Formulation of the Partition Function

Conventional lattice methods can be adapted to treatment of a system of rigid, rodlike particles, or molecules, by the device illustrated in Fig. 1 The particle shown in Fig. la is oriented at an angle t with respect to the preferred axis of the surrounding domain. One of the principal axes of the cubic lattice is aligned with this axis. The particle comprises x isodiametric segments, each of a size that will occupy one cell of the lattice it follows that x so defined is also the axial ratio of the particle. In order to accommodate the particle on the lattice when it is oriented as in Fig. 1 a, we imagine it to be subdivided into y sequences of segments as represented in Fig. 1 b, with each sequence oriented parallel to the preferred axis. The parameter y serves as a measure of disorientation of the particle with respect to the domain axis see Fig. lb. 

The analysis depends on evaluation of the number Vj of situations (i.e., appropriate sequences of empty lattice sites) accessible to molecule j with orientation yj when j — 1 molecules have been added previously to the space, or lattice, in which the mixture is confined. This quantity can be formulated as the product of the number of lattice sites available to the initial segment of the chain and the probabilities that the sites required for each successive segment of the cham are unoccupied and, hence, accessible. The number of eligible locations for the initial segment of the chain is just the total number Hq — x(j — 1) of vacant sites, where no is the total number of lattice sites. In formulating the probability that the site required by the second seg- 

Accessibility of the site required for the first segment of the following sequence is not contingent on vacancy of a preceding site. Hence, the a priori probability Pj of a vacancy is required. It is just the volume fraction of vacancies, i.e.. 

The number fraction Nj exceeds the a priori probability Pj of a vacancy to the extent that y is less than x compare Eqs. (1) and (2). Hence, Vj is maximal for yj = 1 and it decreases with increase in yj, i.e., with disorientation of rod j. 

The steric or combinatory part of the partition function for the system follows as the product of the Vj for each of the Op rodlike molecules incorporated in the lattice i.e., 

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Comparing the expression (4.131) with the canonical formulation of the partition function... 

The framework of the Flory theory [104,105-117] can be summarized as follows. A chain segment has the length equal to its diameter, what makes possible placement at a given lattice site of either a segment or a solvent molecule. The objective is the formulation of the partition function Z. One makes the standard approximation... 

P ) in the approximate P formulation of the partition function will arise, and therefore the properties calculated will also be affected. Despite these facts, several options that seek to reduce the error 0(P " ) have been proposed and, together with a number of efficient simulation implementations, make the finite-P discretized formulations extremely useful. 

Another way of formulating this problem is to use derivatives of the partition function without a weight function. This is done with the following relationships ... 

Quite similar equations can be formulated for AG and AH by use of the partition function f of the activated complex. It follows from equations (6) and (7) that AEp can only be evaluated if the partition functions and AEz are available from spectroscopic data or heat capacity measurements. However, if AG = AH, the entropy change AS equals zero, and if AEz also equal to zero, either AG or AH can then be identified with the potential energy change. If... 

In formulating the terms of the partition function, it is convenient to group them according to the maximum number of residues permitted in a strand, which is denoted by/. The ultimate interest is in the limiting conformational behaviour as / approaches n. 

The theory of IEs was formulated by Bigeleisen and Mayer.9 The IE on the acid-base reaction of Equation (1) is defined as the ratio of its acidity constant KA to the acidity constant of the isotopic reaction, Equation (2). The ratio KJ KA is then the equilibrium constant XEIE for the exchange reaction of Equation (3). That equilibrium constant may be expressed in terms of the partition function Q of each of the species, as given in Equation (4), which ignores symmetry numbers. 

The symbol ( )n denotes one mole of domains of any spin state with n molecules in each domain. Interactions between the domains, irrespective of the spin state, are neglected. The total energy of an individual complex molecule is considered to be composed of electronic and intramolecular vibrational contributions. In terms of the partition functions for these contributions the equilibrium constant K = xhs/O-xhs) may be formulated as... 

Let us remind that the problem of the partition function calculation for a closed polymer chain with topological constraints is usually formulated as integration over the set fl of closed paths from a fixed topological class, or with fixed value of the topological invariant ... 

To introduce the formulation, we consider the exact connection between the unperturbed and perturbed systems. We focus on the Helmholtz free energy, A, which is the quantity of interest at constant N, T, and V, where N is the number of particles, T is the temperature, and V is the volume of the system the alternative case (constant N, T, and P), which leads to the Gibbs free energy, can be treated similarly. The Helmholtz free energy for the potential energy function V(r A) can be written in terms of the partition function Zxas... 

