Diffusion ideal system

This expression can be used to describe both pore and solid diffusion so long as the driving force is expressed in terms of the appropriate concentrations. Although the driving force should be more correctly expressed in terms of chemical potentials, Eq. (16-63) provides a qualitatively and quantitatively correct representation of adsorption systems so long as the diffusivity is allowed to be a function of the adsorbate concentration. The diffusivity will be constant only for a thermodynamically ideal system, which is only an adequate approximation for a limited number of adsorption systems. 

A similar method of analysis of transient state diffusion kinetics has been propos-ed 144,1491 based on the consideration that, in any experiment, the kinetic behaviour of the system represented by S(X), DT(X) will generally deviate from that of the corresponding ideal system represented by Se, De in either of two ways (i) ideal kinetics is obeyed, but with a different effective diffusion coefficient D , where n = 1,2,... denotes a particular kinetic regime (Dn is usually deduced from a suitable linear kinetic plot) or (ii) ideal kinetics is departed from, in which case one is reduced to comparison between the (non-linear) experimental plot and the corresponding calculated ideal line. 

The action of diffusion is illustrated in Figure 3.5 for a system consisting of air, acetone and water. Let us assume we have a column in which fresh water continuously flows down the walls in the form of a film or thin layer. If we introduce a gaseous mixture of acetone and air to the column, the acetone will diffuse into the water phase. For the purposes of this example, assume the air to be insoluble in water. In our idealized system a stagnant... 

Pressure-dependent sorption and transport properties in polymers can be attributed to the presence of the penetrant in the polymer. Crank (32) suggested in 1953 that the "non-ideal" behavior of penetrant-polymer systems could arise from structural and dynamic changes of the polymer in response to the penetrant. As the properties of the polymer are dependent on the nature and concentration of the penetrant, the solubility and diffusion coefficient are also concentration-dependent. The concentration-dependent sorption and transport model suggests that "non-ideal" penetrant-polymer systems still obey Henry s and Fick s laws, and differ from the "ideal" systems only by the fact that a and D are concentration dependent,... 

For k>kf the adspecies mass transfer process is described by the diffusion Eq. (63). If the species migration in the subsurface region and the exchange with the gaseous phase occur fast, then k — l, therefore the boundary condition comprises the 3rd kind condition. Otherwise, it would be necessary to take into account the temporal evolution of the species in subsurface layers k , and the kinetic equations for these layers can contain the time derivatives. Most works devoted to mass transfer problems and also to the surface segregation of the alloy components [155,173]. The boundary conditions in the non-ideal systems are discussed in Ref. [174]. They require the use of equations for the pair functions of the type d(6,Jkq)/dx — 0. When describing the interphase boundary motion, the 3rd kind boundary conditions are also possible, although the 1st and the 2nd kind conditions are used more often. The latter are mainly applied to the description of many problems with species redistribution in the closed volume [175],... 

A more rigorous approach to calculating the diffusion coefficients has been adopted by Kikuchi [165], A binary substitution alloy (s = 3) has been considered with the vacancy mechanism of atom migration. He was the first to take account of the temporal correlations and to obtain expressions for the correlation cofactor fc in the non-ideal systems. The derived coefficients satisfy Onsager s reciprocal relations. 

The essential difference between distillation and absorption is that in the former, the vapour has to be introduced in each stage by partial vapourisation of the liquid, which is therefore at its boiling point, whereas in absorption the liquid is well below its boiling point. In distillation, there is diffusion of molecules in both directions, so that for an ideal system, equimolar counter-diffusion exists. In absorption, gas... 

Bosse [48] proposed a new model to predict binary Maxwell-Stefan diffusion coefficients Dij, based on Eyrings absolute reaction rate theory [49]. A correlation from Vignes [50] which was shown to be valid only for ideal systems of similar-sized molecules without energy interactions [51] was extended with a Gibbs-excess energy term... 

Other definitions of chemical diffusion coefficients were also suggested for various particular cases (e.g., see [iii, vi-viii]). In all cases, however, their physical meaning is related either to the ambipolar diffusion or to diffusion in non-ideal systems where the activity coefficients differ from unity. 

