K A Beaver. D.A. Evans Curtin-Hammett: Limiting Cases Chem 206 Case 2. Curtin-Hammett Conditons k1, k, < kA, kB: If the rates of reaction are much slower than the rate of To relate this quantity to AG values, recall that AG =-RT In Keg or Keg interconversion, (AGAB is small relative to AG, and AG2), then the ratio of e aRT, k, =e G tRT, and k2 =e" GRT. Substituting this into the above A to B is constant throughout the course of the reaction Combining terms 问“e,dm=ore . AAG/RT Curtin-Hammett Principle: The product composition is not solely dependent on relative proportions of the conformational isomers in the substrate; it is controlled by the difference in standard Gibbs energies of the respective transition states major Rxn Coord minor The Derivation Within these limits, we can envision three scenarios Using the rate equations PA/ KiA and d=kalB] we can write If both conformers react at the same rate, the product distribution will be the same as the ratio of conformers at equilibrium d(PBl k2[B d(Pal (2) If the major conformer is also the faster reacting conformer, the product from the major conformer should prevail, and will not reflect the Ince A and B are in equlIbrium, we ca3ne丙 equilibrium distribution If the minor conformer is the faster reacting conformer, the product apPAl Integrating, we get Pel k2 ratio will depend on all three variables in eq(2), and the observed product distribution will not reflect the equilibrium distribution PAl k This derivation implies that you could potentially isolate a product and b are in uilibrium we mus which is derived from a conformer that you cant even observe in the of the conformers as well as the equilib ground state! ing the productPB (2) (3) Using the rate equations = (4) [PB] [PA] A B k2 [B] [B] [A] [PB] [PA] [PB] [PA] PA kB k1 kA k2 d[PA] d[PB] = k1[A] d[PB] d[PA] k1[A] = k2 [B] or Since A and B are in equilibrium, we can substitute Keq = k1 = k2 Keq e -DG 2 /RT e -DG 1 /RT (e-DG°/RT) e -DG 2 /RTe -DG°/RTe DG 1 = /RT e -(DG2 + DG°-DG1 = )/RT or e = -DDG/RT A B PB [PB] [PA] PA K. A. Beaver, D. A. Evans Curtin - Hammett: Limiting Cases Chem 206 k1, k2 << kA, kB: If the rates of reaction are much slower than the rate of interconversion, (DGAB ‡ is small relative to DG1 ‡ and DG2 ‡ ), then the ratio of A to B is constant throughout the course of the reaction. DG° DDG ‡ DG1 ‡ DG2 ‡ Rxn. Coord. Energy major minor d[PA] dt = k1[A] and d[PB] dt = k2[B] we can write: d[PB] d[PA] k1 = k2 Keq k1 = k2 Keq Integrating, we get To relate this quantity to DG values, recall that DG o = -RT ln Keq or Keq = e -DG°/RT, k1 = e-DG 1 ‡/RT, and k2 = e-DG 2 ‡/RT. Substituting this into the above equation: Where DDG ‡ = DG2 ‡+DG°-DG1 ‡ The Derivation: Curtin - Hammett Principle: The product composition is not solely dependent on relative proportions of the conformational isomers in the substrate; it is controlled by the difference in standard Gibbs energies of the respective transition states. When A and B are in rapid equilibrium, we must consider the rates of reaction of the conformers as well as the equilibrium constant when analyzing the product ratio. (1) Case 2: Curtin-Hammett Conditons Within these limits, we can envision three scenarios: • If the major conformer is also the faster reacting conformer, the product from the major conformer should prevail, and will not reflect the equilibrium distribution. • If both conformers react at the same rate, the product distribution will be the same as the ratio of conformers at equilibrium. • If the minor conformer is the faster reacting conformer, the product ratio will depend on all three variables in eq (2), and the observed product distribution will not reflect the equilibrium distribution. This derivation implies that you could potentially isolate a product which is derived from a conformer that you can't even observe in the ground state! Combining terms: DGAB ‡ slow slow fast