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Energy dependence of the cross section for CH3I + K → CH3 + KI near threshold
CilEMICAL
Volume 36, number 2
ENERGY
DEPENDENCE
OF THE CROSS
Hyung Kyu SfiIN De~a~lmcrlr of Chemistry **. University
of Nevada,
1 March 1976
PIiYSKCS Li?MERS
SECTION
Reno.
FOR CH,I
Newada 89507.
f K-P CH, + KI NEAR THRESHOLD*
USA
Rcccivcd 28 October 1375
The cnqy dependence of reaction UOM sections 0 ofCf131 f K - Ct13 + KI nc31 threshold has been investigated by use of an idealized model of the !iurd-sphere collision of CHJ-1 with K in which CH3-I is assumed to undergo harmonicvihrationai motion. Near lhrashold only ;I small fraction of trajectories enters the rcnction rqrion, hut they all pass on to reaction. As energy increases. o increases very sharply bccausc snort trajectories cntcr the reaction region. At still hi$lcr energies a becomes small bccausc a put of such trujcctories reflects from the rcpulsivc wall back to tllc rerctaut region. A maximum value of Q is found to be 35 X2 at 3.2 kcal/molc.
In a recent letter [1] we reported the calculation of reaction cross sections for Cl-l31 t K -+ CH3 + KI using the model of hard-sphere interaction between CH, and I and between I and K in a collinear collision. The method gave a simple cxprcssion of the rcaclion probability as a function of the incident angle of
trajectories, which determines the translational and vibrational energies of the “three-particle” system. It was shown that the method gives rcactioll cross sections w!lich are in good agreement with molecular beam experiments by Gersh Ed Bernstein [2] in the post-maximum energy retion, but is not capable of explaining the appearance of the maximum at low energies. In the mc:thod tlic square-well potential is assumed for CH,-1 and all trajectories which enter the reaction region near the threshold energy lead to reaction. However, the number ofsucb trajectories can be very small heca~~se the entrance channel cannot have a wide open square well shape in reality due to the vibrational motion of CH,--1. In this letter we shall modify the square-well potential with harmonic vibrational motion and show the appearance of the reaction cross section maximum can be predicted in
and cross section, and the comparison of the result with experiment in the post-maximum energy region bavc already been discussed in detail in ref. [ I], WC shall proceed directly to the introduction of the harmonic-oscillatory motion of CH3-I in the model and the calculation of reaction cross sections at low cncrgies in what follows. In ref. [ 1 ] we used the classical model of the hardsphere contact between the atoms K and I and betwccn CH3 and I us shown in fig. la, where regions I, II, and III are the reactant valley, the reaction region, and the product valley, respectively. Obviously, the square-well model show in fig. 1 b is a high!y idealized picture for the intramolecular motion of the CH,-I bond. A realistic potential energy should show minimum values somewhere on line ab and in regi&l II, the Iiric connecting such minimum values in the region
the idealized model for CH31-k K + CH3 + KI. Since methods for !he calculation of reaction probability * This work was supported by 1hc U.S. Air Force Office of Scientific Rc~arch, Grant AFOSR-72-2231. ** Theoretical Chemistry Group Contribution No. 1069.
Fig. 1. (a) Skewed potential energy surface for CH3-I + K -+ Cl13 + KI. Line ob is the entrance channel and hc is the exit channel. (b) Cross section of the entrance chamel in the wuarc-well potential model. (c) hlodification of the cross scction (b) with *ACvibrztiond motion of the (X3-1 bond.
253
Volume 38, number 2
CffEhilCAL PfIYSICS LETTERS
being the reaction path, and show &t-zp~~~e~~i~Ienergy rapidly rising in the corner regions of n and b shown in fig. fc. l%efefc3re, the ir~lproveiner~t is to construct the polcntid cncrm SUifaCe of region Ii sllch that tile cross section of the entrance channel takes the shape shown
in fig.
