Totally absorbing boundary

In the equations written so far, the reaction between the particles of the geminate pair is not accounted for. This can be performed by introducing a boundary condition at a distance R, which is called the reaction distance. If the reaction on encounter is very fast, one can use the totally absorbing boundary condition... 

It can be shown that for the totally absorbing boundary condition, and for R r, the long-time behavior of the pair survival probability W(l) is described by... 

The methods used in geminate recombination to account for the reaction are also applicable to bulk ion recombination. In the simplest case of fully diffusion-controlled recombination, the reaction can be represented by the totally absorbing boundary condition... 

Using the totally absorbing boundary condition (47), the Debye [68] rate coefficient is recovered as... 

Both Hong and Noolandi [72] and Rice et al. [73] inverted Laplace transforms of order s 1/2 for small s to get the term in f 1/2. The nature of higher-order time dependence was not discussed. For the totally absorbing boundary condition (47), with Rice et al. showed... 

It can be seen that within the steady state limit the encounter distance is reduced by a factor of (Kact + 47r > fl)/Kact in comparison to Smoluchowski s totally absorbing boundary. 

Artificial boundaries have been defined as boundaries at which the occupation probability of one or more sites obeys special equations, not covered by the analytic expression r(n) and g(n) that apply to the other n. The variety of possible artificial boundaries is, of course, endless. A restricted class are the pure boundaries, defined as those in which only the end site requires a special equation. Another subdivision is in reflecting boundaries those which conserve total probability and absorbing boundaries, at which probability disappears. The latter definition requires comment. 

For c= 1 one obtains (7.1b) as a special case, distinguished by the fact that the probability for stepping into limbo is the same as for the other steps to the left. We shall refer to this special case of an absorbing pure boundary as totally absorbing. It is particularly important in connection with first-passage problems, see chapter XII, but it is by no means the only possible absorbing boundary. 

Measurements were recorded only after total absorbance values at several wavelengths were constant for at least two hours. For biphasic dispersions this was done at several temperatures and a solubility phase boundary was taken to be the temperature at which the absorbance due to scattering falls sharply to 0.02 or less. This is the synthetic method for determining solubilities (25). ... 

Fig. 23. Total internal redection occurs at boundary between denser, transparent phase (characterized by and rarer, absorbing phase (characterized by n )...

A realistic boundary condition must account for the solubility of the gas in the mucus layer. Because ambient and most experimental concentrations of pollutant gases are very low, Henry s law (y Hx) can be used to relate the gas- and liquid-phase concentrations of the pollutant gas at equilibrium. Here y is the partial pressure of the pollutant in the gas phase expressed as a mole fraction at a total pressure of 1 atm x is the mole fraction of absorbed gas in the liquid and H is the Henry s law constant. Gases with high solubilities have low H value. When experimental data for solubility in lung fluid are unavailable, the Henry s law constant for the gas in water at 37 C can be used (see Table 7-1). Gas-absorption experiments in airway models lined with water-saturated filter paper gave results for the general sites of uptake of sulfur dioxide... 

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