Statistical thermodynamics Gibbs energy

A rams, D. S., and J. M. Prausnitz, "Statistical Thermodynamics of Liquid Mixtures A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems," AIChE J., 1975, 21, 116. 

Structural and molecular biologists often study the temperature dependence of the equilibrium position of a reaction or process. The Gibbs free energy undoubtedly provides the correct thermodynamic criterion of equilibrium. An understanding of this parameter can be achieved from either a macroscopic level (classical thermodynamics) or a molecular level (statistical thermodynamics). Ultimately, one seeks to understand the factors influencing AG° for a specific reaction. 

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

Abrams, D.S. and Prausnitz, J.M., Statistical thermodynamics of liquid mixtures a new expression for the excess Gibbs energy of partly or completely miscible systems, A. I. Chem. E. /., 21 (1975) 116-128. 

Chapter 9 deals with the general principles of computational thermodynamics, which includes a discussion of how Gibbs energy minimisation can be practically achieved and various ways of presenting the output. Optimisation and, in particular, optimiser codes, such as the Lukas progranune and PARROT, are discussed. The essential aim of these codes is to reduce the statistical error between calculated phase equilibria, thermodynamic properties and the equivalent experimentally measured quantities. 

In this chapter, we discuss classical non-stoichiometry derived from various kinds of point defects. To derive the phase rule, which is indispensable for the understanding of non-stoichiometry, the key points of thermodynamics are reviewed, and then the relationship between the phase rule, Gibbs free energy, and non-stoichiometry is discussed. The concentrations of point defects in thermal equilibrium for many types of defect structure are calculated by simple statistical thermodynamics. In Section 1.4 examples of non-stoichiometric compounds are shown referred to published papers. 

TABLE 13.3 T-dependent Equilibrium Constant (KT), Gibbs Free Energy of Reaction (AGT), and Overall Entropic Shift (AAG = AG12oo — AG90o) for the Water Gas Shift Reaction (cf. Tables 13.1, 13.2, and Text), as Determined from Theoretically ( B3LYP ) or Empirically ( Hill ) Evaluated Statistical Thermodynamic Formulas Versus Experiment ( Exp. )... 

Statistical thermodynamics can provide explicit expressions for the phenomenological Gibbs energy functions discussed in the previous section. The statistical theory of point defects has been well covered in the literature [A. R. Allnatt, A. B. Lidiard (1993)]. Therefore, we introduce its basic framework essentially for completeness, for a better atomic understanding of the driving forces in kinetic theory, and also in order to point out the subtleties arising from the constraints due to the structural conditions of crystallography. 

Although the statistical approach to the derivation of thermodynamic functions is fairly general, we shall restrict ourselves to a) crystals with isolated defects that do not interact (which normally means that defect concentrations are sufficiently small) and b) crystals with more complex but still isolated defects (i.e., defect pairs, associates, clusters). We shall also restrict ourselves to systems at some given (P T), so that the appropriate thermodynamic energy function is the Gibbs energy, G, which is then constructed as... 

In this section two prediction techniques are discussed, namely, the gas gravity method and the Kvsi method. While both techniques enable the user to determine the pressure and temperature of hydrate formation from a gas, only the KVSI method allows the hydrate composition calculation. Calculations via the statistical thermodynamics method combined with Gibbs energy minimization (Chapter 5) provide access to the hydrate composition and other hydrate properties, such as the fraction of each cavity filled by various molecule types and the phase amounts. 

Section 5.1 presents the fundamental method as the heart of the chapter— the statistical thermodynamics approach to hydrate phase equilibria. The basic statistical thermodynamic equations are developed, and relationships to measurable, macroscopic hydrate properties are given. The parameters for the method are determined from both macroscopic (e.g., temperature and pressure) and microscopic (spectroscopic, diffraction) measurements. A Gibbs free energy calculation algorithm is given for multicomponent, multiphase systems for comparison with the methods described in Chapter 4. Finally, Section 5.1 concludes with ab initio modifications to the method, along with an assessment of method accuracy. 

The statistical thermodynamic method and the Gibbs energy minimization presented in this chapter represents the state-of-the-art for the prediction of the following types of phase equilibria ... 

From elementary statistical thermodynamics we know that the Gibbs free energy of the system is... 

The treatments in the preceding sections have been pretty abstract, and it may be hard to understand statements like Thus, if G can be determined as a function of T, P, and ,, all of the thermodynamic properties of the system can be calculated (which appeared after equation 2.5-9). However, there is one case where this can be demonstrated in detail, and that is for a monatomic ideal gas (Greiner, Neise, and Stocker, 1995). Statistical mechanics shows that the Gibbs energy of a monatomic ideal gas without electronic excitation (Silbey and Alberty,... 

Statistical mechanics is often thought of as a way to predict the thermodynamic properties of molecules from their microscopic properties, but statistical mechnics is more than that because it provides a complementary way of looking at thermodynamics. The transformed Gibbs energy G for a biochemical reaction system at specified pH is given by... 

The thermophysical properties necessary for the growth of tetrahedral bonded films could be estimated with a thermal statistical model. These properties include the thermodynamic sensible properties, such as chemical potential /t, Gibbs free energy G, enthalpy H, heat capacity Cp, and entropy S. Such a model could use statistical thermodynamic expressions allowing for translational, rotational, and vibrational motions of the atom. 

In addition to these complications in interpreting H/D exchange data, it must be bom in mind that hydrogen exchange provides a static measure of protein flexibility proteins in solution exist as an ensemble of different conformations. The population of each conformation is determined by its Gibbs free energy according to the standard statistical thermodynamic relationship... 

After the seminal work of Guggenheim on the quasichemical approximation of the lattice statistical-mechanical theory[l], various practical thermodynamic models such as excess Gibbs energies[2-3] and equations of state[4-5] were proposed. However, the quasichemical approximation of the Guggenheim combinatory yields exact solution only for pure fluid systems. Therefore one has to resort to numerical procedures to find the solution that is analytically applicable to real mixtures. Thus, in this study we present a new unified group contribution equation of state[GC-EOS] which is applicable for both pure or mixed state fluids with emphasis on the high pressure systems[6,7]. 

Note now that Hg is an arbitrary constant that may be assigned any convenient value. By contrast, S° is determined, at least in principle, by the methods of statistical thermodynamics, so that S is uniquely determined. We may now combine the above quantities to find the Gibbs free energy ... 

Ideal-gas tables of thermodynamic properties derived from statistical mechanics are based on the thermodynamic temperatures (as well as on the values of the physical constants used) and are hence independent of any practical temperature scale. The enthalpy of formation, Gibbs energy of formation, and logarithm of the equilibrium constant might depend on temperature-adjusted data. 

The boiling point is calculated from the adopted thermodynamic functions and the chosen enthalpy of sublimation at 298.15 K so that the Gibbs energy functions calculated by integration of the crystal liquid data and by statistical methods from the gas phase are equal at the boiling point. 

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