Dahm Equation

The K-M equation was derived for a matrix of infinitesimally small particles. Because the numerator of the K-M function is the absorption coefficient,/(i ) varies linearly with the concentration of each component of the sample. However, it is known that the K-M equation is only an approximation. The Dahm Equation (3.83) may be expected to give a more exact solution for diffuse reflectance in... 

The Dahm equation is frequently expressed in terms of an Absorption-Remission function A R, T) which has as one of its characteristics that it has the same value for any thickness of a sample. 

We will begin our series of examples by assuming that we are mixing particles that have a drastically different absorption fraction but the same remission fraction. The plot in Figure 3.6a shows how f(Roo) (denoted there as K-M) and a/r vary as a function of the particle fraction of graphite in a mixture of graphite (infinite absorption) and NaCl (zero absorption), with each assumed to have a remission fraction of 0.04. The value of 0.04 was chosen because it is the specular reflectance at normal incidence from the surface of a planar sample with a refractive index of 1.5. The value of f(Roo) was calculated from the Dahm Equation (3.84). Because the function A(/ , T) is constant for... 

FIGURE 3.11 Reflectance measured (diamonds) by image analysis using visible light compared to reflectance calculated (line) from the Dahm equation for mixtures of wheat and rape seed meal. (Reproduced from D. J. Dahm and K. D. Dahm, Near-Infrared Technology in the Agriculture and Food Industries, 2nd edn. (R Williams and K. Norris, eds.), St. Paul, MN, pp. 1-17 (2001) by permission of American Association of Cereal Chemists, Inc. Copyright 2001). 

In the model used to fit these experimental data, we assumed that the remission at the left-hand side of the chart is from an opaque layer, although the opaque layer of pure rape seed is almost certainly more than one particle thick because of the effect of voids. In a sample with voids, an opaque layer must be at least two particles thick. The fractions for a hypothetical layer were calculated from the assumed characteristics of the individual components using Equation (3.84) and Equation (3.85). The Roo values for an infinitely thick sample making up the line in Figure 3.12 were calculated by applying an inverse form of the Dahm equation... 

These assigned values may be quite unreasonable for remission from a layer that contained nothing but those particles. This is a curve fitting exercise. We are manipulating four parameters to fit a line. There are many combinations of these four parameters that give the same line. Presumably, none of the sets will be the properties of a single layer of particles, because we have not included voids in the model, and a layer of particles, one particle thick, would certainly contain voids. Nonetheless, this exercise illustrates the capability of the Dahm equation and the representative layer theory to describe particulate systems, even when there is very high absorption, a place where there... 

We believe these examples show both the usefiilness of the representative layer theory in describing particulate samples. Further we have illustrated the applicability of the Dahm equation to situations where continuous theories do not work well. Finally, we have tried to make clear that, however useful the theory may be in explaining what we observe, we have not created an easy way to determine particle properties, or to correct simply absorption data so it will be linear with analyte concentration. 

Beginning with the properties of the individual particles in a mixture, the representative layer theory gives a way of calculating the properties of a layer of particles. The merging of the continuous and discontinuous approaches, as embodied in the Dahm equation, gives a way to calculate the spectroscopic properties of a sample from that of such a layer. 

Kawano et al. used a microelectrode technique and the Dahms-Ruff equation to explore the diffusion mechanism of the iodide/triiodide redox couple and explain the physical diffusion accompanying an exchange reaction [22]. They showed that when high concentrations of iodide and tiiiodide are added to a RTCL as the solvent, the diffusion coefficient derived from the exchange reaction, expressed by / + —> /f + / , become significant and superior to the simple physical dif-... 

H+ (H3O"1") discharge occurs and the pH at the electrode surface increases. Dahms and Croll [68] derived an equation to evaluate the surface pH as a function of the bulk pH. Haris [69] calculated the change in surface pH during the electrodeposition of a bivalent metal. 

Dahms (1968) and Botar and Ruff (1985) studied exchange reactions such as those represented by Equation (2.6), stating that such processes can be described in terms of a second-order reaction kinetics, so that the apparent diffusion coefficient, D pp, measured in electrochemical experiments (e.g., CA) under diffusion-controlled conditions, can be expressed by ... 

Under these circumstances, the apparent rate at which Q appears to move through the film from electrode to the outer boundary of the film depends upon the rate of the electron-transfer reaction between P and Q. Considerations of analogous reactions in homogeneous solution showed that such a process is equivalent to diffusion (76, 77). The apparent diffusion coefficient observed for a species. Dp, is composed of contributions from the physical movement of the species (governed by its translational diffusion coefficient, D) and the electron-transfer process. When bimolecular kinetics apply and the species can be considered as points, then Dp can be estimated from the Dahms-Ruff equation. 

FIGURE 8-14b The velocity component field (x.t) along the local scalar gradient vector direction (x,t) obtained vis the scalar transport equation (Dahm et al. 1992). (Reprinted by permission of American Institute of Physics.)... 

The redox species can move about their equilibrium position of the irreversible attachment with the polymer (in the three-dimensional network the redox species are either covalently or electrostatically bound), which is referred to as bounded diffusion. In the opposite extreme (free diffusion), rapid molecular motion thoroughly rearranges the molecular distribution between successive electron hops, thus leading a mean-field behavior. The mean-field approximation presupposes that k > k, and leads to Dahms-RufF-type behavior for freely diffusing redox centers, but the following corrected equation should be applied [28] ... 

Kubelka-Munk solution [25] While the Schuster solution is for one particle, Kubelka-Munk generalized the absorption and scattering phenomena to the whole sample (K and S) and solved the radiation transfer equation in a different way than Schuster. More information about the derivation is provided in Griffiths and Dahm [23]. The solution is... 

Recently, the discontinuous approach has been applied by Dahm and Dahm [62,63] to the two-flux results obtained from the radiation transfer equation. This has resulted in being able to recast the results obtained from the continuous approach in terms of a layer of particles. This has been termed the representative tayer theory. In the discussion of deviations from the K-M equation, we gave without proof an analogous mathematical expression reached by the discontinuous treatment (Equation (3.65)). In this section we give the derivation of some of the more important formulas in the discontinuous treatment. 

Dahm and Dahm considered what happens when the thickness of the sample is doubled or halved, and computed the total backward flux, i co for an infinitely thick sample by an iterative solution of the latter set of equations. They showed that if the sample consists of an infinite number of layers, each with a forward flux of U, a backward flux of r and absorption of ai. 

Low values for the positive and negative charge carrier mobility have been reported for solid helium (Dahm, 1986). The dependence of the mobilities on temperature is thermally activated (see Figure 7). Increase of external pressure reduces the mobilities. These experimental facts indicate that an ionic-type transport occurs in this solid. The diffusion coefficients, D , of the carriers follow Equation 111 in... 

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