Classical statistical mechanics thermodynamic functions

The evaluation of the free energy is essential to quantitatively treat a chemical process in condensed phase. In this section, we review methods of free-energy calculation within the context of classical statistical mechanics. We start with the standard free-energy perturbation and thermodynamic integration methods. We then introduce the method of distribution functions in solution. The method of energy representation is described in its classical form in this section, and is combined with the QM/MM methodology in the next section. 

Thermodynamic effects of directional forces in liquid mixtures.— The theory applied to pure liquids in the last two sections can be generalized to liquid mixtures and can be used to discuss the effects of directional forces on the thermodynamic functions of mixing. Classical statistical mechanics leads to a complete expression for the free energy of a multicomponent system in terms of the intermolecular energies Ust for all pairs of components s and t. Each Ust can be expanded in the general manner (2.1), so that it is separated into a spherically symmetric part and various directional terms. 

Phase transitions in statistical mechanical calculations arise only in the thermodynamic limit, in which the volume of the system and the number of particles go to infinity with fixed density. Only in this limit the free energy, or any thermodynamic quantity, is a singular function of the temperature or external fields. However, real experimental systems are finite and certainly exhibit phase transitions marked by apparently singular thermodynamic quantities. Finite-size scaling (FSS), which was formulated by Fisher [22] in 1971 and further developed by a number of authors (see Refs. 23-25 and references therein), has been used in order to extrapolate the information available from a finite system to the thermodynamic limit. Finite-size scaling in classical statistical mechanics has been reviewed in a number of excellent review chapters [22-24] and is not the subject of this review chapter. 

The classical translational and rotational contributions to thermodynamic functions must be corrected by choosing the correct divisors. This yields the same results as in quantum statistical mechanics. The classical formulas for vibration are numerically inadequate, even with the correct divisors, and we do not attempt to use classical statistical mechanics for electronic motion. There is nothing to be gained by using classical statistical mechanics for a dilute gas. 

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. 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

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. 

Applications in statistical mechanics are based on constructing expressions for Q(N, V, E) (and other partition functions for various ensembles) based on the nature of the interactions of the particles in a given system. To understand how thermodynamic principles arise from statistics, however, it is not necessary to worry about how one might go about computing Q(N, V, E), or how Q might depend on N, V, and E for particular systems (classical or quantum mechanical). It is necessary simply to appreciate that the quantity Q(N, V, E) exists for an NVE system. 

It is evident that Equation (4.11) is of a very general mathematical form (i.e. a hyperbolic function). At low 6 it reduces to Henry s law at high surface coverage, a plateau is reached as 6—>1. Other equations of the same mathematical form as Equation (4.11) have been derived from a classical thermodynamic standpoint (Brunauer, 1945) and by application of the principles of statistical mechanics (Fowler, 1935). 

H-function in the statistical mechanics of molecular collision, and the excess entropy (S — Seq) for adiabatic systems in classical thermodynamics. 

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. 

In Section 2.5.3 we derived the semiclassic expression for the ceinonical partition function [see Eq. (2.110)] based on the assumption that at sufficiently high temperatures we may replace the Hamiltonian operator by its classic analog, the Hamiltonian function [see Elq. (2.100)]. In this section we will sketch a more refined treatment of the semiclassic theory developed in Section 2.5 originally due to Hill and presented in detail in his classical work on stati.stical mechanics [326]. Because of Hill s clear and detailed exposition and because we need the final result mainly as a justification to treat confined fluids by means of classic statistical thermodynamics, we will just briefly outhne the key ideas of Hill s treatment for reasons of completeness of the current work. 

In this chapter, we first present some of the notation that we shall use throughout the book. Then we summarize the most important relationship between the various partition functions and thermodynamic functions. We shall also present some fundamental results for an ideal-gas system and small deviations from ideal gases. These are classical results which can be found in any textbook on statistical thermodynamics. Therefore, we shall be very brief. Some suggested references on thermodynamics and statistical mechanics are given at the end of the chapter. 

In section 1.2, we introduced the quantum mechanical partition function in the T, V, N ensemble. In most applications of statistical thermodynamics to problems in chemistry and biochemistry, the classical limit of the quantum mechanical partition function is used. In this section, we present the so-called classical canonical partition function. 

The fundamental problem in classical equilibrium statistical mechanics is to evaluate the partition function. Once this is done, we can calculate all the thermodynamic quantities, as these are typically first and second partial derivatives of the partition function. Except for very simple model systems, this is an unsolved problem. In the theory of gases and liquids, the partition function is rarely mentioned. The reason for this is that the evaluation of the partition function can be replaced by the evaluation of the grand canonical correlation functions. Using this approach, and the assumption that the potential energy of the system can be written as a sum of pair potentials, the evaluation of the partition function is equivalent to the calculation of... 

