Diffusional models

Kramers HA. 1940. Brownian motion in a field of force and the diffusional model of chemical reactions. Physica7 284-304. 

Figure 12. An example of use of to assess the rate and depth of sediment mixing from a core on the slopes of the Bahamas (Henderson et al. 1999b). The exponential decrease in Pbxs seen in the upper 6 cm of the sediment reflects decay of °Pb as it is mixed downward. The diffusional model of mixing described in the text indicates a mixing rate, D, of 51 cm kyr for this core. The two circled points at greater depth reflect rapid injection of surface material to depth in a process known as conveyor-belt feeding (Robbins 1988 Smith et al. 1997).

If a diffusional model with AHm < kT is appropriate, then the time th between electron hops must approach the period cor of the optical mode vibrations that trap or correlate the electrons. With an wj = 10" s, the hopping time Th would be short relative to the time scale of Mossbauer spectroscopy, ca. 10" s. We can therefore anticipate an isomer shift for the octahedral-site iron that is midway between the values typical for Fe ions and Fe " " ions. From Table 1 and Eq. (1), we can predict a room-temperature isomer shift of 6 0.75 mm/s wrt iron. Consistent with this prediction is the... 

The high-temperature transport data has been well rationalized 0 on the basis of a diffusional model with AH = 0.16 eV for all x, which is an agreement with localized Fe " -ion configurations responsible for a cooperative Jahn-Teller distortion below Tn. The ratio (1 - c)/c obtained from Eq. (15) for the Seebeck coefficient, with a spin-degeneracy factor P = 2, gives the factor c(l — c) entering the conductivity expression... 

Carroll, K. M., Harkness, M. R., Bracco, A. A. Balcarcel, R. R. (1994). Application of a permeant/polymer diffusional model to the desorption of polychlorinated biphenyls from Hudson River sediments. Environmental Science Technology, 28, 253-8. 

R. G. Stehele and W. I. Higuchi. Diffusional model for transport rate studies across membranes. J. Pharm. Sci. 56 1367—1368, 1967. 

Kramers, H. A., Brownian Motion in a Ffeld of Force and the Diffusional Model of Chemical Reactions, Physica, 7, 284 (1940). 

A quantitative criterion is developed for the applicability of a previous diffusional model for cell deposition on surfaces, using as a starting point a modified Fokker-Planck equation. 

For a condition between complete mixing and plug flow, diffusional models such as that of Bennett and Grimm (1991) are used. For conservative work, the complete mixing model is appropriate. 

Higuchi, W.I. Diffusional models useful in biopharmaceutics. Drug release rate processes. J. Pharm. Sci. 1967, 56, 315-324. 

Freeman and Skrivan (5) have studied the decomposition rate of ammonia and methane in a plasma jet (Reaction 8) and have shown it to be rate limited by a diffusion process. The apparatus used has been fully characterized and shown to be a very good fit for a diffusional model. 

A Diffusional Model for Nafion. Several structural models of Nafion have been proposed these have been based on a variety of transport and spectroscopic properties of the polymer (17-20). 

Thus far we have only transcribed Kuhn s theory from the diffusional model to the energy barrier model. 

The slow variable models are ultimately based on purely macroscopic measurements. For example Onsager s diffusional model of Eq. (A.53) derives from Eq. (A. 15), which in turn is based on macroscopic phenomenology. Similarly, Onsager s Langevin model of Eq. (A.52) derives from Eq. (A.28), which in turn is suggested by the empirical friction law —for a macroscopic particle. 

Quantitative calculations of the CIDNP effect can be performed as described in Section II.B.5. One has to set up the nuclear spin system of the intermediate radical pair or biradical, choose a diffusional model, compute reaction probabilities for every nuclear spin state by solving the stochastic Liouville equation numerically or approximately, establish a correlation between the nuclear spin states in the paramagnetic intermediates and the nuclear spin states in the products to obtain the populations of the latter, and finally apply Eq. 61 or the formalism of the preceding section to get line intensities. This approach, which for all but the simplest systems is impracticable except on a computer, is often necessary with the usual uncertainty of the parameters entering the calculations of the radical pair mechanism, a reasonable accuracy can be expected. However, qualitative relationships between signal intensities, especially signal phases, and parameters of the reaction mechanism as well as magnetic properties of the intermediates are... 

Pb removal from water by means of a Na-exchanged phillipsite-rich tuff was studied, using a diffusional model. The model is based on the linear driving force (LDF) approximation and takes into account both fluid-particle and intra-particle resistances to diffusion, making no specific assumption on their relative magnitude or on the form of the ion exchange isotherm. 

An Important point concerning the diffusional model is that lifetime and static quenching measurements are equivalent. Thus, in the scheme, eqs. 12 to 16, if fluorescence decay time is monitored, the appropriate equation yielding kq is eq. 22 ... 

Schlosshauer, M., Baker, D. Realistic protein-protein association rates from a simple diffusional model neglecting long-range interactions, free energy barriers, and landscape ruggedness. Protein Sci. 2004,13,1660-9. 

Furthermore, the rate ratio passes through a minimum as increases. The good fit of the two experimental points to the curve for q = 1 seems to support the diffusional model for ke in the case of vinyl chloride and also indicates that the values of the various kinetic constants applied are reasonable. It is interesting to note that with a value of q = 0, i.e. the dependmcy predicted by the propagational model, no reduction in rate should be observed on addition of large particles, irrespective of the value of N /N. ... 

Models Considering a Membrane Diffusion Model. Sobotka et al. (1982) claimed that the empirical time delay models previously described do not properly consider the physical nature of the electrode response. Instead, they insisted that models based on Pick s second law are superior and encouraged their use to more accurately model system dynamics. Sobotka et al. (1982), in their review, presented many of the different diffusional models that have been developed and discussed their usefulness. 

Kn>l. Knudsen diffusional model the vapour molecules collide more frequently with the pore walls than with each other. The flux (7 ) is proportional to the pore radius ... 

The general kinetic-diffusional model has two limiting cases ... 

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