Chemical equilibrium statistical treatment

The papers of Wagner and Schottky contained the first statistical treatment of defect-containing crystals. The point defects were assumed to form an ideal solution in the sense that they are supposed not to interact with each other. The equilibrium number of intrinsic point defects was found by minimizing the Gibbs free energy with respect to the numbers of defects at constant pressure, temperature, and chemical composition. The equilibrium between the crystal of a binary compound and its components was recognized to be a statistical one instead of being uniquely fixed. 

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. 

Chapter 2 contains several applications of these tools to very simple systems. Except for section 2.10, the material presented here is contained in most standard introductory textbooks in statistical thermodynamics. Section 2.10 is a detailed treatment of a chemical equilibrium affected by the adsorption of a ligand. The results of this section are applied mainly in Chapter 3, but some more general conclusions also appear in later chapters such as 5, 7, and 8. 

First, they are often obtained from incorrect statistical treatments. When the primary experimental quantities are the equilibrium constants, which are measured at different temperatures, any error that makes AH° greater also makes AS greater, as indicated by Equation 1.89. The propagation of errors will tend to distribute enthalpy and entropy estimates along a line characterized by a slope equal to the harmonic mean of the experimental temperatures [85]. This artefact has been pointed out many times [86-88]. Several correct statistical treatments have been advanced [85-89]. For example, the fair value of the correlation coefficient (0.951) of the enthalpy-entropy correlation (plotted in Figure 1.2) for the complexation of seven amines with h has been taken to imply a chemical causation [90]. A correct statistical treatment shows [85] that the 95% confidence interval for p is (850, 147) and includes the harmonic mean of the experimental temperatures, 298 K. Thus, the hypothesis that the observed enthalpy-entropy compensation is just a consequence of the propagation of experimental errors cannot be rejected at the 5% level of significance. 

The motion of particles in a fluid is best approached tlirough tire Boltzmaim transport equation, provided that the combination of internal and external perturbations does not substantially disturb the equilibrium. In otlier words, our starting point will be the statistical themiodynamic treatment above, and we will consider the effect of botli the internal and external fields. Let the chemical species in our fluid be distinguished by the Greek subscripts a,(3,.. . and let f (r, c,f)AV A be the number of molecules of type a located m... 

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. 

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

The DSP approach nicely answers the controversial question about which substituent parameters should be employed to correlate pKa data for 4-substituted pyridinium ions. Statistically, the best correlation is given by Eq. (9), which has values to measure the resonance contribution of a substituent, a result in keeping with chemical intuition. This correlation is statistically superior to a Hammett treatment, where both resonance and inductive effects of a group are combined into a single parameter, p or ap.53,54 Moreover, now it is possible to rationalize why a simple Hammett treatment using ap works so well. Equation (9) reveals that the protonation equilibrium is much more sensitive to an inductive effect (p, — 5.15) than to a resonance effect (p = 2.69). Hence, substituent parameters, such as erp, which are derived from a consideration of the dissociation constants for benzoic acids where resonance contributions are small serve as a useful approximation. The inductive effect is said to have a larger influence on pKa values for pyridinium ions than for benzoic acids because the distance between the substituent and the reactive site is shorter in the pyridine series.53... 

During and after World War II, Horiuti continued his research in chemical kinetics and its applications. His results were compiled in a voluminous paper entitled A Method of Statistical-Mechanical Treatment of Equilibrium and Chemical Reactions (1948). This method is applicable both to heterogeneous and homogeneous systems. Horiuti and his co-workers further attempted to apply the method to the study of a number of chemical syntheses and reactions, such as ammonia synthesis and ethylene hydrogenation. Nearly all of his research papers were published in the Journal of the Institute for Catalysis, of which he was the chief editor. 

The treatment of statistical thermodynamics has been extended to include the calculation of equilibrium constants for simple chemical reactions. At the end of the book, new sections on photophysical kinetics, electrochemical kinetics, and a brief chapter on polymers have been added. 

The treatments of chemical kinetics within the frame of the Arrhenius and the Eyring approaches were essentially based on the postulates of classical statistical equilibrium thermodynamics. It was assumed that a chemical system must pass through the sequence of equilibrium states. The principle of microscopic reversibility holds true all the way from the initial to final products. This implies that the pathways of the forward and backward reactions coincide. We have mentioned above that there exists a method of verification of the validity of thermodynamic equations used for the determination of the reaction enthalpy change. The heat production, AH, can be measured directly using a calorimetric technique. This cannot be done for the activation energy, It is necessary, therefore, to scrutinize the applicability of the conventional approaches of physical chemistry for a description of biochemical processes. 

In the above formal development, we found that Gibbs stability condition from equilibrium thermodynamics and Prigogine s stability condition from nonequilibrium thermodynamics for a chemically reactive system emerge from the statistical mechanical treatment of nonequilibrium systems. Unlike the stability conditions of Gibbs and Prigogine, the inequalities (384), (385), (396), and (397) are not postulates. They are simple consequences of the statistical mechanical treatment. Moreover, these inequalities apply to both equihbrium and nonequilibrium systems. 

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