Equation statistical thermodynamic derivation

In the following sections we will see how temperature, entropy, and free energy are statistical properties that emerge in systems composed of large numbers of particles. In Chapter 12, the appendix to this book, we dig more deeply into statistical thermodynamics, derive a set of statistical laws that are used in this chapter, and show how Equation (1.1) - the fundamental equation of macroscopic thermodynamics - is in fact a statistical consequence of more fundamental principles operating at the microscopic level. 

For example, Lu and Jiang showed [100], based on a statistical thermodynamic derivation in which some of the parameters of the model were calibrated by using experimental data, that the approximation in Equation 6.20 can be made for polymers with a vinyl-type chain backbone, where Coo(Tg) is the value of the characteristic ratio at Tg. C, which will be discussed in Chapter 12, depends both on polymer chain stereoregularity and on the temperature. Direct applications of the method which will be presented in Chapter 12 for its prediction as a function... 

In the statistical thermodynamic derivation of the equations of state, the L-J potential is usually written as... 

At first the BET equation was derived from the kinetic considerations analogous to those proposed by Langmuir while deriving the monomolecular adsorption isotherm. First, statistical thermodynamic derivation was carried out by Cassie [123]. Lately, a slightly modified derivation has been proposed by HiU [124-126], Fowler and Guggenheim [127]. 

Jaroniec presented a statistical thermodynamic derivation of the Jovanovic equation using generalized ensemble theory. The final result may be written in the form ... 

The partition function is defined in terms of the different possible energies of the individual particles in a system. The developers of statistical thermodynamics derived their equations without an understanding of the quantum theory of nature. But now, we recognize that atomic and molecular behavior is described by quantum mechanics, and our development of statistical thermodynamics must recognize that. It is why we have put off a discussion of statistical thermodynamics until after our treatment of quantum mechanics. 

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

Thermodynamically Consistent Isotherm Models. These models include both the statistical thermodynamic models and the models that can be derived from an assumed equation of state for the adsorbed phase plus the thermodynamics of the adsorbed phase, ie, the Gibbs adsorption isotherm,... 

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. 

Data for a large number of organic compounds can be found in E. S. Domalski, W. H. Evans, and E. D. Hearing, Heat capacities and entropies in the condensed phase, J. Phys. Chem. Ref. Data, Supplement No. 1, 13 (1984). It is impossible to predict values of heat capacities for solids by purely thermodynamic reasoning. However, the problem of the solid state has received much consideration in statistical thermodynamics, and several important expressions for the heat capacity have been derived. For our purposes, it will be sufficient to consider only the Debye equation and, in particular, its limiting form at very low temperamres ... 

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... 

Fig. 1. Equations of state derived from various statistical thermodynamic theories...

Adsorption isotherms may be derived from a consideration of two-dimensional equations of state, from partition functions by statistical thermodynamics, or from kinetic arguments. Even though these methods are not fundamentally different, they differ in ease of visualization. We consider examples of each method in Sections 9.3 and 9.4. 

Our approach until now has been to discuss adsorption isotherms on the basis of the equation of state of the corresponding two-dimensional matter. This procedure is easy to visualize and establishes a parallel with adsorption on liquid surfaces (Chapter 7) however, it is not the only way to proceed. In the following section we consider the use of statistical thermodynamics in the derivation of adsorption isotherms and examine some other approaches in subsequent sections. 

It should be apparent — since an adsorption isotherm can be derived from a two-dimensional equation of state —that an isotherm can also be derived from the partition function since the equation of state is implicitly contained in the partition function. The use of partition functions is very general, but it is also rather abstract, and the mathematical difficulties are often formidable (note the cautious in principle in the preceding paragraph). We shall not attempt any comprehensive discussion of the adsorption isotherms that have been derived by the methods of statistical thermodynamics instead, we derive only the Langmuir equation for adsorption from the gas phase by this method. The interested reader will find other examples of this approach discussed by Broeckhoff and van Dongen (1970). 

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. 

Until now, we have focused our attention on those adsorption isotherms that show a saturation limit, an effect usually associated with monolayer coverage. We have seen two ways of arriving at equations that describe such adsorption from the two-dimensional equation of state via the Gibbs equation or from the partition function via statistical thermodynamics. Before we turn our attention to multilayer adsorption, we introduce a third method for the derivation of isotherms, a kinetic approach, since this is the approach adopted in the derivation of the multilayer, BET adsorption isotherm discussed in Section 9.5. We introduce this approach using the Langmuir isotherm as this would be useful in appreciating the common features of (and the differences between) the Langmuir and BET isotherms. 

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. 

