Equilibrium model assumptions

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

A plot of the initial reaction rate, v, as a function of the substrate concentration [S], shows a hyperbolic relationship (Figure 4). As the [S] becomes very large and the enzyme is saturated with the substrate, the reaction rate will not increase indefinitely but, for a fixed amount of [E], it reaches a plateau at a limiting value named the maximal velocity (vmax). This behavior can be explained using the equilibrium model of Michaelis-Menten (1913) or the steady-state model of Briggs and Haldane (1926). The first one is based on the assumption that the rate of breakdown of the ES complex to yield the product is much slower that the dissociation of ES. This means that k2 tj. 

However, we have to reflect on one of our model assumptions (Table 5.1). It is certainly not justified to assume a completely uniform oxide surface. The dissolution is favored at a few localized (active) sites where the reactions have lower activation energy. The overall reaction rate is the sum of the rates of the various types of sites. The reactions occurring at differently active sites are parallel reaction steps occurring at different rates (Table 5.1). In parallel reactions the fast reaction is rate determining. We can assume that the ratio (mol fraction, %a) of active sites to total (active plus less active) sites remains constant during the dissolution that is the active sites are continuously regenerated after AI(III) detachment and thus steady state conditions are maintained, i.e., a mean field rate law can generalize the dissolution rate. The reaction constant k in Eq. (5.9) includes %a, which is a function of the particular material used (see remark 4 in Table 5.1). In the activated complex theory the surface complex is the precursor of the activated complex (Fig. 5.4) and is in local equilibrium with it. The detachment corresponds to the desorption of the activated surface complex. 

The solution to either Eq. (9.9) or (9.10) has not one wave but two or more one for the adsorbed phase and a second in the gas phase. In many cases these waves move coincidentally but that assumption should not be invoked in all cases. The speed of these waves can be derived from the original pde together with the form of the adsorption equilibrium model. 

In this summary, the local thermal equilibrium model has been used to derive the energy equation. This model is much simpler than the two-phase model however, the local thermal equilibrium model is most likely not adequate to describe the transport of energy when the temperature of the fluid and solid are undergoing extremely rapid changes. Although such extremely rapid temperature changes are not expected, in most RTM, IP, and AP processes the correctness of the local thermal equilibrium assumption can be verified by following the procedure discussed by Whitaker [28]. 

Assumption (f) of isothermal flow means that the method is different to the homogeneous equilibrium model (which assumes adiabatic flow). The difference between the two assumptions is usually small. The isothermal flow assumption gives a slightly simpler method and yields a conservative low value of G for relief sizing purposes. The DIERS Project Manual1111 gives the alternative version of Tangren et al/s method, which assumes adiabatic flow and is therefore equivalent to the HEM. 

A discussion of the different types of assumption that can be made in two-phase flow models is given in Chapter 9. DIERS[8] recommended the use of the homogeneous equilibrium model (HEM) for relief sizing, and so, preferably, a code which implements the HEM should be chosen. The model will need to incorporate sufficiently non-ideal modelling of physical properties and provision for multiple line diameters and potential choke points, as required by the application. 

Partition coefficients are used to describe the distribution of nonpolar organic compounds between water and organisms. It can be viewed as a partitioning process between the aqueous phase and the bulk organic matter present in biota (Schwarzenbach et al. 1993). The premise behind the use of equilibrium models is that accumulation of compounds is dominated by their relative solubility in water and the solid phases, respectively. Equilibrium models, therefore, rely on the following assumptions (Landrum et al. 1996) ... 

Many attempts have been made to quantify SIMS data by using theoretical models of the ionization process. One of the early ones was the local thermal equilibrium model of Andersen and Hinthome [36-38] mentioned in the Introduction. The hypothesis for this model states that the majority of sputtered ions, atoms, molecules, and electrons are in thermal equilibrium with each other and that these equilibrium concentrations can be calculated by using the proper Saha equations. Andersen and Hinthome developed a computer model, C ARISMA, to quantify SIMS data, using these assumptions and the Saha-Eggert ionization equation [39-41]. They reported results within 10% error for most elements with the use of oxygen bombardment on mineralogical samples. Some elements such as zirconium, niobium, and molybdenum, however, were underestimated by factors of 2 to 6. With two internal standards, CARISMA calculated a plasma temperature and electron density to be used in the ionization equation. For similar matrices, temperature and pressure could be entered and the ion intensities quantified without standards. Subsequent research has shown that the temperature and electron densities derived by this method were not realistic and the establishment of a true thermal equilibrium is unlikely under SIMS ion bombardment. With too many failures in other matrices, the method has fallen into disuse. 

