Calculation of V-T and V-V Energy Transfer Probabilities in Methanol and Deuterated Methanols

ifI44taM .T, PI/IgB, UII, 187-110 (11110)

Calculation of V-T and V-V eDern traDefer probabilitiet iD methaDol and deulerated methaDol,

Y V Chllollloplloti R .. o

Department of Ohomical Engmo •• 'mg, lndilln InBtitute of TeehnololY.

Kanpur-208016

"""_at.

^Normal coUl'dinato 8"aly ... of DlIIthBllol and deuterate.1 methMlolB IS oarried out oall1UJliug " valene" foroc field to •• tim .. te the broatlUng .pharo p ... meters of the vibrational D'odoo. The vibration.trBnSl .. tion (V-T) energy _sCer probabiliti .. of th. low lying vibrational modes, and the vibration·

vibration (V -V) anergy t .... .,tion probabilitie. in methanol ""d denterated ,oot.hauols are ca.loulate,l in the tempor"turo range 300-1000 K, using the I!lehw ... tz, Blawsky .... d Herzfeld (BSH) breathing sphero theory. The V.T unergy trallllfer pro ... eo are ",ore effioient in CH"OH ""d CD,OH compared to CH.OD and CD.OD respectively, The V-V tr .... aition probabilities o.ro 8bnOst temperature independent for pro ... w.th omaller 6111, and ...

• trougly temperature dependent for p.o ... w,th larger 6E, An the vibra- 1.iona.lll1Odos

or

a lovel, either low 1ying Vibrational level 0" upper Vibratioll&l level, equilibrate Cast through V-V pro... The probabilities for the de·

uxoitBtion of a mode from the upper VibratIonal level by .. V·V- pro.- coupled with a simuUanoottl ~xcit"tion DC a mode from the low lyinl! Vibrationa.l modo is low hy eevoral orders DC magnitude. Once the Vibrationa.l JDoilea of metbaawl or deuterated motbanolB are eelaotIvely exoited, the 1081 of "ibratlcmal enerllY can he oxpected to bo low,

1. latroduedoll

The energy t .... ll~f~l' 11l'()"(lK80~ dmiug' \Uoleou/lu: oollisionH in glioses and ga.s mixtures playa. domina,nt lolA ill oholUlcal relloctioill1. isotope sepllorllotion. through laser exoitation u.nd ill tho dosign of ga.~ llloSOl'ti, The imPOI'ta.nCO ef the know·

ledge of onorgJ' tra.nsfor 1'lIotO>! ill tho deSIgn of huel' Ilyatems OWl be viowed from the impa,ct tha.t such 110 knowle~e made on tile CO~ lallal output pow~r, From

OJ. lovel of 30 mw in pure COs, tho power output ba.s inorellobOd to a few ki}owa.tt/I m mixtures of CO~·N2-He (FIIok>11964. Fa.tAl ot a/1964, 1965. Moeller and Rigdon 1965. Gerry 1970, Judd 1973).

TJlid powor enhlUloelRent is clo~ely related to ~he e1Iioient Vibratioll·Vibrllotion (V-V) OncIgy tra.nsfel· in CO.-Na Bond tho Vibra.tion:Tra,nslation (V·T) energy tra.nsfer in CO2·Btl. With tho developmont of la.sers, the 168IIr ipitiated chemioliU l'sactiollS have blOen widely studied by Ka,rlov (1974), StephellSon and Freud (1976), PJ:eseB M al (19'77) and Herma.n ot cd (19'76, 1m). On(l(t

tM

l'elloCtants Ilre 18'7 18

188 Y TI O/uJ[a.pati Roo

selectively excited using II

J~SOl'

I'a.dilltioll, tho ijnorgy gots

r.,diatrj~1led ~oug

the trlUlsllltion$l IUld v~ious interna.l modoR of t,hc molocull!~ dunng, l,\oUllIlOIIJI kn 1 d f th V V /l,Jld ".T enorgy trallsfor rates IIro requll'od for II

&Uda ow-agoD e ' ' , '

d ^..,J. d' f 'lln reaotion meohIInlSm, Vurmg the p~~. ~vlll'al

oomplete un er~.an wg a ." , . '

tJ' .• ' ^'II I' d m' tl,n-ol('ou!aT onefO"y trlll/ilfor JlI gllses h4il l(JCfllVod

years Ie mtol'mo eOll til' an ,"".' . ", , '

muoh IIttelltioll.

