Wagners Equation

Considered here is the case where the interaction of gaseous oxygen with the oxide lattice can be represented by a chemical reaction of the form  

1 The notation adopted for defects is from Kroger and Vink [48]. 

The single particle flux of charge carriers, with neglect of cross terms between fluxes, is given by 

The charge carrier diffusing more rapidly causes a gradient in the electrical potential V((), in which the transport of carriers with opposite charge is accelerated. At steady state, no charge accumulation occurs. The fluxes of ionic and electronic defects are therefore related to each other by the charge baleince 

The limits of integration are the oxygen partial pressures maintained at the gas phase boundaries. Equation (10.10) has general validity for mixed conductors. To carry the derivation further, one needs to consider the defect chemistry of a specific material system. When electronic conductivity prevails, Eqs. (10.9) and (10.10) can be recast through the use of the Nemst-Einstein equation in a form that includes the oxygen self-diffusion coefficient Dg, which is accessible from ionic conductivity measurements. This is further exemplified for perovskite-type oxides in Section 10.6.4, assuming a vacancy diffusion mechcinism to hold in these materials. 

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Wagner equation Wagner number Wakamatsu reaction Waldhof fermentor Walkman Wallace plasticity Wallach procedure Wall baffles Wallboard Wall geometries Wallpaper paste Wallpaper pastes Wallpapers Wall plaster Walnut oil... 

The accurate representation of vapor pressure data over a wide temperature range requires an equation of greater complexity. The Wagner equation (eq. 22) expresses the reduced vapor pressure as a function of reduced temperature TIT/... 

The other modern equation used for correlation is the modified and linearized Wagner "" equation (2-30), which has the advantage that it will match critical data exactly, although it cannot be extrapolated above the critical point. The equation is also included with coefncients to several hundred compounds in the Technical Data Book—Petroleum Refining. 

The Dunwald—Wagner equation, based on the application of Ficks second law of diffusion into or out of a sphere (radius r) [477], can be written... 

Dunwald—Wagner equation, 70 dyes, effect on decomposition of Ag2C204, 265 -, AgN3, 266... 

The following table provides a numerical listing of the P-p-T surface for carbon dioxide in the region of interest for supercritical fluid extraction and chromatography. These data were calculated using an empirical equation of state (the Schmidt-Wagner equation),12 the parameters of which were determined from a fit of experimental P-V-T (pressure-volume-temperature) measurements.3 8 Note that the pressures are tabulated in bars for convenience. The appropriate SI unit of pressure is the megapascal (1 bar = 0.1 MPa). 

The equilibrium conditions in electrochemical systems are usually expressed in terms of electrochemical potentials. For non-equilibrium systems, the gradient of chemical and/or electrochemical potential is a driving force for flux of particles i. See also Wagner equation, - Wagner factor and ambipolar conductivity, -+ On-sager relations. 

Similar approaches are used for most steady-state measurement techniques developed for mixed ionic-electronic conductors (see -> conductors and -> conducting solids). These include the measurements of concentration-cell - electromotive force, experiments with ion- or electron-blocking electrodes, determination of - electrolytic permeability, and various combined techniques [ii-vii]. In all cases, the results may be affected by electrode polarization this influence should be avoided optimizing experimental procedures and/or taken into account via appropriate modeling. See also -> Wagner equation, -> Hebb-Wagner method, and -> ambipolar conductivity. 

See also - chemical potential, -> electrolytic domain, - gas titration coulometric, - Wagner equation. 

Wagner equation — denotes usually one of two equations derived by -> Wagner for the flux of charged species Bz under an -> electrochemical potential gradient, and for the - electromotive force of a -> galvanic cell with a mixed ionic-electronic -> conductor [i-v] ... 

An important form of the second formula, which is also denoted as the Wagner equation, can be obtained by integration ... 

Sometimes the relationship between the rate of the parabolic metal-oxidation process (dnox/dt), limited by oxygen transport through a dense oxide film, is also called the Wagner equation [vi] ... 

Concepts of local equilibrium and charged particle motion under - electrochemical potential gradients, and the description of high-temperature -> corrosion processes, - ambipolar conductivity, and diffusion-controlled reactions (see also -> chemical potential, -> Wagner equation, -> Wagner factor, and - Wagner enhancement factor). 

Theoretical and semi-empirical equations were derived for gas emulsions (as well as for suspensions of non-conducting spherical particles and O/W emulsion), specifying Eq. (8.33) . A relation for coefficient B can be derived from Maxwell-Wagner equation [45,46]... 

In any case, the new parameter of the generalized invariant has now to be obtained from additional experiments in elongational flows. But the Wagner equation can now be written in the unified form ... 

This section presents different results obtained using the Wagner equation with the form of the damping function of equation (33) together with the generalized invariant of equation (25). It is primarily devoted to comparisons between predictions and experimental results in shear and uniaxial elongational flows described in paragraph 3 for LD. 

These constitutive equations differ in their mathematical form the Wagner equation is an integral equation whereas the Phan Thien Tanner model is a differential one. 

The Wagner equation finds its theoretical basis in the derivation of the more general K-BKZ equation. Unfortunately, it loses part of its original thermod3mamic consistency since, for simplification purposes, only the Finger strain measure is taken into account. Doing so, it is no more derivable from any potential function and additionally it does not predict second normal stress differences any more. 

Data from the triple point to the critical point can be correlated with either a modified form of the Wagner equation [Wagner, W, "A New Correlation Method for Thermoctynamio Data Applied to the Vapor-Pressure Curve of Argon, Nitrogen, and Water, J.T.R. Watson (trans. and ed.), lUPAC Thermodynamic Tables Project Centre, London, 1977 Ambrose, D.,/. Chem. Thermodyn., 18 (1986) 45 Ambrose, D., and N. B. Ghiassee,/. Chem. Thermodyn., 19 (1987) 903, 911]... 

Generally, E in Eq. (2-22) is assigned a value of 6, but values of 2 or 1 have also been used, particularly when correlating low-temperature data. While the Wagner equation can be used to correlate most fluids over the whole liquid range, a fifth term is often required for alcohols [Poling, B. E., Fluid Phase Equil., 116 (1996) 102] ... 

C. Wagner, Equations for transport in solid oxides and sulfides of transition metals. Prog. Solid State Chem., 10(1) (1975) 3-16. 

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