Uptake rates, Diffusion

Sorption Rates in Batch Systems. Direct measurement of the uptake rate by gravimetric, volumetric, or pie2ometric methods is widely used as a means of measuring intraparticle diffusivities. Diffusive transport within a particle may be represented by the Fickian diffusion equation, which, in spherical coordinates, takes the form... 

Figure 11. Uptake rates of inorganic Hg (a) and of methylmercury (b) by a marine alga as a function of the octanol-water distribution ratio of the Hg-species under various conditions of pH and chloride concentrations. The neutral species HgCl and CH5HgClH diffuse through the membranes. Reprinted with permission from [79] Mason, R. P. et al. (1996). Uptake, toxicity, and trophic transfer in a coastal diatom , Environ. Sci Technol., 30, 1835-1845 copyright (1996) American Chemical Society...

Model calculations have demonstrated that active cells are surrounded by zones containing substrate concentrations lower than those of the bulk liquid [12-14], This concentration gradient results from the dynamic interplay between the rates of substrate uptake and diffusion through the diffusion layer surrounding the cell (see [15] for details). Boone et al. [13] developed a model using spherical coordinates that allows calculation of the diffusive substrate flux to a suspended spherical cell. In their model calculations, the cell surface concentration was set to arbitrary values between zero and about half of the bulk concentration. It... 

Micelles forming above the c.m.c. incorporate hydrophobic molecules in addition to those dissolved in the aqueous phase, which results in apparently increased aqueous concentrations. It has to be noted, however, that a micelle-solubilised chemical is not truly water-dissolved, and, as a consequence, is differently bioavailable than a water-dissolved chemical. The bioavailability of hydrophobic organic compounds was, for instance, reduced by the addition of surfactant micelles when no excess separate phase compound was present and water-dissolved molecules became solubilised by the micelles [69], In these experiments, bacterial uptake rates were a function of the truly water-dissolved substrate concentration. It seems therefore that micellar solubilisation increases bioavailability only when it transfers additional separate phase substrate into the aqueous phase, e.g. by increasing the rates of desorption or dissolution, and when micelle-solubilised substrate is efficiently transferred to the microorganisms. Theoretically, this transfer can occur exclusively via the water phase, involving release of substrate molecules from micelles, molecular diffusion through the aqueous phase and microbial uptake of water-dissolved molecules. This was obviously the case, when bacterial uptake rates of naphthalene and phenanthrene responded directly to micelle-mediated lowered truly water-dissolved concentrations of these chemicals [69]. These authors concluded from their experiments that micellar naphthalene and phenanthrene had to leave the micellar phase and diffuse through the water phase to become... 

Four strategies are generally employed to demonstrate mass transfer limitation in aquatic systems. Most commonly, measured uptake rates are simply compared with calculated maximal mass transfer rates (equation (17)) (e.g. [48,49]). Uptake rates can also be compared under different flow conditions (e.g. [52,55,56,84]), or by varying the biomass under identical flow conditions (e.g. [85]). Finally, several recent, innovative experiments have demonstrated diffusion boundary layers using microsensors [50,51]. Of the documented examples of diffusion limitation, three major cases have been identified ... 

Figure 11. Relative uptake rates as a function of carrier number in the diffusion-limited case. Calculations were performed according to the equations of Berg and Purcell [35] for a carrier radius of 1 nm on a cell with a radius of 1 pm...

The shape and length of the wave that propagates in the bed is related to the mathematical form of the uptake rate expression that is substituted into the last term in Eq. (9.10). Equation (9.15) is the pde describing diffusion of an adsorbate from the gas phase into the adsorbent This is but one of many diffusion based uptake models that might be substituted into the uptake rate term of Eq. (9.9) or (9.10). 

Equation 9.15, when solved for the case of macro-pore diffusion gives us, in the low loading Hmit, the famihar relationship that mass uptake is proportional to the square root of time. The same relationship can be derived for micro-pore diffusion as well. The solution to this equation can be used with the appropriate particle sizes to estimate diffusivity from uptake rate measurements. 

In this section we deal with the following two approximations for adsorption uptake rate and how to go about calculation of the relationship between the mass transfer coefficients (k, kK) that appear and the solutions to diffusion problems. 

