Transition State Theory k1 A P k1 A [TS]‡ P k‡ Previously, ETS E TS n −(E −E )/ k T −ε / k T TS = e TS A B = e B nA EA A But this is a microscopic equation. For thermodynamic analysis, need reaction coordinate functions of whole systems. Transition State Theory Analogous expression in the bulk would be: NTS −(E −E )/ RT −ΔE ‡ / RT ETS = e TS A = e E TS NA But this equation only describes zero-point energy difference. Not correct. EA A reaction coordinate Transition State Theory E Analogous expression in the bulk would be: NTS −(E −E )/ RT −ΔE ‡ / RT TS = e TS A = e NA But this equation only describes zero-point energy difference. Only correct at zero K. A Instead, need to express in terms of reaction coordinate distribution of molecules in vibrational modes (that depends on T). Transition State Theory E Analogous expression in the bulk would be: no vibrational states in RC “mode” NTS −(E −E )/ RT −ΔE ‡ / RT TS = e TS A = e NA But this equation only describes zero-point energy difference. Only correct at zero K. A Instead, need to express in terms of reaction coordinate distribution of molecules in vibrational modes (that depends on T). Challenge: How to deal with that funny reaction-coordinate vibrational mode? Can’t describe energetics until we do. Transition State Theory: Zero-Point Energies and Partition Functions Define “partition function” Q as E distribution of energy states for each degree of freedom ETS,0K TS Q defines how these states are filled at temperature T. Q = qvibqrotqtrans = (qvib1qvib2···)··· E A A,0K NA NTS reaction coordinate QA = QTSqvibRC = NA,0K NTS,0K Reaction coordinate mode can’t be represented by partition function. So we include extra factor qvibRC. Transition State Theory: Zero-Point Energies and Partition Functions k N qvibRCQTSNTS,0K E ‡ 1 TS K = ‡ = = k NA QANA,0K E TS,0K TS ‡ QTS −ΔE0K / RT = qvibRC e QA −ΔG‡ / RT = qvibRCe E A A,0K (with ΔG‡ representing free energy of activation for any temperature) reaction coordinate Transition State Theory: Zero-Point Energies and Partition Functions k ‡ 1 = q e−ΔG / RT k ‡ vibRC E ‡ ⎛ kBT ⎞ −ΔG‡ / RT k‡ k1 = k ⎜ ‡ ⎟e k1 ⎝ hν ⎠ standard vibrational reaction coordinate partition function k T ‡ We assume k‡ = ν‡. k = B e−ΔG / RT 1 h Transition State Theory: The Eyring Equation k T ‡ k = B e−ΔG / RT rxn h ΔG‡ = ΔH‡ –TΔS‡, so k T ‡ ‡ Eyring equation. k = B eΔS / Re−ΔH / RT rxn h 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 state Transition State Theory: Eyring Plots kBT ΔS‡ / R −ΔH ‡ / RT k = e e can also be written: rxn h ⎛ k ⎞ ⎛ k ΔS‡ ⎞ ΔH ‡ ⎛ 1 ⎞ ln⎜ rxn ⎟ = ⎜ln B + ⎟ - ⎜ ⎟ (Has form ⎜ ⎟ y = mx + b.) ⎝ T ⎠ ⎝ h R ⎠ R ⎝T ⎠ slope = -ΔH‡/R ‡ intercept = ln(kB/h) + ΔS /R ) T / k k ln( Different from Arrhenius plot— vertical axis ln(k/T). T 1/T Transition State Theory: Eyring and Arrhenius ‡ ‡ Has Arrhenius form! kBT ΔS / R −ΔH / RT krxn = e e But, E and ΔH vary h a (slightly) differently w/ T. Fix this by re-writing as ek T ‡ ‡ k = B eΔS / Re−(ΔH +RT )/ RT rxn h Ea A Makes intuitive sense. ekBT ΔS‡ / R ‡ A = e ; Ea = ΔH + RT Better to graph than h transform. Transition State Theory: Activation Parameters What does ΔH‡ tell us about a reaction? ΔH‡ dominates ΔG‡. ‡ ΔH (kcal/mol) t1/2 (for ΔS‡ = 0) 5 30 ns 10 2.6 μs 15 12 ms 20 57 s chemists work here. 25 3.2 days 30 41 years Transition State Theory: Activation Parameters What does ΔS‡ tell us about a reaction? Illustrates structure of transition state. Obvious examples: Cl + HH Cl H H Cl H + H ΔS‡ = -12 cal/mol·K (e.u., entropy units) O O O O O O O 2 O O O ΔS‡ = +10 cal/mol·K.