§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.

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): 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: 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. decomposition asymptote When r = r0, Vr (r = r0) = -De r, Vr (r) = 0 r > r0, interatomic attraction, r < r0, interatomic repulsion. Zero point energy: E0 = De-D0

J. Comp. Chem., 2011, 32, 5: 797-809   

V = V(rAB, rBC, rAC ) = V(rAB, rBC , ) For triatomic system A + BC  AB + C A B C rAB rBC rAC A B C rBC rAB  V = V(rAB, rBC, rAC ) = V(rAB, rBC , ) For triatomic system, the potential is a four-dimension function.

A B C rBC rAB  = 180 o V = V(rAB, rBC) 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. A B C rBC rAB  = 180 o 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

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

Intermolecular potential energy surface for CS2 dimer A three dimentional plot of fitted potential energy surface along with its contour plot. The position of selected configurations have been shown on the contour plot. The distance R fixed at 9.33 au and the angle γ fixed at 0. The contours are symmetric with respect to the dashed line. The energy labels are in atomic unit. [Color figure can be viewed in the online issue, which is available at wiley onlinelibrary.com.] © This slide is made available for non-commercial use only. Please note that permission may be required for re-use of images in which the copyright is owned by a third party. 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 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() According to equilibrium assumption

According to statistical thermodynamics, K can be expressed using the molecular partition function. 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.

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

For a general elementary reaction statistical expression for the rate constant of TST For a general elementary reaction 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  HHF  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. 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 1013. The pre-exponential factor depends on the standard entropy of activation and related to the structure of activated complex.

Example: John C. Polanyi 1986 Noble Prize Canada 1929/01/23 ~ 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 Hydrolysis of ethyl acetate 2.0  10-5 5.0  10-4 Decomposition of HI 0.5 0.15 Decomposition of N2O 1 John C. Polanyi 1986 Noble Prize Canada 1929/01/23 ~