Ic. Then,
fc3r exifmple,
a reactant
trajcc-
s2 of the entrance channel from the far right of region I with an energy ET cannot proceed to region II. In ref. [I 3 such a trajectory is considered to be reactive. In fig. lc the shaded area thus represents the portion of the potential energy surface whicfa is not nvaifablo for reaction. Since no Iowcitergy trajectories can Ic;ld to reaction, even if they satisfy the condition that f+ > Vgr, the fraction of trajectories which finally reach the product chnnnel will be reduced by the filctor that is detern~ir~cd by the shaded portion of line a& for the energy region from ET -‘:Vrl to Ei-. Above L::, = Ki, the trajectories are not affected by such inaccessible regions and they ail reacfl region II. A portion of such higfl-energy trajectories will hit the repulsive wall (Iincs QI-anclfor ua of fig. I a) and cross back to the entrance chaimul, whjlc Llie remaining portion will cross the tory
which
arrives
nt the corrIer
points
s1 and
exit ~~~~llne1 (line bc of fig. 1a) and pass 011to ~-em tion. In Wlii3t. folIoWs, we shall put this i&n into a qIlanform and explain the appcarznce of the reaction probability (or cross section) maximum for CIiji + K + CH, + KI at IOW dfisioi~ energies. titatke
An
approach
to p:oduce
the I3otential
energy
sur-
cross section at the entrance chtlnneI ab which takes the shape sftown in fig. ic is to introduce the flarrnonic-oscili;ltor iutramolccular I’unction 1’0:CH3--I; Face
Y(Q) = 5 Mw” (Q-F~~)~, whcrcr2 is the distaixe beI and ihc WII~W of rnesS of CXI~ (SW reC tl]).
LW~XII
fn tig.
la,
the ‘2 coodin&
is
converted into the rec-
tangular system_v as
where
7 = mc/(mB
i
mC);
here, A, B, and C, rcspcc-
tively, stand for K, I, and CE,. Since WCate intcresteci in de tcrmining the fructioa of’the entiancc &a~nel (fine ~5) which is not accessible to.incident trajectories, we can USCthe potential function &XZ by elrher V(r$ or V(J). We shall dcterminc the curvitture of the ~oteni~~l enerp surface in region II (i.e., the corer regions shown in Fig. lc) in accordiiuce with the ~~~tr~~olecuI~r function V(r2) f‘clrCH-+. The vibrational frequency c3f the bond is [3] Y = W/~R = 254
1 hlarch 1976
533 cm- f and the equ~~ibriun~ separation is 2.139 a [4]. For VII = 1.5 kcal/molc and r2e = 2.139 A, the result of the square-well mode1 presented in ref. It ] shows good agrccrncnt with experiment above .&Y-r = 4 kcd[moIe. The irnportancc of finite curvature of the hrtrmttnic potential energy surfilcc on the reaction probability starts to show up below this encrm. Thercfore, in fig. 2, the cross section of the potential energy surface along the channel ~6 is modified with V(r2) for ~5~ < 4 kcalfmole, and is approximated with a multi-step surface. The ordinate refers to the translationa1 cncrw Eei. and the abscissa stands for the fractional departure of CHg--I from the equilibrirnn vilfue. Since r-z0 = 2.139 A, the modification with multi-steps up to 4 kcallmolc corresponds to the consideration of the hnrmonii; interaction poterlriol with the rt 6% deviation of the CH,- I band. Then, the modification of the potential surface in this fashion should result in decreased reaction probabilities at lower energies, since a portion of the entrance chnnnei is no longer accessible to low-lying trajectories. On the other hand, for I~i~~l-Iyirl~trajectories (I$ > 4 kcal/mnle) such :zmodification does not kad to a change in the reaction probability obtained front the square-well model. Most of. the high energ trajectories wt&A~ enter rcgian II with smdl incident angles wiII rcflcct back from the rcpulsive wrtII (fine oc or on of fig. I a) yielding no reaction. Only n small fraction of such trajectories will proeccd to the product valley. Therefore, at some irlterrx~cdi~tc energy, the reaction probability is expcctcd io take a maximum
value.
