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Theory on rates of solution reactions influenced by slow fluctuations in viscous solvents, and its experimental confirmation
Theory OQ Rates of Solution Reactlonr Influenced by Slow Fluctuations In WKOUOSolvents, and ita experimental Confirmation
Institute of MsteriaidsSclence, Unfvareity of Tsukubs, Tsukuba, Ibaraki MS, Jepan Abstract
I. Introduction The mosttraditioualtheoryfor chemicalmactionfatesis the transitionstatetheory(TSJ
distributionof pop&ions in tbe mactantstateis alwaysrraintainedin thermalcqsouilibrium in the courseof the reactim. Then,alsotbe populationof nactaatsin the transitionstateis always maintainedinthcmtalequilibrirnnwiththoseiathenacrantstruc,rurdtherateco~~caobe calculatedfrom~populatianatthe~ti~stltte,w~bisdetermined~bythe~of the free-energystufaee,without knowledgeon dynamicsof fluctuationsfor realizingthe transitionstate.In thisassmption,therefore,the rateconstantshouldnot dependon bow fast fluctuationsam in the reactantstate.Moleculararrangements of the solute-solventsystemin solutioncan takevariousconformations.Tbey fluctuatefrom time to time due to shakingand/or dampingby miqroscopicmotionsof solventmolecules.Sincethe sj5eedof theseconformational tluctuationsdecceasesastbevisc~Gty qofsoiventaincreases, q- canberegardedasameasureof tbe speed& the solutionreactionsmentionedearlier,the rate constantobsen~I decreasesas qincreases,thatis,as fi-~decnases.This~thattherateoo~tofthesesoiution reactionsdependson the speedof Toual fl&oos in the solute-sq,lvent system,aud hencetbgttheSereactionsarewo~~O~locatedoutsiderhefrpneWnrkOf~ TST. To be mom exact,thesenacrionsarecouirokl by slow speedsof thesefluctuations. Thefusttheorygivingthe rl-i~~decreaseoftherateconsutntistheKssrmers~ presentedas early as in 1940.He explicitly treateddynamicalprocessesof fluctuationsin the l.caalmtstate,not lirawin$O prkwi the therulaiequili~utn distihutiorl thereii His reactiofl schemecan he understoodin Fig.1 which shows,along a reactioncoczrdiie X. a double-well potentialW(x) composedof a feac&ntand a productwell with a transition-statebakr ktween them. Reactiontakesplaceasa mwlt of diffusive Brownian motionsof reactantssurmounting / 0167-732~S~.§O 8 1995 Eisevier Science B.V. All rights reserved. sSDlO167-7322 (95) 00903-5 ,’
mall height (of about 15 decmaae8as~incmases.To be mme exact, bwever, &,, often decmyes more slowly tbau cm y’as kobs-q-u withOeac1. thatcxpectedfromtkKramersthcory. of kdn, approximatelytwo currentsof theories Kmmerstheory. the was initiated by Grate and Hynes7 felt by reacta~ shoulddecreasewhen they sur.ldwuII ~~~‘~c~.~~~t.~~~~ ~expectedfromtbeKmmemtlmqba+ljeuceits qdepeuq expectedfhmthaKcamerstkxy.Ttushaoffkquency(that
~~f~~~~g~~~~~~~~~rng
time (r) depeudeutstochasticmotiousof the positionX(f) of a reactantalong the reaction coordinateX in the field of the double-wellpoteatiakW(x), 1s d2X(r) dWtXO)l - J~_$-t,~dr’ -77 = - dxw
+ R(r) ,
(2)
) (24) npresentsthe frlctioaal tllemoe 1 of e+(2) deaxibes friaiand fofw whwe! of the teaaant velocity #f(r~dz’ in the pat at tail of the frictional me* function t(7) at z~ t - I’(5 0). origin of fnkmscopic motionsof solvent ttxdeculcs mast be nlatd to eachother. This relation is known aa
at the point S, which thereforenp~sentsthe kcation itkm state.Mramolecular vibrationalfluctuationsalongthe caordinateq on the ordinatean much fasterthan diffusive Brown&a nucnlationsof intemlolecularar-
Pigum 2. Two-diKnensional pcbtemialsurfacefor Kaaion in the Sumi-Marcusscheme dOngthCiCJOOdb%tCXfOf diffusive Brownian motions andtbecomdiuateqfosmuch fastcfintremolecubuvibraaional motions.afxi an exampleof mctive trajectoriesthere.
