Near equilibrium assumption

Analytical chromatographic options, based on linear and nonlinear elution optimization approaches, have a number of features in common with the preparative methods of biopolymer purification. In particular, both analytical and preparative HPLC methods involve an interplay of secondary equilibrium and within the time scale of the separation nonequilibrium processes. The consequences of this plural behavior are that retention and band-broadening phenomena rarely (if ever) exhibit ideal linear elution behavior over a wide range of experimental conditions. First-order dependencies, as predicted from chromatographic theory based on near-equilibrium assumptions with low molecular weight compounds, are observed only within a relatively narrow range of conditions for polypeptides and proteins. 

We next discuss the implications of fast variable physics for Kramers model in a bit more detail. The main point is that for fast variable processes the near equilibrium assumption of the slow variable models is strongly violated. This implies especially that the driving force for typical liquid phase reactions is not supplied by the potential of mean force, as is required for the validity of Eq. (3.41). 

These comments are closely related to questions of the validity of the frequency-dependent friction concept [13]. Namely, the use of frequency-dependent friction to correct Eq. (3.27) for finite solvent response times has physical content only for low-frequency processes, because if the solute s frequencies are too high the near equilibrium assumption is violated and lT(5 x) no longer drives the solute s motion. 

A major criticism of the stochastic probability approach is that relatively slow secondary reactions, for which the near-equilibrium assumption does not apply, cannot be accommodated. In this situation, it is necessary to derive and solve simultaneous partial differential equations for mass conservation and obtain expressions for the first and second moments of the elution profile and the concomitant plate height arising from slow kinetics of secondary equilibrium. If, once again, the process can be represented as involving the reversible binding of two forms, the resolution of the interconverting species can be given by [59]... 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

Specifically, the major topics covered by this review are (1) a brief discussion of models of element release rates based on diffusion and on TST, and (2) glass-water reactions that dominate near equilibrium. We will discuss these themes with the assumption of some general understanding of chemical kinetics, but the concepts should be comprehensible to readers from outside this field as well. 

The assumption of equilibrium sorption has been supported by the long anticipated residence times in an installed barrier (e.g., days for a barrier thickness of 1-2 m), and by batch kinetic data reported by Cantrell (1996) that indicate near-equilibrium is achieved on the order of one day. Similar assumptions have been applied to the analysis of GAC barriers (e.g., Schad and Gratwohl, 1998). Cantrell (1996) also observed linear isotherms for Sr concentrations below approximately 0.1 mg/L, although this result should be viewed as particular to the specific experimental conditions. Although these results lend support to the simplified modeling approach, more data are clearly needed to better evaluate the key assumption of linear equilibrium sorption. 

As discussed previously, in many propulsion systems the recovery of a large fraction of the dissociation energy in the nozzle expansion through recombination is difficult to achieve. While the assumption of frozen flow with respect to recombination reactions appears necessary for many heat transfer rocket nozzle expansions, it is possible that condensation phenomena are sufficiently rapid to provide near equilibrium flow with respect to phase changes. For this special possibility, phase equilibrium in the presence of frozen dissociation, it has been shown theoretically (48) that the performance in terms of specific impulse of propellants containing light metallic elements can exceed the performance of hydrogen. 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

Inequality 11 was substituted into Equation 8, together with reasonable values of other parameters and 0.1 cm < a < 0.2 cm as a was found to be in this study. This leads to the conclusion that, for systems in which r is less than 0.1 cm, the local equilibrium assumption is applicable (i.e., q is sufficiently small) when D is nearly equal to as observed in the experiments. In soils in which the exchanging particles are not spherical, r would represent approximately the mean diffusion path within clay aggregates or within clay coatings on coarse particles. 

For soils without appreciable clay aggregation, the experimental results and theoretical analysis described here indicate that when diffusion is rate-limiting, the ratio of the hydrodynamic dispersion coefficient to the estimated soil self-diffusion coefficient may be a useful indicator of the applicability of the local equilibrium assumption. For reacting solutes in laboratory columns such as those used in this study, systems with ratios near unity can be modeled using equilibrium chemistry. 

Higher-order approximations have not been proven to be useful when the Navier-Stokes equations are invalid, the most satisfactory procedure is to solve the Boltzmann equation by methods that do not rely on the assumption of near-equilibrium flow. 

Instantaneous equillibrium between sorbed and soluble phases is Inaccurate desorption is slower than adsorption. "Macropore" flow has been defined as rapid and deep Infiltration of rainfall at the onset of a storm as the rain follows preferential flowpaths In the soil. Hence, not all the aldicarb In the surface zones will "see" the water as It flows by, and desorb according to the equilibrium assumption. It Is difficult to determine which, If any, of these possibilities explain the malntainance of high aldicarb concentrations near the soil surface. Nonetheless, the high assigned partition coefficient Is the way In which the model artificially duplicates this behavior. 

Most applications of coupled models use the local equilibrium assumption. It is well known that most heterogeneous and redox reactions in the low temperature, near-surface systems are kinetically controlled (Hunter et al., 1998). However, we lack both theory and data to model kinetic reactions satisfactorily. In addition, inclusion of kinetic reactions makes the results even more difficult to comprehend completely and hence more costly. 

Strictly speaking, soils are always nonequilibrium systems. With care, however, a partial equilibrium or steady state can be attained by assuming that the soil solids do not change. This is the usual assumption in cation exchange and adsorption studies. Kittrick and co-workers were able to obtain near-equilibrium measurements of some soil minerals in studies requiring -several years. From the resulting ion activities in solution, they were able to calculate some of the equilibrium constants used for the mineral stability diagrams shown later in this book. 

One of the conclusions that can be drawn at present is that previous calculations of mitochondrial and cytosolic metabolite concentrations from total cell amounts (for a review of the calculation methods, see [5]) are incorrect, presumably because the assumptions on which these calculations were based (near-equilibrium of certain enzyme reactions) appear to be invalid [33-35]. 

The free enthalpy as condition of equilibrium dG = 0 arises naturally, if the system is in contact with a thermal reservoir and a pressure reservoir. The constraint dn" -I-dn" = 0 arises from the conservation of mass and the assumption that chemical reactions are not allowed. Near equilibrium, a plot of the chemical potentials looks like that in Fig. 6.11. 

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