.3 The material balance equatin Net change f [J] due t diffusin, cnvectin, and reactin [ ] [ ] [ ] d J J J n = D v k [ J ] fr n - th reactin dt x x hard t slve in general slve it numerically If there is n cnvectin [ ] [ J] d J dt kt [ J] = [ Jd ] e where [ J ] [ J ] d d [ ] = D k J x = A is in the absence f reactin n x 0 4Dt π Dt e [ J ] d = A x 0 4Dt n e π Dt (015) Chemical Kinetics by M Lim 1
Transitin state thery (TST) Activated cmplex (AC) frms between R s as they cllide and begin t assume the nuclear and electrnic cnfiguratins characteristic f P. Change in PE assciated with frmatin f the AC: E a Using the cncepts f statistical thermdynamics Partitin functin, q: average f state that thermally accessible t a mlecule at T f the system. qjm, Mlar Gibbs energy GJ, m( T) = GJm, ( 0) ln (16.51) N where q : J, m aa + b cc + dd K = standard mlar partitin functin f J T be calculated using infrmatin abut energy levels f the mlecule c ( q, N ) ( q, N ) a ( qa, m NA) ( q, m NA) where C m A D m A E0 e b ( ) ( ) E ( A) E ( ) E E C + E D 0 0 0 0 d (16.5a) 0 (015) Chemical Kinetics by M Lim A
.4 The Eyring equatin ( ) C ( )( p ) A p p p [ ][ ] [ ][ ] k A C P v k A Rapid equilibrium between K + = pj = = C = K A p A+ and C p p [ ] C J [ A][ ] v = k A = k C = k K A p k = [ ][ ] [ ][ ] p kk p 1 k = κν ( transmissin ceff )( vibratinal frequency alng the r.c. ) (015) Chemical Kinetics by M Lim 3
.4 The Eyring equatin K = Nq qq A C E0 A e where ( ) ( ) ( ) E E C E A E 0 0 0 0 kt q q q q hν C C C C ( : fr all ther mdes) Nq kt K = K where K = e hν qq A A C E0 Hard t calculate q C aa + b cc + dd kt k = kk = ( κν ) K p p hν kt kt k = κ K = κ C:the Eyring equatin K h p h where K K p C K = c ( q, N ) ( q, N ) a ( qa, m NA) ( q, m N A) C m A D m A E0 e b ( ) E ( D) E ( A) E ( ) where E E C + 0 0 0 0 0 (it requires structure f the AC) (015) Chemical Kinetics by M Lim 4 q d (16.5a) 1 1 kt = = ν 1 e hν hν 1 1 + kt vib h kt
.4(d) The cllisin f structureless particles A and are structureless atms (nly E ) m m A =, = Λ =, = 3 3 J m ΛA Λ π mkt p J IkT Vm q = ( and ) Λ E C 3 trans Ert h kt kt = κ = κ h p h p qaq kt = κ h C trans V V h q q V Nq k K e rna 3 3 NAΛ AΛ IkT 3 p e Λ V C m h kt ΛΛ A IkT = κ NA e h Λ C h = κπ 8kT e πµ 3 E 0 A C E0 E E 0 (015) Chemical Kinetics by M Lim 5 0
.4(e) Observatin and manipulatin f the AC The develpment f femtsecnd pulsed laser: - can bserve shrt-living intermediates (ex, activated cmplex) in real time. - can cntrl the utcme f chemical reactins. ICN + hν I CN I CN I + CN 600 pm (05 fs) Femtchemistry ν [ ] * IH CO + h HOCO HO + CO (015) Chemical Kinetics by M Lim 6
.5 Thermdynamic aspects Statistical thermdynamics versin f TST: needs the structure f the AC (barely available) empirical apprach in terms f thermdynamic functin (utilizing the cncepts f an equilibrium between Rs and the AC) G = ln K, G = H T S k kt = κ h kt h p p K kt = κ h p G ( κ is absrbed int the S term) S R H = e e e E a = ln k T = + Ea = H + H fr reactins in sln r unimlecular reactin (015) Chemical Kinetics by M Lim 7
ln = E a k ln A H = ln A E = + H ( a ) kt S H kt S R = ln + k e e h p R = h p kt S ln A = ln + h p R = k T h p SR A e e H ( ) < + S 0 A C even mre negative The mre cmplex the steric requirements f the encunter, the m S steric factr, P = e S re negative the value f, and the smaller the value f. S steric R P (015) Chemical Kinetics by M Lim 8
Crrelatin analysis (especially fr rganic reactins in slutins) ln K vs ln k rg vs G Fr similar rganic reactins in slutins r G is related t when verall reactins mechanism cntrlled by linear G free energy relatin ( LFER) ln K ln k an elementary reactin (015) Chemical Kinetics by M Lim 9