.i&)*
&
LL+-.@E2 axIIax k&r dx I P(X;f)- kwYX;r)-
(4)
of the lasttermon therightThe essentialdifferencebetwaaneqs.(I) and(4) is thepresence
handsideof q.(4),andit describeslossof reactantsby reactionsduringdifksion alongX. Thediffusii constantD is n asinverselyproportionalto thesolventviscosityq as before,while k(x) is independentof q, dctenninedby intramolecular vibrationalmotions. It was shown by thepresentauthorllthatthis Sumi-Mmusmodelgivesasa rateconstant
k- l/(k& +k/‘) ) with k, >o (
(5)
where km represents the TSTqxsctcd
rate constant independent of D (= q-l), while kf(> 0) represents a partinfluencedby the speedof conformational fIuctuationswhichbecomesIower andslowerasthesolventviscosity,q incmases. To be moteexact,k@ecreases with r~ as
A/ w #I-= I with Ocacl~
(6)
This fractiofd-powerdependence of &/on t7.l canbe obtainedin the naturalsituationthatk(x) hasa nonvanishing width in X. When km a &, in the low viscosityregion,eq. (5) reducesto k - km. recoveringtheresultof theTST. When k, cx km in the high viscosityregion, on theotherhand,eq.(S) reducesto k-5 kf 0~1]-*. andthefateconstantdecfeases moreslowly thanq.‘, asobservedin photoisomenzation of stilbenesin solver&. Therateconstantof photoinduced knneri&on of stilbene4in solvents,however,always decreases as ‘I in-. Thisseemstomeanthattherateconstantissolargebecauseofalow ~i~tof~~~-~~~~~~’ beencontrolledin solventsby a slowspeedof conformational fluctuationsin the solute-solvent system.This situationitselfis veryinteresting, but is not mostcoLLyGnicM loomthestandpointof investigating thegeneral expnssionfor therateconstantof soltion reactions.More adequate is a situationthattherate constantcanbe described by theTST in tbesmall q region,but decreases with q in thelarge r~region,enteringthenon-TSTregime.Sucha situationwasrecentlyreportedby Asanoandhis collaborator@ with:thepressure dependence of rateconstants of thermalZB3isomerization of substituted azobenzenesand N-benzylidcneanili. Their d&a enables us to determine which current of theoriesinitiatedby GmteandHynesor by Sumiand Marcus is more appropriate
surmountingover a small potentialbarrier,the mole&e dropsto ap intermediatestateat the iowest point of the S1 suffece.Then, the moleculemakesa transitionfo the top of the So surface, from which, abng the So surface,about half of it r&x to the 2 form and the remaininghalf relaxtotheoriginalEform.rnthisprocess l.surmoUnting over the potential barrier on the St surface is rate Eiting. Then, measuringa rise of pop&t&n in t&e2 form we can determinethe r&e constantof the barrier surmounting,as performedfor stilbenes.Sincethe 2 form is metastableon the S, surface.higher than the E fotm, it gradually transformsitself to the E form, SW mountingover a potential barrier with height d&own in Fig.3. This processis denotedby 2. Process1 is very fast becauseof a low barrier on tbe St surface.In stilbenes,the barrier height is about IS kJ/mol and this processtakesabout 100ps after photoexcitation. as met+ tioned in Section I. Process2 is, on the other hand.slow becauseof a high barrier on the So
X=N
--
Xed h \ d
NM*
NQ?
NM02
Figure 3. Mokcular structureof DBNA and DNAB. and their photocyclecomposed of E/Z isomerization(process1,)and Z/E isomerizationon the ground @ii>-and the excited (S Q-statepotentialsurfaces.
for DBPJA
encebetween DBNA inTFB andDNAB in GTA in theactivationvolumeintheTSTngime arises
100
‘1 10
0 0 0 0000 ” 0 Ooo . ..***.* QO l * ‘@* It@& (Sd’)
0
i
00
.