WC shall discuss the calculatiork of reaction cross
sections in tfli: nei$iborhood
of the maximum using t11r:values ot’ Yll = 1.5 kcaI/rnolc and G -I 4.3 A as in ref. [I] with the modification of the potential eners surface in re_r$on II witfi the potential function Y(rz) = $ ML? (r2-1=,_,)2 such that the Hltriw33 channel tnkos tftc sfqx showr~ in f2g. 2. When the square-weil potential is used, the renction probability is unity for i.e., the cross section is ET less than 2.6 kc#n&?,
o=&
z 58 A” as indicated by the broken curve in fig. 3, but the reaction probability P can now hccomc very small at such energies due to the fepufsivc wnlI which appears in the corner regions (set fig. 2). At f+ = 2 kc~~~nlole, 2’= I .O from the square-well model, but the harmonic model of fig. 2 gives P = 0.43 indicating that the repulsive potentiiif. at the corner regions blocks 57% of the incident trajectories at the entrance
Volume
38. number
2
CflEhllCAL
PllYSICS
I March
LEITISRS
1976
5 ET
4 3
2 vr,
I+.
-‘~_‘_-
‘__-i__-
_‘-i*__
.
_j&__“‘__2b
-
__-___
A
ET (kcdhde) IGg. 2. Cross section of the cntritncc clx~nricl calculatal with the llarmonic-vilrr:lti~nIIl motion of the C’Iix--I bond. The ;~rea under the multi-step curve is inaccessible to inciticnt trajcctorim.
clianncl: the cross section is then 25 A’. When i:‘.t- rises to 3 kcol/rnolc, we find P = 0.60 and u = 35 Ay2 21sshown in fig. 3, whereas the squarcwcli modci of Icf. [I] gives
0.79. For & > 4 kcal/mole, the rcpulsivc corner does not affect the incident trjectories. Fig. 3 shows that tha reaction cross section decrcrms very sharply with lowcriug E: . below E, 1- = 3 kcai/rnole. For cxa~npie, at E.r = 1.: kcai/mole, u = 15 K’. Such a “normal”, Rrrhenius-like positive energy dependence near the threshold has nlso been predicted by LaBuddc ct al. 141, by Bunker and Goring [5], and recently by Eu and Liu [ti] _ In fig. 3, the maximum appears at f$ z 3.2 kcai/moie, while the ohscrved moiccuiar-beam datil [2,7] shows that it appears at x 4 kCi~i/IIlole. It is clltiIcly possible to obtain :I better :Igrcemcnt hetwecn the observed and calculated energies at which tllc Inaxiniunl
apprs
by choosing
different
values
of
but such a procedure may only obscure the intent of the present work which is to show the appCi3rarKX of the cross section maximum in the idealized mode1 VII,
when the potcritial energy surface at the entrance channci is modified with multi-steps of the harmonic oscii-
lation of CH3-I. The extension
of the above approach to other reactions is straightforward. One such reaction is CH,I +
Rb + CH, + RbI for which Litvak et al. [St found the energy dcpendencc of reaction cross sections which is similar to that of the CH3 + K system at higher enurgics. However, their results which wcrc obtained for ~5.~> 2.7 kcti/mole (i.e., 0.12 cV) do not exhibit a cross-section
maximum.
If the maximum
is to occur,
!i
I?p. 3. Depcndeoce of’the reaction cross section on tllc tmnshtional cncrpy. Tha solid curve ic the rcsuit of the present appreach. ‘171~broken curve is the result shown in ref. [I] which is obtained with the squsrc-wull potential at thic cntrancc channcl; SW fig. lb. ICxpcrirnenta! data: o Gcrsh and Bernstein, o
Ilarris and
Wilson.
