Statistical Kinetic Theories

The reaction rate can be obtained by trajectory analysis by examining a very large number of trajectories. This can be prohibitively expensive for most reactions. An alternative approach is to use statistical theories that deal with a large ensemble of 

Standard representation of the TS in organic chemistry textbooks is the point of maximum energy on the reaction coordinate. More precise is the definition provided in Section 1.6 the TS is the col, a point where aU the gradients vanish, and all of the eigenvalues of the Hessian matrix are positive except one, which corresponds to the reaction coordinate. In statistical kinetic theories, a slightly different definition of the TS is required. 

Phase space is a 6N- 12 dimensional representation of the atomic (3n - 6) coordinates and their associated (3N- 6) momenta. Reactive trajectories in phase space move from reactant to product. The TS is the hyperplane such that all trajectories that cross this plane do so only once. In other words, trajectories that cross this plane from the reactant side will go on to products without ever turning back and recrossing the plane toward reactant. Given this definition, the rate of reaction is the number of tfajectories that cross the plane per unit time.  

In order to remove the need for explicit trajectory analysis, one makes the statistical approximation. This approximation can be formulated in a number of equivalent ways. In the microcanonical ensemble, all states are equally probable. Another formulation is that the lifetime of reactant (or intermediate) is random and follows an exponential decay rate. But perhaps the simplest statement is that intramolecular vibrational energy redistribution (IVR) is faster than the reaction rate. IVR implies that if a reactant is prepared with some excited vibrational mode or modes, this excess energy will randomize into all of the vibrational modes prior to converting to product. 

RRKM theory assumes both the statistical approximation and the existence of the TS. It assumes a microcanonical ensemble, where all the molecules have equivalent energy E. This energy exceeds the energy of the TS (Eq), thanks to vibration, rotation, and/or translation energy. Invoking an equilibrium between the TS (the activated complex) and reactant, the rate of reaction is 

---

Substances at high dilution, e.g. a gas at low pressure or a solute in dilute solution, show simple behaviour. The ideal-gas law and Henry s law for dilute solutions antedate the development of the fonualism of classical themiodynamics. Earlier sections in this article have shown how these experimental laws lead to simple dieniiodynamic equations, but these results are added to therniodynaniics they are not part of the fonualism. Simple molecular theories, even if they are not always recognized as statistical mechanics, e.g. the kinetic theory of gases , make the experimental results seem trivially obvious. 

Statistical mechanics and kinetic theory, as we have seen, are typically concerned with the average behaviour of an ensemble of similarly prepared systems. One usually hopes, and occasionally can demonstrate, that the variations of these properties from one system to another in the ensemble, or that the variation with time of the properties of any... 

Ernst M FI 1998 Bogoliubov-Choh-Uhlenbeck theory cradle of modern kinetic theory Progress in Statistical Physics ed W Sung et al (Singapore World Scientific)... 

Theoretical equation forms may be derived from either kinetic theory or statistical mechanics. However, empirical and semitheoretical equations of state have had the greatest success in representing data with high precision over a wide range of conditions (1). At present, theoretical equations are more limited in range of appHcation than empirical equations. There are several excellent references available on the appHcation and development of equations of state (2,3,18,21). 

When experimental data are not available, methods of estimation based on statistical mechanics are employed (7,19). Classical kinetic theory suggests a contribution to CP of S R for each translational degree of freedom in the molecule, a contribution of S R for each axis of rotation, and of R for each vibrational degree of freedom. A cmde estimate of CP for small molecules can be obtained which neglects vibrational degrees of freedom ... 

The kinetic theory attempts to describe the individual molecules energies and interactions statistical thermodynamics attempts to fundamentally develop the equation of state from considerations of groupings of molecules. These approaches are complementary in many ways (3,123,124). A weU-referenced text covering molecular thermodynamics is also available (125). 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

The static properties of an isolated chain constitute a good starting point to study polymer dynamics many of the features of the chain in a quiescent fluid could be extrapolated to the kinetics theories of molecular coil deformation. As a matter of fact, it has been pointed out that the equations of chain statistics and chain dynamics are intimately related through the simplest notions of graph theory [16]. 