The prerequisites of the evaluation of data characteristic of intracrystalline processes in the case of zeolite sorbents are discussed, along with the conditions under which diffusion can be compared to self-diffusion. Selected results of investigations carried out in the author s laboratory are given in order to demonstrate the consistency of sorption kinetic data with intracrystalline mobility data of single components on molecular sieves (HS). Various types of surface barrier which may influence the uptake rate are also described. 

If a NS monocrystal takes up a single component from a fluid phase and Intercrystalline transport does not influence the uptake rate, one should be aware of the possibility that, besides intracrystalline diffusion, the following processes may either contribute or even govern the uptake rate ... 

Henry s law equilibrium. If there is no reaction in the liquid phase (or it is slow relative to uptake and diffusion), the gas-liquid system eventually comes to equilibrium, which can usually be described by Henry s law discussed earlier. This does not reflect a lack of uptake of the gas at equilibrium but rather equal rates of uptake and evaporation i.e., it is a dynamic equilibrium (see Problem 12). The equilibrium between the gas-and liquid-phase concentrations is characterized by the Henry s law constant, H (mol L-1 atm-1), where II = [X /I. ... 

Finally, the agitation rate does not affect the uptake rate if the particle diffusion controls the process. However, the latter criterion may be not safe the agitation in solution may have attained its limiting hydrodynamic efficiency, so that a change in the agitation rate has no effect on the uptake rate even in film diffusion-controlled systems. 

In the above equations, H0, //p, and H are the plate-height contributions due to the finite particle size, solid diffusion, and liquid-film diffusion, respectively. CGS units are used in these equations. Obviously, the bigger the height of the plate, the higher the resistance to the diffusion and the lower the uptake rate. 

Calculation of a Theoretical Rate of Uptake Where Diffusion Through the Air is Rate Determining. Assuming that diffusion of the oxide vapors through the air is rate controlling, it should be possible to use Maxwell s equation to calculate the rate of uptake of the oxide vapors under varying conditions of pressure and temperature. The use of Maxwell s equation requires knowledge of the interdiffusion constant ( Di.2 ) of the oxide vapor in air. The interdiffusion constants of the vapor species studied here are not known, but they may be estimated by the Stefan-Maxwell equation (14) ... 

Except for an initial period where the experimentally measured uptake rate is about twice the calculated rate, the shapes of the two curves correspond quite well. This supports the hypothesis that the uptake rate, where the vapor condensation coefficient is small, is determined by the combined effect of a slow rate of surface reaction and slow diffusion of the condensate into the interior of the substrate. 

Wall-coated flow tube reactors have been used to study the uptake coefficients onto liquid and solid surfaces. This method is sensitive over a wide range of y (10" to 10 1). For liquids this method has the advantage that the liquid surface is constantly renewed, however if the uptake rate is fast, the liquid phase becomes saturated with the species and the process is limited by diffusion within the liquid, so that corrections must be applied [70,72,74]. Many experiments were designed to investigate the interaction of atmospheric species on solid surfaces. In this case the walls of the flow tube were cooled and thin films of substrate material were frozen on the wall. Most of the reaction probabilities were obtained from studies on flow tubes coated with water-ice, NAT or frozen sulfate. Droplet train flow tube reactors have used where liquid droplets are generated by means of a vibrating orifice [75]. The uptake of gaseous species in contact with these droplets has been measured by tunable diode laser spectroscopy [41]. 

In unbuffered systems, kinetic limitation may come either from slow transport to the cell surface or from slow dissolution of solid species resulting in depletion of the bulk medium. Simple calculations show (Table 1) that, in the open ocean, steady state diffusion of such elements as zinc of iron to the cell surface may match the uptake rate of fast growing algae. This is in accordance with the postulate that, in a stable environment, all potentially limiting elements should effectively be co-limiting primary production and that the average uptake rates should match the diffusion rates (Morel and Hudson, 1985). 

The expression DA/L has units of cm3 s 1 and represents the diffusive uptake rate of the sampler under ideal conditions. 

For indoor environments the effects of changes in temperature and pressure on diffusive uptake rate will be insignificant compared with other sources of error. High humidity can affect the sorption capacity of hydrophilic sorbents such as charcoal. This will reduce the time before saturation of the sorbent occurs. 

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