has to bc at still lower energies (
iowcr energies shows the appearance of such a maximum. If I/,, is taken as 0.5 kcal/molc, it appears at 1.7 kcai/moic (O.G73 eV); the reaction cross section then rapidly decrcascs with further lowering of the translational cncrgy. At higher cncgies, whcrc modification of the potential erierw surfiicc is not important, the prt-sent approach gives the relative cross section which is in fair agrecmcnt with the cxpcrirnental data. 111suInm;lry, WI: may stztc that the idenlizcd hardsphere collision model can give both the cnerey depcndence of reilction cross sections at higher oncrgies and the appcar:uice of the cross section niaxiniuni at iowcr cncrgics which agree with experiment for Cl-131
+ K + CH, + Ki. The maximum appears when the square-well potential is modified with the harmonic oscillation for CIi,-I. Examination of the motion of individual trajectories in the reaction region shows that at low energies fewer trajectories enter the region through the entrance chznnzl when the modification is made, but all such trajectories reach the product valley without hitting the rcpulsivc wall 0~. However, some of them hit the repulsive wail oa; such trajecto-
ries correspond to the situation that the CH,-t bond is being compressed when the atom K approaches from the right. As the encrpy is increased, some trajectories are reflcctcd from the repulsive walI of the 25s
Volume 38. nurnbar 2
CHEbflC‘~L
reaction regiiqn (line oc and/or oa) and cross back to the reactant valIey through the entrance channei (line ab), thus crossing the channel twice. As the energy continues.to
increase,
more trajectories
return
to the
reactant valley, and only a small fraction of trajectories which’ entered region II through the channel in of B in fig. In proceed io reaction. tire neighborhood Therefore, the reaction cross section becomes small a? higher energies. The present study is based on an idealized collision model, but it reveals some important information on reactive collisions. It is particularly interesting to find that the essence of collision dynamics can be understood in such basic terms.
256
PliYSfCS
1 Xfnrch 1976
LtTnTRS
References [ 1 j H.K. Shin. Chcm. L’hys. Letters 34 (1975) [ 21 hl.E. Ccrsh and R.B. lbrnztein, J. Chcm.
546.
Phys. 55 ( 197 1)
4661; 56 (1972) 6131. [ 3) Ci. Herzbcrg, Eleclrunic spectra of polyatomic molecules (Van Nostrand, Princeton, 1967) p. 621. [4] K.A. LnBuddc, P.J. KUIIIZ, K.B. Bcrnstsin and RX. Levine,
[S]
Chcm. Phys. 6286. D.L. Bunker
Letters and EA.
19 (1973)7: Goring,
(1972) 521. [Gj B.C. Ku and W.S. Liu, J. Chcm. [7] [8]
J. Chcm. Chern.
I’hys.
P_hys. 59 (1973) Lcttcrs
1.5
Phys. 63 (1975) 592. R.M. Hxris and J.1:. Wilson, J. Chem. Phys. 54 (1971) 2088. H.E. Litvak, A.G. Ureiirl and R.H. Hcrnstcin. J. Chcm. Pirys. 61 (1974) 4091.
Volume 36, number 2
ENERGY
DEPENDENCE
OF THE CROSS
Hyung Kyu SfiIN De~a~lmcrlr of Chemistry **. University
of Nevada,
1 March 1976
PIiYSKCS Li?MERS
SECTION
Reno.
FOR CH,I
Newada 89507.
f K-P CH, + KI NEAR THRESHOLD*
USA
Rcccivcd 28 October 1375
The cnqy dependence of reaction UOM sections 0 ofCf131 f K - Ct13 + KI nc31 threshold has been investigated by use of an idealized model of the !iurd-sphere collision of CHJ-1 with K in which CH3-I is assumed to undergo harmonicvihrationai motion. Near lhrashold only ;I small fraction of trajectories enters the rcnction rqrion, hut they all pass on to reaction. As energy increases. o increases very sharply bccausc snort trajectories cntcr the reaction region. At still hi$lcr energies a becomes small bccausc a put of such trujcctories reflects from the rcpulsivc wall back to tllc rerctaut region. A maximum value of Q is found to be 35 X2 at 3.2 kcal/molc.
