Local Equilibria

The great value of kinetic theory is that it frees us from many of the constraints of the equilibrium model and its variants (partial equilibrium, local equilibrium, and so on see Chapter 2). In early studies (e.g., Lasaga, 1984), geochemists were openly optimistic that the results of laboratory experiments could be applied directly to the study of natural systems. Transferring the laboratory results to field situations, however, has proved to be much more challenging than many first imagined. 

The above property of the chemical potential offers a convenient device for discussing the relaxational dynamics of near-equilibrium systems. By near-equilibrium one usually has in mind a system where each part, has reached equilibrium locally, but long wavelength modes have not been completely relaxed. This allows one to define a. local chemical potential /t(x) whose spatial variation serves as the driving force towards global equilibrium. 

Among the basic concepts to be introduced are ionic equilibrium, local equilibrium, local electro-neutrality, etc. 

In equilibrium, the morphology of a surface is determined by the anisotropy of the surface energy. While the surface morphologies that we observe in most practical situations are not in global equilibrium, local equilibrium can be achieved at the intersection of two facets or at the point were a grain boundary intersects the surface. Observations of surfaces in local equilibrium give us the... 

When each catalyst particle is in adsorption equilibrium locally with the gas phase, the axial-dispersion term of the emulsion phase gas in the equations of continuity is ( f -I- m Egs) where and are the... 

Other than the particle dimension d, the porous medium has a system dimension L, which is generally much larger than d. There are cases where L is of the order d such as thin porous layers coated on the heat transfer surfaces. These systems with Lid = 0(1) are treated by the examination of the fluid flow and heat transfer through a small number of particles, a treatment we call direct simulation of the transport. In these treatments, no assumption is made about the existence of the local thermal equilibrium between the finite volumes of the phases. On the other hand, when Lid 1 and when the variation of temperature (or concentration) across d is negligible compared to that across L for both the solid and fluid phases, then we can assume that within a distance d both phases are in thermal equilibrium (local thermal equilibrium). When the solid matrix structure cannot be fully described by the prescription of solid-phase distribution over a distance d, then a representative elementary volume with a linear dimension larger than d is needed. We also have to extend the requirement of a negligible temperature (or concentration) variation to that over the linear dimension of the representa-... 

Appendices follow Chapter 6. In Chapter 2, it has been pointed that local entropy may be expressed in terms of same independent variables as if the system were at equilibrium (local equilibrium). The limitations of Gibbs equation have been discussed in Appendix I. At no moment, molecular distribution function of velocities or of relative positions may deviate strongly from their equilibrium form. This is a sufficient condition for the application of thermodynamics method. Some new developments related to alternative theoretical formalism such as extended irreversible are discussed in Appendices II and III. 

It is usually assumed in a dynamic system that even though the system as a whole is not in equilibrium, local equilibiimn holds at every point. The path of a dynamic process can then be shown on an equilibrium phase diagram as it evolves, such as the sequence of compositions shown by the arrows in Figure 6.21. Even supersaturation, discussed in the next section, can be treated this way. 

P(2) Design for structural stability taking account of second order effects shall ensure that, for the most unfavourable combinations of actions at the ultimate limit state, loss of static equilibrium (locally or for the structure as a whole) does not occur or the resistance of individual cross-sections subjected to bending and longitudinal forces are not exceeded. 

Table 17.6 Estimate of the equilibrium localization of fillers in polymer blends by calculation of a wetting coefficient, coa, from Young s equation. Here, 1 and 2 are two polymer components in blend and p represents solid filler...

For a gas, only the latter effect should be expected to be important. Although we are rather concerned here with the case of liquids, the consideration of gases is very instructive so as to highlight the microscopic origin of viscosity. Besides, gases and liquids are very similar in that their velocity distributions are both Maxwellian at equilibrium (local equilihrium in a layer). 

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. 

The microscopic contour of a meniscus or a drop is a matter that presents some mathematical problems even with the simplifying assumption of a uniform, rigid solid. Since bulk liquid is present, the system must be in equilibrium with the local vapor pressure so that an equilibrium adsorbed film must also be present. The likely picture for the case of a nonwetting drop on a flat surface is... 

An explicit expression for the coefficient of shear viscosity can be obtained by assuming the system is in local themiodynamic equilibrium and using the previously derived expression for X and v. Thus we obtain... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

Assuming that additive pair potentials are sufficient to describe the inter-particle interactions in solution, the local equilibrium solvent shell structure can be described using the pair correlation fiinction g r, r2). If the potential only depends on inter-particle distance, reduces to the radial distribution fiinction g(r) = g... 

In a microscopic equilibrium description the pressure-dependent local solvent shell structure enters tlirough... 

Frankenthal R P and Kruger J (eds) 1984 Equilibrium Diagrams of Localized Corrosion Proc. vol 84-9 (Pennington, NJ Electrochemical Society)... 

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