The first two of the above sections are a simplification and slight expansion of the derivation from the review article by van der Waals and Platteeuw (1959). They were written assuming that the reader has a minimal background in statistical thermodynamics on the level of an introductory text, such as that of Hill (1960), McQuarrie (1976), or Rowley (1994). The reader who does not have an interest in statistical thermodynamics may wish to review the basic assumptions in Sections 5.1.1 and 5.1.4 before skipping to the final equations and the calculation prescription in Section 5.2. 

The derivation is a primary example of application of first principles in statistical thermodynamics, to link both microscopic and macroscopic domains for practical applications. For the reader s convenience, Table 5.1 gives the nomenclature used in Sections 5.1.1 and 5.1.2 as well as a listing (in parentheses) of the equations in which each term first appears. 

A statistical thermodynamic equation for gas adsorption on synthetic zeolites is derived using solid solution theory. Both adsorbate-adsorbate and adsorbate-adsorbent interactions are calculated and used as parameters in the equation. Adsorption isotherms are calculated for argon, nitrogen, ammonia, and nitrous oxide. The solution equation appears valid for a wide range of gas adsorption on zeolites. 

Schirmer et al. (7.) indicate that the constants and E j may be derived from physical or statistical thermodynamic considerations but do not advise this procedure since theoretical calculations of molecules occluded in zeolites are, at present, at least only approximate, and it is in practice generally more convenient to determine the constants by matching the theoretical equations to experimental isotherms. We have determined the constants in the model by a method of parameter determination using the measured equilibrium data. Defining the entropy constants and energy constants as vectors... 

The kinetic derivation has the disadvantage that it refers to a certain model. The Langmuir adsorption isotherm, however, applies under more general conditions and it is possible to derive it with the help of statistical thermodynamics [8,373], Necessary and sufficient conditions for the validity of the Langmuir equation (9.21) are ... 

The first goal of this article is to determine the conditions under which the transition from a liquid-expanded (LE) to a liquid-condensed (LC) phase is horizontal or inclined, A simple criterion involving the ranges of attractive and repulsive interactions is suggested, which can explain qualitatively the nature of the transition. Then, a thermodynamic approach for. systems that exhibit inclined transitions is presented, followed by the derivation on the basis of statistical thermodynamics of a two-dimensional equation of state for the surface pressure against the surface area per molecule. 

The entropy of a mobile adsorption process can be determined from the model given in [4], It is based on the assumption that during the adsorption process a species in the gas phase, where it has three degrees of freedom (translation), is transferred into the adsorbed state with two translational degrees of freedom parallel to the surface and one vibration degree of freedom vertical to the surface. From statistical thermodynamics the following equation for the calculation of the adsorption entropy is derived ... 

Transport Limitation For the estimation of the mass transport limitation, Equation (20) has an important drawback. In many cases neither the rate constant k nor the reaction order n is known. However, the Weisz-Prater criterion, cf. Equation (21), which is derived from the Thiele modulus [4, 8], can be calculated with experimentally easily accessible values, taking < < 1 for any reaction without mass transfer limitations. However, it is not necessary to know all variable exactly, even for the Weisz-Prater criterion n can be unknown. Reasonable assumptions can be made, for example, n - 1, 2, 3, or 4 and / is the particle diameter instead of the characteristic length. For the gas phase, De can be calculated with statistical thermodynamics or estimated common values are within the range of 10-5 to 10 7 m2/s. In the liquid phase, the estimation becomes more complicated. A common value of qc for solid catalysts is 1,300 kg/m3, but if the catalyst is diluted with an inert material, this... 

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. 

Adsorption from solution is an exchange process. Consequences of this "first law" pervade all attempts to define individual (or partial) isotherms. Any assumption made on the adsorption of component 1 involves an assumption regarding component 2 deriving an equation for 1 implies deriving an equation for 2. This is (or should be) reflected in all models, and all thermodynamics and statistical thermodynamics should be consistent with this principle. 

In this paper, a modified HK method is presented which accounts for spatial variations in the density profile of a fluid (argon) adsorbed within a carbon slit pore. We compare the pore width/filling pressure correlations predicted by the original HK method, the modified HK method, and methods based upon statistical thermodynamics (density functional theory and Monte Carlo molecular simulation). The inclusion of the density profile weighting in the HK adsorption energy calculation improves the agreement between the HK model and the predictions of the statistical thermodynamics methods. Although the modified Horvath-Kawazoe adsorption model lacks the quantitative accuracy of the statistical thermodynamics approaches, it is numerically convenient for ease of application, and it has a sounder molecular basis than analytic adsorption models derived from the Kelvin equation. 

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