Diffusion constant, spherical

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

In this exercise we shall estimate the influence of transport limitations when testing an ammonia catalyst such as that described in Exercise 5.1 by estimating the effectiveness factor e. We are aware that the radius of the catalyst particles is essential so the fused and reduced catalyst is crushed into small particles. A fraction with a narrow distribution of = 0.2 mm is used for the experiment. We shall assume that the particles are ideally spherical. The effective diffusion constant is not easily accessible but we assume that it is approximately a factor of 100 lower than the free diffusion, which is in the proximity of 0.4 cm s . A test is then made with a stoichiometric mixture of N2/H2 at 4 bar under the assumption that the process is far from equilibrium and first order in nitrogen. The reaction is planned to run at 600 K, and from fundamental studies on a single crystal the TOP is roughly 0.05 per iron atom in the surface. From Exercise 5.1 we utilize that 1 g of reduced catalyst has a volume of 0.2 cm g , that the pore volume constitutes 0.1 cm g and that the total surface area, which we will assume is the pore area, is 29 m g , and that of this is the 18 m g- is the pure iron Fe(lOO) surface. Note that there is some dispute as to which are the active sites on iron (a dispute that we disregard here). 

Size of the particle. Assuming a spherical shape, the value of the rotational diffusion constant Dp can provide an average radius using the following relationship ... 

Based on the magnitude of diffusion constants for globular (i.e., spherical) protein molecules, one can estimate that proteins like hemoglobin (mass = 68 kDa diffusivity = 6.2 m /s X 10 ) can readily diffuse 20-30 micrometers within 10-20 seconds. By contrast, low-molecular-weight metabolites, such as glycine (mass = 75 diffusivity = 95 m /s X 10 ) and arginine (mass = 174 diffusivity = 58 m /s X 10 ), will rapidly traverse these distances in a few seconds or less. 

A3.3 Three-dimensional diffusion using spherical coordinates with constant D... 

Therefore, the effective collective diffusion constant for an infinitely long cylinder is 2/3 of that of a spherical gel. 

Fig. 23. Position dependence of the effective collective diffusion constant normalized by the collective diffusion constant of spherical gels. D0 = (K + 4ji/3)/f. At the boundary, the values for sphere, cylinder, and disk are 1, 2/3, and 1/3, respectively...

The effective diffusion constant is reduced by a factor of 1/3 compared with that for a spherical gel. 

Figure 23 shows the effective collective diffusion constant normalized by the diffusion constant for a spherical gel as a function of the position from the center of the cylinder. From r/a = 0, i.e., on the cylinder axis, De/D0 decreases gradually and approaches the value 2/3 and 1/3, respectively for a cylindrical and disk gels. 

While this equation is thought to overestimate the diffusion-limited rate constant slightly, it is a good approximation. If the diffusing particles are approximately spherical, diffusion constants DA and DB can be calculated from Eq. 9-25, and Eq. 9-28 becomes Eq. 9-29. 

The steady-state solution for diffusion through spherical shells with boundary conditions dependent only on r may be obtained by integrating twice and determining the two constants of integration by fitting the solution to the boundary conditions. 

A is the wavelength of the laser in vacuum and q is the magnitude of the so called scattering vector. In turn, for spherical particles, the diffusion constant is related to the particle diameter through the Stokes-Einstein equation ... 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

Diffusion measurements " have been made on starch acetates dissolved in such solvents as acetic acid and acetic anhydride, and molecular weight values up to 224,700 have been indicated. Unfortunately, diffusion constants of starch fractions have been investigated only to a very limited extent. Diffusion measurements probably would provide evidence of value in the determination of the physical shape of the molecules of the two starch fractions the rate of diffusion of the amylopectin acetate should conform most nearly to the normal hydrodynamic predictions for spherical macromolecules. 

At what pressure does the mean free path of krypton (Kr) atoms (d = 3.16 X 10 ° m) become comparable with the diameter of the 1-L spherical vessel that contains them at 300 K Calculate the diffusion constant at this pressure. 

If the molecular weight of a protein is known from measurements of osmotic pressure, of sedimentation and diffusion, of sedimentation equilibrium, of light scattering, or by any other inethod, and if its partial specific volume is also known, then the frictional ratio fjf may be determined either from its sedimentation constant or its diffusion constant. The frictional coefficient / is that characteristic of a spherical unhydrated molecule of the same molecular weight and partial specific volume as the protein under consideration. If r is the radius of tliis hypothetical sphere and rj the viscosity of the medium, then... 

Here, F is the volume fraction of the solute. The Einstein coefficient V is 2.5 for spheres and is always larger for molecules vhich deviate from the spherical shape, v is of course a function of the parameter a. It approaches a lower limit at very high velocity gradient and an upper limit v, at zero velocity gradient. Most relatively small proteins with absolute dimensions less than 500 or 600 A in any direction have sufficiently high rotary diffusion constants so that the measured value of V may be taken as equal to for the velocity gradient in almost any capillary viscometer. The rest of this discussion of viscosity will be... 

To address these questions, we follow a simplified spherical macroion that is allowed to move concomitantly with lipid diffusion. To do so, we extend our model to include protein diffusion and performed CH-DMC (see Section 2.4) calculations. We studied the same mixed membranes considered in Figure 4, focusing on two typical cases. In the first, the model protein has a diffusion constant much larger than that of lipids in the unperturbed (bare) membrane, with a ratio D = 10 between the two, while in the second, the diffusion constant is comparable to that of the lipids, and D = 2 (see Eq. 10). As we show, these two scenarios lead to different lipid and protein diffusion characteristics. 

In these studies the rate of the mass and contact diameter of water and -octane drops placed on glass and Teflon surfaces were investigated. It was found that the evaporation occurred with a constant spherical cap geometry of the liquid drop. The experimental data supporting this were obtained by direct measurement of the variation of the mass of droplets with time, as well as by the observation of contact angles. A model based an the diffusion of vapor across the boundary of a spherical drop has been considered to explain the data. Further studies were reported, where the contact angle of the system was 9 < 99°. In these systems, the evaporation rates were found to be linear and the contact radius constant. In the latter case, with 9 > 99°, the evaporation rate was nonlinear, the contact radius decreased and the contact angle remained constant. 

This reduces the half-time for a one half mil diameter specimen to 34.2 hours still lending a formidable time barrier to detailed studies. However, spherical polymer particles, of reasonable uniform size distribution, are rather readily available down to as low as 0.1 pm diameter. This reduces the half time down to 7.66 seconds and brings transport studies of even the lowest diffusion constant polymers84 within easy experimental reach. Studies of this kind were pioneered by Berens82,204 and later by Enscore et al.73,74 with n-hexane in polystyrene. The elegant studies of Berens of the transport of vinyl chloride monomer in polyvinyl chloride have been summarized in a recent review by Berens and Hopfenberg205 and only the main conclusions will be presented here. 

Assuming a spherical shape for the fluorescent molecule, the degree of change in the rotational Brownian motion is given by Eq. (3.25), where v is the volume of the spherical molecule, r)0 is the solvent viscosity, r is the fluorescence lifetime of the chromophore, and T is the temperature. The values of r0 and r/v can be obtained from a plot of Mr versus T/rj0. Thus, if the fluorescence lifetime of the chromophore is known, it is possible to determine the hydrodynamic volume of the rotating molecule and its rotational diffusion constant D,. This data treatment is known as the Perrin-Weber approximation,25 after the two scientists who first derived the equations in the case of protein chromophores. 

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