Diffusion Associated with Chemical Reaction

Credit Carter HG, Kibler KG Langmuir-type model for anomalous moisture diffusion in composite resins. 12(2) 118-131, copyright 1978 hy Journal of Composite Materials, Reprinted 

Typical weight gain data for two phase diffusion are shown in Fig. 3.2. 

In several circumstances when a solid is exposed to a specific harsh environment a chemical reaction takes place that transforms the atomic or molecular structure of the sohd and splits it into two phases. Such events may occur within polymers exposed to acids, glass fibers exposed to water, silicon exposed to oxygen, etc. 

In this case, the reaction-modified phase arises within the region containing the diffusing molecules, while the unreacted phase remains outside the region where diffusion takes place. These distinct phases are separated by a moving reaction front. 

In the one-dimensional case, this process is described by Fig. 3.3 sketched above (Ghez 1988 Cussler 2009). 

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Other growth processes are also derived from theory. They include those associated with chemical reactions to form condensed species taking place either at the particle surface or within the particle volume. The growth by surface reactions or a vapor diffusion-limited process is... 

We now pass to the explicit calculation of entropy production. We shall consider here the very important special case in which mechanical and thermal equilibrium are already established. Mechanical equilibrium excludes the production of entropy by viscous flow, while uniformity of temperature, which is necessary for thermal equilibrium, excludes the internal production of entropy arising from the transport of heat between two regions at different temperatures. Similarly we assume that diffusion equilibrium has been attained within each phase of the system. The only production of entropy which can take place in a system of this kind is that associated with chemical reactions, with the transport of matter from one phase to another, or in general with any change which can be expressed in terms of a reaction co-ordinate... 

Diffusion in biological stem is, however, mostly associated with chemical reactions. In such situations the simplest assumption that can be made regairding the diffusing stem is that within the diffusing space the rate of utilization of the diffusing solute is constant. l.e. independent of time and place. Equation (5) in such conditions will take the form... 

The method described here to account for variations of porosity and mass diffusivity associated with precipitation/dissolution of reactive solids provides coupling of chemical reactions to mass transport. The method does not allow the physically impossible exceedance of available pore volume by precipitated solids. 

A to products by considering mass transfer across the external surface of the catalyst. In the presence of multiple chemical reactions, where each iRy depends only on Ca, stoichiometry is not required. Furthermore, the thermal energy balance is not required when = 0 for each chemical reaction. In the presence of multiple chemical reactions where thermal energy effects must be considered becanse each AH j is not insignificant, methodologies beyond those discussed in this chapter must be employed to generate temperature and molar density profiles within catalytic pellets (see Aris, 1975, Chap. 5). In the absence of any complications associated with 0, one manipulates the steady-state mass transfer equation for reactant A with pseudo-homogeneous one-dimensional diffusion and multiple chemical reactions under isothermal conditions (see equation 27-14) ... 

Here we have used the approximation that can be replaced by Dj y and that variations of D y can be ignored within the averaging volume. The fact that only a single tortuosity needs to be determined by equations 1.152 and 1.153 represents the key contribution of this study. It is important to remember that this development is constrained by the linear chemical kinetic constitutive equation given by equation 1.113. The process of diffusion in porous catalysts is normally associated with slow reactions and equation 1.93 is satisfactory however, the first-order, irreversible reaction represented by equation 1.113 is the exception rather than the rule, and this aspect of the analysis requires further investigation. The influence of a non-zero mass average velocity needs to be considered in future studies so that the constraint given by equation 1.97 can be removed. An analysis of that case is reserved for a future study which will also include a careful examination of the simplification indicated by equation 1.117. 

