Concentration, component, diffusion

Diffusion is the molecular transport of mass without flow. The diffu-sivity (D) or diffusion coefficient is the proportionality constant between the diffusion and the concentration gradient causing diffusion. It is usually defined by Fick s first law for one-dimensional, binary component diffusion for molecular transport without turbulence shown by Eq. (2-149)... 

As shown in Fig. 33, the decreasing mechanism of this fluctuation is summarized as follows At a place on the electrode surface where metal dissolution happens to occur, the surface concentration of the metal ions simultaneously increases. Then the dissolved part continues to grow. Consequently, as the concentration gradient of the diffusion layer takes a negative value, the electrochemical potential component contributed by the concentration gradient increases. Here it should be noted that the electrochemical potential is composed of two components one comes from the concentration gradient and the other from the surface concentration. Then from the reaction equilibrium at the electrode surface, the electrochemical potential must be kept constant, so that the surface concentration component acts to compensate for the increment of the concen-... 

Another important leakage mechanism is a concentration-driven diffusive flux in contrast with the pressure-driven hydrodynamic flux considered previously. The flux of species A relative to the average molar velocity of all components is given by Fick s first law. 

As discussed in Section 8.2 the relation between the chemical diffusion coefficient and diffusivity (sometimes also called the component diffusion coefficient) is given by the Wagner factor (which is also known in metallurgy in the special case of predominant electronic conductivity as the thermodynamic factor) W = d n ajd In where A represents the electroactive component. W may be readily derived from the slope of the coulometric titration curve since the activity of A is related to the cell voltage E (Nernst s law) and the concentration is proportional to the stoichiometry of the electrode material ... 

In a mixture of two gases A and B provided that the total pressure, and hence the total molar concentration is constant, d[A]/dx and d[B]/dx are equal and opposite and A and B tend to diffuse in opposite directions. In equimolar counter-diffusion, the two components diffuse at equal and opposite rates =- ba- Equation (108) can then be integrated... 

If initial concentration gradients are in two complementing components, such as FeO and MgO (with the sum of FeO and MgO molar concentrations being a constant), and all other components have uniform concentration, the diffusion between the two components may be treated as interdiffusion, or effective binary diffusion. 

If the diffusion of a minor or trace element can be treated as effective binary (not uphill diffusion profiles) with a constant effective binary diffusivity, the concentration profile may be solved as follows. The growth rate u is determined by the major component to be n D ff, and is given, not to be solved. Use i to denote the trace element. Hence, w, and Dt are the concentration and diffusivity of the trace element. Note that Di for trace element i is not necessarily the same as D for the major component. The interface-melt concentration is not fixed by an equilibrium phase diagram, but is to be determined by partitioning and diffusion. Hence, the boundary condition is the mass balance condition. If the boundary condition is written as w x=o = Wifl, the value of Wi must be found using the mass balance condition. In the interface-fixed reference frame, the diffusion problem can be written as... 

Diffusion from spherical silicate samples can be studied readily by observing the loss of volatile components of the silicate as a function of time. Where the sphere is initially uniform in composition and subsequent vaporization allows one to assume a zero surface concentration of the vaporizing component thereafter, the solution to the differential equations and boundary conditions governing concentration-independent diffusion is given by... 

For elemental solids and stoichiometric compound crystals, the primary influence of irradiation on their kinetic behavior is due to the increase in Acv(s Ac,). We would expect the enhancement in the component diffusion to be in proportion to the increase in the (average) defect concentrations, thus influencing all homogeneous, inhomogeneous, and heterogeneous solid state reactions. 

During self-diffusion in a pure material, whether a gas, liquid, or solid, the components diffuse in a chemically homogeneous medium. The diffusion can be measured using radioactive tracer isotopes or marker atoms that have chemistry identical to that of their stable isotope. The tracer concentration is measured and the tracer diffusivity (self-diffusivity) is inferred from the evolution of the concentration profile. 

Generally, a set of coupled diffusion equations arises for multiple-component diffusion when N > 3. The least complicated case is for ternary (N = 3) systems that have two independent concentrations (or fluxes) and a 2 x 2 matrix of interdiffusivities. A matrix and vector notation simplifies the general case. Below, the equations are developed for the ternary case along with a parallel development using compact notation for the more extended general case. Many characteristic features of general multicomponent diffusion can be illustrated through specific solutions of the ternary case. 

Depending on the conditions of the diffusion experiment, the diffusivities defined by eq 1 are given particular names. For one-component diffusion under the influence of a concentration gradient, eq 1 reduces to... 

The description of diffusion may be complex in mixtures with more than two components. Diffusion coefficients in multicomponent mixtures are usually unknown, although sufficient experimental and theoretical information on binary systems is available. The Maxwell-Stefan diffusivities can be estimated for dilute monatomic gases from D k Dkl when the Fick diffusivities are available. The Maxwell diflfusivity is independent of the concentration for ideal gases, and almost independent of the concentration for ideal liquid mixtures. The Maxwell-Stefan diffusivities can be calculated from... 

When a component of a continuous fluid phase (a liquid solution or a gas mixture) is present in nonuniform concentration, at uniform and constant temperature and pressure and in the absence of external fields, that component diffuses in such a way as to tend to render its concentration uniform. For simplicity, let the concentration of a given substance be a function of only one coordinate x, which we shall take as the upward direction. The net flux Z] of the substance passing upward past a given fixed point Aq (i.e., amount per unit cross-sectional area per unit time) is under most conditions found to be proportional to the negative of the concentration gradient ... 

The mass transfer equation is written in terms of the usual assumptions. However, it must be considered that because the concentration of the more abundant species in the flowing gas mixture (air), as well as its temperature, are constant, all the physical properties may be considered constant. The only species that changes its concentration along the reactor in measurable values is PCE. Therefore, the radial diffusion can be calculated as that of PCE in a more concentrated component, the air. This will be the governing mass transfer mechanism of PCE from the bulk of the gas stream to the catalytic boundaries and of the reaction products in the opposite direction. Since the concentrations of nitrogen and oxygen are in large excess they will not be subjected to mass transfer limitations. The reaction is assumed to occur at the catalytic wall with no contributions from the bulk of the system. Then the mass balance at any point of the reactor is... 

Diffusion in multicomponent system is difficult to analyze. Transport of one component is affected by the presence of the other component due to their mumal interaction. This results in the coupling of fluxes. Thus, single-component diffusion equation cannot be used to predict diffusion in a multicomponent system. Greenlaw et al. [38] proposed a simple relationship in which the diffusion coefficients for components i and j are interdependent on both component concentrations ... 

At low concentrations, the diffusion coefficient is independent of the component concentration. At high concentrations, the diffusion coefficient depends strongly on the concentration of the solute, especially if the molecular volume and viscosity of the pure solute differ strongly from those of the solvent. Since a concentrated solution is no longer ideal, we can write... 

Consider multi-component diffusion of gases A and B through stagnant gas C (Gianakopulos, 1972 Cutlip and Shacham, 1999).[23] The governing equations for concentration of A and B are ... 

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