Dr M. BROUARD. 5. Thermodynamic formulation of Transition State Theory Entropy of activation. Thermochemical kinetics. CHEMICAL REACTION RATES

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1 CHEMICAL REACTION RATES Dr M. BROUARD Trinity Term 2003 A. Bimlecular Reactins 5 Lectures 1. Intrductin Simple cllisin thery. Ptential energy curves and surfaces. The reactin crdinate and barriers t reactin. Types f ptential energy pathway. 2. Cllisin Thery Cllisin versus reactin crss-sectins. Dependence f reactin crss-sectins cllisin energy and reactant state. Crss-sectins as average ver reactant rientatin and impact parameter. Micrscpic t macrscpic rate cnstants. 3. Transitin State Thery Reactin crdinate and the Transitin State. The quasi-equilibrium assumptin. Cmparisn with simple cllisin thery. 4. Applicatins f Transitin State Thery The pre-expnential factr and the temperature dependence f reactin rate cnstants. Kinetic istpe effects. Quantum mechanical tunnelling. Transitin state spectrscpy. 5. Thermdynamic frmulatin f Transitin State Thery Entrpy f activatin. Thermchemical kinetics. B. Unimlecular Reactins 3 Lectures 1. Thermally Activated Unimlecular Reactins The Lindemann and Lindemann-Hinshelwd theries f unimlecular reactin rates and their limitatins. Types f ptential energy surface. Energy dependent reactin rates. Strng cllisin assumptin. 2. Energy Dependent Reactin Rates RRK thery fr the energy dependent activatin and reactin rate cnstants. Intramlecular vibratinal redistributin. 3. RRK Thery f Unimlecular Reactins and Beynd Thermally and Chemically activated reactins. Transitin state (and RRKM) thery f unimlecular reactins in the high pressure limit. 1

2 Suggested Reading A. General Texts 1. P.W. Atkins Physical Chemistry, OUP, 6 th Editin (1998), Chapters 25 and K J Laidler, Chemical Kinetics, Harper and Rw, 3 rd Editin (1987), Chapters 5 and M J Pilling and P W Seakins, Reactin Kinetics, Oxfrd University Press, (1995), Chapters 3 t M J Pilling and I W M Smith (Ed), Mdern Gas Kinetics, Blackwells (1987), Chapters A.3, B.1, B.2 and C J L Steinfeld, J S Francisc and W L Hase, Chemical Kinetics and Dynamics, Prentice Hall (1989), Chapters 6, 8, 9 and M. Bruard, Reactin Dynamics, OUP Primer (1998), Chapters 1, 2, 5 and 6. B. Mre Detailed Texts 1. R D Levine and R B Bernstein, Mlecular Reactin Dynamics, OUP (1987), Chapters 1, 4, 5 and R B Bernstein, Chemical Dynamics via Mlecular Beam and Laser Techniques, OUP (1982), Chapters 3, 4, 6 and P J Rbinsn and K A Hlbrk, Unimlecular Reactins, Wiley- Interscience (1972), Chapters 1 and J H Beynn, J R Gilbert, Applicatins f Transitin State Thery t Unimlecular Reactins, Wiley and Sns (1984). 2

3 Prblems The prblems given belw are nt all shrt and straightfrward, and this is t be expected in a curse which tries t link nn-trivial thery with experimental data. The prblems invlving calculatins f rate cnstants are necessarily quite tricky, and fr these sme hints are given. Mst prblems are lnger than yu wuld expect t find in a Finals General Paper. Bimlecular Reactins 1. (a) The fllwing table cntains data n the prperties f the reactants and transitin state (TS) fr the reactin D + H 2 DH + H. btained frm an ab initi ptential energy surface. Like the ther istpic variants f this reactin, the TS is linear. Parameter Reactants (H 2 ) Transitin State r e, DH /Å 0.93 r e, HH /Å Ptential energy/kj ml Frequencies/cm 1 Stretch Bend (dubly degenerate) 870 Use the data t evaluate the cnventinal transitin state thery (TST) rate cnstant at 300 K. Hints i. Only a rati f rtatinal partitin functins is required: the mment f inertia f the TS may be calculated frm the expressin fr a linear ABC mlecule with equal bnd lengths given in Atkins, Physical Chemistry r the lectures. ii. The expnential term in the rate cnstant expressin, ɛ 0, refers t the difference in zer-pint energies, which is nt quite the same as the ptential energy barrier, V. (b) The experimental rate cnstant fr this reactin at 300 K is cm 3 mlecule 1 s 1, which is very clse t that btained by quantum mechanical calculatin. Why is the TST answer yu btain in (a) significantly lwer than this value? The quasi-classical trajectry calculated rate cnstant fr the same ptential energy surface as used in the quantum mechanical calculatins is cm 3 mlecule 1 s 1 at 300 K. Suggest a reasn why this classical result is much clser t experiment than the TST calculated value. (c) Based n yur answer t part (a), and the TST results given in the lecture curse fr the reactin H + D 2 HD + D, evaluate the istpe effect predicted by TST. Cmpare this value with that btained by cnsidering vibratinal zer-pint energy differences alne (i.e. ignring the differences in A- factrs). On the basis f TST, what is the dminant cntributin t the istpe effect fr this reactin at 300 K? (d) The experimental istpe effect fr the tw reactins at 300 K is k D+H2 k H+D2 14 at 300 K. Why is the experimentally derived kinetic istpe effect larger than the ne predicted by TST? 3

