Diffusion of a Multispecies Component

There are two methods to write the diffusion equation for a multispecies component. One is to write the diffusion equation for the conserved component, and then relate the species concentrations by the reaction(s). Using one-dimensional H2O diffusion as an example, the diffusion equation is Equation 3-22a  

That is, the concentration change of the component with time is due to the diffusive flux of the two species. A factor of is required for the OH species because two OH groups convert into one H2O molecule through the following reaction (Reaction 1-10)  

The second equation depends on whether equilibrium is rapidly reached locally between the species. If it is, then the equilibrium condition provides the third equation  

The equilibrium constant K depends on temperature and an approximate expression is /C = exp(1.876 3110/T), where Tis in K. Equations 3-78, 3-80a, and 3-80b can be solved simultaneous numerically (e.g., Zhang et al., 1991a,b) for three unknowns, CH20 CH20m 1-oh- If equilibrium is not reached, then 

Equation 3-80b would not be applicable and the third equation would involve the reaction and diffusion of a species, which is similar to the second way to mathematically describe the diffusion of a multispecies component. 

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The total concentration (w) of a multispecies component is independent of species interconversions of the t)q)e of Reaction 3-81, but is affected by the diffusion flux of individual species. Because each species may have a distinct dif-fusivity, the diffusion equation for w may be written as... 

In a silicate melt or aqueous solution, a component may be present in several species. The species may interconvert and diffuse simultaneously. For example, the H2O component in silicate melt can be present as at least two species, molecular H2O (referred to as H20m) and hydroxyl groups (referred to as OH) (Stolper, 1982a). The diffusion of such a multispecies component is referred to as multispecies diffusion (Zhang et al., 1991a,b). Starting from Equation 3-5d, the one-dimensional diffusion equation for this multispecies component can be written as... 

In summary, the diffusion behavior of both H2O and CO2 demonstrates the importance of understanding the role of speciation in diffusion, and the very different consequences due to that role. Diffusion of a single-species component (such as Ar) usually does not depend on its own concentration (when the concentration is low), but depends on the melt composition. For a multispecies component, speciation affects the diffusion behavior. For H2O, speciation makes the diffusion behavior very complicated even at low H2O concentrations, total H2O diffusivity still depends on H2O content (because the species concentrations are not proportional), in addition to the dependence on melt composition. If species concentrations are proportional to each other and hence to the total concentration of the component, then the diffusivity is independent of the concentration of the component, as in the case of CO2 diffusion. Many multispecies components probably satisfy this condition that the concentrations of... 

If a diffusion component is present as two or more different species, the diffusion of the component is often referred to as multispecies diffusion (Zhang et al., 1991a,b). Multispecies diffusion is distinguished from multicomponent diffusion in that in the former case, the multiple species are from one component. 

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