F. W. Sears, An Introduction to Thermodynamics, The Kinetic Theory of Gases, and Statistical Mechanics, 2nd edn., Addison-Wesley, Reading, Massachusetts, 1966. 

Hinshelwood (51) used reasoning based on statistical mechanics to show that the energy probability factor in the kinetic theory expressions (e E,RT) is strictly applicable only to processes for which the energy may be represented in two square terms. Each translational and rotational degree of freedom of a molecule corresponds to one squared term, and each vibrational degree of freedom corresponds to two squared terms. If one takes into account the energy that may be stored in 5 squared terms, the correct probability factor is... 

The development of theoretical chemistry ceased at about 1930. The last significant contributions came from the first of the modern theoretical physicists, who have long since lost interest in the subject. It is not uncommon today, to hear prominent chemists explain how chemistry is an experimental science, adequately practiced without any need of quantum mechanics or the theories of relativity. Chemical thermodynamics is routinely rehashed in the terminology and concepts of the late nineteenth century. The formulation of chemical reaction and kinetic theories take scant account of statistical mechanics and non-equilibrium thermodynamics. Theories of molecular structure are entirely classical and molecular cohesion is commonly analyzed in terms of isolated bonds. Holistic effects and emergent properties that could... 

In this section, using the representation theory introduced before, we analyse the structure of statistical mechanics and kinetic theory for bosons starting from Eq. (44). We consider that Eq. (44) describes the evolution of an ensemble of quantum particles specified through the density operator p such that the entropy is given by(A.E. Santana et.al., 1999 A.E. Santana et.al., 2000)... 

C. V. Heer, Statistical Mechanics, Kinetic Theory, and Stochastic Processes,... 

A simplified approach is statistical rate theory (transition state theory) with the help of which the overall rate constant k(T) may be obtained from potential energy surface (PES) in a single jump averaging out all of the intermediate details. It is generally not possible to extract microscopic details such as energy-dependent cross sections from conventional kinetics experiments. The preferable approach is to calculate microscopic quantities from some model and then perform the downward averaging for comparison with measured quantities. 

Abstract The statistical thermodynamic theory of isotope effects on chemical equilibrium constants is developed in detail. The extension of the method to treat kinetic isotope effects using the transition state model is briefly described. 

EXPLORING THE ORBITAL DECOMPOSITION OF THE KINETIC THEORY WITH STATISTICAL ATOMIC MODELS... 

Figure 13.3. A P- V-T surface for a one-component system in which the substance contracts on freezing, such as water. Here Tj represents an isotherm below the triple-point temperature, 72 represents an isotherm between the triple-point temperature and the critical temperature, is the critical temperature, and represents an isotherm above the triple-point temperature. Points g, h, and i represent the molar volumes of sohd, hquid, and vapor, respectively, in equilibrium at the triple-point temperature. Points e and d represent the molar volumes of solid and liquid, respectively, in equihbrium at temperature T2 and the corresponding equilibrium pressure. Points c and b represent the molar volumes of hquid and vapor, respectively, in equilibrium at temperature and the corresponding equihbrium pressure. From F. W. Sears and G. L. Sahnger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. 3rd ed., Addison-Wesley, Reading, MA, 1975, p. 31.

Parallel with the phenomenological development, an alternative point of view has developed toward thermodynamics, a statistical-mechanical approach. Its philosophy is more axiomatic and deductive than phenomenological. The kinetic theory of gases naturally led to attempts to derive equations describing the behavior of matter in bulk from the laws of mechanics (first classic, then quanmm) applied to molecular particles. As the number of molecules is so great, a detailed treatment of the mechanical problem presents insurmountable mathematical difficulties, and statistical methods are used to derive average properties of the assembly of molecules and of the system as a whole. 

---