl//cZ as A function of the dimensionless pressure for the two collision mechanisms. The marked diffcrerice in the ratio at, low pressures, especially for fairly simple molccidcs, indicatcs that one roult3 rim such n reaction iienrly
/
0.16
-
0.14 0.12
10-3
0.10 I
10-1
10 102 10s k,o!ll/k, m.
1
10'
104
104
Fig. 4.-Rst.e conshnt of the rextion having the higher activation energy Z!CTSUS roduced prcvsiirr. Strong collision mcchanism. Froin t o p to bottom s = 5 , 7, 0, 15; n2* nl*= 6 ; e-8 = 0.6; vp/vl = 1.13. 1
0.08
0.M 0.04
0.02
IO-'
:8 10-2
10-9 io -2 io-* 1 io 102 103 Fig. 7.--Quarituni yiclrl wrws pressurr. The model has 3 degrees of freedom, rup ( - 0 ) = 03, photo-rseitation is to the 8th stste, the lowest reactive state is the (ith, arid the ratio of the Kmsel frqriency factor t o the ratc constnnt for fluorescrncr (msuined independent, of vibrational encrgv) \\:IS taken iis 1.0.
4 10-3
10-4 I
IO 102 103 104 104 101 kioilllki m. Fig. 5.-Ratc constant of the reaction having the higher activation cnergy oersus reduced prrssurc.. Othcr &ita slime as Fig. 2. 10-1
'
cehmoi
"itomt
";n:
~4
8
0.027 0.081 0.243 0.729
P. 10
J
io
2
10-1
1 10 kioN/kI
102
104 104
1113
Fig 0 - I h t r o of tho rato constant lor the rrac t i o n of louer activation energy t o that for the reartiori of highcr activation energy 5s a function of reduced pieswrr I'pprr riirvr, strpwifie mcchanisxn, s = 15; lower ciirve, strong collision mechanism, 8 = 17. nz* ni* = 5 , e-8 = 0 0 ; Y,/Y, .=. 1.13.
-
to completion and then just mensure thc ratio of products as SL function of pressure, provided the products did not themselves react. This would pcrmit one to work conveniently at pressures lotwr than are readily accessible when the rate constants themselves must be measured. III. Photochemical Bireactions A possible mechanism for certain photochemical reactions is Aground +An* I A k * + M +A,* 4-M a i k Ak* +(Products)l Ak* +Agroiind ( f
Aground represents
c,l hYk)
(4)
dk
a reactant molecule in its ground
Big. 8.--Hatio of quantum yiclds as a function of diinensionless pressure. lttotni is the total nurriber of levels considered in the cnlculation, %l$ght is the level to which photoexcitation O C C I I ~ R , n,* is the minimum nuniber of quanta of vihrationnl cncrgy required t o produce (Product)l, V I is t,he freqriency factor for reaction 1, and vfluor is the rate constant for flriorescencc from any Icvcl.
/
I
/"
/----....---------8
-
L-_
0.009 0.027 0.081 0.243
Fig. 9.-Ratio
P. of qiiantiim yields as a fiinrtion of dimensionless pressure.
clcctronic state (we are not specifying thc vibrational lcvels of this Rtatc cxplicitly); A,* is the vibrational lovcl (or lcvds) of thc clcctimic state to which reactant molcoulcs &re pllolo-excited ; 1 i8 thc intensity of light absorbed which iriduccs this transition; arid dk is the probability of fluorescence and/or internal coilversion from thc lcth vibrational level of the excited electronic state, assumed to be independent of prcssure. The other symbols are essentially thosc previoiisly defincd. We set up the rate equations, invoked the steady-state assumption, solved the resulting system of linear inhomogeneous equations, and calculated qiiantim yields as functions of prcssure for one stepwise and onc strong collision model. The microscopic decomposition rate used was calculated by Kassel theory; the same formula was used as was employed in the iinimolecular thermal reactions calculation discussed above. The stcpwise model employed collisional transition rate constants identical to those described above. The strong collision modcl used rate constants ct,j = a, a constant, for transitions from highcr to lower energy states. The “upward” transition rate constants thcn are dcfincd by microscopic reversibility. The collision mechanisms in both cases met the requirements of microscopic reversibility, in contrast to some other models which have been treated recently.3at13The neglect of the upward transition rates introduces little change in the general appearance of plots of quantum yield us. pressure; however, in some cases this approximation appreciably changes the temperature dependence of the quantum yield. Figure 7 shows plots of quantum yield vs. a reduced pressure (on a log scale) for the strong (1) and stepwise (2) models. A simple shift of the pressure scale essentially superimposes the two curves, in ngrcement with Porter and Connelly’s conclusion that measurements of quantum yields must be quite precise to distinguish bctwccn the two collision mechanisms. Photochomical reactions in which one excited electronic state can dccompose by tn*o or more diffcrent, modes to yield distinct product$, howcvw, may well he a much more profitable means to gcttirig information about collisional energy transfer. For a strong collision mochanism and with photo-excitation to a range of levcls over which the microscopic decomposition rates do riot vary by much, it can be shown readily that the ratio of quantum yields is essentially indcpendent of pressure. One must include the proviso that fluorescence or internal conversion occiir to an oxtent which permits the ready return of clcctronically excited, vibrationally deactivated molecules to the ground electronic state. On the other hand, if a stepwise collisional vibrational energy process is operative, then, as is (13) J. T. Dubois, J . Chem. P h y s , 83, 229 (1960).
5t 0.001 0.003 0.009 0.027
Fig. 10.-Ratio
1’. of quantum yiclds as a function of dimensionlcss pressure.
50 40
’
230 20
lo
t
:
L..
\ L A -
0.001 0.003 0.009 0.027
P. Fig. 11.-Ratio
of quantum yields as a function of dimensionless pressure.
intuitively evident, the ratio of quantum yields is quite markedly dependent on pressure. The quantum yield of the reaction having the higher activation energy decreases the more rapidly as the pressure increases, provided that the vibrational eriergy of the state to which excitation occurs is large compared to skT. This effect is demonstrated in Fig. 8, I), 10 and 11. The mechanism employed herc is identical to the stepwise mechanism earlier described except that a second Kasxltype microscopic decomposition rate has bccn included, corrrsponding to the second mode of reaction. Similar plots for the strong; collision mechanism would, of courso, be essentially st>raight horizontal lincs. The fly ill the ointment hero, of course, is the dificulty of establishing beyond all reasonable doubt that reaction is occurring from only one excited electronic stntc. Studies of such reactions would, however, provide an interesting check on the rcsults obtained by Rabinovitch and his coworkers,14 in which the decomposition of “hot” secondary butyl radicals was found to lend support to the strong collision hypothesis. (14) R. E Harrington, B. S. Rabinovitch and M. R. Hoare, iWd., 33, 744 (1960).