‘*.
00 e
200 04 ’
400 600 Pressure (MPa)
800
Figure4. Prcssundependence of therateconsiantkds of thermalZJEisomerhtion of DBNA in TFB [part(a)] andDNAB in GTA [pm (b)) at temperatutzs around300K.
model which gives q.(5) and (6). More detaileddismssionscan be fowl in Raf. 14. Next, let us discusswhetherthe &oteHynes theory canbe fitted to the rate constantsk&. In this thoery,&&km is t&ted to the frequency(that is, the speed)~4with which the mutant passes.by diffusive Brownian motions,throughthe transition-state-barrierregion in Fig.1, as
DNAi3inGTAat298K (a)
lti
10'
10' vfscosity
100 rl (Pas)
10'
ld
(W”
Figure 6. W.:osity dependence of Sfor DBNA inTFBat 278 K and DNAB in GTA at 298 K.
I
For example, NC). , J.M. H-P DE, Khoshtasiyaaad P. Ulstrup, Eur. I. Bimkmit. MO* 423 (1991). Asa~~M~.waavoraad0.R~~,Ace.chcm.RQg.23.294(1990). For example. P-3. Steiaimeh es ol., Bbdtem. Jo. 3988 (1991). As a review, D.H. Wakleck, C&em. Res. 91.415 (1991). Ear ~xJ%@% Ba, Busaceges. pbyr. Cbem. 95. no.3 (lQQl), Rate Pmcesses in Dissipative Systems. H.A. Kmtefs, Fhyrica 7,284 (1940). RE. @me aad J.T. Hyaex, J. Chem. Phyx. 73.2715 (1980), aad 74‘4465 (1981). 8, Begchi arid D.W. Ox&&y. J. Cham. P&S. 78.2735 (1983); G. RothMerger, D.K. Negus and R.M. fid%~, J. chum. Php. 79, SMO (1983); S.K. Kii and O.R. Fle!!;;rlg. J. Phys. Chem. 92, 2168 (1988); 8t N. Sivakue%ar,EA. Holmrg and D.H. Waldeck. 3. Cbem. Phvs. 90.2305 (1989); stc also. BS a raview article, 8. Bag&i, Iateraat. Rev. Phys. Chcm. 6, I (1987). N.G. VM K8mpg. S(vc&u%& Pmcesses in Physics und Chemistry (NordvHollrmd: Amsterdam. 1992). H. Sum! and B.A. Marcus. 3. Chem. Phvs. 84.4894 f1986). N. Agmon aad J.Y. Hopfieid, J. Chem. lihys. j8.6945 (1983). H. Sumi, J. Phys. Chem. 95. 3334 (1991). K. Cosstick, T. Asaao aad N. Ohao, High Pressure Res. 11.37 (1992). T Aaaxo. Ii. Puruta end H. Sami, J. Am. Cheat. Sot. 116.5345 (1894). T. Asaao et d, I. Org. C&n. 58.4418 (1993). and Cartier papers cited therein. T. Asane and T. Okada, J. Drg. Chem. gl, 4454 (1986), aad earlier paperscited therein. H. Sti aad T. Asatto, C&m. Phys. Lea. (lQQ3). and J. Chem. Phys. 102. no.24 (IQQS), in press. R. Zwaazig, J. Stat. Phys. 9.215 (1973); E. Corms, BJ. West and K. Lindenberg, 1. Chem. Phys. 82, 2708 (1985); Bt B.J. Germer, K.R Wilson aad AT, H;ncs, J. Chem. whys. 96.3537 (1989). F.J. Berm+, F. Batallaa. I.L. Martinez. N. Ciarcia-Heraandezaad 8. Eaciso, J. Phys. Condens.Matter 2. 6659 (19QOhB E.G.D. Cohen, P. Westermdjs aed I M. de S~heppcr,Phys. Rev. Len. 99.2872 (1987). J. True, Ber. Buaaenga. Phys. Chem. 95,228 (199t), and earlier paperscited therein.
