Debye equilibrium model

The BMREP and SDM currently use the Davies technique for activity coefficient prediction. The Davies technique is a combination of the extended Debye-Huckel equation (6) and the Davies equation (7). The Davies technique (and hence both equilibrium models) is accurate up to ionic strengths of 0.2 molal and may be used for practical calculations up to ionic strengths of 1 molal (8). Ion-pair equilibria are incorporated for species that associate (e.g., 1-2 and 2-2 electrolytes). The activity coefficients (y ) are calculated as a simple function of ionic strength (I) and are represented as ... 

Correlations for the determination of the dissociation equilibrium constants and solubility values for SO2 and CO2 as functions of temperature as well as the equations for activity coefficients are given in Ref. [70], Thermodynamic non-idealities are taken into account depending on whether species are charged, or not. For uncharged species, a simple relationship from Ref. [102] is applied, whereas for individual ions, the extended Debye-Hiickel model is used according to Ref. [103]. 

Do activities and solubilities calculated from this alternate approach differ from those calculated with association-equilibrium extended-Debye-Hiickel models under conditions of interest in FGD wet scrubbing ... 

Although reasonable activation energies may be obtained, it is still often difficult to predict accurate thermal rate constants. One reason for this is the deviation of the real system from ideality, which introduces parameters that are not computationally well-defined in the conventional MO-TST approach. Recall that the quasi-equilibrium constant in Equation (6) is in terms of activities. Thus, the rate constant equation (Eqn. 12) is a function of activity coefficients of reactants and transition states, and these coefficients cannot be computed with the usual Debye-Huckel model. 

Most models use dilute solution assumptions and former models [93] assumed unit values [2,19,93,94] of all activity coefficients (except in some cases for H+ ions). Bernhardsson et al. [4] reached a compromise by using two sets of equilibrium constants, one for dilute solutions and the other for concentrated solutions. Other authors used Debye-Huckel, the truncated Davies model [96,98], or the B-dot Debye-Huckel model [98] to derive the activity coefficients in concentrated solutions. 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Figure 8.8 shows the resulting saturation indices for halite and anhydrite, calculated for the first four samples in Table 8.8. The Debye-Hiickel (B-dot) method, which of course is not intended to be used to model saline fluids, predicts that the minerals are significantly undersaturated in the brine samples. The Harvie-Mpller-Weare model, on the other hand, predicts that halite and anhydrite are near equilibrium with the brine, as we would expect. As usual, we cannot determine whether the remaining discrepancies result from the analytical error, error in the activity model, or error from other sources. 

The important message from Einstein or Debye models is that vibrations of atoms in a crystal contribute to Entropy S and to Heat Capacity C therefore they affect the thermodynamic equilibrium of a crystal by modifying both the Eree energy F, which... 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

The Gibbs phase rule is the basis for organizing the models. In general, the number of independent variables (degrees of freedom) is equal to the number of variables minus the number of independent relationships. For each unique phase equilibria, we may write one independent relationship. In addition to this (with no other special stipulations), we may write one additional independent relationship to maintain electroneutrality. Table I summarizes the chemical constituents considered as variables in this study and by means of chemical reactions depicts independent relationships. (Throughout the paper, activity coefficients are calculated by the Debye-Hiickel relationship). Since there are no data available on pressure dependence, pressure is considered a constant at 1 atm. Sulfate and chloride are not considered variables because little specific data concerning their equilibria are available. Sulfate may be involved in a redox reaction with iron sulfides (e.g., hydrotroilite), and/or it may be in equilibrium with barite (BaS04) or some solid solution combinations. Chloride may reach no simple chemical equilibrium with respect to a phase. Therefore, these two ions are considered only to the... 

When the displacements of the nuclei are considered in terms of the phonon modes of the crystal, the reorganization energy can be expressed in terms of the relative shifts of the equilibrium positions of acoustic vibrations of acoustic vibrations. Using the Debye model and assuming that the friction is independent of frequency, the reorganization energy is proportional to the friction coefficient, Et = AI2>7Ty]0 2DQ (see Section 2.3). 

Studying the ESPT of hydroxy aromatic sulfonates, Huppert and co-workers [40-44] suggested an alternative model based on the geminate proton-anion recombination, governed by diffusive motion. The analysis was carried out by using Debye-Smoluchowskii-type diffusion equations. Their ESPT studies in water-methanol mixtures showed that solvent effects in the dissociation rate coefficient are equal to the effects in the dissociation equilibrium constant [45], 4-Hydroxy-1-naphthalenesulphonate in a water-propanol mixture as the solvent system has been found to behave somewhat differently than water-methanol or water-ethanol media, with a possible role of a water dimer [46,47],... 

