Elementary partition functions

Elementary partition functions A2.4.1. Vihration partition function 

For each degree of freedom of vibration, there is a vibration frequency v, (in s ), and the respective partition function is written  

This measure has two limit values that are ofteu used (see sectiou A2.4.4)  

When the molecule has n degrees of freedom of vibration, the overall partition function is  

---

We notice that in most the cases, the contributions of the elementary partition functions are of the same order of magnitude. Table A2.1 summarizes these... 

Table A2.1. Order of magnitude of elementary partition functions by degrees of freedom...

Partition functions are very important in estimating equilibrium constants and rate constants in elementary reaction steps. Therefore, we shall take a closer look at the partition functions of atoms and molecules. Motion, or translation, is the only degree of freedom that atoms have. Molecules also possess internal degrees of freedom, namely vibration and rotation. 

By applying the machinery of statistical thermodynamics we have derived expressions for the adsorption, reaction, and desorption of molecules on and from a surface. The rate constants can in each case be described as a ratio between partition functions of the transition state and the reactants. Below, we summarize the most important results for elementary surface reactions. In principle, all the important constants involved (prefactors and activation energies) can be calculated from the partitions functions. These are, however, not easily obtainable and, where possible, experimentally determined values are used. 

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. 

The most accurate theories of reaction rates come from statistical mechanics. These theories allow one to write the partition function for molecules and thus to formulate a quantitative description of rates. Rate expressions for many homogeneous elementary reaction steps come from these calculations, which use quantum mechanics to calculate the energy levels of molecules and potential energy surfaces over which molecules travel in the transition between reactants and products. These theories give... 

With the introduction of the lattice structure and electroneutrality condition, one has to define two elementary SE units which do not refer to chemical species. These elementary units are l) the empty lattice site (vacancy) and 2) the elementary electrical charge. Both are definite (statistical) entities of their own in the lattice reference system and have to be taken into account in constructing the partition function of the crystal. Structure elements do not exist outside the crystal and thus do not have real chemical potentials. For example, vacancies do not possess a vapor pressure. Nevertheless, vacancies and other SE s of a crystal can, in principle, be seen , for example, as color centers through spectroscopic observations or otherwise. The electrical charges can be detected by electrical conductivity. 

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. 

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... 

The classical approximation for the rotational density of states of a molecule is familiar from elementary statistical mechanics, where it is common to assume that the rotational states form a continuum in calculating the rotational partition function. For the external rotations of most molecules this approximation is very good. For example the classical approximation for the rotational partition function of an asymmetric top is... 

The TST was developed originally by Eyring and others on the basis of statistical mechanics [see, e.g., Lasaga (1983) or Moore and Pearson (1981)]. The fundamental result is a bimolecular rate constant for an elementary process expressed in terms of (1) the total molecular partition functions per unit volume iqi) for reactant species and for the activated complex species (q ), and (2) tlie difference in zero-point potential energies between the activated complex and reactants (Eq) ... 

The partition functions of a chain or of a set of chains are expanded in powers of the effective two-body interaction (or eventually in powers of the effective two-body and three-body interactions). The elementary tool is the diagram made of a set of polymer lines (on which correlation points are fixed) and of interaction lines joining interaction points on the polymer lines. Each term in the expansion corresponds to a given number of interaction lines. The calculation of such a term is performed by summing up all the contributions of the diagrams associated with this number of lines. 

Note that for any particular hydrogen molecule, the allowed values of J are either even or odd, but not both. This limitation, which allows the occupation of only one-half of the possible states, effectively divides the partition function by a factor of two. This division by two was accomplished in our elementary argument by introducing the symmetry factor. 

Consider a homogeneous elementary reaction in a dilute gas phase, where the critical complex as well as the constituent gas molecules of the phase behaves statistically independent. The in Eq. (II.3.f) is expressed according to Eq. (11.17) as p =Qt/C where is the partition function per unit volume of a single critical system and (7 is its concentration. Since, however, we are dealing with a single critical system in the assembly, C is simply the reciprocal of volume

[Pg.17]

From elementary statistical thermodynamics we know that the equilibrium constant can be written in terms of the partition functions of the individual molecules taking part in a reaction. These quantities represent the sum over all energy states in the system—translational, rotational, vibrational, and electronic. The probability that a molecule will be in a particular energy state, f ,-, is given by the Boltzmann law,... 

We can still neglect the vibrational portion of the partition function and the portion for the electronically excited states. In the rotation portion of the partition function a symmetry number enters. This emerges because certain symmetries in transitions are not permitted. The entropy for a symmetrical molecule is thus as smaller, as more symmetrical such a molecule is, with otherwise same characteristics. We experience here a strange contradiction If the elementary particles would be freely mobile in a molecule, then we would expect that they distribute equally. That means an asymmetrical molecule should want itself to convert into a more symmetrical molecule. On the other hand, the law of symmetry in entropy tells to us that an asymmetrical molecule has larger entropy. 

In the last section, we demonstrated the potential of determining the rate constant of an elementary reaction by calculating the energies and the partition function of the reactants and the transition state. We discussed that these parameters can be obtained directly through MO calculations if the reactant and transition state configurations are known. How are these configurations determined In this section, we discuss some of the most common ways to determine these configurations from the PES. 

Molecules CH4,02, and so forth are packages of electric charge. Quantum mechanics and statistical mechanics describe them using Hamiltonians, wave functions, and partition functions. At the same time, elementary models count on formula diagrams to portray the Angstrom scale. The gains lie in immediacy and chemical intuition. Hence, the compounds of Bunsen burners and stoves are represented in digital terms ... 

In this section we first review briefly the formal connection between properties of the one- and two-particle Green s function and elementary excitations. Determining the partition function we then work out interrelations between the Kohn-Sham equation of density-functional theory and the general many-body perturbation theory for the exact exchange-correlation energy. 

---