PO 11 12
13 14 IS 16 t; 19 20
Institute of MsteriaidsSclence, Unfvareity of Tsukubs, Tsukuba, Ibaraki MS, Jepan Abstract
I. Introduction The mosttraditioualtheoryfor chemicalmactionfatesis the transitionstatetheory(TSJ
distributionof pop&ions in tbe mactantstateis alwaysrraintainedin thermalcqsouilibrium in the courseof the reactim. Then,alsotbe populationof nactaatsin the transitionstateis always maintainedinthcmtalequilibrirnnwiththoseiathenacrantstruc,rurdtherateco~~caobe calculatedfrom~populatianatthe~ti~stltte,w~bisdetermined~bythe~of the free-energystufaee,without knowledgeon dynamicsof fluctuationsfor realizingthe transitionstate.In thisassmption,therefore,the rateconstantshouldnot dependon bow fast fluctuationsam in the reactantstate.Moleculararrangements of the solute-solventsystemin solutioncan takevariousconformations.Tbey fluctuatefrom time to time due to shakingand/or dampingby miqroscopicmotionsof solventmolecules.Sincethe sj5eedof theseconformational tluctuationsdecceasesastbevisc~Gty qofsoiventaincreases, q- canberegardedasameasureof tbe speed& the solutionreactionsmentionedearlier,the rate constantobsen~I decreasesas qincreases,thatis,as fi-~decnases.This~thattherateoo~tofthesesoiution reactionsdependson the speedof Toual fl&oos in the solute-sq,lvent system,aud hencetbgttheSereactionsarewo~~O~locatedoutsiderhefrpneWnrkOf~ TST. To be mom exact,thesenacrionsarecouirokl by slow speedsof thesefluctuations. Thefusttheorygivingthe rl-i~~decreaseoftherateconsutntistheKssrmers~ presentedas early as in 1940.He explicitly treateddynamicalprocessesof fluctuationsin the l.caalmtstate,not lirawin$O prkwi the therulaiequili~utn distihutiorl thereii His reactiofl schemecan he understoodin Fig.1 which shows,along a reactioncoczrdiie X. a double-well potentialW(x) composedof a feac&ntand a productwell with a transition-statebakr ktween them. Reactiontakesplaceasa mwlt of diffusive Brownian motionsof reactantssurmounting / 0167-732~S~.§O 8 1995 Eisevier Science B.V. All rights reserved. sSDlO167-7322 (95) 00903-5 ,’
mall height (of about 15 decmaae8as~incmases.To be mme exact, bwever, &,, often decmyes more slowly tbau cm y’as kobs-q-u withOeac1. thatcxpectedfromtkKramersthcory. of kdn, approximatelytwo currentsof theories Kmmerstheory. the was initiated by Grate and Hynes7 felt by reacta~ shoulddecreasewhen they sur.ldwuII ~~~‘~c~.~~~t.~~~~ ~expectedfromtbeKmmemtlmqba+ljeuceits qdepeuq expectedfhmthaKcamerstkxy.Ttushaoffkquency(that
~~f~~~~g~~~~~~~~~rng
time (r) depeudeutstochasticmotiousof the positionX(f) of a reactantalong the reaction coordinateX in the field of the double-wellpoteatiakW(x), 1s d2X(r) dWtXO)l - J~_$-t,~dr’ -77 = - dxw
+ R(r) ,
(2)
) (24) npresentsthe frlctioaal tllemoe 1 of e+(2) deaxibes friaiand fofw whwe! of the teaaant velocity #f(r~dz’ in the pat at tail of the frictional me* function t(7) at z~ t - I’(5 0). origin of fnkmscopic motionsof solvent ttxdeculcs mast be nlatd to eachother. This relation is known aa
at the point S, which thereforenp~sentsthe kcation itkm state.Mramolecular vibrationalfluctuationsalongthe caordinateq on the ordinatean much fasterthan diffusive Brown&a nucnlationsof intemlolecularar-
Pigum 2. Two-diKnensional pcbtemialsurfacefor Kaaion in the Sumi-Marcusscheme dOngthCiCJOOdb%tCXfOf diffusive Brownian motions andtbecomdiuateqfosmuch fastcfintremolecubuvibraaional motions.afxi an exampleof mctive trajectoriesthere.
