Equilibria from Atomic Structures

A major goal in chemistry is to predict the equilibria of chemical reactions, the relative amounts of the reactants and products, from their atomic masses, bond lengths, moments of inertia, and their other structural properties. In this chapter, we consider reactions in the gas phase, which arc simpler than reactions in liquid phases. In Chapter 16 we will consider solvation effects. 

First consider the simplest equilibrium between two states, A and B, A B. 

Tv o-state equilibria include chemical isomerization, the folding of a biopoly-mers from an open to a compact state, the binding of a ligand to a surface or a molecule, the condensation of a vapor to a liquid or the freezing of a liquid to a solid. 

The equilibrium constant K is the ratio of the numbers or concentrations of particles in each of the two states at equilibrium. To make it unambiguous whether K is the ratio of the amount of A divided by the amount of B or B divided by. 4, the arrow in Equation (13.1) has a direction. It points to the final 

The quantity that predicts chemical equfiibria is the chemical potential. For equilibria at fixed temperature and pressure, the appropriate extremum function is the Gibbs free energy dG = -SdT + Vdp + pAdN -t- usdN, where N, and Nb are the numbers of particles in the tw o states, and pa and ps are their chemical potentials. At constant T and p, the condition for equilibrium is 

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To determine K from experiments, you need some way of detecting the numbers of particles of types A, B, and C at equilibrium, and you need to know the stoichiometric coefficients, a, b, and c. Equation (13.18) allows you to predict chemical equilibria from atomic structures. 

A remarkable achievement of statistical mechanics is the accurate prediction of gas-phase chemical reaction equilibria from atomic structures. From atomic masses, moments of inertia, bond lengths, and bond strengths, you can calculate partition functions. You can then calculate equilibrium constants and their dependence on temperature and pressure. In Chapter 19, we will apply these ideas to chemical kinetics, which pertains to the rates of reactions. Reactions can be affected by the medium they are in. Next we will develop models of liquids and other condensed phases. 

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. 

The atomic temperature factor, or B factor, measures the dynamic disorder caused by the temperature-dependent vibration of the atom, as well as the static disorder resulting from subtle structural differences in different unit cells throughout the crystal. For a B factor of 15 A2, displacement of an atom from its equilibrium position is approximately 0.44 A, and it is as much as 0.87 A for a B factor of 60 A2. It is very important to inspect the B factors during any structural analysis a B factor of less than 30 A2 for a particular atom usually indicates confidence in its atomic position, but a B factor of higher than 60 A2 likely indicates that the atom is disordered. 

The above stm study also discovered a facile transport of surface gold atoms in the presence of the liquid phase, suggesting that the two-step mechanism does not provide a complete picture of the surface reactions, and that adsorption/desorption processes may have an important role in the formation of the final equilibrium structure of the monolayer. Support for the importance of a desorption process comes from atomic absorption studies showing the existence of gold in the alkanethiol solution. The stm studies suggest that this gold comes from terraces, where single-atomic deep pits are formed (281-283). 

In the equilibrium structure, the main VB structure is the covalent CH bonds structure (I) as expected. The second most important are those where one of the CH bonds is connected with a covalent bond and the other with an ionic bond made by electron transfer from the hydrogen atom to the carbon atom, (II) and (IV). In contrast, the contribution from the structures that describe electron transfer from the carbon atom to a hydrogen atom is small and negative. The contribution from the HH bond structure (VIII) and ionic structures, (IX) and (X), is very small. The total occupation number of CH bonds is 0.9654, while that of HH bond is -0.0147. This indicates almost no bond formation between two hydrogen atoms in the equilibrium structure. 

We show by X-ray diffraction in the Bragg-Brentano geometry that the atomic stmcture of lead sulfide nanoparticles in thin films prepared by wet chemical method is different from the B stmcture, which is the equilibrium phase for bulk single-crystalline PbS. The atomic structure of nanoparticles can be desaibed by the cubic space group Fm-3m with both tetrahedral and octahedral coordinations for sulfur atoms. 

Once the model atomic structure of an amorphous oxide adsorbent is created, one may proceed to simulate physical adsorption on (or in) this material. The peculiarity of oxide adsorbents (compared to carbon adsorbents for example) is that one has to take account of the highly inhomogeneous electrostatic field at their surfaces. The problem of the reliable calculation of the effect this field upon the adsorption energy is not yet totally resolved. However, an effective adsorption potential is typically used in such situations, with parameters that are adjusted by, for example, fitting the calculations to the temperature dependence of the experimental Henry s Law constants. Such potentials generally give reasonable values for other simulated equilibrium and kinetic adsorption properties. There is even an indication that the effective parameters of the gas-solid adsorption potential are sometimes transferrable from one (oxide) adsorption system to another. 

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