T1leorio~

11BVO beeu dnvc1opod 't,o prodlOt tl1e

on6rg~

transfol' probllbilitieB whicll ta.kc into u,coount oitllel'. tho. short l'lIngo I'epulsivo forces (Sohwartz et al 1962) or tho jong rllngo Ilttl'IlotJVO t,)J'co~ only (Sharma IUld lIrau 1969). Tho theory of Schwart.z, S[lIw~ky and H~rzfcld (1952) (SSR) 11^a8 b~lI 8uoci}ssfully applied by DipkolJH aJld Rip~mollti (1961) to predict. tho onergy trBollSfer proba.hilitioB in diatomio molooules OY01' It wido rango of tl'mpOl'aturo.

Ta.nczos (1956) has ext"nded tho thoory for polyatomio Iuolocules lJY iptroducing a brBathiI;Jg sphere approxiula.tiou in whioh it is 118sumod that tho mo*ion of each surflloou atom for all normal modes is din,eted ra.dially, that is

alon~

tho moat effectiVe direction for onorgy t,raustol'. ISSH·Tal1czoB method has bd!l11 widely used for th~oretical clliculatioll of the trallsit,loll probabiJitieR (StrottoJl 1965, Lambert et al 1970, Yonozllowa "lid J<'uono 1974). Yardley IJ,nd Moots (1968) applied the SSH·brc~tbing sphero theory fo]' both iuter lind intraUlolornla.r V·V onergy trallsfer llroeosBe~ Huccessfully. This mothod is IIpplied to ostima.te tho V·T and V· V onergy trllnRfor probabilities in moth8ol101 8ol1d dOllwratod methanols in the temprratUl'll r .. nge 300-1000 K a.nd tho I'tlsults arc preRcnted in this pllpol' 2. Method of calculation

Aooording to th9 modifiud SSH·broatl\ing sphoro thoory (Yllrdlc,y IlJld Mooro 1968) the probllbility,

pl:Ha,

b), that during Il. bintll'y colliSIon the Vibrlltiollal mode 'a' of ono m<:>jecu]o chllngl's it .. quantum atato fl'om i to j, while simllltllll6' omly th" qna.ntnUl st,ato of mode 'h' in til" uthor molec\1h' rhHongcB from k to I is given by

P(a,t:~ b)

=

P.(a)PG(b)u,(a)Ul(b)

I

U/_I(a) /21 Uk_I(II) 1 ~ in which the intogral I is givOJl by

oxp(X)(pv/kT) (-pVI ) d {l-exp(X)}" axp 2kT II

x = 4:~v {1-. (l+2~ r }

whore Po is the sterie f~tor, 111, tho dogmleraoy of the stfloW J

I

U'-J

I

ⁱ the ma.trix element, p the rodueed mass of tho coJJis.ion pair, T the temperature,

All

tho energy oxcbtlnged botwoon thQ Vibra.tiona) a.1Id trruu;\a.tiona\ degrooA of froodOln:

AE

=

hv.(i-jHhvD(k-l)

139

WAll1'e "., and. "" .are "the frequenciOB of the Vibrationq,] modea (II iB th . t 1

" t t' . ,

e m anno

0-

oulM .orce oons MI In l'(lolprocallength units for th~ r p ] . .

• v e \l Rive ,exponential poton-

tial fnnotlOn

v=

l'o"xp(-ar)

f: j<l the

Lennar~-Jonc~

^well

dep~h

and v is the relative v(llocity of the collision pa.rtners, Tho mtegral I has to bo evah~q,ted w'th th I I"

• J . 0 OWer mnt VIII," as !/lero

for e:ll!otlwrmUI proceSSeS (!J.E

>

0) and v - {2 'EI )1" d h .

• 111'" - U '" lOr en ot ermlO proOOIlllBS

(AE

<

0). Tho matrIX el!}mont is given by

, U,_J

,2 =

1T' UI'~J ,"

"

whoro

I

Ut_Jrl , iH the matrix eloment for tho modo n. The mOotl'ix elements WIlt'6 evaluated by Rapp Oond. Sharp (1963). Tn p80rticular for th" 1-0 trMlsition, the mOotrix elemont lUI-on '" is relOoted to the br680thing sphere pa.ramet.er

< An"

>

of St.re~ton (1966) and is given by IU

,2 _

2a;"<A,,'> It

I_on .- 1 61To ...

