Transition State TheoryPA[TS]‡k1k‡PAk1Ereaction coordinateATSPreviously,TkTkEEnnBBATS//)(ATSeeε−−−==EAETSBut this is a microscopic equation.For thermodynamic analysis, need functions of whole systems.Transition State TheoryEreaction coordinateATSRTERTEENN//)(ATS‡ATSeeΔ−−−==EAETSAnalogous expression in the bulk would be:But this equation only describes zero-point energy difference. Not correct.Transition State TheoryEreaction coordinateATSRTERTEENN//)(ATS‡ATSeeΔ−−−==Analogous expression in the bulk would be:But this equation only describes zero-point energy difference. Only correct at zero K.Instead, need to express in terms of distribution of molecules in vibrationalmodes (that depends on T).Transition State TheoryEreaction coordinateATSRTERTEENN//)(ATS‡ATSeeΔ−−−==Analogous expression in the bulk would be:But this equation only describes zero-point energy difference. Only correct at zero K.Instead, need to express in terms of distribution of molecules in vibrationalmodes (that depends on T).Challenge: How to deal with that funny reaction-coordinate vibrational mode?Can’t describe energetics until we do.no vibrationalstates in RC “mode”Transition State Theory:Zero-Point Energies and Partition FunctionsEreaction coordinateATSEA,0KETS,0KDefine “partition function” Q as distribution of energy states for each degree of freedomQ defines how these states are filled at temperature T.Q = qvibqrotqtrans= (qvib1qvib2···)···A,0KAANNQ =TS,0KTSvibRCTSNNqQ =Reaction coordinate mode can’t be represented by partition function. So we include extra factor qvibRC.Transition State Theory:Zero-Point Energies and Partition FunctionsEreaction coordinateATSEA,0KETS,0KA,0KATS,0KTSvibRCATS‡‡NQNQqNNkkK ===1RTGRTEqQQq/vibRC/ATSvibRC‡‡0KeeΔ−Δ−==(with ΔG‡representing free energy of activation for any temperature)Transition State Theory:Zero-Point Energies and Partition FunctionsRTGqkk/vibRC‡‡eΔ−=1Ereaction coordinateRTGhTkkk/‡B‡‡eΔ−⎟⎠⎞⎜⎝⎛=ν1standard vibrationalpartition functionk1k‡We assume k‡= ν‡.RTGhTkk/B‡eΔ−=1Transition State Theory:The Eyring EquationRTGhTkk/Brxn‡eΔ−=RTHRShTkk//Brxn‡‡eeΔ−Δ=ΔG‡= ΔH‡–TΔS‡, soEyring equation.Relates rate constant to T in a physical way.ΔH‡: Enthalpy required to reach transition state (like Ea)ΔS‡: Entropy lost/gained getting to transition stateTransition State Theory:Eyring Plots⎟⎠⎞⎜⎝⎛Δ⎟⎟⎠⎞⎜⎜⎝⎛Δ+=⎟⎠⎞⎜⎝⎛TRHRShkTk 1‡‡Brxn-lnlnRTHRShTkk//Brxn‡‡eeΔ−Δ=can also be written:(Has formy = mx + b.)kTln(k/T)1/TDifferent from Arrhenius plot—vertical axis ln(k/T).slope = -ΔH‡/Rintercept = ln(kB/h) + ΔS‡/RTransition State Theory:Eyring and ArrheniusRTHRShTkk//Brxn‡‡eeΔ−Δ=Has Arrhenius form!But, Eaand ΔH vary (slightly) differently w/ T.Fix this by re-writing asRTRTHRShTkk/)(/Brxn‡‡eee+Δ−Δ=AEa;ee/B‡RShTkAΔ=RTHE +Δ=‡aMakes intuitive sense.Better to graph than transform.Transition State Theory:Activation ParametersWhat does ΔH‡tell us about a reaction?ΔH‡dominates ΔG‡.ΔH‡(kcal/mol)(for ΔS‡= 0)t1/25101520253030 ns2.6 μs12 ms57 s3.2 days41 yearschemists work here.Transition State Theory:Activation ParametersWhat does ΔS‡tell us about a reaction?Illustrates structure of transition state.Obvious examples:HHCl HClHCl HH++ΔS‡= -12 cal/mol·K (e.u., entropy units)OOOOOOOOOO2ΔS‡= +10