Diffusion thermodynamic factor

The performance of adsorption processes results in general from the combined effects of thermodynamic and rate factors. It is convenient to consider first thermodynamic factors. These determine the process performance in a limit where the system behaves ideally i.e. without mass transfer and kinetic limitations and with the fluid phase in perfect piston flow. Rate factors determine the efficiency of the real process in relation to the ideal process performance. Rate factors include heat-and mass-transfer limitations, reaction kinetic limitations, and hydro-dynamic dispersion resulting from the velocity distribution across the bed and from mixing and diffusion in the interparticle void space. 

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 ... 

When it comes to the equilibration of water concentration gradients, the relevant transport coefficient is the chemical diffusion coefficient, Dwp. This parameter is related to the self-diffusion coefficient by the thermodynamic factor (see above) if the elementary transport mechanism is assumed to be the same. The hydration isotherm (see Figure 8) directly provides the driving force for chemical water diffusion. Under fuel-cell conditions, i.e., high degrees of hydration, the concentration of water in the membrane may change with only a small variation of the chemical potential of water. In the two-phase region (i.e., water contents of >14 water molecules... 

The quantum thermodynamic factor S is the quantum correction to the Kramers-Grote-Hynes classical result in the spatial diffusion limited regime, derived by Wolynes " ... 

In comparison with the qualitative description of diffusion in a binary system as embodied by Eqs. (11), (12) or (14), the thermodynamic factors are now represented by the quantities a, b, c, and d and the dynamic factors by the phenomenological coefficients which are complex functions of the binary frictional coefficients. Experimental measurements of Dy in a ternary system, made on the basis of the knowledge of the concentration gradients of each component and by use of Eqs. (21) and (22), have been reviewed 35). Another method, which has been used recently36), requires the evaluation of py from thermodynamic measurements such as osmotic pressure and evaluation of all fy from diffusion measurements and substitution of these terms into Eqs. (23)—(26). 

These assumptions, however, oversimplify the problem. The parent (A,B)0 phase between the surface and the reaction front coexists with the precipitated (A, B)304 particles. These particles are thus located within the oxygen potential gradient. They vary in composition as a function of ( ) since they coexist with (A,B)0 (AT0<1 see Fig. 9-3). In the Af region, the point defect thermodynamics therefore become very complex [F. Schneider, H. Schmalzried (1990)]. Furthermore, Dv is not constant since it is the chemical diffusion coefficient and as such it contains the thermodynamic factor /v = (0/iV/01ncv). In most cases, one cannot quantify these considerations because the point defect thermodynamics are not available. A parabolic rate law for the internal oxidation processes of oxide solid solutions is expected, however, if the boundary conditions at the surface (reaction front ( F) become time-independent. This expectation is often verified by experimental observations [K. Ostyn, et al. (1984) H. Schmalzried, M. Backhaus-Ricoult (1993)]. 

A/iAg as a function of time with a single and spatially fixed sensor at , or one can determine D with several sensors as a function of the coordinate if at a given time [K.D. Becker, et al. (1983)]. An interesting result of such a determination of D is its dependence on non-stoichiometry. Since >Ag = DAg d (pAg/R T)/d In 3, and >Ag is constant in structurally or heavily Frenkel disordered material (<5 1), DAg(S) directly reflects the (normalized) thermodynamic factor, d(pAg/R T)/ In 3, as a function of composition, that is, the non-stoichiometry 3. From Section 2.3 we know that the thermodynamic factor of compounds is given as the derivative of a point defect titration curve in which nAg is plotted as a function of In 3. At S = 0, the thermodynamic factor has a maximum. For 0-Ag2S at T = 176 °C, one sees from the quoted diffusion measurements that at stoichiometric composition (3 = 0), the thermodynamic factor may be as large as to 102-103. 

The primary difference between D and D is a thermodynamic factor involving the concentration dependence of the activity coefficient of component 1. The thermodynamic factor arises because mass diffusion has a chemical potential gradient as a driving force, but the diffusivity is measured proportional to a concentration gradient and is thus influenced by the nonideality of the solution. This effect is absent in self-diffusion. 

Inherent structure analysis of diffusion via molecular dynamics of a deeply supercooled binary Lennard-Jones fluid have provided renewed impetus to the decisive role played by thermodynamic factors [52,53]. The location of the mode crossover temperature and the onset of super-Arrhenius behavior were related to the static structure of the liquid via the potential energy hypersurface [52,53],... 

