Diffusion isothermal-isobaric

The type of treatment described here was originally introduced by Scott and Dullien [4], who confined attention to isothermal isobaric diffusion in binary mixtures. Similar equations were independently published shortly after by Rothfeld [5], and the method was later extended to multi-component mixtures by Silveston [6], Perhaps the most complete exposition is given by Mason and Evans [7],... 

The simplest system used for diffusional analysis is that of an isothermal, isobaric binary system where the micromolecular solvent (H20) is designated as component 1 and the solute as component 2. Thus, for concentration gradients of these components, we may measure the net flux of solute across an arbitrary plane or boundary due to the relaxation of the concentration gradient. The interdiffusional flux in a binary liquid mixture is commonly described as mutual diffusion. 

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

The Kirkendall effect in metals shows that during interdiffusion, the relaxation time for local defect equilibration is often sufficiently short (compared to the characteristic time of macroscopic component transport) to justify the assumption of local point defect equilibrium. In those cases, the (isothermal, isobaric) transport coefficients (e.g., Dh bj) are functions only of composition. Those quantitative methods introduced in Section 4.3.3 which have been worked out for multicomponent diffusion can then be applied. 

In 2002, Morrow and Maginn presented an all-atom force field for [C4mim][PF6] using a combination of DFT calculations (B3LYP/6-311+G ) and CHARMM 22 parameter values [13]. MD simulations were carried out in the isothermal-isobaric ensemble at three different temperatures. The calculated properties contained infrared frequencies, molar volumes, volume expansivities, isothermal compressibilities, self-diffusivities, cation-anion exchange rates, rotational dynamics, and radial distribution functions. These thermodynamic properties were found to be in good agreement with available experimental values [13]. 

The basic empirical relation to estimate the rate of molecular difussion, first postulated by Fick (1855) and, accordingly, often referred to as Fick s first law, quantifies the diffusion of component A in an isothermal, isobaric system. According to Fields law, a species can have a velocity relative to the mass or molar-average veloc-... 

Write down Fick s first law for diffusion in a binary, isothermal, isobaric mixture. 

At about the same time that Maxwell and Stefan were developing their ideas of diffusion in multicomponent mixtures, Adolf Fick and others were attempting to uncover the basic diffusion equations through experimental studies involving binary mixtures (Fick, 1855). The result of Fick s work was the law that bears his name. The Fick equation for a binary mixture in an isothermal, isobaric system is... 

Here we deduce bounds on the directions of irreversible transfers across the interface in Figure 7.4. We consider six processes workfree constant-mass heat transfer, adiabatic constant-mass work, isobaric constant-mass heat transfer, isothermal constant-mass work, isothermal-isobaric diffusion, and adiabatic-workfree diffusion. 

Isothermal-isobaric single-component diffusion. Now consider both phases to contain samples of the same pure component, and let the interface between them be thermally conducting, movable, and permeable. Adjust the reservoirs so the phases have the same T and P then we have T = TP = constant and P = pP = constant. The process is diffusion of the pure component across the interface. Under these restrictions, the combined laws (7.2.13) reduce to... 

For isothermal-isobaric diffusion of pure substance 1, the combined laws (7.2.13) reduce to... 

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. 

In addition, the relationship between the composition gradients and the molar flow densities must be considered. Indeed, for negligible surface diffusion, the dusty-model equations for isothermal/ isobaric diffusion and reaction processes become ... 

Composttton Dependence of Dab The diffusion coefficient of a binary solution is a flinction of composition. If the tnie driving force for ordinaiy diffusion is the isothermal, isobaric gradient of chemical potential, then the binary diffiisivity can be written in the form... 

The Nemst-Planck equation is often employed by practitioners because of its similarity to Pick s law and its convenient separation of diffusion and migratiOTi terms. It should be borne in mind, however, that the theory is inconsistent with the basic requirements of irreversible thermodynamics [6, 7]. Nemst-Planck theory uses n + k properties to characterize transport in an isothermal, isobaric n-species system containing... 

Therefore from eq.(1.7) we derive that in the case of a multicomponent system in isothermal isobaric conditions and with gradients of chemical potentials, the forces which drive the diffusion process are the gradients of chemical potentials of the various components. 

Using eq.(1.2),the phenomenological relationships for isothermal isobaric diffusion process in an r - components system is... 

For simplicity, isothermal and isobaric conditions are assumed here, i.e.,Vf = Vp = 0.) It is easy to verify that the sum all of the mass-diffusive fluxes j is zero, as it must be. 

In Eqns. (4.41) and (4.42), we should have marked z and c with an index k, designating the chemical nature of the diffusing particles (components). This is necessary since diffusion of particles of the sort k occurs in a solvent and the system consists of at least two components. In the previous section, we showed that under isothermal and isobaric conditions, the diffusive flux of particles of type k in the solvent is... 

In a binary mixture, diffusion coefficients are equal to each other for dissimilar molecules, and Fick s law can determine the molecular mass flows in an isotropic medium at isothermal and isobaric conditions. In a multicomponent diffusion, however, various interactions among the molecules may arise. Some of these interactions are (i) diffusion flows may vanish despite the nonvanishing driving force, which is known as the mass transfer barrier, (ii) diffusion of a component in a direction opposite to that indicated by its driving force leads to a phenomenon called the reverse mass flow, and (iii) diffusion of a component in the absence of its driving force, which is called the osmotic mass flow. 

Assume that a simple film model exists for the mass transfer, equilibrium is established at the gas-liquid interface, and the diffusion occurs at isobaric and isothermal conditions. Also assume that neither helium nor argon is absorbed so that N2=N3 = 0. Then, the Maxwell-Stefan equations for the diffusion of argon and helium are... 

A heterogeneous reaction A -> 2B with nth order kinetics. /rA = k( A (n > 0) takes place on a catalyst surface. The component A with initial concentration CA0 diffusses through a stagnant film on the catalyst surface at isothermal and isobaric conditions. Assume one-dimensional diffusion, and determine the concentration profile of component A within the film of thickness 8 if the k is constant. 

Isothermal and isobaric conditions. Pick law describes diffusion in the membrane. External mass transfer limitations are negligible. The model is solved for given reactant concentrations at the opposite membrane sides. 

The equation (2.338) for vanishing average molar velocity u = 0, can also be simplified, when it is applied to binary mixtures of ideal gases. As a good approximation, at low pressures generally up to about 10 bar, the diffusion coefficient is independent of the composition. It increases with temperature and is inversely proportional to pressure. The diffusion coefficients in isobaric, isothermal mixtures are constant. In this case (2.338) is transformed into the equation for c = const... 

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