Statistical Thermodynamical Background

This chapter will show how equilibrium concentrations and reaction rate constants depend on microscopic properties of the reacting molecules, such as bond energies, vibrational frequencies, and rotational moments, which can be studied spectroscopically. A short introduction to the necessary statistical-thermodynamical background is given. Finally the theoretical rate expressions are used to discuss the rate constants of some elementary surface reactions. 

Chemical Kinetics of Solids covers a special part of solid state chemistry and physical chemistry. It has been written for graduate students and researchers who want to understand the physical chemistry of solid state processes in fair depth and to be able to apply the basic ideas to new (practical) situations. Chemical Kinetics of Solids requires the standard knowledge of kinetic textbooks and a sufficient chemical thermodynamics background. The fundamental statistical theory underlying the more or less phenomenological approach of this monograph can be found in a recent book by A. R. Allnatt and A.B. Lidiard Atomic Transport in Solids, which complements and deepens the theoretical sections. 

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. 

With this in mind, this chapter will be set up system-wise, working from the more simple towards the more complex monolayers and inserting a section on surface rheology as soon as it is needed. In order to conUiin the treatment within reasonable limits, we shall mostly restrict ourselves to binary systems. Each section starts, where possible and appropriate, with the required (thermodynamic, statistical, electrostatic,. ..) background. Recall from sec. 3.1 that, at equilibrium, surface equations of state are the same as for Langmuir monolayers. 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

Abstract Fluctuation Theory of Solutions or Fluctuation Solution Theory (FST) combines aspects of statistical mechanics and solution thermodynamics, with an emphasis on the grand canonical ensemble of the former. To understand the most common applications of FST one needs to relate fluctuations observed for a grand canonical system, on which FST is based, to properties of an isothermal-isobaric system, which is the most common type of system studied experimentally. Alternatively, one can invert the whole process to provide experimental information concerning particle number (density) fluctuations, or the local composition, from the available thermodynamic data. In this chapter, we provide the basic background material required to formulate and apply FST to a variety of applications. The major aims of this section are (i) to provide a brief introduction or recap of the relevant thermodynamics and statistical thermodynamics behind the formulation and primary uses of the Fluctuation Theory of Solutions (ii) to establish a consistent notation which helps to emphasize the similarities between apparently different applications of FST and (iii) to provide the working expressions for some of the potential applications of FST. 

The previous background material covered many aspects of thermodynamics, statistical thermodynamics, and solution thermodynamics. At this point, we have all we need to derive the main expressions provided by FST. There are many derivations of the principal expressions in the literature, including a matrix approach that is general for any number of components, all of which involve a series of thermodynamic manipulations (Kirkwood and Buff 1951 Hall 1971 O Connell 1971b Valdeavella, Perkyns, and Pettitt 1994 Ben-Naim 2006 Kang and Smith 2008 Nichols, Moore, and Wheeler 2009). We will not use that type of approach here as our primary concern is binary and ternary solutions for which a more transparent and, in our opinion, simpler approach is available. 

In order that the reader may appreciate the nature of thermodynamic quantities, thermodynamic data, and their uses, it is necessary for him to know the present state of thermodynamic theory and be aware of sources of information on the fundamental background of classical and statistical thermodynamics. This Chapter does not aim to give a short course in thermodynamics nor does it review in detail the history of the foundation of the subject with which such names as Joseph Black (1728—-99), Count Rumford (1753—1814), Sadi Carnot (1796—1832), James Joule (1818—89), and Lord Kelvin (1824—1907) are associated. 

As already mentioned, this book is primarily written for students of physics and physicists wishing to enter into polymer science for the first time. Interested macromolecular chemists and chemical engineers may also find it useful. The prerequisite for an understanding is not a special one, all that is needed is a background in phenomenological and statistical thermodynamics on the level of the respective courses in physical chemistry, together with the related mathematical knowledge. 

The formalism of the thermodynamics of solutions was described in Chapter 3. In this chapter we shall revisit the topic of solutions and apply statistical mechanics to relate the thermodynamic properties of solutions to atomistic models for their structure. Although we will not give a rigorous presentation of the methods of statistical mechanics, we need some elements of the theory as a background for the solution models to be treated. These elements of the theory are presented in Section 9.1. 

S. Adair, H. S. Sinuns, K. Linderstrom-Lang, and, especially, J. Wyman. These treatments, however, were empirical or thermodynamic in content, that is, expressed from the outset in terms of thermodynamic equilibrium constants. The advantage of the explicit use of the actual grand partition function is that it is more general it includes everything in the empirical or thermodynamic approach, plus providing, when needed, the background molecular theory (as statistical mechanics always does). 

In 1923, Peter Debye and Erich Hiickel developed a classical electrostatic theory of ionic distributions in dilute electrolyte solutions [P. Debye and E. Hiickel. Phys. Z 24, 185 (1923)] that seems to account satisfactorily for the qualitative low-ra nonideality shown in Fig. 8.3. Although this theory involves some background in statistical mechanics and electrostatics that is not assumed elsewhere in this book, we briefly sketch the physical assumptions and mathematical techniques leading to the Debye-Hiickel equation (8.69) to illustrate such molecular-level description of thermodynamic relationships. 

See, e.g., J. M. Seddon and J. D. Gale, Thermodynamics and Statistical Mechanics (The Royal Society of Chemistry, London 2001), for friendly background reading. 

Much background information about the notion of interfacial tension, its thermodynamical and statistical interpretation and a number of other aspects have already been dealt with in Volumes I and II We start by briefly reviewing these. 

In this article we shall focus on recent work involving dilute aqueous surfactant solutions. As a background the thermodynamics and statistics of these solutions will be discussed first (Section II). The distribution of substrate molecules in microheterogeneous solution is considered in Section m. It is decisive for the kinetics of elementary photochemical reactions (Section IV), which depend on the peculiar colloidal solution structure. Effects of the microscopic environments on photochemical reactions are treated in Section V. Finally, the use of known photochemical systems as probes for studying details of the structure of surfactant solutions will be considered in Section VI. 

This chapter is devoted to the presentation of the computational methods used for the determination of the thermodynamic and the kinetic data of the compound considered in this work. We give in a first part a theoretical background of the methods used in the present work Electronic Structure Theory (ah initio, and Density Functional Theory), Statistical Mechanics theory. Group Additivity method, and multifrequency Quantum Rice-Ramsperger-Kassel theory (QRRK). 

---