Equation (2.71) can be compared with Eq. (2.46) for the thermal conductivity of gases, and with Eq. (2.19) for the viscosity. For binary gas mixtures at low pressure, is inversely proportional to the pressure, increases with increasing temperature, and is almost independent of the composition for a given gas pair. For an ideal gas law P = cRT, and the Chapman-Enskog kinetic theory yields the binary diffusivity for systems at low density... 

This method provides the exact solutions for ideal systems at constant temperature and pressure. It is successful in describing diffusion flow in (i) nearly ideal mixtures, (ii) equimolar counter diffusion where the total flux is zero (Nt = 0), (iii) diffusion of one component through a mixture of n — 1 inert components, and (iv) pseudo-binary case and the diffusion of two very similar components in a third. 

Individual component efficiencies can vary as much as they do in this example only when the diffusion coefficients of the three binary pairs that exist in this system differ significantly For ideal or nearly ideal systems, all models lead to essentially the same results. This example demonstrates the importance of mass-transfer models for nonideal systems, especially when trace components are a concern. For further discussion of this example, see Doherty and Malone (op. cit.) and Baur et al. [AIChE J. 51,854 (2005)]. It is worth noting that there exists extensive experimental evidence for mass-transfer effects for this system, and it is known that nonequilibrium models accurately describe the behavior of this system, whereas equilibrium models (and equal-efficiency models) sometime... 

The diffusion limit will obscure very fast rates of electron transfer (/cobs = for et d) [16]. Even if electron transfer is slow with respect to diffusion ( obs = et), work accompanies the formation of the precursor complex and/or separation of the successor complex (this is especially prevalent when the reactants and/or products are charged). Work term contributions to the observed rate of reaction can overwhelm the intrinsic factors that govern the electron transfer event [19]. For this reason, excluding special circumstances [20-26], intermolecular reactions are not ideal systems for examining the mechanistic details of electron transfer. 

The intermolecular dipole-dipole relaxation in liquids is both of reorienta-tional and translational origin. The theoretical models have divided the problem into the translational diffusion of monatomic particles, and the reorientational motion is included as off-center effects. For that reason, the dipole-dipole relaxation in the idealize system of spherical particles has been simulated [25-27]. The simple theories were found to perform well if only the correct radial distribution was taken into account. MD simulations of more realistic systems of liquids... 

Greenlaw FW, Shelden RA, and Thompson EV. Dependence of diffusive permeation rates on upstream and downstream pressures V. Experimental results for the hexane/heptane (ideal) and toluene/ethanol (non-ideal) systems. J. Memb. Sci. 1977 2(4) 333-348. 

For non-ideal systems, on the other hand, one may use either D12 or D12 and the corresponding equations above (i.e., using the first or second term in the second line on the RHS of (2.498)). In one interpretation the Pick s first law diffusivity, D12, incorporates several aspects, the significance of an inverse drag D12), and the thermodynamic non-ideality. In this view the physical interpretation of the Fickian diffusivity is less transparent than the Maxwell-Stefan diffusivity. Hence, provided that the Maxwell-Stefan diffusivities are still predicable for non-ideal systems, a passable procedure is to calculate the non-ideality corrections from a suitable thermodynamic model. This type of simulations were performed extensively by Taylor and Krishna [96]. In a later paper, Krishna and Wesselingh [49] stated that in this procedure the Maxwell-Stefan diffusivities are calculated indirectly from the measured Fick diffusivities and thermodynamic data (i.e., fitted thermodynamic models), showing a weak composition dependence. In this engineering approach it is not clear whether the total composition dependency is artificially accounted for by the thermodynamic part of the model solely, or if both parts of the model are actually validated independently. 

The Nernst-Planck model is based on limiting laws for ideal systems. It accounts only for diffusion and electric transference of ions, not for electroosmotic solvent transfer in the ion-exchanger phase, swelling or shrinking of the ion-exchange material, variations of activity coefficients and diffu-sivities, and possible slow structural relaxation of the exchanger matrix. It also postulates the existence of individual diffusion coefficients for ions. 

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