Statistical thermodynamics state functions

For book-keeping purposes the production of entropy during chemical change is considered as reducing the useful energy of the system by disorderly dispersion. In many cases this waste can be calculated statistically from the increase in disorder. To be in line with other thermodynamic state functions, any system is considered to be in some state of disorder at all temperatures above absolute zero, where entropy vanishes. 

In Section 2.1, we remarked that classical thermodynamics does not offer us a means of determining absolute values of thermodynamic state functions. Fortunately, first-principles (FP), or ab initio, methods based on the density-functional theory (DFT) provide a way of calculating thermodynamic properties at 0 K, where one can normally neglect zero-point vibrations. At finite temperatures, vibrational contributions must be added to the zero-kelvin DFT results. To understand how ab initio thermodynamics (not to be confused with the term computational thermochemistry used in Section 2.1) is possible, we first need to discuss the statistical mechanical interpretation of absolute internal energy, so that we can relate it to concepts from ab initio methods. 

We will begin with a necessary (but nonchemical) review of some statistics that we later apply to gaseous systems. (We use gases almost exclusively in our discussion of statistical thermodynamics.) We will see how we can separate, or partition, a system into smaller units and define an important quantity called a partition function. In time, we will see that the partition function is related to the thermodynamic state functions that define our system. 

Statistical thermodynamics therefore gives expressions for all of the basic thermodynamic state functions, and they all depend on the partition function q. 

The importance of the partition function in statistical thermodynamics is that if we know q, we can determine thermodynamic properties. Indeed, almost all of the thermodynamic state functions can be written in terms of the change of the partition function as some state variable, T or V, changes. (Only A and G depend directly on q, and on the natural logarithm of q at that This fact does not obviate the discussion to follow.)... 

Chapter 17 introduced some of the basic concepts that led to the development of a statistical approach to energy and entropy. This is statistical thermodynamics. By the end of the chapter, equations were applied to monatomic gases, and thermodynamic state functions—mostly entropy—were calculated whose values were very close to experimental values. Also, in some of the exercises you were asked to derive some simple expressions that were also derived from phenomenological thermodynamics. For example, we know from early chapters in this book that the equation AS = i In (V2/V1) is applicable for an isothermal change in volume of an ideal gas. We can also get this expression using the Sackur-Tetrode statistical thermodynamic expression for S. These correspondences are just two examples where phenomenological and statistical thermodynamics are consistent with each other. That is, they ultimately make the same predictions about the state functions of a system, and how they change with a process. 

In thermodynamic integrations and multiple-step thermodynamic perturbations it is necessary to define a path that connects the initial and final states. This is done by formulating a A-dependent Hamiltonian such that 3. = 0 and k = 1 define the initial and final states, respectively. If only the free energy difference is needed, any path can be taken since the free energy is a thermodynamic state function. In practice, however, calculated systematic and statistical errors will depend on the particular path chosen. 

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. 

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

To understand how collision theory has been derived, we need to know the velocity distribution of molecules at a given temperature, as it is given by the Maxwell-Boltzmann distribution. To use transition state theory we need the partition functions that follow from the Boltzmann distribution. Hence, we must devote a section of this chapter to statistical thermodynamics. 

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. 

The density of states is the central function in statistical thermodynamics, and provides the key link between the microscopic states of a system and its macroscopic, observable properties. In systems with continuous degrees of freedom, the correct treatment of this function is not as straightforward as in lattice systems - we, therefore, present a brief discussion of its subtleties later. The section closes with a short description of the microcanonical MC simulation method, which demonstrates the properties of continuum density of states functions. 

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. 

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. 

The partition function is the central feature of statistical thermodynamics. From the partition function the various thermodynamic variables such as entropy, enthalpy, and free energy may be evaluated. It is also possible, in principle, to deduce the equation of state for a system from the partition function. 

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. 

To a chemist the entropy of a system is a macroscopic state function, i.e., a function of the thermodynamic variables of the system. In statistical mechanics, entropy is a mesoscopic quantity, i.e., a functional of the probability distribution, viz., the functional given by (V.5.6) and (V.5.7). It is never a microscopic quantity, because on the microscopic level there is no irreversibility. ... 

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. 

In some polymer-nonpolar solvent systems, % has been calculated as a function of concentration on the basis of the statistical-thermodynamical theory called the equation of state theory [13,14]. This semiempirical theory takes into account not only the interaction between solute and solvent, but also the characteristics of pure substances through the equations of state of each component. At present, however, we cannot apply this approach to such a complex case as the NIPA-water system. Thus, at the present stage, we must regard % as an empirical parameter to be determined through a comparison between calculated and experimental results. The empirical estimation of % for the NIPA-water system will be described in the next section. 

Thermodynamics is based on the atomistic view, that is, that matter consists of elementary particles such as atoms and molecules that cannot be divided into smaller units. The three different states of matter are the result of the simultaneous interaction of a very large number, usually N = Na =6.02x 1023, of elementary particles. Thus, the macroscopic behavior of an ensemble of particles can be mathematically described as a state function that can be related to the individual behavior on a molecular scale, leading to the scientiLcally rigorous framework of statistical thermodynamics (Gcpel and Wiemhcfer, 2000). 

To illuminate the binding situation we can retreat to the statistical thermodynamics point of view. Experimentally accessible state functions (AG, AH, AS, ACp etc.) relate to the differences of average states in the ensembles of all constituents of the system including counterions, buffers, solvents etc. before... 

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) ... 

It has been shown that under standard conditions most singledomain globular proteins exhibit a folding/unfolding behavior consistent with the two-state mechanism (Freire and Biltonen, 1978a Privalov, 1979). From a statistical thermodynamic standpoint the implication is that the population of partially folded intermediate states is negligible and the partition function reduces to two terms ... 

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