Tho bl·Ua.thillg sphere a.mpliLude factol'~

<An">

can be eV801uatod following HlP msthod of normal coordina.t,· ana.lysiH by Stratton (1965) and are given by

<A"z>

=

N.1L-1H(L')-I]"" 1

Whm'!1 N. is tho uumbel' of Hurfo,cu a.toms MId L is t.ile tranBforma.tion matrix between the intornal oool'din80t.es R !lind the norma.I coordina.tes Q, exprellsed in ma.trix notation 6~

R=LQ H=BM-IB'

where M iM 80 diagona.l ma.trix of the o,tomic ma.saet! and B is the t,ransforma,tion ma,trix between the ca.rtoRi80n diaplaroment. coordin80t.e~ X 8o11d the internal ooordinat.es E, defined l)y

E=BX

The alem.mtB of tho H matrix (']1,11 bo ea.sily obta,med from the Wils0lll! 0 matrix whioh is defined as 0 = BM-l B, The transformll.tion mlLtrix Loan J:1G detor- mined by moa.p.a of the OF ma.trix method of Wilson, D~oiuf.\ a.nd' ~$a (1955) Md 1101'0 to lie" norma.lized by tl\l' requirement, LL' = G,

3.

Details

Ta.na.ka., Ku.rllttl!o1li l\ud Mizu/lhima. (1957) Miermined the Urey-Brad1ey foroe oonstants of methanol, but they were reswicted to th" A' vibra.tions only.

Ma.rgottin-Maolou (1960) oarried out a. llormal ooordinarte aI\a.l~ of methlW>l

140 Y V Ohalapati RtJO

and deutera.ted methllollols with Il Vw.anoe foroll field IIoIld determined the relev.a.nt force 1J0nstants, Zerbi, Overend a.nd Crawford (1963) oamed out more refinlld ow.culationa of the Uray-Bra.dely forer. oonstllollts by a. lelltst squares 1;eehniquo using vibra.tional frequency data from motl'.llonol Bond dentorated motha.nols IItnd confirmed that the vibrational IIJ111ignment. of Margottin-){lIoolou is consistent..

In this work tho vibrBttional Bona.lyais iii oarried ont a.ssuming the vw'ence foroe field and the relovllnt foron oonst.Btntll and

f

ma.trix are ta.lcen from Margottin- Ma.clau (1960). Tho goometl'ica,l pa.rametors 1l80q. ill the present cw.oulllttions IIr!'

! = 0·956

A.

B ~ ]·427

,A,

and Y

=

^1'096

A

in agreement wit.h VonklltteswarJu a.nd Gordy (1955) and Z"rbi et al (196S) and 1111 the IIngles aro ta.ken 1,0 hl'tetrll- hedra.l Tho relevllnt intornlll cOOl'liinll.tes arc Rhowl1 in Figure I. Tho int.flrnlll

Energy 'ransjeT pTobabiUUu ' 1'1

eool'dinllotes a.re ta.ken in linear combma.tion a.s 8Ylllmo~ry coordina.tea in agroo- mont with Ma.rgottin-Maclou (1960). Throughout the calculations tho torsiona.l IIlode Va is. ignorml. The deletion of tho torsional mode does not a.ffect the ca.kula.tions, btICo,URU this mode is not coupled, with t,he othCl" modcH. Tho vibra.tional frequency of this mode, expressed in Wa.Vll numbers, (,ha.nges from 270 om-I for CRsOR to 19(; om-¹ for CDsOD llond is far ~ltpara.tcd from tho othoT vibra.tional modeH. Thc neglect of such mod(l in the calculatlOn~ iR in agreelllent with the Vibrational analysis of methyl amine carried out by Wn et al (1961) With t,he negloct of the t.orsionRoI modo, t.ho J'e8\dting ,,1!wen symmetry ooordinRota~

1101'1' :

A' VibmliollR

8. = A'Y

y'2 I (A(J.-AfJs)

142 Y 11 Ollalapcm' BfJ6

f.

a ••

alt. aDd duca •• foa

The secular oquations aro solved by the method of' Miyazawa. (1,958) on,

mli

l401/70U digital oomputer to obta.in the norm~ized transformatIon m",trlll: L.

