State Properties from Statistical Thermodynamics

two approaches to calculate state properties are presented. First, a fundamental formulation based on statistical thermodynamics is introduced. In the next section, a noncomputationally demanding, semiempirical-estimation-based approach is 

The coverage-dependent enthalpy of formation of the adsorbates, including the zero-point energy correction, is defined as follows  

In Equation (8.18), Hr(Tg) is the enthalpy of formation in gas phase at T, which is obtained from standard thermodynamic databases. The value of AH depends on the method used for thermodynamic consistency at the enthalpic level. When Equation (8.17) is used, the DFT-predicted heat of adsorption is assumed for. . On the other hand, in the method of Blaylock et al. [9], thermodynamically corrected values such as those presented in Table 8.4 are used. is the heat capacity at constant pressure of a species on the surface. The zero-point energy correction is given by 

To compute the entropy and heat capacity from statistical thermodynamics, one has to consider the type of adsorption, depending on how strongly the adsorbate binds to the surface. In general, adsorbates that bind weakly to surfaces, for example, closed-shell adsorbates such as H O and CO, have a low barrier for surface diffusion, which makes them highly mobile on the surface. On the other hand, strongly bound adsorbates have a high barrier for surface diffusion and are assumed to be immobile on the surface. As a result of differences in adsorption, the statistical 

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This volume also contains four appendices. The appendices give the mathematical foundation for the thermodynamic derivations (Appendix 1), describe the ITS-90 temperature scale (Appendix 2), describe equations of state for gases (Appendix 3), and summarize the relationships and data needed for calculating thermodynamic properties from statistical mechanics (Appendix 4). We believe that they will prove useful to students and practicing scientists alike. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

All previously discussed methods are primarily based on phenomenological considerations, in contrast to chapter six (K. Binder et al.), which starts from statistical thermodynamics. This section reviews the state of the art in fields of Mraite-Carlo and Molecular Dynamics simulations. These methods are powerful tools for the prediction of macroscopic properties of matter from suitable models for effective interactions between atoms and molecules. The final chapter (G. Sadowski) makes use of the results obtained with simulation tools for the establishment of molecular-based equations of state for engineering applications. This approach enables the description and in some cases even the prediction of the phase behavior as a function of pressure, temperature, molecular weight distribution and for copolymers also as a function of chemical composition. 

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. 

A method for the estimation of thermodynamic properties of the transition state and other unstable species involves analyzing parts of the molecule and assigning separate properties to functional groups (Benson, 1976). Another approach stemming from statistical mechanics is outlined in the next section. 

The skeptical reader may reasonably ask from where we have obtained the above rules and where is the proof for the relation with thermodynamics and for the meaning ascribed to the individual terms of the PF. The ultimate answer is that there is no proof. Of course, the reader might check the contentions made in this section by reading a specialized text on statistical thermodynamics. He or she will find the proof of what we have said. However, such proof will ultimately be derived from the fundamental postulates of statistical thermodynamics. These are essentially equivalent to the two properties cited above. The fundamental postulates are statements regarding the connection between the PF and thermodynamics on the one hand (the famous Boltzmann equation for entropy), and the probabilities of the states of the system on the other. It just happens that this formulation of the postulates was first proposed for an isolated system—a relatively simple but uninteresting system (from the practical point of view). The reader interested in the subject of this book but not in the foundations of statistical thermodynamics can safely adopt the rules given in this section, trusting that a proof based on some... 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

Based upon experimentally observed spectroscopic data, statistical thermodynamic calculations provide thermodynamic data which would not be obtained readily from direct experimental measurements for the species and temperature of interest to rocket propulsion. If the results of the calculations are summarized in terms of specific heat as a function of temperature, the other required properties for a particular specie, for example, enthalpy, entropy, the Gibb s function, and equilibrium constant may be obtained in relation to an arbitrary reference state, usually a pressure of one atmosphere and a temperature of 298.15°K. Or alternately these quantities may be calculated directly. Significant inaccuracies in the thermochemical data are not associated generaUy with the results of such calculations for a particular species, but arise in establishing a valid basis for comparison of different species. 

Two important objectives of statistical mechanics are (1) to verify the laws of thermodynamics from a molecular viewpoint and (2) to make possible the calculation of thermodynamic properties from the molecular structure of the material composing the system. Since a thorough discussion of the foundations, postulates, and formal development of statistical mechanics is beyond the scope of this summary, we shall dispose of objective (1) by merely stating that for all cases in which statistical mechanics has successfully been developed, the laws quoted in the preceding section have been found to be valid. Furthermore, in discussing objective (2), we shall merely quote results the reader is referred to the literature [3-7] for amplification. 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

In Chapter 2, we developed statistical thermodynamics as the central theory that enables ns in principle to calculate thermophysical properties of macroscopic confined flriids. A key feature of statistical thermodynamics is an enormous reduction of information that takes place as one goes from the microscopic world of electrons, photons, atoms, or molecules to the macroscopic world at which one performs measurements of thermophysical properties of interest. This information reduction is effected by statistical concepts such as the most probable distribution of quantum states (see Section 2.2.1). 

The previous summary provides the basic relationships, derived from the first and second laws, used for the manipulation of available experimental data. However, statistical thermodynamics is then required to develop expressions for the thermodynamic properties in terms of the fluctuating quantities of interest here. First, we will use statistical thermodynamics to provide the characteristic thermodynamic potentials in terms of the appropriate partition function, which will involve a sum over the microscopic states available to the systan. Second, we will provide relevant expressions for the fluctuations nnder one set of variables, which can then be used to rationalize the thermodynamic properties of a system characterized by a different set of variables. 

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