The input of the problem requires total analytically measured concentrations of the selected components. Total concentrations of elements (components) from chemical analysis such as ICP and atomic absorption are preferable to methods that only measure some fraction of the total such as selective colorimetric or electrochemical methods. The user defines how the activity coefficients are to be computed (Davis equation or the extended Debye-Huckel), the temperature of the system and whether pH, Eh and ionic strength are to be imposed or calculated. Once the total concentrations of the selected components are defined, all possible soluble complexes are automatically selected from the database. At this stage the thermodynamic equilibrium constants supplied with the model may be edited or certain species excluded from the calculation (e.g. species that have slow reaction kinetics). In addition, it is possible for the user to supply constants for specific reactions not included in the database, but care must be taken to make sure the formation equation for the newly defined species is written in such a way as to be compatible with the chemical components used by the rest of the program, e.g. if the species A1H2PC>4+ were to be added using the following reaction ... 

The thermodynamic model of Krissmann [53] was used in the calculations of these experiments, though this was limited by the phase equilibrium (Eq. (3)) and the reaction equilibrium (Eq. (4)). Calculation of the activity coefficients of the H+ ions and HSOj was performed according to the extended Debye-Hiickel theory, using the approximation of Pitzer... 

Figure 7-1. Upper panel TDSS functions for MCP calculated at TDDFT level using the Debye (full line) and the fit (dotted line) models for b(o>). Lower panel tme dependent evolution of the excited state free energy 9k i) (in eV) of MCP calculated at TDDFT level using fit model for s(ai). For the latter, we also report the transition energies towards the vertical ground state at the various times. All the energies are referred to the ground state equilibrium free energy...

Various theories have been proposed and tested to explain salt effect in vapor-liquid equilibrium, including models based on hydration, internal pressure, electrostatic interaction, and van der Waal forces. Although the electrostatic theory of Debye as modified for mixed solvents has had limited success, no single theory has yet been able to account for or to... 

Temperature Effects. The temperature range for which this model was assumed to be valid was 0°C through 40°C, which is a range covering most natural surface water systems (28). Equilibrium constants were adjusted for temperature effects using the Van t Hoff relation whenever appropriate enthalpy data was available (23, 24, 25). Activity and osmotic coefficients were temperature corrected by empirical equations describing the temperature dependence of the Debye-Huckel parameters of equations 20 and 21. These equations, obtained by curve-fitting published data (13), were... 

The difference between the extended Debye-Hiickel equation and the Pitzer equations has to do with how much of the nonideahty of electrostatic interactions is incorporated into mass action expressions and how much into the activity coefficient expression. It is important to remember that the expression for activity coefficients is inexorably bound up with equilibrium constants and they must be consistent with each other in a chemical model. Ion-parr interactions can be quantified in two ways, explicitly through stability constants (lA method) or implicitly through empirical fits with activity coefficient parameters (Pitzer method). Both approaches can be successful with enough effort to achieve consistency. At the present, the Pitzer method works much better for brines, and the lA method works better for... 

In spite of the partial success in theoretical description, we believe that more realistic models are needed for the theory to have a predicting power. For example, measurements usually take place in the presence of a large excess of simple electrolyte. The electrolyte present is often a buffer, a rather complicated mixture (difficult to model perse) with several ionic species present in the system. Note that many effects in protein solutions are salt specific. Yet, most of the theories subsume all the effect of the electrolyte present into a single parameter, the Debye screening parameter n. In the case of the Donnan equilibrium we measure the subtle difference between the osmotic pressures across a membrane permeable to small ions and water but not to proteins. We believe that an accurate theoretical description of protein solutions can only be built based on the models which take into account hydration effects. 

The simplest approach conventionally employed to describe the grain screening in colloidal plasmas is the Debye-Hiickel (DH) approximation, or, its modification for the case of the grain of finite size, the DLVO theory [6,7], The DH approximation represents the version of Poisson-Boltzmann (PB) approach linearized with respect to the effective potential based on the assumption that the system is in the state of thermodynamical equilibrium. The DH theory yields the effective interparticle interaction in the form of the so-called Yukawa potential which constitutes the basis for the Yukawa model. 

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