Statistical mechanics vibrational partition function

Again, for detail, see a statistical mechanics text. For a general N-atomic molecule, there are (3N — 6) normal vibrations >j (3N — 5) for a linear molecule) and the corresponding vibrational partition function is... 

The rates and mechanisms of chemical reactions can be predicted, in principle, by the standard methods of statistical thermodynamics, in terms of the partition functions of reactants and the transition-state complex. However, the range of applicability of this transition-state (absolute rate) theory is severely limited by the fact, that an evaluation of the vibrational partition function for the transition state requires a detailed consideration of the whole PES for the reaction. Thus, a calculation of the absolute rate constants is possible only for relatively simple systems. This indicates a need for more approximate, empirical methods of treating chemical reactions and formulating the reactivity theory, which would allow... 

Statistical mechanics is that branch of physical chemistry dealing with partition functions. We appeal to statistical mechanics for two results, discussed in Appendix 6.A.. First, a partition function of a molecule can be expressed as a product of the partition function for the motion of the center of mass and an internal partition function. Second, the internal partition function can be approximately represented as a product of contributions from each bound mode. To apply the theory we need to compute partition fimctions (or to look them up in the JANAF tables ). In what follows, we are extremely cavalier and take all rotational partition functions to be equal, and larger by an order of magnitude than all vibrational partition functions that we also take to be equal to one another. This provides a reasonable order of magnitude but is otherwise not a recommended procedure. 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

In order to calculate thermodynamic activation parameters, we need to know how to evaluate the translational, rotational, and vibrational parts of the partition functions. This can be accomplished by means of the standard formulas of statistical mechanics (see, for example, Dole, 1954). 

The statistical mechanical contribution to transition state theory uses partition functions. These are statistical mechanical quantities made up from translational, rotational, vibrational and electronic terms, though the electronic terms can normally be ignored if the reaction occurs in the ground state throughout. 

Statistical mechanics provides a bridge between the properties of atoms and molecules (microscopic view) and the thermodynmamic properties of bulk matter (macroscopic view). For example, the thermodynamic properties of ideal gases can be calculated from the atomic masses and vibrational frequencies, bond distances, and the like, of molecules. This is, in general, not possible for biochemical species in aqueous solution because these systems are very complicated from a molecular point of view. Nevertheless, statistical mechanmics does consider thermodynamic systems from a very broad point of view, that is, from the point of view of partition functions. A partition function contains all the thermodynamic information on a system. There is a different partition function... 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

A further complication associated with the application of molecular mechanics calculations to relative stabilities is that strain energy differences correspond to A (AH) between conformers with similar chromophores (electronic effects) and an innocent environment (counter ions and solvent molecules), whereas relative stabilities are based on A (AG). The entropy term, TAS, can be calculated by partition functions, and the individual terms of AS include vibrational (5vib), translational (5 trans) and rotational (Arot) components, and in addition to these classical terms, a statistical contribution (5stat). These terms can be calculated using Eqs. 3.40-3.43tl21]. 

Statistical mechanical approaches apply mainly to deductions about structure and are the basis of interpretations of the entropy of ions in solution and the solution s heat capacity. The entropy of a system can be calculated if the partition functions of the ions and the water molecules surrounding them are known." The partition functions (translatory, rotational, and vibrational) can be obtained from textbook material by assuming a structure of the ion-solvent complex. By comparing calculations based on various assumptions about stmcture with the values obtained from experiments, certain stmctures can be shown to be more likely (those giving rise to the calculations that match the experiment), others less probable, and some so far from the experimental values that they may be regarded as impossible. 

In chapter 1.3 a number of examples of elaborations have already been given, mostly using lattice statistics. All of them Involve a "divide and rule" strategy, in that the system (i.e. the adsorbate) is subdivided into subsystems for which subsystem-partition functions can be formulated on the basis of an elementary physical model. For instance, in lattice theories of adsorption one adsorbed atom or molecule on a lattice site on the surface may be such a subsystem. In the simplest case the energy levels, occurring in the subsystem-partition function consist of a potential energy of attraction and a vibrational contribution, the latter of which can be directly obtained quantum mechanically. Having... 

Evaluation of the entropy change in adsorption by the statistical mechanics approach can be found in numerous sources. Making use of Refs. [22,23], we will outline what is required for the present purpose. The partition function for a gaseous molecule qm is the product of the translational component qlr and the internal components rotational rot vibrational qviu and electronic qt. ... 

The molecular properties, such as geometry, vibrational frequencies, and rotational constants, are needed to compute thermodynamic properties such as enthalpy, entropy, and Gibbs free energy through calculation of the partition functions of the substances using statistical mechanics methods. 

The advantage of this formulation is that the partition functions for all compounds featuring in the reaction can be calculated using statistical mechanics for vibrational and rotational motion of mechanical systems. While this is still a difficult problem, a detailed consideration of different reacting systems yields a mechanistic insight in how the reaction occurs on a molecular level. 

Although we have derived this equation assuming that the reaction coordinate for atom transfer corresponds to translational motion, the same expression is obtained if the reaction coordinate is assumed to be vibrational motion. According to Eq. (37), the reaction rate constant may be calculated using the relevant molecular partition functions, known from statistical mechanics, remembering that Oahb clude the translational motion... 

The equilibrium constant K can be treated in two different but equivalent ways. In the first procedure, the methods of statistical mechanics are used. Equilibrium constants can be calculated reliably in terms of partition functions for the molecules involved, and the same can be done for the particular equilibrium constant KK What are required are the masses of the reactant molecules and the activated complex, their moments of inertia, and their vibrational frequencies. Eor the reactant molecules this is usually straightforward. These parameters are also known for an activated complex if a reliable potential energy sur-... 

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