Statistical thermodynamics chemical equilibrium

In this equation, we are labeling each molecular partition function with the label of the relevant component. Also, we are reminded that each component is occupying the same volume and has the same temperature (otherwise the system is not at equilibrium), but that each component has its own characteristic amount at equilibrium. Statistical thermodynamics gives an expression for the chemical potential of a component ... 

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. 

The understanding of isotope effects on chemical equilibria, condensed phase equilibria, isotope separation, rates of reaction, and geochemical and meteorological phenomena, share a common foundation, which is the statistical thermodynamic treatment of isotopic differences on the properties of equilibrating species. For that reason the theory of isotope effects on equilibrium constants will be explored in considerable detail in this chapter. The results will carry over to later chapters which treat kinetic isotope effects, condensed phase phenomena, isotope separation, geochemical and biological fractionation, etc. 

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

Now that we have considered the calculation of entropy from thermal data, we can obtain values of the change in the Gibbs function for chemical reactions from thermal data alone as well as from equilibrium data. From this function, we can calculate equilibrium constants, as in Equations (10.22) and (10.90.). We shall also consider the results of statistical thermodynamic calculations, although the theory is beyond the scope of this work. We restrict our discussion to the Gibbs function since most chemical reactions are carried out at constant temperature and pressure. 

The statistical thermodynamic approach to the derivation of an adsorption isotherm goes as follows. First, suitable partition functions describing the bulk and surface phases are devised. The bulk phase is usually assumed to be that of an ideal gas. From the surface phase, the equation of state of the two-dimensional matter may be determined if desired, although this quantity ceases to be essential. The relationships just given are used to evaluate the chemical potential of the adsorbate in both the bulk and the surface. Equating the surface and bulk chemical potentials provides the equilibrium isotherm. 

Thus, lattice defects such as point defects and carriers (electrons and holes) in semiconductors and insulators can be treated as chemical species, and the mass action law can be applied to the concentration equilibrium among these species. Without detailed calculations based on statistical thermodynamics, the mass action law gives us an important result about the equilibrium concentration of lattice defects, electrons, and holes (see Section 1.4.5). 

Transition-state theory is based on the assumption of chemical equilibrium between the reactants and an activated complex, which will only be true in the limit of high pressure. At high pressure there are many collisions available to equilibrate the populations of reactants and the reactive intermediate species, namely, the activated complex. When this assumption is true, CTST uses rigorous statistical thermodynamic expressions derived in Chapter 8 to calculate the rate expression. This theory thus has the correct limiting high-pressure behavior. However, it cannot account for the complex pressure dependence of unimolecular and bimolecular (chemical activation) reactions discussed in Sections 10.4 and 10.5. 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modem ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic stmcture picture of molecular and supramolecular interactions. 

Chemical solid state processes are dependent upon the mobility of the individual atomic structure elements. In a solid which is in thermal equilibrium, this mobility is normally attained by the exchange of atoms (ions) with vacant lattice sites (i.e., vacancies). Vacancies are point defects which exist in well defined concentrations in thermal equilibrium, as do other kinds of point defects such as interstitial atoms. We refer to them as irregular structure elements. Kinetic parameters such as rate constants and transport coefficients are thus directly related to the number and kind of irregular structure elements (point defects) or, in more general terms, to atomic disorder. A quantitative kinetic theory therefore requires a quantitative understanding of the behavior of point defects as a function of the (local) thermodynamic parameters of the system (such as T, P, and composition, i.e., the fraction of chemical components). This understanding is provided by statistical thermodynamics and has been cast in a useful form for application to solid state chemical kinetics as the so-called point defect thermodynamics. 