.5(b) Reactin between ins [ ][ ] v = k C = k A a γ C C K a a γγ A A K γ [ ][ ] [ ][ ] [ ][ ] C C K = = = K C = A A A 0 kk k k = Kγ Kγ Recall that lgγ = Az lg k = lg k lg K 0 0 J J : Debye-Huckel limiting law ( ) 0 = lg k A za + z za + z I 0 = lg k + AzAz I k lg = AzAz I " kinetic salt effect" k γ I lg K γ = lgγ lgγ lgγ Highly charged inic cmplex frm tw less highly charged ins is favred by a high I because the new in has a denser inic atmsphere and interacts with that atmsphere mre strngly C A (015) Chemical Kinetics by M Lim 10 γ
The dynamics f mlecular cllisins.6 Reactive cllisins Rate cnstant: an average ver events in which Rs in different initial states evlve int Ps in their final states Fr a full picture f the reactin: state-specific rate cnstant is needed can be btained frm crssed mlecular beams Methds fr examining the energy distributin in the prduct - Infrared chemiluminescence - Laser-induced flurescence (LIF) - Multiphtn inizatin (MPI) - Reactin prduct imaging - Resnant multiphtn inizatin (REMPI) (015) Chemical Kinetics by M Lim 11
.6(b) State-t-state dynamics Recall that Can be written 0 ( ) ( ) ( ) k = N σ ε v ε f ε dε A rel k = N σv A rel State-t-state rate cnstant fr state n t m, k nm k = σ v N nm nm rel A where σ nm is the state-t-state crss-sectin nm, ( ) ( ) ( ) k = knm T fn T f T : ltzman factr fr the initial state n at a temperature T n (015) Chemical Kinetics by M Lim 1
.7 Ptential energy surfaces (PES) PE as a functin f the relative psitins f all the atms taking part in the reactin H + H H + H H + H - H H - H + A C A H -------------- H -- H A C R A R C H C (015) Chemical Kinetics by M Lim 13
.7 Ptential energy surfaces (PES) Varius trajectries (C: the path f the least PE) the saddle pint the saddle pint H + H H H H + H A C H -------------- H H A C R A R C A C Transitin state: a set f cnfiguratins thrugh which successful reactive trajectries must pass (015) Chemical Kinetics by M Lim 14
.8 Sme results frm experiments and calculatins Experimental cntrl: adjust relative speed f apprach: E t adjust degree f vibratin: E vib then, detect E vib f P (015) Chemical Kinetics by M Lim 15
.8 Sme results frm experiments and calculatins The directin f attack The directin f separatin 1. Sticky cllisin: P in randm directin. Cllisin lasts < 1 ps: the directin f attack is preserved The anistrpy f the PE changes as H appraches H with different angles f attack (015) Chemical Kinetics by M Lim 16
.8(b) attractive and repulsive surfaces Attractive surface high E trans is effective, P with high E vib Repulsive surface high E vib is effective, P with high E trans ex, F + HCl Cl + HF (015) Chemical Kinetics by M Lim 17
.8(c) Classical trajectries Calculate, using CM, the trajectries f the atm taking place in a reactin frm a set f initial cnditins, such as velcities, relative rientatins, and E internal s f the reacting particles. The initial values used fr the E internal reflect the quantizatin f E elec, E vib, and E rt but the features f QM are nt used explicitly in the calculatin f the trajectry cf, quantum mechanical scattering thery Direct mde prcess Cmplex mde prcess Cllisin cluster has a lng life time ex, KCl + Nar Kr + NaCl H -------------- H -- H A C R A R C (015) Chemical Kinetics by M Lim 18
.8(d) QM scattering thery Cmplete QM calculatins f time-dependent wavefunctins and rate cnstants are very nerus because it is necessary t take int accunt all the allwed electrnic, vibratinal, and rtatinal states ppulated by each atm and mlecule in the system at a given temperature. (015) Chemical Kinetics by M Lim 19