The rates and mechanisms of chemical reactions can be predicted, in principle, by the standard methods of statistical thermodynamics, in terms of the partition functions of reactants and the transition-state complex. However, the range of applicability of this transition-state (absolute rate) theory is severely limited by the fact, that an evaluation of the vibrational partition function for the transition state requires a detailed consideration of the whole PES for the reaction. Thus, a calculation of the absolute rate constants is possible only for relatively simple systems. This indicates a need for more approximate, empirical methods of treating chemical reactions and formulating the reactivity theory, which would allow... 

The Ross melting rule [155] is expressed instead in terms of a threshold value of an excess free energy. It is based on a postulate of invariance of a scaled form of the partition function. A useful formulation of the statement requires the further assumption of a cell model for the solid. In terms of the free energy, the Ross melting rule has... 

Expression (2) contains a term, GMm, not present in (3). The reason is that in dealing with free energies we also have to consider entropic contributions. A formal derivation will not be reported here in short, it consists in a formal definition of the partition function of the whole solution, in its factorization into M and 5 components, in the introduction of the continuous distribution of the solvent, and then in the use of standard formulations of statistical mechanics to get free energy (Gibbs or Helmholtz) contributions for the M portion. The formal treatment can be found e.g. in the Ben-Naim s books [14, 15], and the application to our model in ref. [8]. 

In the statistical thermodynamic formulation (see Appendix A) the Henry constant is given by the ratio of the partition functions per unit volume for the adsorbed and vapor phases, with due correction for the difference in potential energy (/, //p. For an inert gas there are no internal degrees of freedom so, assuming classical behavior, this ratio of partition functions is equivalent to the configuration integral and one has the simple result... 

Au contraire to the empirical equation of Tait for EOS predictions, theoretical models can be used but generally require an understanding of forces between the molecules. These laws, strictly speaking, need be derived from quantum mechanics. However, Lenard-Jones potential and hard-sphere law can be used. The use of statistical mechanics is an intermediate solution between quantum and continuum mechanics. A canonical partition function can be formulated as a sum of Boltzmann s distribution of energies over all possible states of the system. Necessary assumptions are made during the development of the partition function. The thermodynamic quantities can be obtained by use of differential calculus. For instance, the thermodynamic pressure can be obtained from the partition function Q as follows ... 

Gordon et al. ° have obtained a discrete form of the partition function of Fixman for an isolated coil. Using some elegant mathematics involving the Mobius inversion, they have shown that the so called excluded volume can be re-included in an almost Markovian theory. The result is a convergent series for the expansion factor of a polymer coil and a clean formulation for the partition function for an isolated coil which, for all but very short chains, may be better than any previously available. 

Methanol.—The problem of internal rotation in methanol is complex and the calculation of the thermodynamic functions proved troublesome because at low temperatures the molecule falls outside the limits of validity of the original treatment. A special method was developed making possible the separation of the partition functions for internal rotation from that for overall rotation. This method is correct for all temperatures of practical importance, but a more general formulation has been given, - by which the tabulated thermodynamic functions have been calculated. 

The main task in the computation of thermodynamic functions is the calculation of the partition function, denoted Z. In our case, its basic form can be formulated directly, as... 

Instead of formulating the reaction rate expression in terms of molecular partition functions, it is often convenient to employ an approach utilizing pseudo thermodynamic functions. From equation 4.3.29, the second-order rate constant is given by... 

The molecular approach, adopted throughout this book, starts from the statistical mechanical formulation of the problem. The interaction free energies are identified as correlation functions in the probability sense. As such, there is no reason to assume that these correlations are either short-range or additive. The main difference between direct and indirect correlations is that the former depend only on the interactions between the ligands. The latter depend on the maimer in which ligands affect the partition function of the adsorbent molecule (and, in general, of the solvent as well). The argument is essentially the same as that for the difference between the intermolecular potential and the potential of the mean force in liquids. 

The most accurate theories of reaction rates come from statistical mechanics. These theories allow one to write the partition function for molecules and thus to formulate a quantitative description of rates. Rate expressions for many homogeneous elementary reaction steps come from these calculations, which use quantum mechanics to calculate the energy levels of molecules and potential energy surfaces over which molecules travel in the transition between reactants and products. These theories give... 

A matrix formulation of the time-dependent transition partition function is combined with a generator matrix formalism to permit rapid and accurate calculation of the first and second orientation... 

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