Various isotope effects are the ultimate cause of natural variations in the distribution of the stable isotopes of nitrogen. There are two types of isotope effect (i) physical, and (ii) chemical. Physical effects are associated with processes such as freezing/melting, evap-oration/condensation, adsorbtion, diffusion, etc chemical effects occur in both inorganic and biochemical reactions and are the dominant reason for observed N-isotopic variations in living and sedimentary OM. Isotope effects are manifest as differences in the relative distribution of N and N between reactants and products, the result of such changes is commonly referred to as isotope fractionation. Processes like those outlined above typically have a characteristic fractionation associated with the reaction. The fundamental cause of the... 

The expression for shows, that the entropy flux for open systems consists of two parts the thermal flux associated with the heat transfer, and the flux due to diffusion. The second expression consists of four terms associated with, respectively, the heat transfer, diffusion, viscosity, and chemical reactions. The expression for the dissipative function a has quadratic form. It represents the sum of products of two factors a flux (specifically, the heat flux /, diffusion flux momentum flux n, and the rate of a chemical reaction and a thermodynamic force, proportional to gradient of some intensive variable of state (temperature, chemical potential, or velocity). The second factor can also include external force F]t and chemical affinity Aj. 

These are quite relevant in socio-economic systems. Variation of population in a particular region with time and variation of population density from point to point have their own importance. The first one may be called extensive variable and the second one may be called intensive variable. Spatio-temporal behaviour of population can be studied by the proper choice of variables. Similar phenomena in physico-chemical systems are easy to analyse since the important variables are diffusion coefficients of particular species and those associated with autocatalytic reaction. 

The second part deals with mixing phenomena associated with tnrbulence. In that part, notions relating to turbulence are first presented. The problems associated with dispersion and mixing in connection with chemical reactions are then considered. The key notion, from a fundamental standpoint, regards the interrelation between the phenomena of turbulence and that of molecular diffusion, the latter being the actnal cause for mixing that allows a chemical reaction to occur. 

The electrode reaction rate may be controlled by diffusion or by chemical reaction. The impedances associated with these cannot be derived so simply. All that needs to be said here is that they are not pure resistances, i.e. they include a reactive part which, for convenience, is called pseudocapacity to distinguish it from the true capacity of the double layer. The diffusion impedance is also often referred to as Warburg impedance . 

Several possible models can be discussed for the molecular basis of slow inhibition, but experimental evidence in support of one or the other is still lacking for glycosidases. A reversible chemical reaction at the active site, for example, formation of the cyclic imine 3 or a diffusion-controlled association with a trace of 3 in equilibrium with the 5-araino-5-deoxypyranose 1 can be precluded, because slow inhibition is also observed with 1-deoxynojirimycin and its analogs and with acarbose (see Section II,2,d) and indoli-... 

The problems of the constancy of a and the site of reaction are closely linked. It is very convenient to assume that the charge on the micellar head groups is extensively neutralized by counterions which bind specifically to the micellar surface. In this way micellar stability is associated with a balance between hydrophobic attractions between apolar groups and coulombic repulsions of the ionic head groups which will be reduced by favorable interactions with the counterions in both the Stem and the diffuse Gouy-Chapman layers. It is the behavior of the counterions which is important in considerations of their chemical reactivity. 

The process control of the post-exposure bake that is required for chemically amplified resist systems deserves special attention. Several considerations are apparent from the previous fundamental discussion. In addition for the need to understand the chemical reactions and kinetics of each step, it is important to account for the diffusion of the acid. Not only is the reaction rate of the acid-induced deprotection controlled by temperature but so is the diffusion distance and rate of diffusion of acid. An understanding of the chemistry and chemical kinetics leads one to predict that several process parameters associated with the PEB will need to be optimized if these materials are to be used in a submicron lithographic process. Specific important process parameters include ... 

The second main phenomena prodncing fractionations are kinetic isotope effects, which are associated with incomplete and nnidirectional processes like evaporation, dissociation reactions, biologically mediated reactions, and diffusion. The latter process is of special significance for geological purposes, which warrants separate treatment (Sect. 1.3.3). A kinetic isotope effect also occurs, when the rate of a chemical reaction is sensitive to atomic mass at a particular position in one of the reacting species. 

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