Rate Theories of elementary reaction

2 Transition state theory (TST) for bimolecular reactions Theory of Absolute reaction Rates Theory of activated complex theory

A + B-C  A-B + C During reaction, massive changes of form are occurring, 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

Whether or not the energy change of the reaction can be used to explain the reaction on the basis of thermodynamics? The transition state theory (TST), attempting to explain reaction rates on the basis of thermodynamics, was developed by H. Eyring and M. Polanyi during TST treated the reaction rate from a quantum mechanical viewpoint involves the consideration of intramolecular forces and intermolecular forces at the same time.

According to TST, before undergoing reaction, reactant molecules must 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. Basic consideration

2.1 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): 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:

where D e is the depth of the wall of potential, or the dissociation energy of the bond. r 0 is the equilibrium interatomic distance, a is a parameter with the unit of cm -1 which can be determined from spectroscopy. Morse equation: When r = r 0, V r (r = r 0 ) = -D e r , V r (r  ) = 0 When r > r 0, interatomic attraction exists, r < r 0, interatomic repulsion appears. The equilibrium distance r 0 is the bond length.

Morse curve: the curve obtained by plotting V(r) against r Zero point energy: E 0 = D e -D 0 decomposition asymptote

For triatomic system A + B  C  A  B + C V = V(r AB, r BC, r AC ) = V(r AB, r BC,  ) A B C rABrAB r BC r AC For triatomic system, the potential is a four-dimension function. AB C r BC r AB 

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. V = V(r AB, r BC ) Eyring et al. calculated the energy of the triatomic system: H A + H B H C  H A H B + H C using the method proposed by London. ABC r BC r AB  = 180 o

Schematic of LEP Potential energy surface Contour diagram of the potential energy surface Projection of LEP potential surface

Which way should the reaction follows? Saddle point valley peak reaction path or reaction coordination.

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.

2.2 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(   ) According to equilibrium assumption

According to statistical thermodynamics, K  can be expressed using the molecular partition function. E 0 is the difference between the zero point energy of activated complex and reactants. q is the partition function, f  is the partition function without E 0 stem and volume stem. For activated complex with three atoms, f  can be written as a product of partition function for three translational, three rotational, and five vibrational degrees of freedom.

Only the asymmetric stretching can lead to decomposition of the activated complex and the formation of product. For one-dimension vibrator: For asymmetric stretching

statistical expression for the rate constant of TST For general elementary reaction In which f  ’ can be obtained from partition equation and E 0 can be obtained from potential surface. Therefore, k of TST can be theoretically calculated. Absolute rate theory

For example: For elementary equation: H 2 + F  H  H  F  H + HF Theoretical: k = 1.17  exp(-790/T) Experimental: k = 2  exp(-800/T)

2.3 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. Standard molar entropy of activation, standard molar enthalpy of activation

The thermodynamic expression of the rate of TST is different from Arrhenius equation

According to Gibbs-Holmholtz equation

For liquid reaction: P  V = 0 For gaseous reaction: n is the number of reactant molecules thermodynamic expression of the rate of TST.

is a general constant with unit of s -1 of the magnitude of The pre-exponential factor depends on the standard entropy of activation and related to the structure of activated complex.

Example: reactions P exp(  S/R) (CH 3 ) 2 PhN + CH 3 I 0.5   Hydrolysis of ethyl acetate 2.0   Decomposition of HI Decomposition of N 2 O11 suggests that the steric factor can be estimated from the activation entropy of the activated complex. John C. Polanyi 1986 Noble Prize Canada 1929/01/23 ~