The L matrioos rela.ting the symmetry coordinates S and the normal coordinates Q (oxpl'oBllCld in matrix Jlota.tiOIl S

=

L'fl) for the A' vibrations of motJlanol and doutoratod methanols arH pl'esented in Tablo 1, whilo the matrioes L, for the A" vibrations aro given in Table 2 The 6Iltimated broathing sphere parllol1letors

<A"a>

of tho vibrational modes in muthanol and deuterated methanols 8ol'o presented in Table 3, Yardley and Moore (19B8) obsol"ved that tho VIIIlue of the parameter ^QG whioh best fits the observed data lies in betWelln4'75 X 10⁸ om-¹ and 6·25xlO^B om-I, The vibrational relaxation times of soveral mQleculeH were estimated by Stratton (1965) using tJJ.e BSH theory and it was

shoJn

^that,

the exponential repulsion parameter 01 is abnost independent of U1" type 0' mole·

cule, and th at the repulsion force~ encountered at oollision energies nec~lIs&rY for vibrational enllrgy trllonsfer aro roughly tho Bame for all molecules. F1\rthel' it was also shown that the effeotive 01 is about 5·5 X 10⁸ om-^l for IIoIl the molerlule~.

In the present calculatiolls,

Po

and 01 801'0 80Bsumed to be 2/3 IIond 5·5x10^R crn-¹ respectively ill agreement with Stratton (1:965) and elK

=

507°K (Rirschfolder, Cutiss and Bird 1964). Tn methanol Bond deuteratod methanols, the vibrational modes 08011 be brollodly cla.ssifiod in to two lavoIe-tho low lying vibrationallnvolM

(VI' 1'3' 1'6' v,,

"a,

1'10' I'll' vlZ) an(l higher levels (1'9' 1'4' v •• V,).

.

The tempel'atllro oopendellce~ of tho Vibration-translation unorgy LI'anl!fer probllobiliticll PIO of CHaOR and CHaOD 11.1'0 shown in Figure 2 Tn CR.OR, the vibrational wave uumberll of t,he low lying vibrationa.l modeR IllCrGMII in

tho

orelel'

Tho tranllition probabilities 'plD of thlllle modes in OHaOR illor~aR~ in the reverse order to that of tho magnitude of vibrational wave numbltrR.

PIO(V,)

>

p10(Vl)

>

PlO(Vl0)

>

piO(vs)'

>

pto(vBl

>

PIO(v,)

>

PIO(vlI)' From the variation of the probabilities it can be obsotved that the probability dooreases with increasing AE in general ttxcept for the following OIloatJB.

pto(v,)

>

p10(Vl) and above 500 K PIO(vlQ)

>

^p10(vl)' Such variations in proba.

bility &rise boollouse

<A71>I<Alt> """ <Alol>I<All>

^eo! 3-4. In CRIOD, the vibrational wave numbers are in th" ordllr

The eatima.ted V-T probabilitJeH are found to be in the order

PIO(Va)

>

^PlO(VIO)

>

^PIO(vl)

>

^p10(v,)

>

^pto(va)

>

,PlO(vl)

>

J'lO(Vll)'

J,.ergy

IrfJit&8/er

probabiUUes

,...1.

_{, The}

_~rmation

_{MatJ'ic8a L} fOi' ... ' vibration, (8 .. £0)

148

CllaOHIG)

Q. 0. Q. Q, Q. Q. Q. Q.

(1038)!·) (Jl883) (1466) (3679) (13411) (2960) (1070) (1&.1111) -83210 --01)606 -·08'61 -·031407 -'12041 '01340 -'098/14, -·07710

·OU123 '881120 -'01812 -01206 --01293 -'231167 '00013 --02811 -·00S72 -·12794 --611369 -'23273 -'111887 '0'107 -'00661 --98886 --02372 -·0111116 --04816 1-0246 -·06661 -·00080 -'06366 -·0841111 --47681 -08248 ·00766 -·0995& --51S881i

-·mauo

^{--17826 ·83786}

-·00327 ·241114 -08729 -00424 -·00010 1·0243 -'008114 -'01703

·08886 -·OB834 -·12744 ·17608 -·804811 -·101178 ·568711 ·601711 -·0448' ·0247:1 1-3782 ·018811 - 41887 -0783l4. ·18M7 -·4.0078

CJJ.OD'c)

(1040) (28~) (1478) (2718) (864) (9000) (1280) (14/itl)

