Equilibrium statistical mechanics partition function

Equilibrium vapor pressure of bulk liquid Relative pressure, p/p 0 Statistical mechanical partition function Density in molecules/cubic centimeter Gas constant per mole Number of moles of adsorbed gas Number of moles of adsorbent Isosteric heat of adsorption Differential heat of adsorption Isothermal heat of adsorption Adiabatic heat of adsorption... 

The present state in the theory of time-dependent processes in liquids is the following. We know which correlation functions determine the results of certain physical measurements. We also know certain general properties of these correlation functions. However, because of the mathematical complexities of the V-body problem, the direct calculation of the fulltime dependence of these functions is, in general, an extremely difficult affair. This is analogous to the theory of equilibrium properties of liquids. That is, in equilibrium statistical mechanics the equilibrium properties of a system can be found if certain multidimensional integrals involving the system s partition function are evaluated. However, the exact evaluation of these integrals is usually extremely difficult especially for liquids. 

The calculation above for the conditional probability is hard to perform because it is equivalent to the computation of a partition function. However, similarly to tricks in equilibrium statistical mechanics (the free-energy perturbation method [23]), we can compute the ratios of the conditional probabilities for slightly different Hamiltonians. For example, we may compare the diffusion of different ions a sodium ion and a potassium ion permeating through the gramicidin channel [Koneshan Siva and Ron Elber, Protein, Structure Function, and Genetics, in press]. 

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... 

Before we introduce our restricted partition function formalism, we need to review some salient aspects of the equilibrium statistical mechanics formalism. This will also help with the clarity and continuity of presentation. We will restrict our discussion to the canonical ensemble, but the extension to other ensembles is straightforward and trivial. 

Zwanzig s review is still the best entry point into this field for the serious student of theory. As he points out, the preceding formalism resembles equilibrium statistical mechanics, where thermodynamic properties can be obtained from the partition function. The calculation may be hard, but at least one knows where to start. The analogy breaks down in that a different microscopic property is required for each transport property, whereas all equilibrium properties can be obtained from the partition function. This is because the equilibrium state is unique, while there are many different kinds of nonequilibrium state (including those where the formalism breaks down). The statistical mechanician must pick the right variable and develop the relationship. 

Using the formalism of statistical mechanics, Giddings et al. [135] investigated the effects of molecular shape and pore shape on the equilibrium distribution of solutes in pores. The equilibrium partition coefficient is defined as the ratio of the partition function in the pore... 

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. 

In statistical mechanics the properties of a system in equilibrium are calculated from the partition function, which depending on the choice for the ensemble considered involves a sum over different states of the system. In the very popular canonical ensemble, that implies a constant number of particles N, volume V, and temperature T conditions, the quasiclassical partition function Q is... 

This equilibrium constant is often incorrectly called a partition function - which is in fact a term from statistical mechanics. 

Partition Functions and Statistical Mechanics of Chemical Equilibrium... 

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). 

The Langmuir isotherm can be derived from a statistical mechanical point of view. Thus, for the reaction M + Agas Aads, equilibrium is established when the chemical potential on both phases is the same, i.e., pgas = p,ads. The partition function for the adsorbed molecules as a system is given by... 

We can now utilize some of the statistical mechanics relationships derived in Chapter 8 to find expressions for the free energy and the equilibrium constant in term of the molecular partition functions. From the definition of the free energy (Eq. 9.1) the expression for the enthalpy of an ideal gas (Eq. 8.121), and recalling that Ho = Eq (for an ideal gas), we obtain... 

In all other cases to has the dimensions but not the meaning of a reciprocal frequency (193). The time of adsorption can be calculated by means of statistical mechanics from the partition functions of the gaseous and the adsorbed molecule (193). The equilibrium condition for the adsorption may be written as... 

We have derived a formula for the molecular partition function by considering a system containing many molecules at equilibrium with a heat bath. We can generalize our statistical mechanics by a gedanken experiment of considering a large number of identical systems, each with volume V and number of particles N at equilibrium with the heat bath at temperature T. Such a supersystem is called a canonical ensemble. Our derivation is the same the fraction of systems that are in a state with energy Et is... 

Statistical mechanics, via the partition function, provides a route to the calculation of equilibrium constants. Take as an example this reversible reaction ... 

In the past few years, development of new theories have led to completely new ways of determining free energy changes. Traditionally, the difference in the free energy of two equilibrium state is (AFi 2) and the free energy change of a process can be obtained directly from the statistical mechanical definition of the free energy, F, in terms of the partition function. For the canonical ensemble F = —k T In J = —ksTln Z, where ka is Boltzmann s constant, //(F) is the phase... 

In equation 3.1 the spin terms of the negative species have been canceled out. The quantity 12.43 is obtained from fundamental constants and the translation partition function of the electron. Qim is the ratio of the remaining partition function of the anion to that of the neutral. If the partition function ratio for the anion and neutral are assumed to be the same, this term is zero. With one value of the equilibrium constant the electron affinity of the molecule can be estimated. The statistical mechanical expression for Keq refers to the absolute zero of temperature so that no temperature correction to Ea is necessary. Unfortunately, there were no values for the equilibrium constants or electron affinities. Thus, the value of Keq for one molecule, anthracene, was determined and the electron affinity of other aromatic hydrocarbons referenced to that value. If the partition function ratios are equal,... 

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