In describing thermodynamic and equilibrium statistical-mechanical behaviors of a classical fluid, we often make use of a radial distribution function g r). The latter for a fluid of N particles in volume V expresses a local number density of particles situated at distance r from a fixed particle divided by an average number density p = NjV), when the order of IjN is negligible in comparison with 1. Various thermodynamic quantities are related to g(r). For a single-component monatomic system of particles interacting with a pairwise additive potential 0(r), the relationship connecting the pressure P to g(r) is the virial theorem, ... 

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p s). This corresponds to a set of systems separated by diathermic and permeable walls and allowed to equilibrate. In classical thermodynamics, the appropriate function for fixed p, V, and T is the productpV (see equation (A2.1.37)) and statistical mechanics relates pV directly to the grand canonical partition function ... 

The partition function and the sum or density of states are functions which are to statistical mechanics what the wave function is to quantum mechanics. Once they are known, all of the thermodynamic quantities of interest can be calculated. It is instructive to compare these two functions because they are closely related. Both provide a measure of the number of states in a system. The partition function is a quantity that is appropriate for thermal systems at a given temperature (canonical ensemble), whereas the sum and density of states are equivalent functions for systems at constant energy (microcanonical ensemble). In order to lay the groundwork for an understanding of these two functions as well as a number of other topics in the theory of unimolecular reactions, it is essential to review some basic ideas from classical and quantum statistical mechanics. 

In the classical free statistics, the number of the functional groups on the surface of a tree-like cluster is of the same order of that of the groups inside the cluster, so that a simple thermodynamic limit without surface term is impossible to take. The equilibrium statistical mechanics for the polycondensation was refined by Yan [14] to treat surface correction in such finite systems. He found the same result as Ziff and Stell. Thus the treatment of the postgel regime is not unique. The rigorous treatment of the problem requires at least one additional parameter defining relative probability of occurrence of infra- and intermolecular reactions in the gel. 

The whole of classical thermodynamics is based on the empirical laws which lead to the notions of temperature, internal energy and entropy as functions of state. At no point in the development of these laws, or of their consequences, does any reference need to be made to the idea that matter consists of ultimate particles, i.e. the atomic theory. But thermodynamics gives an insight neither into the possible origin of its laws nor any means of making a direct calculation of the thermodynamic properties of a substance. It is the purpose of statistical mechanics to advance knowledge in both of these directions. 

A rigorous interpretation is provided by the discipline of statistical mechanics, which derives a precise expression for entropy based on the behavior of macroscopic amounts of microscopic particles. Suppose we focus our attention on a particular macroscopic equilibrium state. Over a period of time, while the system is in this equilibrium state, the system at each instant is in a microstate, or stationary quantum state, with a definite energy. The microstate is one that is accessible to the system—that is, one whose wave function is compatible with the system s volume and with any other conditions and constraints imposed on the system. The system, while in the equilibrium state, continually jumps from one accessible microstate to another, and the macroscopic state functions described by classical thermodynamics are time averages of these microstates. 

Parametrization of the thermodynamic properties of pure electrolytes has been obtained [18] with use of density-dependent average diameter and dielectric parameter. Both are ways of including effects originating from the solvent, which do not exist in the primitive model. Obviously, they are not equivalent and they can be extracted from basic statistical mechanics arguments it has been shown [19] that, for a given repulsive potential, the equivalent hard core diameters are functions of the density and temperature Adelman has formally shown [20] (Friedman extended his work subsequently [21]) that deviations from pairwise additivity in the potential of average force between ions result in a dielectric parameter that is ion concentration dependent. Lastly, there is experimental evidence [22] for being a function of concentration. There are two important thermodynamic quantities that are commonly used to assess departures from ideality of solutions the osmotic coefficient and activity coefficients. The first coefficient refers to the thermodynamic properties of the solvent while the second one refers to the solute, provided that the reference state is the infinitely dilute solution. These quantities are classic and the reader is referred to other books for their definition [1, 4],... 

Hargreaves book, which is primarily intended for Higher National Certificate and Bachelor of Science students, presents thermodynamic functions in a pictorial way. Mahan s elementary book is clearly written and gives a classical account of thermodynamic laws, with entropy introduced as a macroscopic quantity. Jancel s book, on the other hand, is highly mathematical and will probably be of interest only to those concerned with the foundations of statistical mechanics. 

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