·342311 -'06005 '02934- -'04081 -·07343 --01297 ·0111144 --08838

·01462 '97970 -01"7 -'10831 -·00628 -23'23 ·00994 -·02826

·03829 -·16806 ·60204 -·31i062 ·06087 -·03810 ·3n39 -1'0601 01088 ·07866 ·04998 ·73932 ·01148'7 --00038 ·00544 -·U292

·03881 'U4753 ·00109 -·10110 '80'777 ·02123 ·17262 ·1"56 'OOlll! ·24130 -'03821 -·03493 ·006111 -1·021? --··004.92 -·01&39 -·16876 -00199 ·10079 '20017 -·181191 ·10367 ·831144 '8'73119

~'09067 ·02787 -1·3806 ·02837 -·0Iu2a - ·078/1R ·39804 -·40951 en.OIl'

(988) (2077) (U3') (1297) (2260) (868) (1047)

·1I88111 '09288 - 051136 -'03100 -·011l22 -·01203 -·00888 ·222611

·02266 -·71084 - 04407 ·00588 -·00012 ·09521 -·001168 '04837 -·22920 -21139 -·04506 -·18970 -·07392 -·03310 -·02709 ·80374 -·08183 ·01286 -·00277 1'02M -'00986 -·00170 ·02616 ·08'29 -'366011 -·10338 '83486 -·10064 -·21719 ·08587 -29740 ·112609 -·00189 -·10310 ·01021 -·00038 -·00189 -·7'7479 -'001133 ·001180 -·011170 ·02604 -18106 -10184 -·BlI913 ·1'7499 -'60609 ·04821 -·06813 -'026115 ·24062 ·00124 101900 -·17817 -·23631 ·181196

en.OD'e'

(988) (2080) (1135) (2724.) (776) (2280) (lOll') (1060) '01331 ·09125 -·00100 ·0478.-. -·01016 ·01174 ·33271 ·11147.

-'00033 -·711180 -'03911 -'011161 -·00010 -·08I1n ·08061 ·011857

·022811 '21838 -·34204 ·26873 '086111 ·03736 -·02402 ·98078

·031199 '02518 ,0087' -·7U84 '01812 '00728 ·00273 ·07821

·141211 -·18133 ·771311 ·011373 ·23152 -003296 -·OOIlSl -'16647 -'00664 -·104011 ·00826 -00309 ·0014.11 ·77469 ·00184 --00178

--B8lIBO ·031118 -17876 -·23327 -·00110' --17293 ·02844. ·064.l7

-·21638 ·001140 -·01113 -'06961 ·12088 ·1114'79 -21866 -·468111 (a) Tho vibl'atioDal wave numbers of methanol &1'$ taken f.om Margottin-MlIClou

(1960) and Zerbi. Ovef8DCi and Crawford (1963).

(b) The lIumbers in p&I'IID~ are the vibratiou1 fNq\18DOilll in am-l. to) The "ibta.tioIlal fre'laenoiea am taken froID BbimallouDhi (11171) ..

Ta .... 2. 'fhe ~rllRllf"rmation _vice$ L for .4" Vj&fMioatl(S _ LQ)

Qo Q.o Q11

CHaOE'·) (3000)(0) (UGO) (1476)

So -1-01135 '()21378 ·O281~

8'0 -0·108424 -·956038 '281057

~fii/ll -0'092998 ·IS3314 HIU880 CH3OD{(J) ₍₂₀₆₀₎ (1160) (111.73)

S. -1'OSSG ·021378 ·02818

SJO ·108424 ·_·956038 ·281067 8" ^{-·092998 ·153314 HIMGS} CDaOB") (2235) (877) (1076)

8. -·781724 ·004223 -·000006

810 ·166172 ·7017967 ·038019

811 -·183120 ·054177 I 18982 CD$OD{" (2228) (892) (1080) 8fJ -·78172( 'U0422:1 -·UOOO06

810 ·166172 ·747967 ·038010

Su -·183120 ·054177 1·1893~

(n) Tho vibl· .. tion,,\ frequencies of methanol 8'''' , .. kon h'om .Mo.l·i-1,,j,tin·Mac!ul1 (1900) and Zerbi, Overen,\ and Crawford (1963)

(b) The numbers in parenthesis are tho vibratJOna.! frequencies III om-I.

(e) Tho vihr .. tional frequencio. nro taken from RhimBnollohi (1972).

