Statistical-mechanic

Statistical mechanics is the branch of physical science that studies properties of macroscopic systems from the microscopic starting point. For definiteness we focus on the dynamics ofan A-particle system as our underlying microscopic description. In classical mechanics the set of coordinates and momenta, (r, p ) represents a state of the system, and the microscopic representation of observables is provided by the dynamical variables, v4(r, p, Z). The equivalent quantum mechanical objects are the quantum state [/ ofthe system and the associated expectation value Aj = /l l/ of the operator that corresponds to the classical variable A. The corresponding observables can be thought of as time averages 

For each of these ensembles of Af systems let fj(M) be the fraction of systems occupying a given microscopic state j. The ensemble probability Pj is defined by Pj = lim// ooj (Af). The macroscopic observable that corresponds to the dynamical variable A is then 

In the grand canonical formulation the sum over j should be taken to include also a sum over number of particles. 

The probability that a system is found in a state of energy Ej is given by 1 

One thing that should be appreciated about the density of states of a macroscopic 

Statistical mechanics is relevant to problems in colloid science at two levels. At the molecular level one is concerned in particular with statistical-mechanical theories of electrolyte solutions, the 

The original theory of double-layer interaction dealt with the problem of the repulsion between two isolated, charged colloidal particles and could, strictly, be applied only to very dilute dispersions. More realistic theories must handle concentrated systems where the interaction between many particles has to be considered. This more complex situation has been, and will continue to be, the subject of important theoretical studies. 

The application of statistical theory to kinetic phenomena such as Brownian motion and rheology is also a field in which renewed activity is developing. 

In many of these areas progress has only become possible through the availability of powerful high-speed computers, which have enabled work to be done that thirty years ago would have been quite impracticable. 

As already indicated in Chapter 7, the introduction of laser technology has already had a major impact on light-scattering methods. These have found particular application in the development of new methods of particle sizing, and several instruments are now available commercially which arc designed for the automatic determination of particle size distributions. These methods are being developed steadily, especially in terms of the associated computer software needed for the rapid analysis of experimental data. In particular, while the measurement of the particle size in monodisperse systems is well established, the mathematical analysis for polydisperse systems and for non-spherical particles presents problems which are not yet fully solved. 

One thing that should be appreciated about the density of states of a macroscopic system is how huge it is. For a system of N structureless (i.e. no internal states) particles of mass m confined to a volume but otherwise moving freely it is given by  

Statistical mechanics is the mathematical means to calculate the thermodynamic properties of bulk materials from a molecular description of the materials. Much of statistical mechanics is still at the paper-and-pendl stage of theory. Since quantum mechanicians cannot exactly solve the Schrodinger equation yet, statistical mechanicians do not really have even a starting point for a truly rigorous treatment. In spite of this limitation, some very useful results for bulk materials can be obtained. 

Statistical mechanics computations are often tacked onto the end of ab initio vibrational frequency calculations for gas-phase properties at low pressure. For condensed-phase properties, often molecular dynamics or Monte Carlo calculations are necessary in order to obtain statistical data. The following are the principles that make this possible. 

Consider a quantity of some liquid, say, a drop of water, that is composed of N individual molecules. To describe the geometry of this system if we assume the molecules are rigid, each molecule must be described by six numbers three to give its position and three to describe its rotational orientation. This 6N-dimensional space is called phase space. Dynamical calculations must additionally maintain a list of velocities. 

An individual point in phase space, denoted by F, corresponds to a particular geometry of all the molecules in the system. There are many points in this phase space that will never occur in any real system, such as configurations with two atoms in the same place. In order to describe a real system, it is necessary to determine what configurations could occur and the probability of their occurrence. 

The probability of a configuration occurring is a function of the energy of that configuration. This energy is the sum of the potential energy from inter- 

The connection between the quantum mechanical treatment of individual atoms and molecules and macroscopic properties and phenomena is the goal of statistical mechanical analysis. Statistical mechanics is the means for averaging contributions to properties and to energies over a large collection of atoms and molecules. The first part of the analysis is directed to the distribution of particles among available quantum states. An outcome of this analysis is the partition function, which proves to be an essential element in thermodynamics, in reaction kinetics, and in the intensity information of molecular spectra. 

Essential to any discussion of probability is its definition. If there are N mutually exclusive and equally likely occurrences, and and only of them lead to some particular result. A, the mathematical probability of A is the ratio n /N. A way to remember this is to think about flipping a coin. There are two mutually exclusive outcomes, heads and tails, and so N = 2. One particular result is tails, and since it is but one of the outcomes, = 1. The probability of tails is, therefore, 1/2. Next, we consider two theorems about relations among probabilities. 