In principle the mobility B and therefore the corrected diffusivity D0 are also concentration-dependent, so Eq. (12) does not necessarily predict quantitatively the concentration dependence of D even for a system where the isotherm obeys the Langmuir equation. Nevertheless, the concentration dependence of B is generally modest compared with that of the thermodynamic factor, so a monatonic increase in diffusivity with adsorbed-phase concentration is commonly observed (Fig. 5). Clearly in any attempt to relate transport properties to the physical properties of the system it is important to examine the corrected, diffusivity D0 (or the mobility B) rather than the Fickian diffusivity, which is in fact a product of kinetic and thermodynamic factors. 

In terms of Eq. (1), the driving force is ApA and the resistance, f2 = L/Pa. Although the effective skin thickness L is often not known, the so-called permeance, PA/L can be determined by simply measuring the pressure normalized flux, viz., Pa/L = [flux of A]/A/j>a, so this resistance is known. Since the permeability normalizes the effect of the thickness of the membrane, it is a fundamental property of the polymeric material. Fundamental comparisons of material properties should be done on the basis of permeability, rather than permeance. Since permeation involves a coupling of sorption and diffusion steps, the permeability is a product of a thermodynamic factor, SA, called the solubility coefficient, and a kinetic parameter, DA, called the diffusion coefficient. 

D0 = RTMa is the intrinsic or corrected diffusion coefficient is the thermodynamic factor... 

Wagner factor — or thermodynamic factor, denotes usually the - concentration derivative of -> activity or - chemical potential of a component of an electrochemical system. This factor is necessary to describe the - diffusion in nonideal systems, where the - activity coefficients are not equal to unity, via Fick s laws. In such cases, the thermodynamic factor is understood as the proportionality coefficient between the selfdiffusion coefficient D of species B and the real - diffusion coefficient, equal to the ratio of the flux and concentration gradient of these species (chemical diffusion coefficient DB) ... 

Another important case relates to ambipolar - diffusion, when a flux of neutral species is driven by a chemical potential gradient In this case, the thermodynamic factor is usually written in similar form (e.g., d[iB/dcB or d In aB/dlncB), and may comprise additional multipliers depending on particular formulae the concentration... 

For nonideal systems, intermolecular interactions may be simplified by introducing the activities into the diffusion potentials. Deviations from ideal behavior can be estimated by the Fick and Maxwell-Stefan dififusivities and the thermodynamic factor. 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

Table 7.2 shows the viscosity, mutual diffusion coefficient, and thermodynamic factor for aqueous solutions of ethylene glycol and polyethylene glycol (PEG) at 25°C the diffusivity decreases considerably with increasing molecular weight, while the viscosity increases. Table 7.2 shows the thermal diffusion ratios for liquids and gases at low density and pressure the thermal diffusion ratios are relatively larger in liquids. 

Table 7.2a. Viscosities, mutual diffusion coefficients, and thermodynamic factors for aqueous solutions of ethylene glycol and PEG at 25°Ca...

We can describe the degree of coupling q and the thermal diffusion ratio of component 1 KTl in terms of the transport coefficients and thermodynamic factor (F)... 

Concentration effects on the heats of transport and the thermal diffusion ratio of chloroform with various alkanes at 30°C and 1 atm are seen in Table 7.6. Table 7.7 shows the experimental heats of transport at various concentrations and at temperatures 298 and 308 K for binary mixtures of toluene (1), chlorobenzene (2), and bromobenzene (3) at 1 atm. The absolute values of heats of transport decrease gradually as the concentrations of the alkane increase. Table 7.7 also contains values of cross coefficients obtained from easily measurable quantities and the thermodynamic factor. 

RT(wk/ck)(5/3x)ck where wk denotes the thermodynamic factor d In ak/d In ck and ak the activity—finally determined by the concentration gradient. If, however, the chemical potential is virtually constant, as it is the case for systems with a high carrier concentration (metals, superionic conductors), zkF(3/3x)< remains as driving force. For the linear approximation to be valid A must be sufficiently small. Experimental experience confirms the validity of Fick s and Ohm s law (that immediately follow from Table 3 for diffusion and electrical transport) in usual cases, but questions the validity of the linear relationship Eq. (96) in the case of chemical reactions. For a generalized transport we will use in the following the relation 173,178,181... 

The chemical diffusion coefficient includes, as we know from the formal treatment in Section VI..3iv., both an effective ambipolar conductivity and an effective ambipolar concentration. The latter parameter is determined by the thermodynamic factor which is large for the components but close to unity for the defects. 

Adsorption and diffusion of linear and branched Ce alkanes in silicalite-1 were investigated by Zhu et al. (34,35). They also developed a mathematical model taking into account the thermodynamical factor for intercrystalline diffusivities, enabling the determination of intracrystalline diffusivity from the uptake curve operated outside the linear adsorption range, van Donk et al. (36) also made transient uptake measurements to investigate the diffusivity of -hexane in Pt/H-mordenite. 

---