Theory, Arrhenius statistical

A good model is consistent with physical phenomena (i.e., 01 has a physically plausible form) and reduces crresidual to experimental error using as few adjustable parameters as possible. There is a philosophical principle known as Occam s razor that is particularly appropriate to statistical data analysis when two theories can explain the data, the simpler theory is preferred. In complex reactions, particularly heterogeneous reactions, several models may fit the data equally well. As seen in Section 5.1 on the various forms of Arrhenius temperature dependence, it is usually impossible to distinguish between mechanisms based on goodness of fit. The choice of the simplest form of Arrhenius behavior (m = 0) is based on Occam s razor. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

Transition state theory, as embodied in Eq. 10.3, or implicitly in Arrhenius theory, is inherently semiclassical. Quantum mechanics plays a role only in consideration of the quantized nature of molecular vibrations, etc., in a statistical fashion. But, a critical assumption is that only those molecules with energies exceeding that of the transition state barrier may undergo reaction. In reality, however, the quantum nature of the nuclei themselves permits reaction by some fraction of molecules possessing less than the energy required to surmount the barrier. This phenomenon forms the basis for QMT. ... 

A unified approach to the glass transition, viscoelastic response and yield behavior of crosslinking systems is presented by extending our statistical mechanical theory of physical aging. We have (1) explained the transition of a WLF dependence to an Arrhenius temperature dependence of the relaxation time in the vicinity of Tg, (2) derived the empirical Nielson equation for Tg, and (3) determined the Chasset and Thirion exponent (m) as a function of cross-link density instead of as a constant reported by others. In addition, the effect of crosslinks on yield stress is analyzed and compared with other kinetic effects — physical aging and strain rate. 

Transition state theory yields rate coefficients at the high-pressure limit (i.e., statistical equilibrium). For reactions that are pressure-dependent, more sophisticated methods such as RRKM rate calculations coupled with master equation calculations (to estimate collisional energy transfer) allow for estimation of low-pressure rates. Rate coefficients obtained over a range of temperatures can be used to obtain two- and three-parameter Arrhenius expressions ... 

Question. Write out the equations for the statistical mechanical form, the thermodynamic form, collision theory and the Arrhenius theory. What are the equivalent terms ... 

Arrhenius theory applies well to solutions of weak acids and bases in water, but fails in the case of strong electrolytes such as ordinary salts. Debye and Hiickel [26] solved this problem assuming complete dissociation, but considering the Coulomb interactions between the ions by a patchwork theory based on both macroscopic electrostatics and statistical mechanics. 

Various quantum-mechanical theories have been proposed which allow one to calculate isotopic Arrhenius curves from first principles, where tunneling is included. These theories generally start with an ab initio calculation of the reaction surface and use either quantum or statistical rate theories in order to calculate rate constants and kinetic isotope effects. Among these are the variational transition state theory of Truhlar [15], the instanton approach of Smedarchina et al. 

It reviews our fast variable theory of liquid phase dynamics [19-26] placing it in its context within statistical mechanics, and also relating it to the recent literature [6,11-17,27,28]. This theory was motivated by and is in accord with the familiar Arrhenius principle that only atypically fast reactant molecules have sufficient energy to surmount chemical activation barriers. 

We have emphasized in this chapter that Arrhenius principle implies that liquid phase chemical reactions occur in a nonclassical fast variable near sudden limit timescale regime rather than in the slow variable near adiabatic regime of standard irreversible statistical mechanics. Despite this, the traditional theories of liquid phase reaction dynamics [7-10] are of the slow variable type. 

From the standpoint of statistics, the transformation Eq. 2.2-19 into Eq. 2.3.b-l and the determination of the parameters from this equation may be criticized. What is minimized by linear regression are the (residuals) between experimental and calculated y-values. The theory requires the error to be normally distributed. This may be true for r, but not necessarily for the group /(Pa - PrPs/I Wa and this may, in principle, affect the values of k, K, K, Ks, — However, when the rate equation is not rearranged, the regression is no longer linear, in general, and the minimization of the sum of squares of residuals becomes iterative. Search procedures are recommended for this (see Marquardt [41]). It is even possible to consider the data at all temperatures simultaneously. The Arrhenius law for the temperature dependence then enters into the equations and increases their nonlinear character. 

The two equivalent adiabatic expressions (106.HI) and (124. HI) represent alternatives of the acciarate formulation of the statistical theory of reaction rates, which rest on two other definitions of the activated complex as a virtual state. In general, they do not involve the Arrhenius exponential factor which includes the classical activation energy. 

Our treatment, based on both the collision and the statistical formulations of reaction rate theory, shows that there exist two possibilities for an interpretation of the experimental facts concerning the Arrhenius parameter K for unimolecular reactions. These possibilities correspond to either an adiabatic or a non-adiabatic separation of the overall rotation from the internal molecular motions. The adiabatic separability is accepted in the usual treatment of unimolecular reactions /136/ which rests on transition state theory. To all appearances this assumption is, however, not adequate to the real situation in most unimolecular reactions.The nonadiabatic separation of the reaction coordinate from the overall rotation presents a new, perhaps more reasonable approach to this problem which avoids all unnecessary assumptions concerning the definition of the activated complex and its properties. Thus, for instance, it yields in a simple way the rate equations (7.IV), corresponding to the "normal Arrhenius parameters (6.IV), which are both direct consequences of the general rate equation (2.IV). It also predicts deviations from the normal values of the apparent frequency factor K without any additional assumptions, such that the transition state (AB)" (if there is one) differs more or less from the initial state of the activated molecule (AB). ... 

It will thus be obvious that both covalent and polar crosslinks contribute to the crosslink network and hence modulus of polyurethane elastomers. The dependence of this modulus or temperature can be divided into contributions from a covalently linked network conforming to the statistical theory of rubber elasticity, and contributions from secondary crosslinks which are assumed to have a temperature dependence governed by the Arrhenius law in which the modulus of elasticity is represented by the equation ... 

As a consequence of the IVR-mediated nature of the multiple-photon excitation process, the vibrational excitation is randomized as the dissociatirai threshold is approached. Hence, the molecule has no memory of the vibrational coordinate that was originally excited. Dissociation therefore occims statistically and can be modelled using the Arrhenius equation or phase-space theories. Mode-selective dissociation is normally not observed. 

---