In a recent letter [1] we reported the calculation of reaction cross sections for Cl-l31 t K -+ CH3 + KI using the model of hard-sphere interaction between CH, and I and between I and K in a collinear collision. The method gave a simple cxprcssion of the rcaclion probability as a function of the incident angle of
trajectories, which determines the translational and vibrational energies of the “three-particle” system. It was shown that the method gives rcactioll cross sections w!lich are in good agreement with molecular beam experiments by Gersh Ed Bernstein [2] in the post-maximum energy retion, but is not capable of explaining the appearance of the maximum at low energies. In the mc:thod tlic square-well potential is assumed for CH,-1 and all trajectories which enter the reaction region near the threshold energy lead to reaction. However, the number ofsucb trajectories can be very small heca~~se the entrance channel cannot have a wide open square well shape in reality due to the vibrational motion of CH,--1. In this letter we shall modify the square-well potential with harmonic vibrational motion and show the appearance of the reaction cross section maximum can be predicted in
and cross section, and the comparison of the result with experiment in the post-maximum energy region bavc already been discussed in detail in ref. [ I], WC shall proceed directly to the introduction of the harmonic-oscillatory motion of CH3-I in the model and the calculation of reaction cross sections at low cncrgies in what follows. In ref. [ 1 ] we used the classical model of the hardsphere contact between the atoms K and I and betwccn CH3 and I us shown in fig. la, where regions I, II, and III are the reactant valley, the reaction region, and the product valley, respectively. Obviously, the square-well model show in fig. 1 b is a high!y idealized picture for the intramolecular motion of the CH,-I bond. A realistic potential energy should show minimum values somewhere on line ab and in regi&l II, the Iiric connecting such minimum values in the region
the idealized model for CH31-k K + CH3 + KI. Since methods for !he calculation of reaction probability * This work was supported by 1hc U.S. Air Force Office of Scientific Rc~arch, Grant AFOSR-72-2231. ** Theoretical Chemistry Group Contribution No. 1069.
Fig. 1. (a) Skewed potential energy surface for CH3-I + K -+ Cl13 + KI. Line ob is the entrance channel and hc is the exit channel. (b) Cross section of the entrance chamel in the wuarc-well potential model. (c) hlodification of the cross scction (b) with *ACvibrztiond motion of the (X3-1 bond.
253
Volume 38, number 2
CffEhilCAL PfIYSICS LETTERS
being the reaction path, and show &t-zp~~~e~~i~Ienergy rapidly rising in the corner regions of n and b shown in fig. fc. l%efefc3re, the ir~lproveiner~t is to construct the polcntid cncrm SUifaCe of region Ii sllch that tile cross section of the entrance channel takes the shape shown
in fig.
Ic. Then,
fc3r exifmple,
a reactant
trajcc-
s2 of the entrance channel from the far right of region I with an energy ET cannot proceed to region II. In ref. [I 3 such a trajectory is considered to be reactive. In fig. lc the shaded area thus represents the portion of the potential energy surface whicfa is not nvaifablo for reaction. Since no Iowcitergy trajectories can Ic;ld to reaction, even if they satisfy the condition that f+ > Vgr, the fraction of trajectories which finally reach the product chnnnel will be reduced by the filctor that is detern~ir~cd by the shaded portion of line a& for the energy region from ET -‘:Vrl to Ei-. Above L::, = Ki, the trajectories are not affected by such inaccessible regions and they ail reacfl region II. A portion of such higfl-energy trajectories will hit the repulsive wall (Iincs QI-anclfor ua of fig. I a) and cross back to the entrance chaimul, whjlc Llie remaining portion will cross the tory
which
arrives
nt the corrIer
points
s1 and
exit ~~~~llne1 (line bc of fig. 1a) and pass 011to ~-em tion. In Wlii3t. folIoWs, we shall put this i&n into a qIlanform and explain the appcarznce of the reaction probability (or cross section) maximum for CIiji + K + CH, + KI at IOW dfisioi~ energies. titatke
An
approach
to p:oduce
the I3otential
energy
sur-
cross section at the entrance chtlnneI ab which takes the shape sftown in fig. ic is to introduce the flarrnonic-oscili;ltor iutramolccular I’unction 1’0:CH3--I; Face
Y(Q) = 5 Mw” (Q-F~~)~, whcrcr2 is the distaixe beI and ihc WII~W of rnesS of CXI~ (SW reC tl]).
LW~XII
fn tig.
