Diffusion cumulative

Some examples are ambipolar diffusion, cumulative rate of topochemical reaction, change in the light velocity when moving from vacuum into the given medium, resultant constant of chemical reaction rate (initial product—intermediary activated complex—final product). 

Fig. 18. Determining the diffusion coefficient from cumulative transmission.

A monolithic system is comprised of a polymer membrane with dmg dissolved or dispersed ia it. The dmg diffuses toward the region of lower activity causiag the release of the dmg. It is difficult to achieve constant release from a system like this because the activity of the dmg ia the polymer is constantly decreasiag as the dmg is gradually released. The cumulative amount of dmg released is proportional to the square root of time (88). Thus, the rate of dmg release constantly decreases with time. Again, the rate of dmg release is governed by the physical properties of the polymer, the physical properties of the dmg, the geometry of the device (89), and the total dmg loaded iato the device. 

The performance of the dmg dehvery system needs to be characterized. The rate of dmg release and the total amount of dmg loaded into a dmg dehvery system can be deterrnined in a dissolution apparatus or in a diffusion ceU. Typically, the dmg is released from the dmg dehvery system into a large volume of solvent, such as water or a buffer solution, that is maintained at constant temperature. The receiver solution is weU stirred to provide sink conditions. Samples from the dissolution bath are assayed periodically. The cumulative amount released is then plotted vs time. The release rate is the slope of this curve. The total dmg released is the value of the cumulative amount released that no longer changes with time. 

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

Usually, simplified representations of the data are used to obtain preliminary structures. Thus, lower and upper bounds on the interproton distances are estimated from the NOE intensity [10], using appropriate reference distances for calibration. The bounds should include the estimates of the cumulative error due to all sources such as peak integration errors, spin diffusion, and internal dynamics. 

Radial motion of fluid can have a significant, cumulative effect on the convective diffusion equations even when Vr has a negligible effect on the equation of motion for V. Thus, Equation (8.68) can give an accurate approximation for even though Equations (8.12) and (8.52) need to be modified to account for radial convection. The extended versions of these equations are... 

Figure 3.42 Evolution of a pulse at the entrance of a micro channel for different diffusion coefficients. Calculated concentration profile (left) and cumulative residence time distribution curve (channel 300 pm x 300 pm x 20 mm flow velocity 1 m s f = 10 s) [27],...

Thus, the rate of change for the cumulative mass of diffusant passing through a membrane per unit area, or the flux of diffusant, j, may be evaluated from the steady-state portion of the permeation profile of a drug, as shown in Eq. (3). If the donor concentration and the steady-state flux of diffusant are known, the permeability coefficient may be determined. 

Note that in the component mass balance the kinetic rate laws relating reaction rate to species concentrations become important and must be specified. As with the total mass balance, the specific form of each term will vary from one mass transfer problem to the next. A complete description of the behavior of a system with n components includes a total mass balance and n - 1 component mass balances, since the total mass balance is the sum of the individual component mass balances. The solution of this set of equations provides relationships between the dependent variables (usually masses or concentrations) and the independent variables (usually time and/or spatial position) in the particular problem. Further manipulation of the results may also be necessary, since the natural dependent variable in the problem is not always of the greatest interest. For example, in describing drug diffusion in polymer membranes, the concentration of the drug within the membrane is the natural dependent variable, while the cumulative mass transported across the membrane is often of greater interest and can be derived from the concentration. 

By integrating the flux with respect to time, we obtain the cumulative mass of diffusing solute molecules per unit area ... 

Koizumi and Higuchi [18] evaluated the mass transport of a solute from a water-in-oil emulsion to an aqueous phase through a membrane. Under conditions where the diffusion coefficient is expected to depend on concentration, the cumulative amount transported, Q, is predicted to follow the relationship... 

Figure 25 Cumulative fraction of the initial donor concentration of [1-blockers that diffused across Caco-2 cell monolayers as a function of donor pH. Transwell systems were used, and stirring was done using a rotary platform shaker. (A), pH 7.4 (B), pH 6.5.

The dynamics of highly diluted star polymers on the scale of segmental diffusion was first calculated by Zimm and Kilb [143] who presented the spectrum of eigenmodes as it is known for linear homopolymers in dilute solutions [see Eq. (77)]. This spectrum was used to calculate macroscopic transport properties, e.g. the intrinsic viscosity [145], However, explicit theoretical calculations of the dynamic structure factor [S(Q, t)] are still missing at present. Instead of this the method of first cumulant was applied to analyze the dynamic properties of such diluted star systems on microscopic scales. 

Our experimental data, when fitted to this equation, yield a value of 1.0 x 10-11 cm2/sec for Di. Least square fit of the same data to Crank s rate equation, which takes into account only the diffusion in the matrix, evaluates Dj to be 2.2 x lO-1 1 cm2/sec. The integral form of Crank s equation (13), giving the cumulative amount released per unit surface area,... 

In many cases, transdermal drug absorption is investigated using a Franz-diffusion cell. The concentrations both in the membrane and the acceptor compartment are assumed to be zero at the start of the experiment. At different time points, the cumulative drug amount per unit area in the receptor q(t) is determined and plotted versus time t (Figure 20.1). After some time, the flux... 

Figure 20.1 Plotting cumulative amount versus time in a Franz-diffusion cell experiment results in a curve approaching a straight line.

At the same time as the basic work was being done on the kinetics and difFusivity effects in coke burning, the kinetics of the processes that determined the CO/CO2 ratio from slow coke was investigated by Weisz (1966). Studies were made of the cumulative CO2/CO ratios for individual, whole, spherical catalyst beads. The results, shown in Fig. 28, scattered very badly. 

The multibaker map preserves the vertical and horizontal directions, which correspond respectively to the stable and unstable directions. Accordingly, the diffusive modes of the forward semigroup are horizontally smooth but vertically singular. Both directions decouple, and it is possible to write down iterative equations for the cumulative functions of the diffusive modes, which are known as de Rham functions [ 1, 29]... 

Figure 8. The diffusive inodes of the multibaker map represented by their cumulative function depicted in the complex plane (ReF, ImF ) versus the wavenumber k.

The cumulative functions of the diffusive modes can be constructed by using Eq. (60). The trajectories start from the border of a disk with an initial position... 

Figure 12. The diffusive modes of the periodic Yukawa-potential Lorentz gas represented by their cumulative function depicted in the complex plane ReFk,hnFk) for two different nonvanishing wavenumbers k. The horizontal straight line is the curve corresponding to the vanishing wavenumber k = 0 at which the mode reduces to the invariant microcanonical equilibrium state.

Figure 6.25 shows the dispersion of the effective diffusion Fq/Q with Fq the first cumulant of the relaxation function. Three branches are visible. A collec-... 

In the case of homogeneous particles the diffusion coefficient D is given by the first reduced cumulant /2K. The magnitude of the scattering vector K is given by ... 

Figure 3. Concentration dependence of the diffusion coefficient of polystyrene in THF at 23 C. (a) collective modes, (b) cumulant values and classical gradient diffusion, (c) cooperative mode. (Reproduced from Ref. 19. Copyright 1985 American Chemical Society. ...

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