.i&)*
&
LL+-.@E2 axIIax k&r dx I P(X;f)- kwYX;r)-
(4)
of the lasttermon therightThe essentialdifferencebetwaaneqs.(I) and(4) is thepresence
handsideof q.(4),andit describeslossof reactantsby reactionsduringdifksion alongX. Thediffusii constantD is n asinverselyproportionalto thesolventviscosityq as before,while k(x) is independentof q, dctenninedby intramolecular vibrationalmotions. It was shown by thepresentauthorllthatthis Sumi-Mmusmodelgivesasa rateconstant
k- l/(k& +k/‘) ) with k, >o (
(5)
where km represents the TSTqxsctcd
rate constant independent of D (= q-l), while kf(> 0) represents a partinfluencedby the speedof conformational fIuctuationswhichbecomesIower andslowerasthesolventviscosity,q incmases. To be moteexact,k@ecreases with r~ as
A/ w #I-= I with Ocacl~
(6)
This fractiofd-powerdependence of &/on t7.l canbe obtainedin the naturalsituationthatk(x) hasa nonvanishing width in X. When km a &, in the low viscosityregion,eq. (5) reducesto k - km. recoveringtheresultof theTST. When k, cx km in the high viscosityregion, on theotherhand,eq.(S) reducesto k-5 kf 0~1]-*. andthefateconstantdecfeases moreslowly thanq.‘, asobservedin photoisomenzation of stilbenesin solver&. Therateconstantof photoinduced knneri&on of stilbene4in solvents,however,always decreases as ‘I in-. Thisseemstomeanthattherateconstantissolargebecauseofalow ~i~tof~~~-~~~~~~’ beencontrolledin solventsby a slowspeedof conformational fluctuationsin the solute-solvent system.This situationitselfis veryinteresting, but is not mostcoLLyGnicM loomthestandpointof investigating thegeneral expnssionfor therateconstantof soltion reactions.More adequate is a situationthattherate constantcanbe described by theTST in tbesmall q region,but decreases with q in thelarge r~region,enteringthenon-TSTregime.Sucha situationwasrecentlyreportedby Asanoandhis collaborator@ with:thepressure dependence of rateconstants of thermalZB3isomerization of substituted azobenzenesand N-benzylidcneanili. Their d&a enables us to determine which current of theoriesinitiatedby GmteandHynesor by Sumiand Marcus is more appropriate
surmountingover a small potentialbarrier,the mole&e dropsto ap intermediatestateat the iowest point of the S1 suffece.Then, the moleculemakesa transitionfo the top of the So surface, from which, abng the So surface,about half of it r&x to the 2 form and the remaininghalf relaxtotheoriginalEform.rnthisprocess l.surmoUnting over the potential barrier on the St surface is rate Eiting. Then, measuringa rise of pop&t&n in t&e2 form we can determinethe r&e constantof the barrier surmounting,as performedfor stilbenes.Sincethe 2 form is metastableon the S, surface.higher than the E fotm, it gradually transformsitself to the E form, SW mountingover a potential barrier with height d&own in Fig.3. This processis denotedby 2. Process1 is very fast becauseof a low barrier on tbe St surface.In stilbenes,the barrier height is about IS kJ/mol and this processtakesabout 100ps after photoexcitation. as met+ tioned in Section I. Process2 is, on the other hand.slow becauseof a high barrier on the So
X=N
--
Xed h \ d
NM*
NQ?
NM02
Figure 3. Mokcular structureof DBNA and DNAB. and their photocyclecomposed of E/Z isomerization(process1,)and Z/E isomerizationon the ground @ii>-and the excited (S Q-statepotentialsurfaces.
for DBPJA
encebetween DBNA inTFB andDNAB in GTA in theactivationvolumeintheTSTngime arises
100
‘1 10
0 0 0 0000 ” 0 Ooo . ..***.* QO l * ‘@* It@& (Sd’)
0
i
00
.
