Ch. 24 Molecular Reaction Dynamics 1. Collision Theory 2. Diffusion-Controlled Reaction

Transcription

1 Ch. 4 Moleculr Rection Dynmics 1. Collision Theory. Diffusion-Controlle Rection Lecture The Mteril Blnce Eqution 4. Trnsition Stte Theory: The Eyring Eqution 5. Trnsition Stte Theory: Thermoynmic Aspects 6. Rective Collisions: will be sippe. 7. Potentil Energy Surfces 8. Some Results from Experiments n Clcultions 9~1. Others: will be sippe. Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-1

2 Rectnt molecules in solution hve to jostle their wy through the solvent, so their encounter frequency is consierbly less thn in gs. However, becuse molecule in solution lso migrtes only slowly wy from loction, two rectnt molecules tht encounter ech other sty ner ech other for much longer thn in gs. The lingering of one molecule ner nother ue to the presence of solvent molecules is clle the cge effect. For the ctivtion energy of rection in solution, we nee to consier the energy of the entire locl ssembly of rectnt n solvent molecules. Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-

3 Suppose tht the rte of formtion of n encounter pir AB is first-orer in ech of the rectnts A n B: A B AB v As you shll see lter, is etermine by the iffusionl chrcteristics of A n B. The encounter pir cn bre up without rection or it cn go on to form proucts P. AB A B v Applying the QSSA, [AB] n AB P v [AB] [AB] t [AB] [AB] 0 [AB] Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-3

4 AB P v [AB] The rte of formtion of proucts is therefore: [P] t [AB] [P] t n A B AB AB A B AB P v v v [AB] [AB] Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-4

5 [P] t n If the rte of seprtion of the encounter pir is much slower thn the rte of prouct formtion, The effective rte constnt is: In this limit, the rection is iffusion-controlle. The rte of rection is governe by the rte t which the rectnt molecules iffuse through the solvent. Ex) Ricl n tom recombintion rections re often iffusioncontrolle ue to the low ctivtion energy. Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-5

6 [P] t n When the ctivtion energy of the rection AB P is lrge, The effective rte constnt is: K where K In this limit, the rection is ctivtion-controlle. The rection procees t the rte t which energy ccumultes in the encounter pir from the surrouning solvent. Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-6

7 For iffusion-controlle rections, [P] t Cf) A B AB v The rte of iffusion-controlle rection is clculte by consiering the rte t which the rectnt to iffuse together. The rte constnt for rection, in which the two rectnt molecules rect if they come within istnce R* (See Justifiction 4.3), is: * 4R DN A where D is the sum of the iffusion coefficients of the two rectnt species in the solution. D D A D B Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-7

8 4R * DN A As shown in Chpt. 1, the iffusion coefficient is relte to the hyroynmic rius by the Stoes-Einstein reltion: D D A D B D T f T 6 For the molecules A n B, of which the hyroynmic rii re R A n R B, respectively, in meium of viscosity : R* R A R B D A T 6R A n Assume tht R A = R B = ½ R*, D B T 6R B D * * T 8RT R DNA 4R N A 3R 3 4 * A D B T 3R * Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-8

9 8RT 3 R: gs constnt. In this pproximte expression, the is inepenent of the ientities of the rectnts, n epens only on T n the viscosity of the solvent. For exmple, the rte constnt for the recombintion of I toms in hexne ( = 0.36 cp) t 98 K is: RT J/K mol (98K) m /mol s g/m s This well grees to the experimentl vlue ( m 3 /mol s). Recll tht 1P 0.1g/m s n 1J 1gm /s Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-9

10 When both iffusion n convection occur, the generlize iffusion eqution (Chpt. 1) is: c t D x c v c x The concentrtion of prticles my chnge s result of chemicl rection. For further refinement of iffusion eqution, the concentrtion chnge by the rection shoul trete. [J] t [J] [J] D v x x Diffusion Convection Rection Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-10

11 Consier smll volume element in chemicl rector (or biologicl cell). During some time intervl, the totl chnge of the number of prticles of some species J must stisfy the following mteril blnce eqution. Net chnge of number of J in the volumeelement Number entering Number leving Number forme by rection Number issppere by rection Accumultion In Out Genertion [J] [J] [J D v ] t x x Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-11

12 [J] [J] [J D v ] t x x The rectnt molecules (J) re forme or isppers ue to chemicl rection when they flow through the volume element. If the rection is pseuofirst-orer, then the net rte of chnge of [J] is: [J] [J] t Therefore, the overll rte of chnge of [J] is: [J] t [J] [J] D v x x Diffusion Convection [J] Rection (For pseuofirst-orer) Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-1

13 [J] t [J] [J] D v x x Diffusion Convection [J] Rection This is clle the mteril blnce eqution. If is lrge, then [J] will ecline rpily. If D is lrge, then the ecline cn be replenishe s J iffuses rpily into the region. The convection term (e.g., cuse by stirring) cn sweep mteril either into or out of the region ccoring to the signs of v n the concentrtion grient ( [J]/ x). Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-13

14 The mteril blnce eqution is secon-orer prtil ifferentil eqution n is fr from esy to solve in generl. As specil cse, consier n unstirre rection vessel (i.e., no convection motion). [J] [J] D [J] t x If the solution of this eqution in the bsence of rection (i.e., = 0) is [J], then the solution in the presence of rection (>0) is [J]*. t t t n e 4Dt 0 where [J] 0 A Dt [J]* [J] e t [J] e Note tht [J] is the solution for system in which initilly lyer of n 0 N A molecules is spre over plne of re A. Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture x x

15 [J]* t t t n e 4Dt 0 where [J] 0 A Dt [J] e t [J] e x This solution revels tht the concentrtion of J ecreses more rpily thn tht (gry lines) in the bsence of rection. Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-15

16 Next Reing: 8 th E: p.880 ~ th E: p.843 ~ 850 Prof. Yo-Sep Min Physicl Chemistry II, Fll 013 Lecture 17-16