Equilibrium diffusional

The problem has been discussed in terms of chemical potential by Everett and Haynes, who emphasize that the condition of diffusional equilibrium throughout the adsorbed phase requires that the chemical potential shall be the same at all points within the phase and since, as already noted, the interaction energy varies wtih distance from the wall, the internal pressure must vary in sympathy, so as to enable the chemical potential to remain constant. 

A second, seemingly less artificial, method would be to add a certain amount of block copolymer to one of the homopolymers and let it diffuse to the interface. This method has not been used to prepare fracture samples with deuterium-labeled block copolymers for several practical reasons Dissolving 5% of copolymer in a sheet of 50x100x2 mm dimensions requires 500 mg of deuterated polymer while a similar interface can be obtained by spin-casting with approximately 5 mg. The time involved to achieve true diffusional equilibrium over millimeter-scale thicknesses is prohibitively long at typical welding temperatures and the presence of a background concentration of deuterium-labeled polymer... 

It is apparent that diffusion is approximetely 10,000 times as fast in air as in water. Since the rate at which a chromatogram can be developed is dependent on the rate at which diffusional equilibrium can be established between the mobile phase and the stationary phase, the rate at which gas chromatograms can be run is much greater than the rate at which liquid chromatograms can be run. Development rates for liquid chromatograms vary from 0.01 to 1.0 ml per min per sq.cm. In gas chromatography, flow rates range from 50 to 1,000 ml per min per sq cm. Because of these rapid flow rates, it is possible to obtain rapid elution of... 

Gradient IPN. In this case, the overall composition or crosslink density of the material varies from location to location on the macroscopic level. One way of preparing these materials involves partial swelling of polymer network I by the monomer II mix, followed by rapid polymerization before diffusional equilibrium takes place. Films can be made with polymer network I predominantly on one surface, and polymer network II predominantly on the other surface with a gradient composition existing throughout the interior. 

Note that diffusional equilibrium occurs only when both terms in (7.3.6) are zero A = 0 and dN = 0. Isothermal-isobaric diffusion cannot occur in the absence of a driving force that is, we carmot have dN 0 with A = 0. However, we can observe metastable equilibria in which a finite driving force exists (A 0), but apparently no diffusion takes place (dNj = 0). As an example, such diffusional metastabilities can occur when the ptue substance can condense into more than one kind of solid phase. Then, on bringing two forms of the solid into contact at different states, the molar Gibbs energies of the two phases differ, but the rate of diffusion in solids can be so small that the metastability may persist over significantly long times. 

Since the diffusional equilibrium criterion (7.3.12) applies separately to each term in (7.3.10), we must have, at equilibrium, dN,- = 0 for every component i. This means that diffusional equilibrium requires not only the absence of diffusion of any component i dNi = 0) but, in addition, the absence of any driving force for diffusion of any component (A,- = 0). We never observe diffusion (dN 0) in the absence of a gradient in the chemical potentials (A, = 0) this cannot occur even if the diffusion is coupled, for a zero driving force for component i disrupts any coupling for that component. 

Thermod3mamic equilibrium encompasses thermal, mechanical, and chemical equilibrium. Chemical equilibrium, in turn, includes both diffusional and reaction equilibrium. In this section we have considered only nonreacting systems, and so, at this point, we have developed only the criteria for thermal, mechanical, and diffusional equilibrium criteria for reaction equilibrium are given in 7.6. 

Can the criteria (7.3.11) for diffusional equilibrium, which require equality of the chemical potentials, be reconciled with the general criterion for isothermal-isobaric equilibrium, namely dG = 0 (7.1.40) ... 

The objective here is to show that the diffusional equilibrium criteria (7.3.11) are a consequence of the more general equilibrium criterion (7.1.40) that applies to any NPT system, including systems containing more than one phase. 

Open-system processes may include chemical reactions, diffusional mass transfer, and energy transfer across system boundaries. All such processes must satisfy the open-system form of the combined laws. When these processes are all complete and equilibrium is established, then it is the equality in (7.5.10) that applies. However, this statement is only a necessary condition it is not sufficient. The necessary and sufficient conditions for equilibrium are that each term in (7.5.10)—including each term in each sum—must be zero. In other words, the system must simultaneously satisfy the criteria already discussed for thermal equilibrium (7.3.3), mechanical equilibrium (7.3.5), diffusional equilibrium (7.3.11) or (7.3.12), and reaction equilibrium (7.6.3). 

In the previous chapter we derived criteria for identifying equilibrium states for example, in a closed system at fixed T and P, the equilibrium state is the one that minimizes the Gibbs energy. That minimization is equivalent to satisfying the equality of component fugacities. More generally, we derived criteria for thermal, mechanical, and diffusional equilibrium in open systems. But although those criteria can be used to identify equilibrium states, they are not always sufficient to answer the question of observability. Observability requires stability. Thermodynamic states can be stable, metastable, or unstable a stable equilibrium state is always observable, a metastable state may sometimes be observed, and an unstable state is never observed. 

Along any pure-fluid, subcritical isotherm, the spinodal separates unstable states from metastable states. At the other end of an isotherm s metastable range, metastable states are separated from stable states by the points at which vapor-Uquid, phase-equilibrium criteria are satisfied. Those criteria were stated in 7.3.5 the two-phase situation must exhibit thermal equilibrium, mechanical equilibrium, and diffusional equilibrium. Since we are on an isotherm, the temperatures in the two phases must be the same, and the thermal equilibrium criterion is satisfied. 

For a pure fluid, diffusional equilibrium wfil occur when there is no net driving force for diffusion of material from one phase to the other. This occurs when... 

To decide among these possibilities we need a stability criterion for mixtures at fixed T, P, and fugacity Equivalently, we can develop the criterion in terms of T, P, and the chemical potential, then convert it to fugacities at the end. Imagine a one-phase binary mixture surrounded by a reservoir that imposes its temperature, pressure, and chemical potential on the system. The latter is accomplished by a semi-permeable membrane that separates the system from the reservoir. The membrane allows molecules of component 1 to pass, but it blocks passage of molecules of component 2. When diffusional equilibrium is established, the value of the chemical potential Gi is the same in the system and in the reservoir. The extensive state of the system is identified by giving values for the fixed quantities T, P, Gj, and N2. 

Island boundary motion (IBM) examines the disappearance of islands in the vicinity of a region having no islands. Masked deposition followed by weak annealing delineates a clean region from an island-covered region. Islands near the boundary slowly lose mass to the clean region by diffusion, while the other islands remain in diffusional equilibrium with each other [95Bekl],... 

The rate of this and many other such phase boundary reactions depends upon the instantaneous state of the surface. For tarnishing processes, this generally means that the rate depends upon the instantaneous activities of the components at the phase boundary as well as upon the temperature. As long as diffusional equilibrium is maintained, and the outer phase boundary reaction alone is rate-controlling, the activity of the metal at the phase boundary between oxidation product and gas is constant and equal to one. There are indications [50] that the electronic defects can particularly influence the rate of dissociation of the gases at the phase boundary between oxidation product and gas. Use is made of this property of solid surfaces in the field of heterogeneous catalysis [4]. Since the defect concentration is determined by the activities of the components in the reaction product, it is understandable that the rate of the phase boundary reaction should, in general, depend upon the component activities in the reaction product at the phase boundary. 

All the equations are valid for arbitrarily large displacement from diffusional equilibrium. The driving force for evolution towards the stationary state, (8.14), is not linearly related to the flux of that evolution, (8.11). 

The derivation starts from the known Gibbsian condition of the diffusional equilibrium of the fluid in the field of external forces... 

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