la,
the ‘2 coodin&
is
converted into the rec-
tangular system_v as
where
7 = mc/(mB
i
mC);
here, A, B, and C, rcspcc-
tively, stand for K, I, and CE,. Since WCate intcresteci in de tcrmining the fructioa of’the entiancc &a~nel (fine ~5) which is not accessible to.incident trajectories, we can USCthe potential function &XZ by elrher V(r$ or V(J). We shall dcterminc the curvitture of the ~oteni~~l enerp surface in region II (i.e., the corer regions shown in Fig. lc) in accordiiuce with the ~~~tr~~olecuI~r function V(r2) f‘clrCH-+. The vibrational frequency c3f the bond is [3] Y = W/~R = 254
1 hlarch 1976
533 cm- f and the equ~~ibriun~ separation is 2.139 a [4]. For VII = 1.5 kcal/molc and r2e = 2.139 A, the result of the square-well mode1 presented in ref. It ] shows good agrccrncnt with experiment above .&Y-r = 4 kcd[moIe. The irnportancc of finite curvature of the hrtrmttnic potential energy surfilcc on the reaction probability starts to show up below this encrm. Thercfore, in fig. 2, the cross section of the potential energy surface along the channel ~6 is modified with V(r2) for ~5~ < 4 kcalfmole, and is approximated with a multi-step surface. The ordinate refers to the translationa1 cncrw Eei. and the abscissa stands for the fractional departure of CHg--I from the equilibrirnn vilfue. Since r-z0 = 2.139 A, the modification with multi-steps up to 4 kcallmolc corresponds to the consideration of the hnrmonii; interaction poterlriol with the rt 6% deviation of the CH,- I band. Then, the modification of the potential surface in this fashion should result in decreased reaction probabilities at lower energies, since a portion of the entrance chnnnei is no longer accessible to low-lying trajectories. On the other hand, for I~i~~l-Iyirl~trajectories (I$ > 4 kcal/mnle) such :zmodification does not kad to a change in the reaction probability obtained front the square-well model. Most of. the high energ trajectories wt&A~ enter rcgian II with smdl incident angles wiII rcflcct back from the rcpulsive wrtII (fine oc or on of fig. I a) yielding no reaction. Only n small fraction of such trajectories will proeccd to the product valley. Therefore, at some irlterrx~cdi~tc energy, the reaction probability is expcctcd io take a maximum
value.
WC shall discuss the calculatiork of reaction cross
sections in tfli: nei$iborhood
of the maximum using t11r:values ot’ Yll = 1.5 kcaI/rnolc and G -I 4.3 A as in ref. [I] with the modification of the potential eners surface in re_r$on II witfi the potential function Y(rz) = $ ML? (r2-1=,_,)2 such that the Hltriw33 channel tnkos tftc sfqx showr~ in f2g. 2. When the square-weil potential is used, the renction probability is unity for i.e., the cross section is ET less than 2.6 kc#n&?,
o=&
z 58 A” as indicated by the broken curve in fig. 3, but the reaction probability P can now hccomc very small at such energies due to the fepufsivc wnlI which appears in the corner regions (set fig. 2). At f+ = 2 kc~~~nlole, 2’= I .O from the square-well model, but the harmonic model of fig. 2 gives P = 0.43 indicating that the repulsive potentiiif. at the corner regions blocks 57% of the incident trajectories at the entrance
Volume
38. number
2
CflEhllCAL
PllYSICS
I March
LEITISRS
1976
5 ET
4 3
2 vr,
I+.
-‘~_‘_-
‘__-i__-
_‘-i*__
.
_j&__“‘__2b
-
__-___
A
ET (kcdhde) IGg. 2. Cross section of the cntritncc clx~nricl calculatal with the llarmonic-vilrr:lti~nIIl motion of the C’Iix--I bond. The ;~rea under the multi-step curve is inaccessible to inciticnt trajcctorim.
clianncl: the cross section is then 25 A’. When i:‘.t- rises to 3 kcol/rnolc, we find P = 0.60 and u = 35 Ay2 21sshown in fig. 3, whereas the squarcwcli modci of Icf. [I] gives
0.79. For & > 4 kcal/mole, the rcpulsivc corner does not affect the incident trjectories. Fig. 3 shows that tha reaction cross section decrcrms very sharply with lowcriug E: . below E, 1- = 3 kcai/rnole. For cxa~npie, at E.r = 1.: kcai/mole, u = 15 K’. Such a “normal”, Rrrhenius-like positive energy dependence near the threshold has nlso been predicted by LaBuddc ct al. 141, by Bunker and Goring [5], and recently by Eu and Liu [ti] _ In fig. 3, the maximum appears at f$ z 3.2 kcai/moie, while the ohscrved moiccuiar-beam datil [2,7] shows that it appears at x 4 kCi~i/IIlole. It is clltiIcly possible to obtain :I better :Igrcemcnt hetwecn the observed and calculated energies at which tllc Inaxiniunl
apprs
by choosing
different
values
of
but such a procedure may only obscure the intent of the present work which is to show the appCi3rarKX of the cross section maximum in the idealized mode1 VII,
when the potcritial energy surface at the entrance channci is modified with multi-steps of the harmonic oscii-
lation of CH3-I. The extension
of the above approach to other reactions is straightforward. One such reaction is CH,I +
Rb + CH, + RbI for which Litvak et al. [St found the energy dcpendencc of reaction cross sections which is similar to that of the CH3 + K system at higher enurgics. However, their results which wcrc obtained for ~5.~> 2.7 kcti/mole (i.e., 0.12 cV) do not exhibit a cross-section
maximum.
