Half-concentration penetration depth

The mathematics of diffusion at flat wall boundaries has been derived in Section 18.2 (see Fig. 18.5a-c). Here, the well-mixed system with large diffusivity corresponds to system B of Fig. 18.5 in which the concentration is kept at the constant value Cg. The initial concentration in system A, CA, is assumed to be smaller than Cg. Then the temporal evolution of the concentration profile in system A is given by Eq. 18-22. According to Eq. 18-23 the half-concentration penetration depth , x1/2, is approximatively equal to (DAt)m. The cumulative mass flux from system B into A at time t is equal to (Eq. 18-25) ... 

The half-concentration penetration depth, z1/2, for a sorbing species is approximately (see Eq. 18-23) ... 

Explain the concept of the half-concentration penetration depth of the wall boundary model. 

Figure 19.8 Diffusivity D and concentration C at wall boundary, (a) Schematic view of a wall boundary. Diffusivity drops abruptly from a very large value DB, which guarantees complete mixing in system B, to the much smaller value Da. The concentration penetrates into system A when time t grows. X(/2(

The time scale xB determines the penetration distance in system A beyond which the influence of the variation of C can be disregarded. For instance, we can use the concept of half-concentration depth (Eq. 18-23) as a measure to assess the penetration distance of the time-dependent variation into system A. Thus ... 

As a predictor of the concentration of cisplatin in normal peritoneal tissues, these data indicate a steady-state penetration depth (distance to half the surface layer concentration) of about 0.1 mm (100 tm). If this distance applied to tumor tissue, penetration even to three or four times this depth would make it difficult to effectively dose tumor nodules of 1- to 2-mm diameter. Fortunately, crude data are available from proton-induced X-ray emission studies of cisplatin transport into intraperitoneal rat tumors, indicating that the penetration into tumor is deeper and is in the range of 1-1.5 mm (10). Such distances are obtained from Equation 9.5 or 9.5 only if k is much smaller than in normal peritoneal tissues — that is, theory suggests that low permeability coefficient-surface area products in tumor (e.g., due to a developing microvasculature and a lower capillary density) may be responsible for the deeper tumor penetration. 

What penetration depth of oxygen would you expect in a sediment, if the removal of oxygen can be described in terms of a half-life of 5 days Sediments have a temperature of 10°C. Their porosity is at 70 percent. For the diffusion coefficients use Tables 3.1 and 3.2. The bottom water shall have an oxygen concentration of 200 imol T. Use an oxygen concentration less than 1 imol T as zero . 

In the course of producing nuclear weapons, an unnamed country (often in trouble with the United Nations) had a small spill of promethium-147 (147Pm) in September 1989. This spill totaled 4.5 microCuries (pCi), covered a soil area of 5 m2. and penetrated to a depth of 0.5 m. In August 1997, the United Nations tested this site, and the concentration of 147Pm was found to be 0.222 pCi/m3 of soil. What is the half-life (in years) of 147Pm ... 

Previous discussion has indicated that unmetabolized small molecular weight, hydrophilic molecules (MW < 500) typically penetrate tissues to (half-surface-concentration) depths that range at steady state from 0.1 to 1 mm. The depth is on the order of 0.1 mm for most tissues of the body, as we have seen in the case... 

Figure 5.75 shows the bulk reaction rate of the biofilm k in dimensionless form as a function of the bulk concentration A with different biofilm characteristic values of A/B (Harremoes, 1978). Even though the depth of penetration may be small compared with the entire film thickness, the effect of internal diffusion resistances is to mask the true zero-order kinetics (see Equ. 5.111), yielding an apparent reaction order of one-half. 

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