Arrhenius Relationship and PlotsRTEaAk−= eobsCan use lnk vs. 1/T plot to obtain A (intercept), Ea(slope)…⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛−=TREAka1ln)ln(obsArrhenius Relationship and Plots…but large factor of error makes this unreliable.RTEaAk−= eobs⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛−=TREAka1ln)ln(obsTheoretical KineticsProblem: Variable components of Arrhenius equation (Ea, A) are not based in chemical theory;Nevertheless, equation works fantastically.Challenge: How do we reproduce form of Arrhenius equation using theory instead of just observation?In other words, how do we express Eaand A in terms of real physical parameters?Collision Theory of ReactionsABrArBTheoretical Basis: Rate parameters are defined by collision frequency and energy.bCollision occurs if b ≤ rA+ rB.ABAnother way of saying it:Maximum radius of a cylinder that will (barely) produce collision:(rA+ rB) = “σ”Area = π(rA+ rB)2= πσ2πσ2Collision Theory of ReactionsABVolume swept in time Δt:πσ2vΔtProbability of collision = 1Frequency of collision (Z)= rate of reaction= # of collision volumes/timeZ = πσ2vΔt/Δt = πσ2vπσ2total speed (A+B)= vIn the real world, cylinder that contains A won’t always contain B; depends on reagent concentration.Z = πσ2v[A][B]Collision Theory of ReactionsBπσ2In the real world, A and B won’t always be pointed right at each other.Solve this by mapping trajectory onto cylinders using “relative” velocity:,BrelμπTkv8=BABAmmmm+=μ(from kinetic theory of gases)],B][A[BμππσTkZ82=μππσTkkBcoll82=ABA(kB=Boltzmannconstant)Collision Theory of Reactions]B][A[BμππσTkZ82=How well does this work?For 2 HI H2+ I2(700 K)calculated Z = 1034molecules/m3sec; actual rate = 1020molecules/m3secOnly one collision in 1014succeeds.Theory: Is because only reactions that have minimum energy “ε” succeds.Collision Theory of ReactionskTnn/totalsuccesseε−=(Maxwell-Boltmannrelationship)kTTkZ/Be]B][A[εμππσ−=82in terms of molecules,RTRTrate/e]B][A[εμππσ−=82in terms of moles.Ereaction coordinateε = ETS– EA&BCollision Theory of ReactionsRTRTk/colleεμππσ−=82This has Arrhenius form!AEaIn practice, this A is much larger than measured A.Theory: Is because ignores orientational bias of reactions.Think SN2 reaction:ClHO-Transition State Theory of ReactionsSaddle point/Transition stateEcoordinate of motioncoordinate of motionClHO-ClHOδ-δ-HOCl-++‡‡Central premise: Pretend TS has the characteristics of a molecule.Transition State Theory of ReactionsIn reality, molecules take many paths near saddle point towards product.We will use one transition state to represent distribution of reaction
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