Igls, March 2003

Statistical Models What do all the abbreviations mean? What assumptions are behind the various models? What can they tell us? Why do we need them? What are the pitfalls? Raphy LevineRob Dunbar Klavs Hansen Olof Echt Catherine Brechignac

Igls, March 2003 The Rate of a Chemical Reaction Reaction rate depends on the concentration of one or more of the reactants Rate = f([A],[B]), normally Rate  [A] m [B] n Rate = k [A] m [B] n : rate equation Overall order of the reaction is p = m+n More generally: with n i the order with respect to the i th component and the overall order of the reaction p =  n i

Igls, March 2003 Elemenatry reactions are described by their molecularity (i.e. the number of reactants that are involved in the reaction step) A  productsunimolecular A + B  products bimolecular etc. We are normally discussing unimolecular first order reactions. Reaction rate depends on only one reactant

Igls, March 2003 Rate coefficient k should be a constant - it should not depend on the concentrations of the species and should be independent of time. But it does depend strongly on temperature. Arrhenius Equation (1889) E act (r) E act (f) HoHo transition state  H o =  H f (products)-  H f (reactants) (most but not all reactions show Arrhenius behaviour)

Igls, March 2003 Relation between cross section and reaction rate expand interaction volume, integrate over all velocities and angular variables, assume Maxwell-Boltzmann velocity distribtion

Igls, March 2003 Normally we do not have information on state-to-state cross sections and detailed potential energy surfaces. Often experimental data just consists of a thermal rate constant. The level of detail in a theoretical model should be consistent with that of the experimental observations to which the model will be applied. One approach is transition state theory. Here, the reactants are assumed to be in thermal equilibrium with the transition state. The theory only takes into account the statistical properties of reactive systems, not the microscopic details. It provides the rate of crossing of the barrier. The statistical assumption is not about the dynamics of the reaction but about the equilibrium nature of the reactants located on one side of the barrier The theory is only applicable when the reactants are in thermal equilibrium

Igls, March 2003 The Point of No Return Transition state theory assumes that there is a coonfiguration of no return. Starting from the reactants, when and if it reaches the critical configuration, it will necessarily proceed to form products. If such a configuration can be identified then it must be crossed by any reactive trajectory and it can only be crossed once. This is the key to the simplicity: it implies that the reaction rate is given by the rate at which the molecules reach (and pass through) this configuration. We do not care how the system approaches the configuration nor what dynamics it undergoes beyond this configuration. Dynamics has been reduced to counting.

Igls, March 2003 E act (f) HoHo transition state When there is an energetic barrier along the reaction path, the location of the barrier is the natural choice for a point of no return. For reactions without an energy threshold, centrifugal barriers can be used to determine the location of the point of no return (identified as the maximum of the effective potential)

Igls, March 2003 E act (f) HoHo transition state For thermal reactants and barriers that are high compared with thermal energy, if the barrier is crossed the motion downhill to the products will not reverse itself At higher energies the barrier can be recrossed. Transition state theory is intended for circumstances where there is not much excess energy available for crossing the barrier. In any case it always provides an upper bound for the reaction rate.

Igls, March 2003 To calculate the reaction rate constant, first compute the rate of barrier crossing then extract the rate constant by dividing by the concentration of the reactants. (see notes!). Important number is the number of internal states of the molecule at the transition state whose energy is in the range density of states of the reactants

Igls, March 2003 Can convert to a temperature dependent rate constant, assuming thermal equilibrium Q: partition function (Qk(T) is the Laplace transform of  (E)k(E))

Igls, March 2003 RRKM (Rice, Ramsperger, Kassel, Marcus) An energy rich molecule is pictured as a collection of coupled harmonic oscillators which exchange energy freely under the assumption that 1. All internal molecular states at energy E are accessible and will ultimately lead to decomposition products 2. Vibrational energy redistribution within the energised molecule is much faster than unimolecular reaction. (RRKM is an extension of RRK theory to explicitly consider vibrational and rotational energies and to include zero-point energies) RRKM : dissociation rates of energy-rich polyatomic molecules dos of the energy-rich molecule. Equilibrium assumption

Igls, March 2003 RRKM: assumes a statistical population of reactant internal states at a fixed total energy. A microcanonical ensemble. The assumption requires that each state has an equal chance of decomposing. A microcanonical ensemble will be maintained as the reactant molecules decompose and therefore the unimolecular decomposition reaction will be described by only one time- independent rate constant k(E). An ensemble of molecules at a fixed temperature T is a canonical ensemble.