Assumptions made in hydrogeochemical modeling programs complicate the transferability to natural systems, e.g. assuming thermodynamic equilibrium. This assumption is often not true especially for redox reactions being dominated by kinetics and catalyzed by microorganisms, and precipitation of certain minerals. Both processes can maintain disequilibria over a long time period. 

We have demonstrated our arguments on the basis of the Ambio (1982) war scenario, using much simplified rainout, aerosol physics and radiative equilibrium models. Our analysis of the amounts of burned materials and smoke production is likewise uncertain and no doubt can be much improved upon, but we have tried to avoid extreme assumptions. It is very hard to carry out sensitivity analyses in this new research field with so many uncertainties. It is quite possible to forward arguments for a much less severe impact of a nuclear war. For instance, one may propose... 

The chemical fractionations observed among chondrites and the compositions of many chondritic components are best understood in terms of quenched equilibrium between phases in a nebula of solar composition (Palme, 2001 Chapters 1.03 and 1.15). The equilibrium model assumes that minerals condensed from, or equilibrated with, a homogeneous solar nebula at diverse temperatures. Isotopic variations among chondrites and their components show that this assumption is not correct and detailed petrologic studies have identified relatively few chondritic components that resemble equilibrium nebular products. Nevertheless, the equilibrium model is invaluable for understanding the chemical composition of chondrites and their components as the solar nebular signature is etched deeply into the chemistry and mineralogy. 

Aim are both sensitive to temperature (e.g.. Figure 3), whereas Xors is moderately sensitive to pressure. The different path calculations show very similar P-T evolution— exhumation by 3 kbar with heating of 20 °C (Figure 5(b) open symbols)—suggesting that the model assumptions are valid, i.e., that chemical equilibrium was attained among minerals, and... 

Thermodyncunic ancdyses, like the one discussed, require very accurate equilibrium data but have the advantage that they are phenomenological no model assumptions are needed. Only after the characteristic functions have been established can models be Invoked for interpretation. It appears that in the literature... 

The solution to (P12) gives us the optimal separation profile as a function of age within the reactor. However, except in the case of reactive phase equilibrium, the assumption of a continuous separation profile is not really required. Furthermore, a continuous separation profile may not be implementable in practice. To address this, we take advantage of the structure of a discretization procedure for the differential equation system. In this case, we choose orthogonal collocation on finite elements to discretize the above model. This results... 

Natural waters obtain their equilibrium composition through a variety of chemical reactions and physicochemical processes. In this chapter we consider principles and applications of two alternative models for natural water systems thermodynamic models and kinetic models. Thermodynamic, or equilibrium, models for natural waters have been developed more extensively than kinetic models. They are simpler in that they require less information, but they are nevertheless powerful when applied within their proper limits. Equilibrium models for aquatic systems receive the greater attention in this book. However, kinetic interpretations are needed in description of natural waters when the assumptions of equilibrium models no longer apply. Because rates of different chemical reactions in water and sediments can differ enormously, kinetic and equilibrium are often needed in the same system. 

Distillation columns are made to separate at least two different components. There are several different types of columns. The assumption of equilibrium between the liquid and vapour that leave each tray is still in common use to model tray distillation. The work of Krishna and Wesselingh shows, however, that non-equilibrium models give results that are very different from those obtained with the equilibrium assumption. 

For a diatomic close to thermal equilibrium the assumption of impulsive collisions with the bath molecules is realistic only when the vibrational jjeriod is longer than the duration of a collision, but this is rarely the case for diatomics vibrating near the potential minimum. However, for realistic potentials the vibrational period lengthens near the dissociation threshold, and it is not so clear that an impulsive model will be quantitatively inaccurate in modeling dissociation, even though it may fail badly in describing vibrational relaxation deep with the well. A stochastic impulsive model of dissociation of a diatomic AB, which uses the zero-frequency frictions describ-... 

An alternative way of relating concentrations (mass fractions) of individual species to/ is the assumption of chemical equilibrium. An algorithm based on minimization of Gibbs free energy to compute mole fractions of individual species from / has been discussed by Kuo (1986). The equilibrium model is useful for predicting the formation of intermediate species. If such knowledge of intermediate species is not needed, the much simpler approximation of mixed-is-burnt can be used to relate individual species concentrations with/. In order to calculate the time-averaged values of species concentrations the probability density function (PDF) approach is used. 

The derivation of the separation conditions is based on the ideal or equilibrium model, i.e., on the assumption that axial dispersion and the mass transfer resistances are all negligible and that the column efficiency is practically infinite. In conventional studies of SMB, it is further assumed that the solid phase flow rate through each column and the void fraction of each column are the same. In the Hnear case, the ratio of the internal flow rate and the solid-phase flow rate can be combined with the slope of isotherm (a,) by using a safety margin, jSy [25,27] ... 

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