T.ble 3. VlbratlOnal energy leYfJla I;:md Bfl~Rt,hIUK ~ph(~l'(' IHlHltlll~t.t:H'H ^of lut'lt.bollul and deutel'Ooted metllBrllo1e..

---

_{CB,OH CH.OD CD,Oli CD.OD}

Level symmetry _lU{rJ)

---

.-

---.-

<..<12> ^ECb) <A2> ^E,b) <A'> ^E'b) <A2>

(cm-') (a.m.u)-l (om-') (a.Dl.u)-' (em-') (a.m.u)-' (onI-') (a.m.u)-

---

'"

^{A.' 1033 0·0622 1040 0·0474 088 0·0535 983 0·1288}

"2 A' 2883 0·2404 2843 0·2384 2077 0'1136 2080 0'1134

"n

^{A' 1455 0'2234 1473 0·2220 1134 0·1664 11:16 0·0502}

'"

^{A' 3679 0·2300 2718 0'1105 3690 0'2297 2724 0·1063}

'"

^{A' la!W 0·2024 864 0·1124 1297 0'0851 776 1·0437}

v" A' 2960 0·2281 3000 0'2281 2260 0·1021 2260 001010

"7 A' 1070 0·2212 1280 0·2130 868 0'1323 1024 0·0317

v,

A' 1425 (1·2173 1456 0·2151 1047 0·0978 1080 0·08110

'"

^{AN 3000 0·2273 2960 0·2273 223" 0'1020 2228 001020}

1'10 A" 1160 0'1985 1160 0·1985 8'17 0·0972 892 0'0972

"u AN 1475 0·2369 1473 0'2369 1075 0·0927 1080 0·0927

""

^{A" 270 213 256 196}

(a) MaT~.Maolou (1980); Zerbi, Overend and O .... wford: (1968).

(b) Shimanouobi (1979).

145

In OH,OD also tlle probability was found to deol-elWle with inoreuing AB except for the mode Vl' Binoe <AlO· > "'" 4 <.All> and AE differs by 120 om-^l only the probability ]'10("J.o) oan be e:pooted to be la.rger thlm P10(v1). The energy tra.nafer probabilities ]'10 for all the low lying vibrationllol modes. except VI' of OH.On q,re am.a.ller compllorad to OR.OB. whioh is in a.greeDl8llt with the e%poeta.- tion bllolllld on AB and ~uoed mass of the oolliBien plloir. In case oiv_a• the vibra- tional wave numbers ohangell from J345 om-l for elisOR to 864 om-l for OR.OD.

Such a ma.rked reduotion in AE mllo\tes Pl"("a> for CHaOD lq,rger than the OOlTOll- pending value in CHIOB.

TI''')

600 SOO

I~

,~~'----~~----~----~~--~~----~

_{~ ~ ~ ~ ~ M}

(T'Ki1l3

. . . 2_ Temperature depllDdomoea of the V-T traD8i1iioD probabllitdell

~. of the low lying vibl"atioDal modeo of CBeOR aDd OB.OD •.

Solid !iDe (OlIoOH). Broken !iDe (ORsOD). plO(-.) aDd plO( ... ) for OReOR are not abo_ in Figure. plO( ... ~ P"'(_1) and P"(",,) Is approximateq- 10 poll' OODt larger than P"'(Pu) and 20 per _ t 1ID&Il"" thaD plO( .. ). P'" (1)1' 'the modea ~. .... _.. and "11 of OB.OD are »ot presented here. lJ1 8"l181"al they .... BIIIIalIer by 1~0 per aent from the Gorresponding valuea in ClIJeOH 1 1"'"( ... CHtOD) )( 10-< ill abDWll.

The probllbilitieB pia for

aD,OR

a.nd

aD.on

are Jlhow.n in]l'iguro S. From the 1Igqre it 011011 be observed that p10 deerelWleB in general with increasing M

19

146 Y V Okalapati Rao

as was observoo previoUllly in the oase of CHaOS: and CHIOD. In ®.OH the wave numbers of the vibrational modes inorease in the order

v, ^<

^"10

^<

^"1

^< V,

< "11 <

"a

< "5.

whereas the probabilities increase in the revBl'IIII order

TI"KI

~.~oo~~~_~~~~fO~~&r~~

__

^~~r-

____

^-;4~=-

______

^~3~~

,ijI.

"""-

Iff '" ... _....

