Statistical thermodynamics defined

For a system consisting of the total number of particles N and maintaining its total energy U and volume V constant, statistical thermodynamics defines the entropy, S, in terms of the logarithm of the total number of microscopic energy distribution states Q N,V,U) in the system as shown in Eq. 3.6 ... 

The treatment of heat capacity in physical chemistry provides an excellent and familiar example of the relationship between pure and statistical thermodynamics. Heat capacity is defined experimentally and is measured by determining the heat required to change the temperature of a sample in, say,... 

For a calculation of d. see R- H. Fowler. Statistical Thermodynamics. Second Edition, Cambridge University Press. 1956. p. 127. In Section 1.5a of Chapter 1 we defined the compressibility and cautioned that this compressibility can be applied rigorously only for gases, liquids, and isotropic solids. For anisotropic solids where the effect of pressure on the volume would not be the same in the three perpendicular directions, more sophisticated relationships are required. Poisson s ratio is the ratio of the strain of the transverse contraction to the strain of the parallel elongation when a rod is stretched by forces applied at the end of the rod in parallel with its axis. 

Thermodynamics describes the behaviour of systems in terms of quantities and functions of state, but cannot express these quantities in terms of model concepts and assumptions on the structure of the system, inter-molecular forces, etc. This is also true of the activity coefficients thermodynamics defines these quantities and gives their dependence on the temperature, pressure and composition, but cannot interpret them from the point of view of intermolecular interactions. Every theoretical expression of the activity coefficients as a function of the composition of the solution is necessarily based on extrathermodynamic, mainly statistical concepts. This approach makes it possible to elaborate quantitatively the theory of individual activity coefficients. Their values are of paramount importance, for example, for operational definition of the pH and its potentiometric determination (Section 3.3.2), for potentiometric measurement with ion-selective electrodes (Section 6.3), in general for all the systems where liquid junctions appear (Section 2.5.3), etc. 

Thus far we have explored the field of classical thermodynamics. As mentioned previously, this field describes large systems consisting of billions of molecules. The understanding that we gain from thermodynamics allows us to predict whether or not a reaction will occur, the amount of heat that will be generated, the equilibrium position of the reaction, and ways to drive a reaction to produce higher yields. This otherwise powerful tool does not allow us to accurately describe events at a molecular scale. It is at the molecular scale that we can explore mechanisms and reaction rates. Events at the molecular scale are defined by what occurs at the atomic and subatomic scale. What we need is a way to connect these different scales into a cohesive picture so that we can describe everything about a system. The field that connects the atomic and molecular descriptions of matter with thermodynamics is known as statistical thermodynamics. 

Statistical thermodynamics is based on a statistical interpretation of how atoms and molecules behave. This statistical nature arises because we have so many atoms and molecules in systems and because matter is intrinsically defined based on probabilities, which is the crux of all quantum mechanics. Rather than delve into the great details of statistical thermodynamics, which would far exceed the scope of this text, we will present its foundations only. 

The molecular modelling of systems consisting of many molecules is the field of statistical mechanics, sometimes called statistical thermodynamics [28,29], Basically, the idea is to go from a molecular model to partition functions, and then, from these, to predict thermodynamic observables and dynamic and structural quantities. As in classical thermodynamics, in statistical mechanics it is essential to define which state variables are fixed and which quantities are allowed to fluctuate, i.e. it is essential to specify the macroscopic boundary conditions. In the present context, there are a few types of molecular systems of interest, which are linked to so-called ensembles. 

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... 

The question raised by Anderson (1970,1971) and Anderson et al (1973) as to whether anion point defects are eliminated completely by the creation of extended CS plane defects, is a very important one. This is because anion point defects can be hardly eliminated totally because apart from statistical thermodynamics considerations they must be involved in diffusion process. Oxygen isotope exchange experiments indeed suggest that oxygen diffuses readily by vacancy mechanism. In many oxides it is difficult to compare small anion deficiency with the extent of extended defects and in doped complex oxides there is a very real discrepancy between the area of CS plane present which defines the number of oxygen sites eliminated and the oxygen deficit in the sample (Anderson 1970, Anderson et al 1973). We attempt to address these issues and elucidate the role of anion point defects in oxides in oxidation catalysis (chapter 3). 

As noted previously in Chapters 3 and 10, statistical thermodynamics relates all thermodynamic observables to the partition function Q. For ease of reference, the definition of Q and the equations defining various thermodynamic variables as a function of Q, some of which have appeared previously, are as follows... 

Elementary and advanced treatments of such cellular functions are available in specialized monographs and textbooks (Bergethon and Simons 1990 Levitan and Kaczmarek 1991 Nossal and Lecar 1991). One of our objectives in this chapter is to develop the concepts necessary for understanding the Donnan equilibrium and osmotic pressure effects. We define osmotic pressures of charged and uncharged solutes, develop the classical and statistical thermodynamic principles needed to quantify them, discuss some quantitative details of the Donnan equilibrium, and outline some applications. 

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. 

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... 

Having determined the magnetic energy levels eafiB) (as eigenvalues of the interaction Hamiltonian) we can proceed with the apparatus of the statistical thermodynamic by defining the (magnetic) partition function... 

From a statistical thermodynamic standpoint, the description of the folding/unfolding equilibrium in proteins requires the specification of the system partition function, Q defined as the sum of the statistical weights of all the possible states of the molecule (see Freire and Biltonen, 1978a) ... 

Statistical thermodynamics has defined, in addition to the particle partition function z, the canonical ensemble partition function Zas follows ... 

Much more can be said about the magnitude of pre-exponcntial factors and activation energies of elementary processes based on statistical thermodynamics applied to collision and reaction-rate theory [2, 61], but in view of the remark above one should be cautious in their application and limit it to well-defined model reactions and catalyst surfaces. 