Theorem 11.1 Probabilities of Mutually Exclusive Events Are Additive 

This follows from the definition of probability. If the probability of some event A is Uy /N and the probability of event B is nJN, the probability that A or B will occur is the sum of the occurrences leading to A or leading to B divided by N, that is, + Wb)/ / the sum of the probabilities. 

Theorem 11.2 The Probability of Simultaneous Occurrence of Unrelated Events Is the Product of the Independent Probabilities of the Events 

The Gibbs free energy is given in terms of the enthalpy and entropy, G — H — TS. The enthalpy and entropy for a macroscopic ensemble of particles may be calculated from properties of the individual molecules by means of statistical mechanics. 

The key feature in statistical mechanics is the partition function Just as the wave function is the corner-stone of quantum mechanics (from that everything else can be calculated by applying proper operators), the partition function allows calculation of alt macroscopic functions in statistical mechanics. The partition function for a single molecule is usually denoted q and defined as a sum of exponential terms involving all possible quantum energy states Q is the partition function for N molecules. 

The partition function may alternatively be written as a sum over all distinct energy levels, times a degeneracy factor g,-. 

Once the partition function is known, thermodynamic functions such as the internal energy U and Helmholtz free energy A may be calculated according to 

Macroscopic observables, such as pressme P or heat capacity at constant volume C v, may be calculated as derivatives of thermodynamic functions. 

It is one of the wonders of the history of physics that a rigorous theory of the behaviour of a chaotic assembly of molecules - a gas - preceded by several decades the experimental uncovering of the structure of regular, crystalline solids. Attempts to create a kinetic theory of gases go all the way back to the Swiss mathematician, Daniel Bernouilli, in 1738, followed by John Herapath in 1820 and John James Waterston in 1845. But it fell to the great James Clerk Maxwell in the 1860s to take 

Four ensembles (A) microcanonical (/rC ). (B) canonical CE). (C) grand canonical GCE). (D) isothermal-isobaric (HE). They are all isolated from the surroundings, but differ among themselves in what can migrate between systems nothing, energy, heat, or molecules. Inspired by Moore [1]. 

For instance, the microcanonical ensemble (fiCE) (Fig. 5.1(A)) consists of an infinite set of replicas of a system with fixed number of particles N, fixed 

5 Lorenzo Romano Amedeo Carlo Bernadette Avogadro, conte di Quaregna e Cerreto (1776-1856). 

PROBLEM 5.1.1. Given the requirement of overall antisymmetry, show that for a system of identical fermions each fermion must have its unique set of quantum number values, different from the values adopted by any other fermion. 

PROBLEM 5.1.2. In a system of identical bosons, any particle can have the same set of quantum number values, as any other particle. 

From the TST expression (12.2) it is clear that if the free energy of the reactant and TS 

Most experiments are performed on macroscopic samples, containing perhaps 10 ° particles. Calculations, on the other hand, are performed on relatively few particles, typically 1-10, or up to 10 in special cases. The (macroscopic) result of an experimental measurement can be connected with properties of the microscopic system. The temperature, for example, is related to the average kinetic energy of the particles. 

The connection between properties of a microscopic system and a macroscopic sample is provided by statistical mechanics. 

At a temperature of OK, all molecules are in their energetic ground state but at a finite temperature there is a distribution of molecules in all possible (quantum) energy states. The relative probability P of a molecule being in a state with an energy e at a temperature T is given by a Boltzmann factor. 

The exponential dependence on the energy means that there is a low (but non-zero) probability for finding a molecule in a high-energy state. This decreased probability for high-energy states is partly offset by the fact that there are many more states with high 

The standard entropy and the molar heat capacity at constant volume of a molecule or a free radical are the sums of the translational (tr), external rotational (rot), vibrational (vib) and electronic (el) terms  

The molar heat capacity at constant pressure can be calculated using the following relationship  

In 1662, Robert Boyle discovered that the volume (V) of a gas in a closed container is inversely proportional to its pressure (P), as long as the temperature (T) is constant. Much later (1802), when temperature could be accurately measured, Joseph-Louis Gay-Lussac was able to show that the constant in Boyle s law (PV = constant) is proportional to the temperature. Sometime later, Benoit Clapeyron wrote the general gas law  

Sections 5.2-5.4 contain a simplified derivation of statistical thermodynamics. There are three fundamental laws. Energy exists in various forms, for example, as heat energy (q) or mechanical work energy (w). The principle of conservation of energy states that the total energy U = q -I- w is constant, but q and w can be transferred into each other. 

The third law of thermodynamics states that entropy is measured as zero at absolute temperature equal to zero. This law follows automatically in statistical mechanics. 

The last section of this chapter treats nonequilibrium statistical mechanics and includes transport processes and molecular dynamics simulations. This section is important for the following chapters, not least for calculating the rate of chemical reactions, including electron transfer reactions. 

