Ch. 24 Molecular Reaction Dynamics 1. Collision Theory

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1 Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic spcts 6. Ractiv Collisions: will b skippd. 7. Potntial Enrgy Surfacs 8. Som Rsults from Exprimnts and Calculations 9~1. Othrs: will b skippd. Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-1

2 ow w ar at th hart of chmistry. Hr w xamin th dtails of what happns to molculs at th climax of ractions. Th simplst quantitativ account of raction rats is in trms of collision thory, which can b usd only for th ractions in th gas phas. Considr th bimolcular lmntary raction, B P v k[][b] whr is th collision cross-sction. v rat of σc σ σ T M T M collision B B [][B] Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-

3 T v k [][B] v σ [][B] M Comparing, k σ Howvr, a collision will b succssful only if th kintic nrgy xcds th activation nrgy (E a ) of th raction. Thrfor th rat constant should also b proportional to a Boltzmann factor of th xponntial form. T M k σ T M E a Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-3

4 ot vry collision will lad to raction, vn if th nrgy rquirmnt is satisfid, bcaus th raction may nd to collid in a crtain rlativ orintation. This stric rquirmnt suggsts that a furthr factor, P should b introducd. k P σ T M E a k stric rquirmnt ncoutr rat nrgy rquirmnt Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-4

5 Rcall that th collision frquncy (z) for lik molculs () is: z σc rl whr is th numbr dnsity of molculs and rlativ man spd. Thrfor, c rl c and Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-5 c m c rl m m whr is th rducd mass. For two idntical molculs, = m/. For two dissimilar molculs, m m B c rl is thir

6 c rl This xprssion also applis to th man rlativ spd of dissimilar molculs. Th total collision dnsity is th collision frquncy multiplid by th numbr dnsity ( ) of molculs: Z 1 z whr th factor of ½ has bn introducd to avoid doubl counting of th collisions. 1 σc rl z σc rl Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-6

7 For collisions of and B molculs, th collision dnsity is: Z σ B c rl ot that th factor of ½ has bn discardd, bcaus w ar considring an molcul colliding with any of th B molculs as a collision. Sinc th numbr dnsity of a spcis J is J = [J], Z Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-7 B B σ [][B] whr is th collision cross-sction, d and d σ d d B ot that d is th radius of.

8 Similarly, th collision dnsity for lik molculs can b xprssd with [], Z 1 σc rl 1 4 σ σ σ 4m kt m Sinc J = [J], Z σ 4kT m [ ] For xampl, in gas (d = 80 pm) at room T and p, Z = /m 3 s Vry larg numbr!!! Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-8

9 ccording to collision thory, th rat of chang in th numbr of molculs pr unit volum pr unit tim can b writtn as th product of this collision dnsity and th probability that th collision occurs with sufficint nrgy: d dt Z whr f is th fraction of collisions that occur with a kintic nrgy along th lin of approach of th atoms in xcss of som thrshold valu E a. By th Boltzmann distribution, f B f E a Sinc J = [J], d[] dt Z B Ea Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-9

10 d[] dt Z B Ea Z B σ [][B] Thrfor, d[] dt σ [][B] Ea σ Ea [][B] It follows that: k σ Ea or k c rl Ea This quation has th rrhnius form, providd th xponntial T dpndnc dominats th T dpndnc of th prxponntial factor. Thrfor, th pr-xponntial factor is a masur of th rat at which collisions occur in th gas. calld frquncy factor. Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-10

11 k Ea σ σ For gas-phas ractions, th simplst procdur for calculating (i.., k) is to us for th valus obtaind for non-ractiv collisions or from tabls of molcular radii. σ d and d d d B << << > Good agrmnt Larg discrpancy Larg discrpancy Raction occurs mor quickly than th collision?? Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-11

12 This larg discrpancy suggsts that th collision nrgy is not th only critrion for raction and that som othr fatur, such as th rlativ orintation of th colliding spcis, is important. Th disagrmnt can b accommodatd by introducing a stric factor (P). Th ractiv cross-sction (*) is xprssd as: * P Thn th rat constant bcoms: Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-1 k Pσ Ea

13 Estimat th stric factor for th raction H + C H 4 C H 6 at 68 K givn that th pr-xponntial factor is dm 3 /mol s. k Ea Pσ 8 kt dm 3 xp Pσ /mol s s a rsult, th stric factor P is th ratio of xp / calc. T 68 K k m m H H m C /mol J/K m H C H 7 kg Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-13

14 d H d C Howvr, bcaus th molculs ar not sphrical, mor approximat collision cross-sction is th avrag of thm. H 4 H C H 0.7 nm 0.64 nm 4 (From Tabl 1.1) 0.46 nm It follows that P = s a gnral guid, th mor complx th molculs, th smallr th valu of P. Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-14

15 In this cas, P = 4.8, indicating an anomalously larg raction cross-sction. * = 4.8 It has bn proposd that th raction procds by a harpoon mchanism. For th raction, th K atom approachs th Br molculs, and whn th two ar clos nough an lctron (th harpoon) flips across to th Br molcul. > ow thy ar ions, and so thr is a Coulombic attraction (th lin on th harpoon). Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-15

16 Undr th Coulombic attraction, th ions mov togthr and th raction tak plac. Consquntly, th harpoon xtnds th cross-sction for th ractiv ncountr. Th valu of P can b stimatd by calculating th distanc at which it is nrgtically favorabl for th lctron to lap from K to Br. Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-16

17 Estimat th valu of P for th harpoon mchanism by calculating th distanc at which it is nrgtically favorabl for th lctron to lap from K to Br. Thr ar thr contributions to nrgy of th procss K + Br K + and Br Ionization (I) of K. Elctron affinity (E a ) of Br 3. Coulombic intraction. Whn th sparation is R, 4 R 0 Th lctron flips across whn th sum of thr contributions changs from positiv to ngativ (i.., whn th sum is zro). Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-17

18 Th nt chang in nrgy whn th transfr occurs at a sparation R is: E I Ea 4 R Sinc th ionization nrgy I is largr than E a, only whn R has dcrasd to lss than som critical valu of R*, E bcoms ngativ. I Ea R* 4 0 t this sparation R*, th ractiv cross-sction is givn by: * R* 0 Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-18

19 * R* I E a R* 4 0 Thus, th stric factor is: P * R* d 40d( I E a ) whr d R K R Br With I = 40 kj/mol, E a ~ 50 kj/mol, and d = 400 pm, P = 4. Good agrmnt with th xprimntal valu of 4.8! Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-19

20 xt Rading: 8 th Ed: p.876 ~ th Ed: p.839 ~ 843 Prof. Yo-Sp Min Physical Chmistry II, Fall 013 Lctur 16-0

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