The statistical thermodynamic treatment of the BET theory has the advantage that it provides a satisfactory basis for further refinement of the theory by, say, allowing for adsorbate-adsorbate interactions or the effects of surface heterogeneity. By making the assumptions outlined above, Steele (1974) has shown that the problems of evaluating the grand partition function for the adsorbed phase could be readily solved. In this manner, he arrived at an isotherm equation, which has the same mathematical form as Equation (4.32). The parameter C is now defined as the ratio of the molecular partition functions for molecules in the first layer and the liquid state. 

Equation (7.8) offers a clear separation of inner-shell and outer-shell contributions so that different physical approximations might be used in these different regions, and then matched. The description of inner-shell interactions will depend on access to the equilibrium constants K. These are well defined, observationally and computationally (see Eq. (7.10)), and so might be the subject of either experiments or statistical thermodynamic computations. Eor simple solutes, such as the Li ion, ab initio calculations can be carried out to obtain approximately the Kn (Pratt and Rempe, 1999 Rempe et al, 2000 Rempe and Pratt, 2001), on the basis of Eq. (2.8), p. 25. With definite quantitative values for these coefficients, the inner-shell contribution in Eq. (7.8) appears just as a function involving the composition of the defined inner shell. We note that the net result of dividing the excess chemical potential in Eq. (7.8) into inner-shell and outer-shell contributions should not depend on the specifics of that division. This requirement can provide a variational check that the accumulated approximations are well matched. 

These comparisons teach us about the performance of this simplest physical theory. An important point is how the iimer shell should be defined to make reasonable statistical thermodynamic predictions. As with the K" (aq) case of Fig. 8.15, a naive eyeball analysis of a radial distribution function might not be the wisest for this assignment. On physical groimds, it has been argued that the inner-shell volume should be chosen aggressively small so that subsequent approximations such as a harmonic approximation for the optimized structure have the best chance of being valid (Pratt and Rempe, 1999). But the discussion of Section 7.4, p. 153, pointed out that this question has a variational answer - see Fig. 7.6,... 

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. 

Statistical thermodynamics of the electric double layer starts with modelling the electrolyte and the Interface. This can be done by specifying all inter-molecular and external Interactions in the phase space as a Hamiltonian. The notion of phase space was defined in sec. 1.3.9a and the Hamiltonian H was introduced in [1.3.9.11. As the kinetic part of the Hamiltonian does not contribute to the configuration Integrals, we sum only over the potential energies of the ions. In the Inhomogeneous system it Is customary to separate the interactions with the charged wall (the external" field) from the interlonlc ones. 

Van der Waals himself defined C through (2.5.141. The modem way is by statistical thermodynamics. We described the principles in chapter 1.3. For the interpretation of C the formalism of distribution functions (sec. 1.3.9) is the most appropriate. In sec. 2.4 we already gave an ab initio derivation for the surface tension, but now concentrate on the interpretation of C. According to van der Waals, for a homogeneous bulk phase... 

The third law, like the two laws that precede it, is a macroscopic law based on experimental measurements. It is consistent with the microscopic interpretation of the entropy presented in Section 13.2. From quantum mechanics and statistical thermodynamics, we know that the number of microstates available to a substance at equilibrium falls rapidly toward one as the temperature approaches absolute zero. Therefore, the absolute entropy defined as In O should approach zero. The third law states that the entropy of a substance in its equilibrium state approaches zero at 0 K. In practice, equilibrium may be difficult to achieve at low temperatures, because particle motion becomes very slow. In solid CO, molecules remain randomly oriented (CO or OC) as the crystal is cooled, even though in the equilibrium state at low temperatures, each molecule would have a definite orientation. Because a molecule reorients slowly at low temperatures, such a crystal may not reach its equilibrium state in a measurable period. A nonzero entropy measured at low temperatures indicates that the system is not in equilibrium. 

In order to determine the statistical thermodynamic probabilities and entropies for the conformational energy surface, a set of "dots" is plotted indicating the angular values of the set of conformers which define the surface. The joystick curser control 1s used to select the set of conformers which occupy a given low energy region. The chosen "dots" are replaced by "asterisks" (to avoid duplication) and the probability and entropy terms are tabulated. Tables of probabilities and entropies may also be produced. 

The factor defined by this, > 0, is the volume viscosity (SI units kg/sm). It has to be determined either experimentally or using methods of statistical thermodynamics, which is only possible for substances with simple molecules. It can be seen that the mean and thermodynamic pressures only strictly agree when C = 0 or the fluid is incompressible, dwi/dxi = 0. 

Statistical thermodynamics uses statistical arguments to develop a connection between the properties of individual molecules in a system and its bulk thermodynamic properties. For instance, consider a mole of water molecules at 25° C and standard pressure (1 bar). The thermodynamic state of the system has been defined on the basis of the number of molecules, the temperature, and the pressure. In order to relate the macroscopic thermodynamic properties such as U, G, H and A to the properties of the individual molecules, one would have to solve the Schrodinger wave equation (SWE) for a system composed of 6 x 10 interacting water molecules. This is an impossible task at present but if it were possible, one would obtain a wave function, I y, and an energy, 6)-, for the system. Moreover,... 

Three other important equations in the statistical thermodynamics of liquids involve the direct correlation function and provide a connection between the intermolecular potential u(r) for two molecules and the potential of mean force W(r). One way of deriving these equations is by writing an approximate expression for the indirect contribution to the pair correlation function g(r). Keeping in mind that W(r) is defined as — lng(r)/p, and defining gind( ) as the contribution to g(r) from indirect interactions, an approximate expression for c(r) is... 

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