Diffusion, coefficients equation

Many polymer properties can be expressed as power laws of the molar mass. Some examples for such scaling laws that have already been discussed are the scaling law of the diffusion coefficient (Equation (57)) and the Mark-Houwink-Sakurada equation for the intrinsic viscosity (Equation (36)). Under certain circumstances scaling laws can be employed advantageously for the determination of molar mass distributions, as shown by the following two examples. 

In considering axial dispersion as a diffusive flow, we assume that Fick s first law applies, with the diffusion or effective diffusion coefficient (equation 8.5-4) replaced by an axial dispersion coefficient, D,. Thus, for unsteady-state behavior with respect to a species A (e.g., a tracer), the molar flux (NA) of A at an axial position x is... 

We have shown that the microscopic expression for the polymer diffusion coefficient. Equation 2, is the starting point for a discussion of diffusion in a wide range of polymer systems. For the example worked out, polymer diffusion at theta conditions, the resulting expresssion describes the experimental data without adjustable parameters. It should be possible to derive expressions for diffusion... 

This is the correct expression for use in the analysis of closed diffusion-cell experiments for the measurement of diffusion coefficients. Equation (48) is known as Fick s First Law of Diffusion. Note that Na = — NB corresponds to saying that w = 0. 

Equation (26) is a differential equation with a solution that describes the concentration of a system as a function of time and position. The solution depends on the boundary conditions of the problem as well as on the parameter D. This is the basis for the experimental determination of the diffusion coefficient. Equation (26) is solved for the boundary conditions that apply to a particular experimental arrangement. Then, the concentration of the diffusing substance is measured as a function of time and location in the apparatus. Fitting the experimental data to the theoretical concentration function permits the evaluation of the diffusion coefficient for the system under consideration. 

On the other hand, if approximate steady-state conditions may be assumed, such that the two chemicals are held at a fixed concentration at some distance from the sensory surface, then the difference between the fluxes to the surface turns out to be directly proportional to the ratio of the diffusion coefficients (equation 21.15). That is, for this case where the two chemicals differ in the... 

GPPS = general purpose polystyrene HIPS = high impact polystyrene a) 1 1 HIPS GPPS diffusion coefficient equation In D = 15.61 - 14 500- b) apparent diffusion coefficient calculated from experimental migration data. (1 /T(K))... 

Equation 9.12 indicates that the diffusion coefficient of an aerosol particle is independent of particle density and hence is independent of particle mass. But is this really so Since particle mass is so much greater than molecular mass and the particles are continually undergoing bombardment by the molecules, one would expect changes in the direction of the particle to be gradual, compared to the rapid changes in direction with molecular diffusion. But if this is true, then particle momentum (mass) should be considered in the particle diffusion coefficient equation. 

Near conditions of chemical equilibrium, averages involving are best computed from the right-hand side of equation (44). Under assumptions 1, 2, and 3 of Section 1.3 and the approximation of equal binary diffusion coefficients, equation (44) becomes... 

Before closing this brief discussion on mass transfer fundamentals, further mention should be made of the diffusion coefficient. Equations for predicting gas diffusivities are given by Fuller and are also given in Perry s Handbook. The orders of magnitude of the diffusivities for gases, liquids, and... 

SOLUTION In preparation for the estimation of the mass transfer coefficients in the liquid phase we must first compute the Maxwell-Stefan diffusion coefficients. Equation 4.2.18 is used for this task as illustrated below. 

The rate of transport of water is thus expressed by equation 8 in terms of an apparent diffusion coefficient, D, and the gradient of the overall concentration of water in the rubber, dC /dx. The theory predicts that the diffusion of water in rubber containing hydrophilic Impurities can be expressed in terms of a concentration dependent diffusion coefficient (equation 9). 

This equation shows that a gradient in concentration c can induce a flow of component/. Dfj is called a cross-diffusion coefficient. Equation (13.6.4a) can be expressed in matrix form... 

By plotting ln (r) against r, all of which is done by computer, rc is found. Assuming the Stokes—Einstein equation for the diffusion coefficient [equation (6.10)J, the hydrodynamic radius, R, of the particles is given by... 

Here, is the limiting current density, L is a characteristic length of the system (such as the diameter of a pipe), Cj, is the bulk concentration of the rate limiting species in the solution, and D is its diffusion coefficient. Equation (4.81) allows us to write ... 

The self-diffusion coefficient, (Equation 5.48), depends on the molecular weight through the chain-length dependence of the hydrodynamic radius R. A common empirical expression for this quantity is... 

In test examples, we calculated the experimental profile by solving direct diffusion problems (Equation 12.78) with a diffusion coefficient (Equation 12.81) for preset parameters (l)°-a(3) . In order to do so, we used the numerical procedure of solving nonhnear parabolic equations described in [68]. After that, the calculated profile was distorted by the addition of a random concentration proportional to... 

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