If the maximum
is to occur,
!i
I?p. 3. Depcndeoce of’the reaction cross section on tllc tmnshtional cncrpy. Tha solid curve ic the rcsuit of the present appreach. ‘171~broken curve is the result shown in ref. [I] which is obtained with the squsrc-wull potential at thic cntrancc channcl; SW fig. lb. ICxpcrirnenta! data: o Gcrsh and Bernstein, o
Ilarris and
Wilson.
has to bc at still lower energies (
iowcr energies shows the appearance of such a maximum. If I/,, is taken as 0.5 kcal/molc, it appears at 1.7 kcai/moic (O.G73 eV); the reaction cross section then rapidly decrcascs with further lowering of the translational cncrgy. At higher cncgies, whcrc modification of the potential erierw surfiicc is not important, the prt-sent approach gives the relative cross section which is in fair agrecmcnt with the cxpcrirnental data. 111suInm;lry, WI: may stztc that the idenlizcd hardsphere collision model can give both the cnerey depcndence of reilction cross sections at higher oncrgies and the appcar:uice of the cross section niaxiniuni at iowcr cncrgics which agree with experiment for Cl-131
+ K + CH, + Ki. The maximum appears when the square-well potential is modified with the harmonic oscillation for CIi,-I. Examination of the motion of individual trajectories in the reaction region shows that at low energies fewer trajectories enter the region through the entrance chznnzl when the modification is made, but all such trajectories reach the product valley without hitting the rcpulsivc wall 0~. However, some of them hit the repulsive wail oa; such trajecto-
ries correspond to the situation that the CH,-t bond is being compressed when the atom K approaches from the right. As the encrpy is increased, some trajectories are reflcctcd from the repulsive walI of the 25s
Volume 38. nurnbar 2
CHEbflC‘~L
reaction regiiqn (line oc and/or oa) and cross back to the reactant valIey through the entrance channei (line ab), thus crossing the channel twice. As the energy continues.to
increase,
more trajectories
return
to the
reactant valley, and only a small fraction of trajectories which’ entered region II through the channel in of B in fig. In proceed io reaction. tire neighborhood Therefore, the reaction cross section becomes small a? higher energies. The present study is based on an idealized collision model, but it reveals some important information on reactive collisions. It is particularly interesting to find that the essence of collision dynamics can be understood in such basic terms.
256
PliYSfCS
1 Xfnrch 1976
LtTnTRS
References [ 1 j H.K. Shin. Chcm. L’hys. Letters 34 (1975) [ 21 hl.E. Ccrsh and R.B. lbrnztein, J. Chcm.
546.
Phys. 55 ( 197 1)
4661; 56 (1972) 6131. [ 3) Ci. Herzbcrg, Eleclrunic spectra of polyatomic molecules (Van Nostrand, Princeton, 1967) p. 621. [4] K.A. LnBuddc, P.J. KUIIIZ, K.B. Bcrnstsin and RX. Levine,
[S]
Chcm. Phys. 6286. D.L. Bunker
Letters and EA.
19 (1973)7: Goring,
(1972) 521. [Gj B.C. Ku and W.S. Liu, J. Chcm. [7] [8]
J. Chcm. Chern.
I’hys.
P_hys. 59 (1973) Lcttcrs
1.5
Phys. 63 (1975) 592. R.M. Hxris and J.1:. Wilson, J. Chem. Phys. 54 (1971) 2088. H.E. Litvak, A.G. Ureiirl and R.H. Hcrnstcin. J. Chcm. Pirys. 61 (1974) 4091.