‘*.
00 e
200 04 ’
400 600 Pressure (MPa)
800
Figure4. Prcssundependence of therateconsiantkds of thermalZJEisomerhtion of DBNA in TFB [part(a)] andDNAB in GTA [pm (b)) at temperatutzs around300K.
model which gives q.(5) and (6). More detaileddismssionscan be fowl in Raf. 14. Next, let us discusswhetherthe &oteHynes theory canbe fitted to the rate constantsk&. In this thoery,&&km is t&ted to the frequency(that is, the speed)~4with which the mutant passes.by diffusive Brownian motions,throughthe transition-state-barrierregion in Fig.1, as
DNAi3inGTAat298K (a)
lti
10'
10' vfscosity
100 rl (Pas)
10'
ld
(W”
Figure 6. W.:osity dependence of Sfor DBNA inTFBat 278 K and DNAB in GTA at 298 K.
I
For example, NC). , J.M. H-P DE, Khoshtasiyaaad P. Ulstrup, Eur. I. Bimkmit. MO* 423 (1991). Asa~~M~.waavoraad0.R~~,Ace.chcm.RQg.23.294(1990). For example. P-3. Steiaimeh es ol., Bbdtem. Jo. 3988 (1991). As a review, D.H. Wakleck, C&em. Res. 91.415 (1991). Ear ~xJ%@% Ba, Busaceges. pbyr. Cbem. 95. no.3 (lQQl), Rate Pmcesses in Dissipative Systems. H.A. Kmtefs, Fhyrica 7,284 (1940). RE. @me aad J.T. Hyaex, J. Chem. Phyx. 73.2715 (1980), aad 74‘4465 (1981). 8, Begchi arid D.W. Ox&&y. J. Cham. P&S. 78.2735 (1983); G. RothMerger, D.K. Negus and R.M. fid%~, J. chum. Php. 79, SMO (1983); S.K. Kii and O.R. Fle!!;;rlg. J. Phys. Chem. 92, 2168 (1988); 8t N. Sivakue%ar,EA. Holmrg and D.H. Waldeck. 3. Cbem. Phvs. 90.2305 (1989); stc also. BS a raview article, 8. Bag&i, Iateraat. Rev. Phys. Chcm. 6, I (1987). N.G. VM K8mpg. S(vc&u%& Pmcesses in Physics und Chemistry (NordvHollrmd: Amsterdam. 1992). H. Sum! and B.A. Marcus. 3. Chem. Phvs. 84.4894 f1986). N. Agmon aad J.Y. Hopfieid, J. Chem. lihys. j8.6945 (1983). H. Sumi, J. Phys. Chem. 95. 3334 (1991). K. Cosstick, T. Asaao aad N. Ohao, High Pressure Res. 11.37 (1992). T Aaaxo. Ii. Puruta end H. Sami, J. Am. Cheat. Sot. 116.5345 (1894). T. Asaao et d, I. Org. C&n. 58.4418 (1993). and Cartier papers cited therein. T. Asane and T. Okada, J. Drg. Chem. gl, 4454 (1986), aad earlier paperscited therein. H. Sti aad T. Asatto, C&m. Phys. Lea. (lQQ3). and J. Chem. Phys. 102. no.24 (IQQS), in press. R. Zwaazig, J. Stat. Phys. 9.215 (1973); E. Corms, BJ. West and K. Lindenberg, 1. Chem. Phys. 82, 2708 (1985); Bt B.J. Germer, K.R Wilson aad AT, H;ncs, J. Chem. whys. 96.3537 (1989). F.J. Berm+, F. Batallaa. I.L. Martinez. N. Ciarcia-Heraandezaad 8. Eaciso, J. Phys. Condens.Matter 2. 6659 (19QOhB E.G.D. Cohen, P. Westermdjs aed I M. de S~heppcr,Phys. Rev. Len. 99.2872 (1987). J. True, Ber. Buaaenga. Phys. Chem. 95,228 (199t), and earlier paperscited therein.
PO 11 12
13 14 IS 16 t; 19 20