Equilibrium relation with partition functions

Several methods have been developed for the quantitative description of such systems. The partition function of the polymer is computed with the help of statistical thermodynamics which finally permits the computation of the degree of conversion 0. In the simplest case, it corresponds to the linear Ising model according to which only the nearest segments interact cooperatively149. The second possibility is to start from already known equilibrium relations and thus to compute the relevant degree of conversion 0. 

It is instructive to illustrate the relation between the partition function and the equilibrium constant with a simple, entirely hypothetical example. Consider the equilibrium between an ensemble of molecules A and B, each with energy levels as indicated in Fig. 3.5. The ground state of molecule A is the zero of energy, hence the partition function of A vnll be... 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

It has been previously noted that the first quantum correction to the classical high temperature limit for an isotope effect on an equilibrium constant is interesting. Each vibrational frequency makes a contribution c[>(u) to RPFR and this contribution can be expanded in powers of u with the first non-vanishing term proportional to u2/24, the so called first quantum correction. Similarly, for rates one introduces the first quantum correction for the reduced partition function ratios, includes the Wigner correction for k /k2 and makes use of relations like Equation 4.103 for small x and small y, to find a value for the rate constant isotope effect (omitting the noninteresting symmetry number term)... 

As seen from our discussion in Chapter 3, which dealt with onedimensional problems, in many relevant cases one actually does not need the knowledge of the behavior of the system in real time to find the rate constant. As a matter of fact, the rate constant is expressible solely in terms of the equilibrium partition function imaginary-time path integrals. This approximation is closely related to the key assumptions of TST, and it is not always valid, as mentioned in Section 2.3. The general real-time description of a particle coupled to a heat bath is the Feynman-Vernon... 

In the second part, using the equilibrium relations for the various components present in the water and oil phases and on the interface and coupling them with mass balances, one could calculate the thicknesses of the water and oil layers as functions of the component concentrations. It was shown that the deviation from the ideal dilution law can be accounted for by the partition of the components between phases. A comparison between the calculated and experimental results was made. 

We should be able to relate S and z because the partition function and Boltzmann distribution were originally derived by combining equation (6.7) for the energy of a system with equation (6.4) for the probability of its equilibrium configuration, W, and we already know that entropy and probability are related by... 

Before the significance of the various factors expressed in the equations for In is discussed further, an alternative method of deriving the equilibrium constant will be considered. In this it is related to the partition functions of the various molecules taking part in the reaction. It will suffice to deal with Kj. ... 

Let us e>q>ress the equilibrium constant of the association reaction using the partition functions of reactants and products in the following form using relation [A.3.10] with Eq=0 since the origins of energies of species A and A2 are the same ... 

Secondly, in the theory of irreversible processes, variation principles may be expected to help establish a general statistical method for a system which is not far from equilibrium, just as the extremal property of entropy is quite important for establishing the statistical mechanics of matter in equilibrium. The distribution functions are determined so as to make thermod5mamic probability, the logarithm of which is the entropy, be a maximum under the imposed constraints. However, such methods for determining the statistical distribution of the s retem are confined to the case of a system in thermodynamic equilibrium. To deal with a system out of equilibrium, we must use a different device for each case, in contrast to the method of statistical thermodynamics, which is based on the general relation between the Helmholtz free energy and the partition function of the system. 

Contrary to Section H.l the atomistic point of view is taken in the following sections, i. e. the heat capacity is considered in relation to the mechanical behavior of the particles of atomic dimensions building up a macroscopic specimen. Owing to the great number of particles and, in consequence, to the lack of information regarding the real mechanical situation only statements are possible that are based on laws of probability. The connection of the theory of probability with mechanics leads to the so-called statistical mechanics. In Section II.2 the fundamental relations between statistical mechanics and equilibrium thermodynamics as well as those between partition function and heat capacity are explained. 

DYNAMICS OF DISTRIBUTION The natural aqueous system is a complex multiphase system which contains dissolved chemicals as well as suspended solids. The metals present in such a system are likely to distribute themselves between the various components of the solid phase and the liquid phase. Such a distribution may attain (a) a true equilibrium or (b) follow a steady state condition. If an element in a system has attained a true equilibrium, the ratio of element concentrations in two phases (solid/liquid), in principle, must remain unchanged at any given temperature. The mathematical relation of metal concentrations in these two phases is governed by the Nernst distribution law (41) commonly called the partition coefficient (1 ) and is defined as = s) /a(l) where a(s) is the activity of metal ions associated with the solid phase and a( ) is the activity of metal ions associated with the liquid phase (dissolved). This behavior of element is a direct consequence of the dynamics of ionic distribution in a multiphase system. For dilute solution, which generally obeys Raoult s law (41) activity (a) of a metal ion can be substituted by its concentration, (c) moles L l or moles Kg i. This ratio (Kd) serves as a comparison for relative affinity of metal ions for various components-exchangeable, carbonate, oxide, organic-of the solid phase. Chemical potential which is a function of several variables controls the numerical values of Kd (41). 

For phase partitioning (equilibrium), the variable of interest is the solute concentration in the first phase that would be in equilibrium with the solute concentration in a second phase. For example, in the distillation example above, each component is partitioned between the vapor and liquid phases. The mathematical description of the equilibrium relationship is usually given as the concentration in one phase as a function of the concentration in the second phase as well as other parameters. Some examples are the Henry s Law relation for the mole fraction of a solute in a liquid as a function of the mole fraction of the solute in the gas phase which contacts the liquid ... 

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