This method was used in early force fields in the 1970s. It worked pretty well, better than any previously existing method. There are, however, some better approximations that can be applied to this type of procedure. The Benson type of method, or the bond energy approach, assumes the additivity of bond energies over a sizable range. But bond energies are made up of component pieces, and, in principle at least, such a broad assumption may not be the best way to proceed. Let us look at this in a little more detail. 

If one looks at a molecule such as heptane, for example, one can add all of the appropriate increments and calculate the heat of formation with acceptable accuracy by the method previously described. But there are a few things that are really not proper about that kind of calculation. Heptane in the gas phase at 25°C (where heats of formation are defined) is actually a complicated mixture (a Boltzmann distribution) of a great many conformations, most of which have different enthalpies and entropies. Additionally, each of these conformations is also a Boltzmann distribution over the possible translational, vibrational, and rotational states. The Benson method works adequately for many cases like this because these statistical mechanical terms can be lumped into the increments and averaged out, and they are not explicitly considered. By adjusting the values of the parameters in Eq. (11.1) or (11.2), much of the resulting error of neglecting the statistical mechanics can be canceled out, or at least minimized, in simple cases. But we would like for this scheme to work for more complex cases too. As the system becomes more complicated, errors tend to cancel out less well. So let us go back and approach this problem in a more proper way. 

In MM4 we differentiate and calculate several different kinds of energies. These include the equilibrium energy (E ) at the bottom of the potential well and also the zero-point energy in the lowest vibrational state. We also calculate the thermal excitation energies at 25°C (and can calculate them for any other desired temperatere). We also explicitly calculate the entropy and hence properly differentiate enthalpy from free energy. All of these things are separately calculated with MM4, so that one may 

This difference between the hypothetical point on the potential surface and the real molecule can be well described by statistical mechanics. With respect to heats of formation, the ways for calculating them (by quantum mechanical methods or by molecular mechanical methods) require either the explicit inclusion of statistical mechanics (the details of this procedure have been spelled out in fulf and will be outlined in the following) or else an implicit inclusion by lumping the statistical effects into the bond energies and hoping for the best, as in the Benson method. 

One of the things that we are most interested in obtaining when we do molecular mechanics calculations (or quantum mechanical calculations, or experimental measurements) on molecules is their conformational energies (and sometimes their thermodynamic functions). These will tell us in detail about the overall molecular structures and energies upon which many properties depend. And with a molecular mechanical model 

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In statistical mechanics (e.g. the theory of specific heats of gases) a degree of freedom means an independent mode of absorbing energy by movement of atoms. Thus a mon-... 

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

The statistical mechanical approach, density functional theory, allows description of the solid-liquid interface based on knowledge of the liquid properties [60, 61], This approach has been applied to the solid-liquid interface for hard spheres where experimental data on colloidal suspensions and theory [62] both indicate 0.6 this... 

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

T. L. Hill, Statistical Mechanics, McGraw-Hill, New York, 1956. 

B. Widom, Structure and Thermodynamics of Interfaces, in Statistical Mechanics and Statistical Methods in Theory and Application, Plenum, New York, 1977, pp. 33-71. 

On compression, a gaseous phase may condense to a liquid-expanded, L phase via a first-order transition. This transition is difficult to study experimentally because of the small film pressures involved and the need to avoid any impurities [76,193]. There is ample evidence that the transition is clearly first-order there are discontinuities in v-a plots, a latent heat of vaporization associated with the transition and two coexisting phases can be seen. Also, fluctuations in the surface potential [194] in the two phase region indicate two-phase coexistence. The general situation is reminiscent of three-dimensional vapor-liquid condensation and can be treated by the two-dimensional van der Waals equation (Eq. Ill-104) [195] or statistical mechanical models [191]. 

MacKenzie and co-workers [79]. Related is a statistical mechanical treatment by Reiss and co-workers [80] (see also Schonhom [81]). 

Density functional theory from statistical mechanics is a means to describe the thermodynamics of the solid phase with information about the fluid [17-19]. In density functional theory, one makes an ansatz about the structure of the solid, usually describing the particle positions by Gaussian distributions around their lattice sites. The free... 

Statistical Thermodynamics of Adsorbates. First, from a thermodynamic or statistical mechanical point of view, the internal energy and entropy of a molecule should be different in the adsorbed state from that in the gaseous state. This is quite apart from the energy of the adsorption bond itself or the entropy associated with confining a molecule to the interfacial region. It is clear, for example, that the adsorbed molecule may lose part or all of its freedom to rotate. 

It is of interest in the present context (and is useful later) to outline the statistical mechanical basis for calculating the energy and entropy that are associated with rotation [66]. According to the Boltzmann principle, the time average energy of a molecule is given by... 

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. 

Although the preceding derivation is the easier to follow, the BET equation also may be derived from statistical mechanics by a procedure similar to that described in the case of the Langmuir equation [41,42]. 