"

"

" ,Us

1~~L-______ ~ __ ~ __ ~~ ____ ~~'~ ____ ~~ ____ ~

0.1 O~l 0·11 0·'3

(T'Kf,/3

FIpa-e 3. Temperature dependence of the V·T tnwaitioa probabilitJee P'O.

of the low Iymg vibr"tioaaJ. modaa of CD.OR and CD.OD.

Solid liaa (CD.OD). Broken linD (OD.OHl.

For CD.OD P'"(.,,) Q:! p1D( .. l ..ad P'"(v,,) is Dot preaeated and P'O( .. l X I&-' is plotted. For CD.OR plD("", Q:! PlO( .. ) and PlD( .. l.!:::! PlO("".). 1""( .. ) and P'"(_.) B1'O not mown.

,Pl0(",)

>

P10(,,1O)

>

PIO("1) > P10(.s)

>

PIO("ll) > PIO("a) > plO("s)' Above 500"K.

,Plo for "B and "a s.re approximately 6-10 peroent larger 1,han those of "1 and "11 respeotively. Tho broathing sphere parlloIlleters for the modes JI. and "a a.re approll:imatoly two times larger than those of "1 and "11 and the differenoe

in

AJil

::=

^{60 om-I. In view of} th~ large <,AI> val1lllS for "@ and ". SUch variation in the probabilitios ~pear to be reasonable. In CD.OD. the vibrational frtlquencioa, QxpresBBd in wave numbers-inorello8Q in the order

"&

<

"10 < VI

<

V, < "s

<

"11 < Va

and the estimated V·T probabilitiIIB inorease in the order

plO(v&)

>

plO("10) > plO("1) > ,Pl0(".} > ,Pl°(l'll)

>

,PlO(",) > .f1O(IIt)

Energy transfer probabiZitiea 147

The proba.bility PlII(v1) ill la.rger than those of VI a.~d VJ.l which oan ~o be expeoted ta.king into account tha.t <A

I>

for ., ill sma.ll by a factor of threc oompa.red to V~ and V11' The V -T ~ergy transition proba.bUitiea PlO for 8,11 the low lying vibrationaJ. modes, eltOopt V1 a.nd V5 of ODaOR a.re high compa.red to CDaOD in lIogI'eement with the earlier obsorvation Plade on CRsOR a.nd CRaOD. The va.lue of <All> in ODaOD is approximately 2·5 times la.rger tha.n the corres·

ponding value in CDsOR, whiob resulted in a bigher Vllolue of PlO(v1 ) in CDaOD.

The energy gap AE for the V6 mode ha.s reduood from a. value of 1297 om-¹ in CDaOR to 776 ('m-¹ in CDaOD, a.nd thia ma.rked decrease in AE will a.ooount for the la.rger proba.bility PlO(V5) in CDaOD oompa.red. to CDaOR.

The vibrlloi;ion·vibrlloijon energy tra.nsfer probllobilitias, P01lO(a, b) in metha.nol and deuterlloted metha.nols IIol'e also estima.ted in the 1iemperlloturo ra.nge 300- lOOooK by applying the SSR brellothing sphere theory identica.lly. The tam- porlloture dependen,\os of the V·V tI·a.nsition probabilitieA for CRsOR a.nd CHaOD are presented in figure 4, whUe thoso of ODaOR Ilond CDsOD a.ra presonted in

1000 900

aoo

'0°"':"

_ _ 1

l' - - - ---

---

10~---~---~----~~~----~~----~ -2

O~ 0·12 0·13

(T'1(f'f.! 0·'4 ^0·15

Ff&are f. Temperature dependence of the V. V energy tranefer probabilities, POlIO, of OH"OB and OH"OD.

Solid line (CBaOH), Broken line (OBaOD).

1. PO,'· (v,., .,);

4.

P.,l. (." .. )

X 10';

'I. POl" ("a. ");

10. Po,'" (v" Va) X 100.

2. POllO ('.' Vl.);

II. p.,l0 (vs, VI);

8. p • ." (v •• v.);

3. p.,l0 ('v •• •• J;

6. Po," (VI • .,);

I). Poi"

c •• , ''')

^lIi 10',

148 Y V Okaw.paei Rlio

figure Ii. Binoa a large nUMber of suob, prooaBBes are possible, the probabUitiel of only 110 few repressntllotive prOO&llSeS 11ft praBBnted In ilhe figures. FrOM

:Figures 4: IIond Ii it oan be olearly observod that the probabilitleJl for the V-V prooeSBlls with waller 6I.E initially deoroaso With inoreasing teMperlloW:re, reach 110 MinimUM vaolue and then increase with increasing tempera.ture. The vllolia.tion in the probability is vory BMa.ll and appoar to bl! IIolmost temporlloture independent.

