Statistical thermodynamics, constant

Statistical thermodynamics tells us that Cv is made up of four parts, translational, rotational, vibrational, and electronic. Generally, the last part is zero over the range 0 to 298 K and the first two parts sum to 5/2 R, where R is the gas constant. This leaves us only the vibrational part to worry about. The vibrational contr ibution to the heat capacity is... 

From the third law of thermodynamics, the entiopy 5 = 0 at 0 K makes it possible to calculate S at any temperature from statistical thermodynamics within the hamionic oscillator approximation (Maczek, 1998). From this, A5 of formation can be found, leading to A/G and the equilibrium constant of any reaction at 298 K for which the algebraic sum of AyG for all of the constituents is known. A detailed knowledge of A5, which we already have, leads to /Gq at any temperature. Variation in pressure on a reacting system can also be handled by classical thermodynamic methods. 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

The freely-jointed chain considered previously has no internal restraint, and hence, its internal energy is zero regardless of its present configuration. The entropy (S) is not constant, however, since the number of available configurations decreases with the chain end separation distance. The variation which follows from chain length change by a small amount (dr) at constant temperature (T) is given by the Boltzmann rule of statistical thermodynamics ... 

One can write for Eq. (7-49) an expression for the equilibrium constant. Statistical thermodynamics allows its formulation in terms of partition functions ... 

By applying the machinery of statistical thermodynamics we have derived expressions for the adsorption, reaction, and desorption of molecules on and from a surface. The rate constants can in each case be described as a ratio between partition functions of the transition state and the reactants. Below, we summarize the most important results for elementary surface reactions. In principle, all the important constants involved (prefactors and activation energies) can be calculated from the partitions functions. These are, however, not easily obtainable and, where possible, experimentally determined values are used. 

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. 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

The statistical thermodynamic approach of Pitzer (14), involving specific interaction terms on the basis of the kinetic core effect, has provided coefficients which are a function of the ionic strength. The coefficients, as the stoichiometric association constants in our ion-pairing model, are obtained empirically in simple solutions and are then used to predict the activity coefficients in complex solutions. The Pitzer approach uses, however, a first term akin to the Debye-Huckel one to represent nonspecific effects at all concentrations. This weakens somewhat its theoretical foundation. 

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. 

The concentration of the activated complex may be calculated by statistical thermodynamics in terms of the reactant concentrations and an equilibrium constant [1, 6], If the reaction scheme is written as... 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

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. 

As in statistical thermodynamics, the entropy is defined as In P. Since the numerator is constant, the entropy is, apart from a constant, equal to... 

This gives a rate coefticient k = k Kg, where these quantities can be calculated from statistical thermodynamics (which we shall not do here). [It can be shown from statistical mechanics that k = kT/ h, where k is Boltzmann s constant and h is Planck s constant,]... 

The Hamaker constants of nonpolar fluids and polymeric liquids can be obtained using an expression similar to Equation (67) in combination with the corresponding state theory of thermodynamics and an expression for interfacial energy based on statistical thermodynamics (Croucher 1981). This leads to a simple, but reasonably accurate and useful, relation for Hamaker constants for nonpolar fluids and polymeric liquids. We present in this section the basic details and an illustration of the use of the equation derived by Croucher. 

Statistical thermodynamics is used to obtain the partition function for species strongly bound to the surface (i.e., chemisorbed species). This approach can be used to derive the Langmuir adsorption isotherm, and to estimate the associated equilibrium constant, discussed in Section 11.5.3. The situation in which the adsorbed species is more weakly bound, and moves freely across the surace is considered in Section 11.5.4. 

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. )... 

The virial isotherm equation, which can represent experimental isotherm contours well, gives Henry s law at low pressures and provides a basis for obtaining the fundamental constants of sorption equilibria. A further step is to employ statistical and quantum mechanical procedures to calculate equilibrium constants and standard energies and entropies for comparison with those measured. In this direction moderate success has already been achieved in other systems, such as the gas hydrates 25, 26) and several gas-zeolite systems 14, 17, 18, 27). In the present work AS6 for krypton has been interpreted in terms of statistical thermodynamic models. 

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. 

The equilibrium constant Kl is then analyzed by the methods of statistical thermodynamics to separate out the contribution of the reaction coordinate from other contributions. The rate constant A is also calculated by statistical thermodynamic methods. These calculations are given in Appendix 1 to this chapter. The results of the analysis are expressed by Equation 2.57, where k is the Boltz-... 

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