Clearly, it is more desirable somehow to obtain detailed structural information on multilayer films so as perhaps to settle the problem of how properly to construct the potential function. Some attempts have been made to develop statistical mechanical other theoretical treatments of condensed layers in a potential field success has been reasonable (see Refs. 142, 143). 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

The total change d.S can be detennined, as has been seen, by driving the subsystem a back to its initial state, but the separation into dj.S and dj S is sometimes ambiguous. Any statistical mechanical interpretation of the second law requires that, at least for any volume element of macroscopic size, dj.S > 0. However, the total... 

On the other hand, in the theoretical calculations of statistical mechanics, it is frequently more convenient to use volume as an independent variable, so it is important to preserve the general importance of the chemical potential as something more than a quantity GTwhose usefulness is restricted to conditions of constant temperature and pressure. 

In passing one should note that the metliod of expressing the chemical potential is arbitrary. The amount of matter of species in this article, as in most tliemiodynamics books, is expressed by the number of moles nit can, however, be expressed equally well by the number of molecules N. (convenient in statistical mechanics) or by the mass m- (Gibbs original treatment). 

The coefficients B, C, D, etc for each particular gas are tenned its second, third, fourth, etc. vihal coefficients, and are functions of the temperature only. It can be shown, by statistical mechanics, that 5 is a function of the interaction of an isolated pair of molecules, C is a fiinction of the simultaneous interaction of tln-ee molecules, D, of four molecules, etc., a feature suggested by the fomi of equation (A2.1.54). 

Given this experimental result, it is plausible to assume (and is easily shown by statistical mechanics) that the chemical potential of a substance with partial pressure p. in an ideal-gas mixture is equal to that in the one-component ideal gas at pressure p = p. 

It seems appropriate to assume the applicability of equation (A2.1.63) to sufficiently dilute solutions of nonvolatile solutes and, indeed, to electrolyte species. This assumption can be validated by other experimental methods (e.g. by electrochemical measurements) and by statistical mechanical theory. 

For those who are familiar with the statistical mechanical interpretation of entropy, which asserts that at 0 K substances are nonnally restricted to a single quantum state, and hence have zero entropy, it should be pointed out that the conventional thennodynamic zero of entropy is not quite that, since most elements and compounds are mixtures of isotopic species that in principle should separate at 0 K, but of course do not. The thennodynamic entropies reported in tables ignore the entropy of isotopic mixing, and m some cases ignore other complications as well, e.g. ortho- and para-hydrogen. 

Substances at high dilution, e.g. a gas at low pressure or a solute in dilute solution, show simple behaviour. The ideal-gas law and Henry s law for dilute solutions antedate the development of the fonualism of classical themiodynamics. Earlier sections in this article have shown how these experimental laws lead to simple dieniiodynamic equations, but these results are added to therniodynaniics they are not part of the fonualism. Simple molecular theories, even if they are not always recognized as statistical mechanics, e.g. the kinetic theory of gases , make the experimental results seem trivially obvious. 

Any detailed discussion of statistical mechanics would be hiappropriate for this section, especially since other sections (A2.2 and A2.3) treat this in detail. However, a few aspects that relate to classical themiodynaniics deserve brief mention. 

It is customary in statistical mechanics to obtain the average properties of members of an ensemble, an essentially infinite set of systems subject to the same constraints. Of course each of the systems contains the... 

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 diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

At its foundation level, statistical mechanics mvolves some profound and difficult questions which are not fiilly understood, even for systems in equilibrium. At the level of its applications, however, the rules of calculation tliat have been developed over more than a century have been very successfLil. 

Conservation laws at a microscopic level of molecular interactions play an important role. In particular, energy as a conserved variable plays a central role in statistical mechanics. Another important concept for equilibrium systems is the law of detailed balance. Molecular motion can be viewed as a sequence of collisions, each of which is akin to a reaction. Most often it is the momentum, energy and angrilar momentum of each of the constituents that is changed during a collision if the molecular structure is altered, one has a chemical reaction. The law of detailed balance implies that, in equilibrium, the number of each reaction in the forward direction is the same as that in the reverse direction i.e. each microscopic reaction is in equilibrium. This is a consequence of the time reversal syimnetry of mechanics. 

Thus many aspects of statistical mechanics involve techniques appropriate to systems with large N. In this respect, even the non-interacting systems are instructive and lead to non-trivial calculations. The degeneracy fiinction that is considered in this subsection is an essential ingredient of the fonnal and general methods of statistical mechanics. The degeneracy fiinction is often referred to as the density of states. 

We first consider tlnee examples as a prelude to the general discussion of basic statistical mechanics. These are (i) non-mteracting spin-i particles in a magnetic field, (ii) non-interacting point particles in a box,... 

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