The transition probabilitieB, POlIO, for the prooesses with larger AE inotease with increasing temperature. Farther it is also observed f;hllot the variation of POliO

with temperature is steep for prOO6lllloli with lliorgor M oompared to prOOOllS88 with wlIoller IJ..E. The trBousition probabilities POlIO, for the l~ doexcitation of a mode from the upper Vibrational level by a V-V prooess ooupled to 0-1 exoitllotion of a Mode from the low lying VibtBotiona! level differ by 4-6 orders

f('1\1 -500

\

~~--~--~~--~--~--~

_{0.1 0.1' 0-12}

ll'Kf'll

. . . . 5. Tempemture depemieDoe of the

V·V

energy traDsfll1' probabilitiN.

Po,"'. of CD.OB IOII.d

onIon.

Solid lillll (CD,ODI, Broken Jina (CD,OBI.

1.

p.,'. ('" '.1

X 10";

4. p.,'. (," .... 1 X 10";

'J.

p.,'" ("".

".1;

10. Po,'" ('" ... 1;

2. Po,'" ('r. ");

II. po,'· ('r. "",1;

8. POl" (",.

'"1

X 10";

11. Po," ('10' ",I;

3. Po," ( .... ",)1 t. POl" (v,.,. -.) D. POl'" ('.'

",11

12. Poi'" (~ .. ).

of magnitude !ram those of the prooosses oOuPling the modes among 'bile upper Vibrational1evels only or the lower Vlbrationallevols only.

Tho magnitudes of the various vibration-vibration energy transfer probabili- ties suggest that

80n

the modes in a. given lovel, either upJ;l6r vibrational level or $e low lying vibrational level are ooupled by fut V-V pllioollSes, and they equilibrate quite rapidly, whoreas oq1lililir8otion of tho modes coupling the low lying vibrational level 8ond. the uppor vibra.tionallevel is very slow. ~e energy gap (expressed in wa.ve numb\ll'S) botwoon the two lowest vibr8otional modes is

"pproximllotely three times lllorger than the vibrllot,ional energy of the lowest vibra- tiona.l mode vu' Because of tho large onorgy ga.p the V-V ell6rgy tr80Dllfar pro- bllobility eoupling tho two lowBs1; vibrationa.l modes can bo expected to be very low. If 80 vibrational modo, either from low lying vibrational level (except the lowest vibrationa.l mode) or upper vibrationa.l lovel, is selootively excited, the loss of vibrational onergy ma.y bo expeoted to bo low, tb.ough the &Dergy gets redistributed rapidly among tho varioua modes of the low lying or upper vibra- tional lovel.

In tllo present analysis vibration· vibration energy transfer processes involv- ing multiql\lmtum number changos, overtones and oombina1;ion levels are not considorod. It CHon bo so,id 1;ha.t tho loss of exoita.tion from the vibra.tional modes of the high~r vibra.tionallevel dooB not OoolU' through tho V-V prOC6BSeB involving single qll80utum number changos, but may be possible through V-V prooes8e8 involvmg multiple qua.ntum number changes.

Reference.

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HlIl'DIan I p. Mariella Jr R P and Javan A 1976 J. Ohem. PIIy •• 65 3798 and 1978 J. Ohem.

PAy •• 88 1070

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PIIy •• 88 2641

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Tanaka C, Kuratlllli K and M1zllBhima S I 1957 SpooWooMm. AotG 9 265 Tanozos F I 1956 J. Ohem. Phy8. 211 439

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Wilson Jr E B, DeoillS J C and CrollB PC 1955 MoJeoul1!Zl' VibmtiOflBMoGraw.Hill, New York Wu E L, Zerbi G, Oalifano S and Crawford Jr B 1961 J. ahem. Phy •. 85 2060

Yonazawa Y and Fueno T 1974 Bull. Ohem. SOIl. Japan 47 1894 Zerbi G, Overend J and Crawford Jr B 11l6S J. ahem. PhyB. 38 122