Partition function, equilibrium phase

Here Zint is the intramolecular partition function accounting for rotations and vibrations. However, in equilibrium, the chemical potential in the gas phase is equal to that in the adsorbate, fi, so that we can write the desorption rate in (I) as... 

We now need an expression for the equilibrium constant between the gas phase and the transition state complex. The reaction coordinate is again the (very weak) vibration between the atom and the surface. There are no other vibrations parallel to the surface, because the atom is moving in freely in two dimensions. The relevant partition functions for the atoms in the gas phase and in the transition state are... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

Smit et al. [19] used the partition function given by (10.4) and a free energy minimization procedure to show that, for a system with a first-order phase transition, the two regions in a Gibbs ensemble simulation are expected to reach the correct equilibrium densities. 

The formulae given in Table 4.1 for the molecular partition functions enable us to write the partition function ratio qheavy/qiight or q2/qi where, by the usual convention, the subscript 2 refers to the heavy isotopomer and 1 refers to the light isotopomer if heavy and light are appropriate designations. Then, a ratio of such partition function ratios enables one to evaluate the isotope effect on a gas phase equilibrium constant, as pointed out above. Before continuing, it is appropriate to... 

Equation 5.19 relates the molecular energy states of the primed and unprimed isotopomers in condensed and vapor phase to VPIE. The correction terms account for the difference between the Gibbs and Helmholtz free energies of the condensed phase, and vapor nonideality. The comparison is between separated isotopomers at a common temperature, each existing at a different equilibrium volume, V or V, and at a different pressure, P or P, although AV = (V — V) and AP = (P — P) are small. Still, because condensed phase Q s are functions of volume, Q = Q(T,V,N), rigorous analysis requires knowledge of the volume dependence of the partition function, and thus MVIE, since the comparisons are made at V and V. That point is developed later. 

Bennett and Barter (1997) discuss the effect of partitioning-dissolution in an aqueous phase of alkylphenol. Specifically, they show that the depletion of this crude oil component affects the chemical composition of the original pollutant. Partitioning at equilibrium can be considered the maximum dissolution value of a compound under optimal solvation conditions. Partitioning-dissolution is obtained by washing the crude oil with saline water at variable temperature and pressure conditions, similar to those in the subsurface. The data reported were obtained using a partition device able to simulate the natural environmental conditions of a crude oil reservoir. The alkylphenol partition coefficients between crude oil and saline subsurface water were measured as a function of variation in pressure, temperature, and water salinity. Preliminary trials proved that the experimental device did not allow alkylphenol losses due to volatilization. 

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... 

The statistical thermodynamic approach to the derivation of an adsorption isotherm goes as follows. First, suitable partition functions describing the bulk and surface phases are devised. The bulk phase is usually assumed to be that of an ideal gas. From the surface phase, the equation of state of the two-dimensional matter may be determined if desired, although this quantity ceases to be essential. The relationships just given are used to evaluate the chemical potential of the adsorbate in both the bulk and the surface. Equating the surface and bulk chemical potentials provides the equilibrium isotherm. 

In order to extend the above theory, we have to make use of the equilibrium partition concepts described in Part II. Here we generally define the equilibrium partition function between any phase B and A by ... 

The equilibrium ratio of sorbate concentrations in the adsorbed and vapor phases is given by the ratio of the relevant partition functions. At sufficiently low concentrations, so that very few cavities are occupied,... 

The first or probability factor is essentially a ratio of partition functions, and represents the integrated equilibrium density of phase points on S per phase point in A. The second or trajectory-corrected frequency factor is the number of successful forward trajectories per unit time and per unit equilibrium density on S. The ratio of this to the uncorrected frequency factor 0) >s represents the number of successful forward trajectories per forward crossing. Anderson called this ratio the conversion coefficient to distinguish it from the transmission coefficent of traditional rate theory (1), which was usually defined rather carelessly and given little attention, because it could not be computed without trajectory information. 

The pronounced discrepancy between the measured dynamic 15 °C-elution curve and its extrapolated reversible-thermodynamic part, shown in Fig. 7, represents a direct proof of the inadequacy of the reversible Eq. (3) in the dynamic region of the column (PDC-effect). Moreover, the experiment shows immediately that the polymer of the mobile phase has to dissolve in the gel layer within the transport zone to a considerably higher extent than is allowed by the partition function (4) in a reversible-thermodynamic equilibrium between the gel and the sol at the same column temperature. As a consequence, a steady state, i.e. a flow-equilibrium, must be assumed in the system sol/gel within the considered transport zone, governing the polymer trans-... 

The grand canonical ensemble is appropriate for adsorption systems, in which the adsorbed phase is in equilibrium with the gas at some specified temperature. The use of a computer simulation allows us to calculate average macroscopic properties directly without having to explicitly calculate the partition function. The grand canonical Monte Carlo (GCMC) method as applied in this work has been described in detail earlier (55). The aspects involving binary fluid mixtures have been described previously in our Xe-Ar work (30). 

Consider the equilibrium between reactants A, B,.. . and products M, N,.. . in the gas phase as well as in solution in a given solvent S (Figure 2.2). The equilibrium constant in the gas phase, K%, depends on the properties of the reactants and the products. In very favourable cases it can be estimated from statistical thermodynamics via the relevant partition functions, but for the present purposes it is regarded as given. The problem is to estimate the magnitude of the equilibrium constant in the solution, A , and how it changes to KP-, when solvent Su is substituted for solvent St. 

The traditional apparatus of statistical physics employed to construct models of physico-chemical processes is the method of calculating the partition function [17,19,26]. The alternative method of correlation functions or distribution functions [75] is more flexible. It is now the main method in the theory of the condensed state both for solid and liquid phases [76,77]. This method has also found an application for lattice systems [78,79]. A new variant of the method of correlation functions - the cluster approach was treated in the book [80]. The cluster approach provides a procedure for the self-consistent calculation of the complete set of probabilities of particle configurations on a cluster being considered. This makes it possible to take account of the local inhomogeneities of a lattice in the equilibrium and non-equilibrium states of a system of interacting particles. In this section the kinetic equations for wide atomic-molecular processes within the gas-solid systems were constructed. 

As shown in most physical chemistry texts (e.g., Refs. 1, 2 see also the discussion in Exp. 48), the molecular partition function q for the reactants and products determines the equilibrium constant Kp of a gas-phase reaction. For reaction (1) the relation is... 

Another option is to derive equations for the pressures at which condensation takes place in narrow capillaries. Let us illustrate this for slits. As before, the characteristic function -kTlnS of the grand canonical partition function equals -pV + y A, with V = Ah. Let liquid and vapour coexist Inside the capillary and assume that we have only these two phases (i.e. the contribution of the (thin) inhomogeneity at the phase boundary to is Ignored). Equilibrium... 

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