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Theory of Absolute reaction Rates Theory of activated complex theory Extensive reading: Levine, pp. 882-889. 23.2 potential-energy surfaces
9.7.1 brief introduction 9.7 Transition state theory (TST) What kind of energy is activation energy? (1) Translational energy; (2) Total energy of molecule; (3) Electron energy of outmost shell; (4) Interatomic potential energy (1) SCT; (2) Arrhenius; (3) Light energy; (4) TST;
9.7.1 brief introduction 9.7 Transition state theory (TST) The transition state theory (TST), attempting to explain reaction rates on the basis of thermodynamics (potential energy the nature of chemical bond), 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.
9.7.1 brief introduction 9.7 Transition state theory (TST) Henry Eyring Polányi Mihály H. Eyring, "The activated complex in chemical reactions", J. Chem. Phys., 1935, 3, 107-115 Statistical Mechanical Treatment of the Activated Complex in Chemical Reactions, J. Chem. Phys. 1935, 3, 399 M.G. Evans, M. Polanyi, "Some applications of the transition state method to the calculation of reaction velocities, especially in solution", Trans. Faraday Soc., 1935, 31, 875-894
9.7.1 brief introduction 9.7 Transition state theory (TST) The calculation of absolute reaction rates is formulated in terms of quantities which are available from the potential surfaces which can be constructed at the present time. The probability of the activated state is calculated using ordinary statistical mechanics. This probability multiplied by the rate of decomposition gives the specific rate of reaction.
9.7.1 brief introduction A + C A + 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 (Is it strange??) H H H H H + H This process can be generalized as: A + -C [AC] A- + C Activated complex / Transition state
9.7.1 brief introduction asic consideration 9.7 Transition state theory (TST) (1) Activated complex is in thermodynamic equilibrium with the molecules of the reactants to determine its concentration. (2) The activated complex is treated as an ordinary molecule except that it has transient existence (3) activated complex decomposes at a definite rate to form the product. r c A Rate equation of TST
9.7.2 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., orn-oppenheimer approximation. The other way is to use empirical equation. The empirical equation usually used for system of diatomic systems is the Morse equation:
(1) Potential energy of diatomic systems
Morse equation: 9.7 Transition state theory (TST) (1) Potential energy of diatomic systems V( r) De{exp[ 2a( r r0 )] 2exp[ a( r r0 )]} The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the QHO (quantum harmonic oscillator) because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone and combination bands. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. Due to its simplicity (only three fitting parameters), it is not used in modern spectroscopy. https://en.wikipedia.org/wiki/morse_potential
(1) Potential energy of diatomic systems decomposition asymptote D e : the depth of the wall of potential/dissociation energy of the bond. r 0 : equilibrium interatomic distance/bond length; a: a parameter with the unit of cm -1 can be determined from spectroscopy. Zero point energy: E 0 = D e -D 0 When r = r 0, V r (r = r 0 ) = -D e r, V r (r) = 0 r > r 0, interatomic attraction, r < r 0, interatomic repulsion.
(1) Potential energy of diatomic systems J. Comp. Chem., 2011, 32, 5: 797-809
(2) Discussion on Triatomic systems A + C A + C C C A r A r AC r C A r A r C V = V(r A, r C, r AC ) = V(r A, r C, ) For triatomic system, the potential is a four-dimension function.
(2) Discussion on Triatomic systems 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 C r A r C V = V(r A, r C ) Eyring et al. calculated the energy of the triatomic system: H A + H H C H A H + H C using the method proposed by London.
(3) Potential surface and projection
(3) Potential surface and projection y-products reactants Schematic of LEP Potential energy surface products Projection of LEP potential surface Contour diagram of the potential energy surface
(4) Reaction coordinate 9.7 Transition state theory (TST) peak peak Which way should the reaction follows? reaction path or reaction coordinate. valley Saddle point
(4) Reaction coordinate 9.7 Transition state theory (TST) Activated complex has no recovery force. On any special vibration (asymmetric stretching), it will undergo decomposition. Whenever the system attain saddle point, with no return. it will convert to product
(4) Reaction coordinate 9.7 Transition state theory (TST)
7.3 Kinetic treatment of the rate constant of TST For reaction: 9.7 Transition state theory (TST) 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 c A According to equilibrium assumption K c cc A A r K c c A k K
7.3 Kinetic treatment of the rate constant of TST According to statistical thermodynamics, K can be expressed using the molecular partition function. K c q f E0 c c q q f f RT A exp A A A 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, two rotational, and four vibrational degrees of freedom.
7.3 Kinetic treatment of the rate constant of TST f f f * ' Only the asymmetric stretching can lead to decomposition of the activated complex and the formation of product. For one-dimension vibrator: f * 1 h 1 exp kt For asymmetric stretching * h k T f kt h
7.3 Kinetic treatment of the rate constant of TST f kt h f ' k kt f ' E0 h f f RT K exp A k kt f ' E0 exp h f f RT A statistical expression for the rate constant of TST For a general elementary reaction k kt f ' E0 exp h f RT i 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
7.3 Kinetic treatment of the rate constant of TST For example: For elementary equation: H 2 + F HHF H + HF Theoretical: k = 1.17 10 11 exp(-790/t) Experimental: k = 2 10 11 exp(-800/t)
7.4 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. k kt f ' E0 exp h f f RT A K f ' f f A exp E0 RT k kt h K G RT ln K G H TS Standard molar entropy of activation, standard molar enthalpy of activation
7.4 Thermodynamic treatment of TST G RT ln K K exp k k kt h k T h 9.7 Transition state theory (TST) K exp G RT G RT G H TS kt S H exp exp h R RT The thermodynamic expression of the rate of TST is different from Arrhenius equation
k kt h K ln k ln ln d ln k dt kt h 1 T 9.7 Transition state theory (TST) 7.4 Thermodynamic treatment of TST K d ln K dt According to Gibbs-Holmholtz equation d ln K U dt RT 2 H U pv d ln K H pv dt RT 2 dln k RT H pv 2 dt RT E a RT 2 d ln k dt E RT H pv a
7.4 Thermodynamic treatment of TST E RT H pv a For liquid reaction: pv = 0 Ea RT H For gaseous reaction: pv nrt (1 n) RT n is the number of reactant molecules E H nrt a k kt S H exp exp h R RT thermodynamic expression of the rate of TST. k kt S n Ea exp e exp h R RT
7.4 Thermodynamic treatment of TST kt S n Ea k exp e exp h RT RT k Aexp Ea RT A k T n e exp h S RT is a general constant with unit of s -1 of the magnitude of 10 13. Z ' kt h k SCT PZ ' exp Ea RT P S exp RT The pre-exponential factor depends on the standard entropy of activation and related to the structure of activated complex.
7.4 Thermodynamic treatment of TST suggests that the steric factor can be estimated from the activation entropy of the activated complex. Example: S P exp RT reactions P exp(s/r) (CH 3 ) 2 PhN + CH 3 I 0.5 10-7 0.9 10-8 John C. Polanyi 1986 Noble Prize Canada 1929/01/23 ~ Hydrolysis of ethyl acetate 2.0 10-5 5.0 10-4 Decomposition of HI 0.5 0.15 Decomposition of N 2 O 1 1
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