A rational deduction of elemental abundance from solar and stellar spectra had to be based on quantum theory, and the necessary foundation was laid with the Indian physicist Meghnad Saha s theory of 1920. Saha, who as part of his postdoctoral work had stayed with Nernst in Berlin, combined Bohr s quantum theory of atoms with statistical thermodynamics and chemical equilibrium theory. Making an analogy between the thermal dissociation of molecules and the ionization of atoms, he carried the van t Hoff-Nernst theory of reaction-isochores over from the laboratory to the stars. Although his work clearly belonged to astrophysics, and not chemistry, it relied heavily on theoretical methods introduced by and associated with physical chemistry. This influence from physical chemistry, and probably from his stay with Nernst, is clear from his 1920 paper where he described ionization as a sort of chemical reaction, in which we have to substitute ionization for chemical decomposition. [81] The influence was even more evident in a second paper of 1922 where he extended his analysis. [82]... 

Refs. [i] Denbigh KG (1987) Principles of chemical equilibrium, 4th edn. Cambridge University Press, New York [ii] Radeva T (2001) Physical chemistry of polyelectrolytes. Marcel Dekker, New York [Hi] Ben-Naim A (1992) Statistical thermodynamics for chemists and biologists. Plenum Press, London... 

This textbook proves how equilibrium thermodynamics, a traditionally difficult subject, can be accurately expressed using basic high school geometry concepts. Specifically, the text deals with classical equilibrium ihermodvnnmics and its modem reformulation in metric geometric terms. Hie author emphasizes applications to chemical and phase equilibria in complex chemical systems, statistical mechanical origins, and extensions to near-equilibrium transport properties. 

Equation (7.8) offers a clear separation of inner-shell and outer-shell contributions so that different physical approximations might be used in these different regions, and then matched. The description of inner-shell interactions will depend on access to the equilibrium constants K. These are well defined, observationally and computationally (see Eq. (7.10)), and so might be the subject of either experiments or statistical thermodynamic computations. Eor simple solutes, such as the Li ion, ab initio calculations can be carried out to obtain approximately the Kn (Pratt and Rempe, 1999 Rempe et al, 2000 Rempe and Pratt, 2001), on the basis of Eq. (2.8), p. 25. With definite quantitative values for these coefficients, the inner-shell contribution in Eq. (7.8) appears just as a function involving the composition of the defined inner shell. We note that the net result of dividing the excess chemical potential in Eq. (7.8) into inner-shell and outer-shell contributions should not depend on the specifics of that division. This requirement can provide a variational check that the accumulated approximations are well matched. 

In Section A.l, the general laws of thermodynamics are stated. The results of statistical mechanics of ideal gases are summarized in Section A.2. Chemical equilibrium conditions for phase transitions and for reactions in gases (real and ideal) and in condensed phases (real and ideal) are derived in Section A.3, where methods for computing equilibrium compositions are indicated. In Section A.4 heats of reaction are defined, methods for obtaining heats of reaction are outlined, and adiabatic flame-temperature calculations are discussed. In the final section (Section A.5), which is concerned with condensed phases, the phase rule is derived, dependences of the vapor pressure and of the boiling point on composition in binary mixtures are analyzed, and properties related to osmotic pressure are discussed. 

Adsorption isotherm equations can in principle be derived by first formulating the chemical potential of the adsorbate p° in terms of a model, then equating p to p. Although it is not impossible to derive expressions for p by thermodynamic means, statistical approaches are more appropriate because in this way the molecular picture can be made explicit. Moreover, adsorbates are not macroscopic systems, which is a prerequisite for applying thermodynamics, and statistical thermodynamics lends itself very well to the derivation of expressions for the surface pressure. Another approach is based on kinetic considerations expressions for the rates of adsorption and desorption are formulated at equilibrium the two are equal. 

In real situations surface and volume changes are often made with systems that are at equilibrium with their environment, characterized by a set of chemical potentials p, rather than keeping In ] fixed, as in [2.2.7 and 8j. In other words, area changes in open systems are considered. In statistical thermodynamics the conversion from closed to open implies the transition from the canonical to the grand canonical ensemble. The characteristic function of the latter is nothing other than the sum of the bulk and surface mechemical work terms (see [1.3.3.12] and [I.A6.23D which are the quantities of interest ... 

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