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§9.7 Transition state theory (TST) Theory of Absolute reaction Rates Theory of activated complex theory A + B-C A-B + C During reaction, energies are being redistributed among bonds: old bonds are being ripped apart and new bonds formed. H + H–H H∙∙∙∙∙∙∙∙∙ H∙∙∙∙∙∙H H∙∙∙∙∙∙H∙∙∙∙∙∙H (activated state) H∙∙∙∙∙∙H∙∙∙∙∙∙∙∙∙∙∙∙H H–H + H This process can be generalized as: A + B-C [ABC] A-B + C Activated complex Transition state The transition state theory (TST), attempting to explain reaction rates on the basis of thermodynamics, was developed by H. Eyring and M. Polanyi during 1930-1935. TST treated the reaction rate from a quantum mechanical viewpoint involves the consideration of intramolecular forces and intermolecular forces at the same time. Basic consideration According to TST, before undergoing reaction, reactant molecules form an activated complex which is in thermodynamic equilibrium with the molecules of the reactants. The activated complexes, the energy of which is higher than both reactants and products, is treated as an ordinary molecule except that it has transient existence and decomposes at a definite rate to form the product. r cAB 7.1 Potential energy surfaces According to the quantum mechanics, the nature of the chemical interaction (chemical bond) is a potential energy which is the function of interatomic distance (r): V V (r ) The function can be obtained by solving Schrödinger equation for a fixed nuclear configuration, i.e., Born-Oppenheimer approximation. The other way is to use empirical equation. The empirical equation usually used for system of two atoms is the Morse equation: Morse equation: V (r ) De{exp[ 2a(r r0 )] 2 exp[ a(r r0 )]} decomposition asymptote When r = r0, Vr (r = r0) = -De r, Vr (r) = 0 where De is the depth of the wall of potential, or the dissociation energy of the bond. r0 is the equilibrium interatomic distance, i.e., bond length, a is a parameter with the unit of cm-1 which can be determined from spectroscopy. r > r0, interatomic attraction, Zero point energy: E0 = De-D0 r < r0, interatomic repulsion. J. Comp. Chem., 2011, 32, 5: 797-809 For triatomic system A + BC AB + C C C rA rBC B A A rAC rAB rBC B B V = V(rAB, rBC, rAC ) = V(rAB, rBC , ) For triatomic system, the potential is a four-dimension function. In 1930, Eyring and Polanyi make = 180 o, i.e., collinear collision and the potential energy surface can be plotted in a three dimensions / coordination system. = 180 o A rAB B rBC C V = V(rAB, rBC) Eyring et al. calculated the energy of the triatomic system: HA + HBHC HAHB+ HC using the method proposed by London. Schematic of LEP Potential energy surface Contour diagram of the potential energy surface Projection of LEP potential surface peak Which way should the reaction follows? peak reaction path or reaction coordinate. valley Saddle point INTERMOLECULAR POTENTIAL ENERGY SURFACE FOR CS2 DIMER Journal of Computational Chemistry Volume 32, Issue 5, pages 797-809, 12 OCT 2010 DOI: 10.1002/jcc.21658 http://onlinelibrary.wiley.com/doi/10.1002/jcc.21658/full#fig10 Activated complex has no recovery force. On any special vibration (asymmetric stretching), it will undergo decomposition. Whenever the system attain saddle point, it will convert to product with no return. 7.2 Kinetic treatment of the rate constant of TST For reaction: The rate of the reaction depends on two factors: 1) the concentration of the activated complex (c) 2) the rate at which the activated complex dissociates into products() r cAB According to equilibrium assumption cAB K cA cB r K cA cB k K According to statistical thermodynamics, K can be expressed using the molecular partition function. cAB q f E0 K exp cA cB qA qB f A f B RT E0 is the difference between the zero point energy of activated complex and reactants. q is the partition function, f is the partition function without E0 stem and volume stem. For activated complex with three atoms, f can be written as a product of partition function for three translational, two rotational, and four vibrational degrees of freedom. f f * f ' Only the asymmetric stretching can lead to decomposition of the activated complex and the formation of product. For one-dimension vibrator: For asymmetric stretching 1 f h 1 exp k T B * h kBT f* kBT h kBT f f ' h kBT f ' E0 k K exp h f A f B RT kBT f ' E0 k exp h fA fB RT statistical expression for the rate constant of TST For a general elementary reaction kBT f ' E0 k exp h fi RT In which f’ can be obtained from partition equation and E0 can be obtained from potential surface. Therefore, k of TST can be theoretically calculated. Absolute rate theory For example: For elementary equation: H2+ F HHF H + HF Theoretical: k = 1.17 1011 exp(-790/T) Experimental: k = 2 1011 exp(-800/T) 7.3 Thermodynamic treatment of TST For nonideal systems, the intermolecular interaction makes the partition function complex. For these cases, the kinetic treatment becomes impossible. In 1933, LaMer tried to treat TST thermodynamically. kBT f ' E0 k exp h fA fB RT kBT k K h f' E K exp 0 fA fB RT G RT ln K G H T S Standard molar entropy of activation, standard molar enthalpy of activation G RT ln K G K exp RT kBT k K h G kBT k exp h RT G H T S S H kBT exp exp h R RT The thermodynamic expression of the rate of TST is different from Arrhenius equation kBT k K h According to GibbsHolmholtz equation kBT ln k ln ln K h d ln k 1 d ln K dT T dT d ln K U dT RT 2 H U PV d ln K H PV dT RT 2 d ln k RT H PV dT RT 2 d ln k Ea RT dT Ea RT H PV 2 Ea RT H PV For liquid reaction: PV = 0 Ea RT H PV nRT (1 n) RT For gaseous reaction: n is the number of reactant molecules Ea H nRT k S H k BT exp exp h R RT thermodynamic expression of the rate of TST. S n k BT Ea k exp e exp h R RT S n kBT Ea k exp e exp h RT RT E k A exp a RT S kBT n A e exp h RT k BT is a general constant with unit of s-1 Z' of the magnitude of 1013. h k SCT Ea PZ ' exp RT S P exp RT The pre-exponential factor depends on the standard entropy of activation and related to the structure of activated complex. S P exp RT suggests that the steric factor can be estimated from the activation entropy of the activated complex. Example: reactions P exp(S/R) (CH3)2PhN + CH3I 0.5 10-7 0.9 10-8 1986 Noble Prize Hydrolysis of ethyl acetate 2.0 10-5 5.0 10-4 Canada Decomposition of HI John C. Polanyi 1929/01/23 ~ Decomposition of N2O 0.5 0.15 1 1