4 2. Try anther TST calculatin, this time f the 300 K rate cnstant fr the reactin H + HBr H 2 + Br. The relevant data are cllected in the table belw: nce again yu may assume the TS t be linear. E 0 is the zer-pint energy difference between the TS and the reactants. Parameter Reactants (HBr) Transitin State r e, HBr /Å r e, HH /Å 1.50 E 0 /kj ml Frequencies/cm 1 Stretch Bend (dubly degenerate) 460 Hint Yu may have difficulty calculating the mment f inertia fr the cmplex (the relevant equatin is in Atkins Physical Chemistry ). Yu shuld get kg m (a) What temperature dependences wuld be expected accrding t TST fr the rate cnstants f the fllwing reactins? (Assume all vibratinal partitin functins are unity.) i. A bimlecular reactin between tw diatmic mlecules frming a square planar TS. ii. A bimlecular reactin between an atm and a diatmic mlecule which react via a linear triatmic TS. iii. A bimlecular reactin between an atm and a nn-linear mlecule giving a nn-linear TS. (b) The reactin 2 NO + Cl 2 2 NOCl is believed t be truely termlecular. The temperature dependence f the rate cnstant takes the frm k(t ) T 3.5 e E 0/RT. Accunt fr this via TST. (c) The rate cnstant fr the reactin OH + H 2 (v = 0, 1) H 2 O + H has been measured fr translatinally and rtatinally thermalized H 2 mlecules in vibratinal levels v = 0, 1. The results are k v=0 (T ) = e 18000/RT cm 3 mlecule 1 s 1 k v=1 (T ) = e 11000/RT cm 3 mlecule 1 s 1 where the activatin energies are in J ml 1. Calculate the rate cnstants k v=0,1 (T ) at five temperatures ver the range K, and cmment n the relative magnitudes f the tw state-specific rate cnstants. Next use these data t calculate the thermal rate cnstant (i.e. that fr a Bltzmann vibratinal distributin in the reactants) ver the same temperature range, and plt the results in Arrhenius frm, cmmenting n the shape f the plt yu btain. Yu may neglect cntributins frm vibratinal levels v 2, and take the energy difference between H 2 (v = 1) and H 2 (v = 0) t be 49.6 kj ml 1. 4

5 4. (a) The xidatin f benzaldehyde by permanganate in slutin is believed t invlve the almst cmplete breaking f the aldehydic C H bnd within the transitin state f the rate determining step in the mechanism fr this reactin. If this is s, hw shuld the rate f this reactin at 300 K change when C 6 H 5 CHO is replaced by C 6 H 5 CDO? The wavenumber f the aldehydic C H vibratin may be taken as 2900 cm 1. (b) The fllwing data were recrded fr tw istpic variants f the reactin Cl + HI HCl + I. Temperature/K k H /k D Calculate the difference in activatin energies fr the tw reactins. Cmment n this result, given that the vibratinal wavenumber f HI is 2308 cm If very little is knwn abut the detailed structure f the transitin state, pre-expnential factrs may still be estimated by cnsidering the entrpy change n frming the transitin state. Cnsider the reactin H + C 2 H 6 C 2 H 7 H 2 + C 2 H 5 in the gas phase: estimate the A-factr at 400 K Hints (a) Prceed by calculating a lwer limit t S, the difference between the standard (p = 10 5 Pa) entrpies f the TS and the reactants. Fr S (i.e. S fr C 2 H 7) use C 2 H 6 as a mdel. (b) The additin f an H atm makes minimal differences t mass and mments f inertia f C 2 H 6, s the cntributins t the standard entrpy f the cmplex frm translatin and rtatin C 2 H 6, but with the may be set equal t their cntributin t S additin f a psitive term which accunts fr the reductin in symmetry n frming the TS. Fr this system, the rati f the apprpriate symmetry numbers fr C 2 H 6 and C 2 H 7 is 6. What shuld this term be? (c) A term needs t be added which accunts fr the increase in electrnic degeneracy in ging frm C 2 H 6 (a singlet) t C 2 H 7 (a dublet). (d) There are changes in vibratinal and internal rtatinal partitin functins in frming the H H C structure. A lwer limit n S and thus n S is btained by neglecting them, i.e. S S C 2 H 6 + tw psitive terms, where the latter terms accunt fr the decrease in symmetry and the increase in electrnic degeneracy. (e) Finally, S fr the H atms is required. Use the Sackur-Tetrde equatin (Atkins, 5 th Ed. p686 ). Hwever, remember that H is a 2 S atm. Cmment n yur answer, given that the experimental A-factr is dm 3 ml 1 s 1. Unimlecular Reactins 1. The table belw shws the variatin f the unimlecular rate cnstant, k uni, with pressure at 763 K fr the ismerizatin f cyclprpane t prpene. 5

