Time-dependent concentration profile

FIG. 18 Chloride concentration profile recorded by a microelectrode probe during the hydrolysis of TPMCl at a DCE drop-aqueous interface (O)- The concentration of TPMCl in the organic phase was 50 mmol dm, the drop time from formation to contact with the probe was 4.80 s, and the final drop radius was 0.55 mm. The solid lines represent theoretical time-dependent concentration profiles, from top to bottom, generated for k = 3.50 x 10 , 3.25 x 10 , and 3.00 x 10 molcm s . A value of 1.8 X 10 cm s was employed for the diffusion coefficient of chloride. (Reprinted from Ref. 73. Copyright 1997, American Chemical Society.)... 

Figure 5 shows the diffusion of a solute into such an impermeable membrane. The membrane initially contains no solute. At time zero, the concentration of the solute at z = 0 is suddenly increased to c, and maintained at this level. Equilibrium is assumed at the interface of the solution and the membrane. Therefore, the corresponding membrane concentration at z = 0 is Kc1. Since the membrane is impermeable, the concentration on the other side will not be affected by the change at z = 0 and will still be free of solute. This abrupt increase produces a time-dependent concentration profile as the solute penetrates into the membrane. If the solution is assumed to be dilute, Fick s second law Eq. (9) is applicable ... 

Figure 5 Diffusion into a semi-infinite membrane. The membrane initially contains no solute. At time zero, the concentration of the solution at z = 0 is suddenly increased to and maintained at cx. This abrupt increase produces time-dependent concentration profiles as the solute penetrates into the membrane.

Figure 6 Unsteady diffusion across a membrane. The membrane is initially free of solute. At time zero, the concentrations on the two sides of the membrane are increased to and maintained at c, and c2. The solute penetrates into the membrane from both sides, resulting in time-dependent concentration profiles within the membrane.

Fig. 3 Time-dependent concentration profiles for solute A Effect of phase ratio, with Vc /VL = 1.0 and 4.0.

PospisU, H., Holzhuetter, H.G. A compartment Model to calculate time-dependent concentration profiles of topically applied chemical compounds in the a nteriro compartments of the rabbit eye. ATLA 29,347-365 (2001)... 

The exact solution for the time-dependence of the current at a planar electrode embedded in an infinitely large planar insulator, the so-called semi-infinite linear diffusion condition, is obtained. Solving the diffusion equation under the proper set of boundary and initial conditions yields the time-dependent concentration profile. 

In these cases, the result of the KMC simulation is the system s time-dependent concentration profile. In DNMR, the system simulated with the KMC method is in macroscopic equilibrium, and the microscopic changes of a single species are simulated. 

The classic problem of diffusion in an infinite medium can be solved by use of a similarity transformation. The associated impedance response is discussed in Section 11.3. A general method for finding the time-dependent concentration profile is presented here in the form of an example. 

We seek an expression for the time-dependent concentration profile. 

Regarding the time depending concentration profiles only a very slight decrease of the contamination was observed interrupted by individual maxima e.g. in march 2003. The tendency to a lower contamination might be attribute to the technical prevention measures performed since 1999. 

In principle, arbitrarily high resolution should be attained by applying sufficiently large gradient intensities g, although the present state of the art of NMR microscopy only allows spatial resolution down to 1 xm [13,14]. One has to note, however, that an increase in the spatial resolution is possible only at the expense of measuring time. Hence for the observation of time-dependent concentration profiles, a compromise between time and space resolution is inevitable. 

In practice, peak profiles are not measured over axial coordinates but detected at the column outlets over time. Therefore, a corresponding standard deviation of the time-dependent concentration profile Ot can be used (Equation 2.33) ... 

In order to assess resuspension at a shallow (0.25 m deep) site in the lake (Fig. 27.1(a)), Eq. (27.32) must be solved for the time-dependent concentration profile, with appropriately calibrated expressions for the erosion flux coefficients and the settling velocity. Pertinent measurements and analysis are described elsewhere. Parameters for simulation of concentration profile are summarized in Table 27.9. The organic-rich bottom is conveniently treated as bed (as opposed to fluid mud) in this analysis. 

Time-dependent concentration profiles and rates of translational diffusion... 

This is Pick s equation in one dimension. Its solution yields the time-dependent concentration profiles shown in Figure 2.7a. Illustrations dealing with Pick s law will appear in Chapter 3. 

As an example of the above approach, Brooke and Rees [95] studied the Sr/Ca-chabazite system. Figure 9 shows their computed time-dependent concentration profiles within the zeolite particles both before and after non-ideal behaviour was taken into account. The effect on Dab of taking non-ideality into account was even more dramatic, with a discontinuity appearing in the plot of dDAB/dCA... 

The diffusion in a semi-infinite slab is schematically sketched in Fig. 2.3-2. The slab initially contains a uniform concentration of solute ci oo. At some time, chosen as time zero, the concentration at the interface is suddenly and abruptly increased, although the solute is always present at high dilution. The increase produces the time-dependent concentration profile that develops as solute penetrates into the slab. 

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