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Atkins & de Paula: Atkins’ Physical Chemistry 9e Chapter 22: Reaction Dynamics Chapter 22: Reaction Dynamics REACTIVE ENCOUNTERS 22.1 Collision theory rate constant, kr encounter rate minimum energy requirement steric requirement. A BP v k r [A][B] 1/ 2 8 RT 1/ 2 1/ 2 1/ 2 c c (T / M ) v (T / M ) N AN B (T / M ) [A][B] M k r (T / M )1/ 2 e Ea / RT k r P (T / M )1/ 2 e Ea / RT 22.1(a) Collision rates in gases collision density, the number of (A,B) collisions in a region of the sample in an interval of time divided by the volume of the region and the duration of the interval: 1 Z AB 8kT N A2 [A][B], Z AA 4kT N A2 [A]2 m A 2 1 2 d 2 d 12 (d A d B ), m A mB m A mB Chapter 22: Reaction Dynamics collisionfrequency, z crel N A crel 8kT 1 2 collisiondensity,Z AA 12 zN A 12 crel N 2A Z AB crel N AN B 1 Z AB collision cross-section 8kT N A2 [A][B] 2 volume of tube N 1 / 1 / crel t z 1 / t crel N 22.1(b) The energy requirement dN A ( )vrel N AN B ( ) 0 when ε ε a dt d[A] ( )vrel N A [A][B] dt d[A] ( )vrel f ( )d N A [A][B] f ( ); Boltzmanndistribution of energy 0 dt k r N A ( )vrel f ( )d 0 Chapter 22: Reaction Dynamics k r N A ( )vrel f ( )d 0 12 v 2 rel vrel , A B d 2 a2 vrel cos vrel 2 d 1 2 2 d 2 a2 2 d 2 a2 2 1 1 A B 2 (vrel , A B ) 2 vrel 2 d2 d amax , above which reactionsdo not occur 1 a amax A B a 2 ( ) amax d 2 2 amax 1 a d 2 , ( ) 1 a a ( ) 0 & a ( ) vrel , A B Chapter 22: Reaction Dynamics k r N A ( )vrel f ( )d 0 f (v)dv 4 2kT 3/ 2 v 2 e v 2 / 2 kT f (v)dv 4 2kT 12 v 2 , dv d /( 2 )1 / 2 3/ 2 2 / kT d e (2 )1/ 2 3/ 2 1 1 / 2 / kT f (v)dv 2 d f ( )d e kT 0 0 0 1 ( )vrel f ( )d 2 kT ( )e / kT d a 3/ 2 0 ax ax e ax 2 1/ 2 k r N A ( )vrel f ( )d N Acrel e Ea / RT 0 e ax 1/ 2 8 1 ( )e / kT d kT kT 0 xeax e dx a C , xe dx a a C a / kT 1 e d (kT ) 2 e 8kT a / kT e ( )vrel f ( )d 1/ 2 2 ( ) 1/ 2 e / kT d a / kT Chapter 22: Reaction Dynamics 22.1(c) The steric requirement steric factor, P = σ*/σ. reactive cross-section, σ*, the area within which a molecule must approach another molecule for reaction to occur. 1/ 2 8kT rate constant from collision theory, k r P N Ae E / RT a harpoon mechanism, a process in which electron transfer precedes atom extraction. (Exercise Example 22.2!) Chapter 22: Reaction Dynamics 22.1(d) The RRK model The Rice–Ramsperger–Kassel model (RRK model), a model that takes into account the distribution of energy over all the bonds in a molecule. E P 1 E s 1 s 1 E kb kb ( E ) 1 E for E E s; the # of modes of motion,E ; energyrequiredfor the bondbreakage,E ; energyavailablein thecollision Lindemann-Hinshelwood mechanism Exp. data for unimolecular isomerization of trans-CHD=CHD RRK model s Chapter 22: Reaction Dynamics 22.2 Diffusion-controlled reactions cage effect, the lingering of one molecule near another on account of the hindering presence of solvent molecules. Chapter 22: Reaction Dynamics 22.2(a) Classes of reaction diffusion-controlled limit, a reaction in which the rate is controlled by the rate at which reactant molecules encounter each other in solution. activation-controlled limit, a reaction in solution in which the rate is controlled by the rate of accumulating sufficient energy to react. A B AB v k d [A][B] AB : encounterpair,d : diffusion AB A B v k d [AB] AB P v k a [AB] a : activatedprocess k d [A][B] d [AB] k d [A][B] k d [AB] k a [AB] 0 [AB] dt k a k d kk d [P] k a [AB] k r [A][B], k r a d dt k a k d When k d k a k r k d : diffusion - controlledlimit When k a k d k r ka kd ka K k d : activation- controlledlimit Chapter 22: Reaction Dynamics 22.2(b) Diffusion and reaction A B AB in solution! c 2 c 3dimension [B] D 2 DB 2 [B] ; diffusion equation (Fick' s second law of diffusion) t x t [B] At steady state; 0 2 [B]r 0; r signifies a quantity that varies with the distance r t d 2 [B]r 2 d [B]r b spherically symmetry system 2 [B]r 0 General solution : [ B ] a r r 2 r r r [ B]r [B] ([B] is bulk value) as r , [ B]r 0 at r R (the distance where reaction occurs) R [B]r 1 [B] r Rate of reaction 4R 2 J ( J : molar flux of B toward A) D [B] d [B]r From Fick' s first law J DB B R dr r R A Rate of reaction 4R DB [B] for all 4R DB [B]N A 4R DB N A [A][B] DB DA DB D A is not stationary d [P] k d [A][B] 4R DN A [A][B] k d 4R DN A dt Chapter 22: Reaction Dynamics By using Stokes- Einsteinequation; kT kT DA DB ( RA , RB ; hydrodynamic radius, ; viscosityof medium) 6RA 6RB RA RB 12 R k d 4R DN A 8RT 3 22.3 The material balance equation Generalized diffusion equation: thediffusion equationincluding convection [J] 2 [J] [J] D 2 v t x x [J] 2 [J] [J] Includingchemicalreaction D 2 v k r [J]; materialbalanceequation t x x No convection; [J] [J]e kr t [J] : for no reaction n0 x 2 / 4 Dt e A(Dt)1/ 2 For generalcases, we can solve thematerialbalanceequation numerically!! No reaction;[J] Chapter 22: Reaction Dynamics TRANSITION STATE THEORY transition state theory (or activated complex theory, ACT), a theory of rate constants for elementary bimolecular reactions. transition state, the arrangement of atoms in an activated complex that must be achieved in order for the products to form. 22.4 The Eyring equation A B C‡ K‡ p J RT [ J ] [C‡ ] C‡ P v k r [A][B] pC ‡ p θ p A pB RT ‡ K [A][B] θ p v k ‡ [C ‡ ] RT kr θ k ‡ K ‡ p Our task!! Chapter 22: Reaction Dynamics 22.4(a) The rate of decay of the activated complex transmission coefficient, κ, the constant of proportionality between the rate of passage of the complex (k‡) through the transition state and the vibrational frequency along the reaction coordinate (‡); k‡ = κ‡. 22.4(b) The concentration of the activated complex θ q θ J N q ‡ A K J,m e r E0 / RT K ‡ θ Cθ e r E0 / RT where p 1 bar & r E0 E0 (C‡ ) E0 (A) E0 (B) qA qB J N A 1 Partition function for specific vibration which leads to product formation; q ‡ 1 e h / kT 1 kT h ‡ kT q h ‡ h ‡ 1 1 kT kT q where qC‡ denotes the partition function for all the other modes of the complex. ‡ C‡ h θ N q kT A ‡ ‡ C‡ r E0 / RT K‡ K K e ( K ‡ ; K ‡ with one vibrational mode of C‡ discarded) ‡ θ θ h qA qB qC ‡ Chapter 22: Reaction Dynamics 22.4(c) The rate constant kr k ‡ RT ‡ kT ‡ ‡ kT RT ‡ K K K ‡ ; Eyringequation pθ h ‡ p θ h C For qCθ‡ , we have to know thesize, shape,and st ructureof act ivatedcomplex verydifficult ! 22.4(d) The collision of structureless particles Vmθ q 3 J J θ J h (2mJ kT )1/ 2 Vmθ RT pθ ~ 2 hcB 2I θ 2 IkT Vm A B C (A B), q corresponds to rotatioalmode q 2 3 C‡ ‡ I r 2 , θ C‡ θ C‡ m A mB , mC‡ m A mB m A mB AB kT kT RT N A 3A 3B 2 IkT r E0 / RT N e kr A 2 θ 3 θ h h p C‡Vm C‡ 1/ 2 k r 8 kT 3 2 IkT r E0 / RT 2 e N Ae a 8kT 2 r E0 / RT r 2 , Ea r E0 N A 2 r e E / RT Chapter 22: Reaction Dynamics 22.4(e) Observation and manipulation of the activated complex Na+I- decay Photoreaction of IH∙∙∙OCO van der Waals complex IH∙∙∙OCO HOCO resembles the activated complex of H + CO2[HOCO] ‡ HO+CO Chapter 22: Reaction Dynamics 22.5 Thermodynamic aspects 22.5(a) Activation parameters Gibbs energyof activation, ‡G RT ln K ‡ kT RT ‡G / RT ‡G ‡ H T‡S kT RT is absorbed into S term ‡ S / R ‡ H / RT kr e k Be e B r h pθ h pθ ln k r ‡ k r Ae Ea / RT Ea RT 2 Ea H 2 RT T ‡ S / R Ea / RT k r e Be 2 ‡ S / R A e Be 2 e Pe ‡ S steric / R correlation analysis, a procedure in which ln K (=-ΔrGθ/RT) is plotted against ln k (proportional to -Δ‡G /RT). liner free energy relation (LFER), a linear relation obtained in correlation analysis; reaction becomes thermodynamically more favorable. Chapter 22: Reaction Dynamics 22.5(b) Reactions between ions kinetic salt effect, the effect of a change in ionic strength on the rate constant of a reaction. ‡ θ a ‡ d [P] [ C ] c k ‡ [C ‡ ] K C K dt aA aB [A][B] C K A B ‡ k ‡ K k r0 k ‡ K when 1 k r0 kr k r K K d [P] k r [A][B] dt From Debye- Huckel limit inglaw (log z z- AI 1/ 2 , A 0.509for aq. at 250 C), log A AzA2 I 1/ 2 log B AzB2 I 1/ 2 log C‡ A( z A z B ) 2 I 1/ 2 log k r log k r0 A z A2 z B2 ( z A z B ) 2 I 1/ 2 log k r0 2 AzA z B I 1/ 2 Exercise Example 22.3! Chapter 22: Reaction Dynamics THE DYNAMICS OF MOLECULAR COLLISIONS 22.6 Reactive collisions 22.6(a) Experimental probes of reactive collisions infrared chemiluminescence, a process in which vibrationally excited molecules emit infrared radiation as they return to their ground states. IR chemiluminescence O+CSCO+S Chapter 22: Reaction Dynamics laser-induced fluorescence (LIF), a technique in which a laser is used to excite a product molecule from a specific vibration–rotation level and then the intensity of fluorescence is monitored. Chapter 22: Reaction Dynamics multiphoton ionization (MPI), a process in which the absorption of several photons by a molecule results in ionization. resonant multiphoton ionization (REMPI), a technique in which one or more photons promote a molecule to an electronically excited state and then additional photons are used to generate ions from the excited state. A laser pulse excites electrons in a semiconductor surface (10 layers C 60 on a Cu(111) substrate) which in turn pass their energy to adsorbed molecules (NO). REMPI measures the motion of the desorbed molecules. Chapter 22: Reaction Dynamics reaction product imaging, a technique for the determination of the angular distribution of products. Reaction products detected in the Streamer Chamber when a 1.1-GeV-per-nucleon beam of holmium-165 collided with a holmium-165 target at the Bevalac. Chapter 22: Reaction Dynamics 22.7 Potential energy surfaces potential energy surface, the potential energy as a function of the relative positions of all the atoms taking part in the reaction. HA + HB-HC HA-HB + HC Chapter 22: Reaction Dynamics saddle point, the highest point on a potential energy surface encountered along the reaction coordinate. HA + HB-HC HA-HB + HC Chapter 22: Reaction Dynamics saddle point, the highest point on a potential energy surface encountered along the reaction coordinate. HA + HB-HC HA-HB + HC Chapter 22: Reaction Dynamics Example of potential energy surfaces. Ultrafast reaction dynamics of the complete photo cycle of an indolylfulgimide studied by absorption, fluorescence and vibrational spectroscopy Chapter 22: Reaction Dynamics 22.8 Some results from experiments and calculations HA + HB-HC HA-HB + HC Chapter 22: Reaction Dynamics HA + HB-HC HA-HB + HC Chapter 22: Reaction Dynamics 22.8(a) The direction of attack and separation 300 Chapter 22: Reaction Dynamics 22.8(b) Attractive and repulsive surfaces attractive surface, a potential energy surface in which the saddle point occurs early on the reaction coordinate. repulsive surface, a potential energy surface in which the saddle point occurs late on the reaction coordinate. H + Cl2 HCl +Cl attractive surface repulsive surface Chapter 22: Reaction Dynamics 22.8(c) Classical trajectories direct mode process, a bimolecular process in which the switch of partners takes place very rapidly. complex mode process, a bimolecular process in which the activated complex survives for an extended period. direct mode process complex mode process