6 k uni /10 4 s p/trr (a) By graphical means, use the data t test Lindemann thery. (b) Estimate (as best yu can) the high pressure limiting rate cnstant, k. (c) Based n yur answer t (b), find a value fr p 1/2. Hence estimate k 1, clearly stating the units emplyed. (d) The critical energy, E 0, fr the reactin is 274 kj ml 1 and the cllisin number under the abve cnditins is Z AM = dm 3 ml 1 s 1. Shw that the apprximate number f active scillatrs, s, required by the Hinshelwd mdel is eight. Cmment n this result. 2. Trans-diphenylbutadiene underges unimlecular phtismerizatin in ne f its lw lying electrnically excited states. The rate cnstants fr ismerizatin (determined frm real-time flurescence lifetime measurements) vary with energy in the excited butadiene in the fllwing way: ɛ/cm k 2 (ɛ)/10 10 s (a) Use the data t make a classical RRK estimate f the number f active scillatrs, s, and calculate the rate cnstant k, given that the critical energy ɛ 0 = 1100 cm 1. (b) Implicit in the free energy flw assumptin, upn which RRK thery (and RRKM thery) is (are) based, is that all vibratinal mdes shuld be active (ie. vibratinal energy is assumed t be distributed randmly amng all the vibratinal mdes f the mlecule). Using yur estimate f k in (a), calculate k 2 (ɛ) assuming all the mdes in diphenylbutadiene are active. Cmment n the answers yu btain. 3. (The fllwing questin is adapted frm the Advanced Physical Chemistry paper, 1994.) Free-flw (statistical) theries f redistributin f energy ften yield an equatin fr the rate f dissciatin f a vibratinally excited mlecule f the frm ( E E0 k(e) = A E ) s 1 where A is a cnstant, E is the ttal internal energy, E 0 is the bnddissciatin energy and s the number f vibratinal mdes. The mlecule ClONO can fragment in tw ways ClONO Cl + NO 2 r H (298) = +72 kj ml 1 (1) ClONO ClO + NO r H (298) = +111 kj ml 1 (2) (a) If a reactin system cntains bth ClO and NO, energy-rich ClONO may be frmed by cmbinatin. Assuming that A is the same fr bth channels, estimate the relative imprtance f fragmentatin int prducts (1) and redissciatin t reactants 6

7 (2) fr a ttal energy f ClO + NO f (i) 10 kj ml 1 abve the threshld fr channel (2); and (ii) 50 kj ml 1 abve the threshld. Reactin A /s 1 at T 1000 K (b) Hw can stabilized ClONO be frmed? (c) What d the data suggest abut the prducts f the reactin between Cl and NO 2? C 3 H 8 CH 3 + C 2 H (d) T what extent can the cnstant A be predicted? Is it really likely t be the same fr differing dissciatin pathways? The RRK rate cnstant k is used as an adjustable parameter t fit limiting high pressure A-factrs, A. Thus the thery prvides n explanatin fr the wide variatin f experimental A values frm ne reactin t anther, fr example: (CHO) 2 2CO + H H + C H (a) Use transitin state thery (which is equivalent t RRKM thery in the high pressure limit) t estimate the entrpy f activatin and the rati f transitin state t reagent partitin functins, q /q A, fr each f the abve reactins. (b) Hw are the values yu btain in (a) related t the nature f the transitin states invlved? 7