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25.2 One step of the manufacturing of silicon solar cells is the molecular diffusion (doping) of elemental phosphorous (P) into crystalline silicon to make an n-type semiconductor. This P-doped layer needs to be at least 0.467 μm into the 200-μm thick wafer. The present diffusion process is carried out at 1000ºC. Data for the total amount of phosphorous atoms loaded into the silicon wafer vs. time at 1000ºC are presented in the figure below. The maximum solubility of phosphorous within crystalline silicon is 1.0 × 1021 P atoms/cm3 at 1000ºC. The square silicon wafer has a surface area of 100 cm2 (10 cm/side). Initially, there is no phosphorous impurity in the crystalline silicon. What is the concentration of P atoms (atoms P/cm3) doped into the silicon at a depth of 0.467 μm after 40 min, based on the data provided?...
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25.3 A preheated piece of mild steel, having an initial concentration of 0.20% by weight of carbon, is exposed to a carbonizing atmosphere for 1.0 h. Under the processing conditions, the surface concentration of carbon is 0.70% by weight. If the diffusivity of carbon through steel is 1.0 × 10−11 m2/s at the process temperature, determine the carbon composition at 0.01 cm, 0.02 cm, and 0.04 cm below the surface.
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25.4 A “drug patch” is designed to slowly deliver a drug (species A) through the body tissue to an infected zone of tissue beneath the skin. The drug patch consists of a sealed reservoir containing the drug encapsulated within a porous polymer matrix. The patch is implanted just below the skin. A diffusion barrier attached to the bottom surface of the patch keeps the surface concentration of the drug dissolved in the body tissue constant at 2.0mol/m3, which is below the solubility limit of the drug in the body tissue. There is no difference in the solubility and diffusion coefficient of the drug between the healthy and infected tissues. The mean distance from the drug patch to the infected area of tissue is 5.0 mm. To be effective, the drug concentration in the tissue must be 0.2mol/m3 or higher when it reaches the infected zone. Determine the time it will take in hours for the drug to begin to be effective for treatment. The effective molecular diffusion coefficient of the drug through the body tissue, both healthy and infected, is 1.0 × 10−6 cm2/s.
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25.5 The contamination of water-saturated soils with toxic organic solvents is an important environmental problem. The organic solvents can dissolve into to the water and diffuse through the water-saturated soil by molecular diffusion, resulting in contamination of the soil and the water. The figure below illustrates a very simplified view of this situation, where liquid tetrachloroethylene (TCE, species A) rests at the bottom of a pore filled with stagnant liquid water. The depth of the organic (TCE) layer is 0.10 cm, and the total length of the pore is 3.1 cm. The top of the pore is “capped” with a clay layer so that the contaminated water is contained within the pore. The diameter of the pore is not known. The density of the liquid TCE is 1.6 g/cm3, and the density of liquid water is 1.0 g/cm3. The molecular diffusion coefficient of TCE in liquid water is 8.9 × 10−6 cm2/s at 293 K. The solubility limit of TCE in water is 1.00 × 10−6 gmole/cm3 of the aqueous phase. The molecular weight of TCE is 166 g/gmole....
a. Initially, there is no TCE dissolved in the water. How long will it take for the TCE concentration in the water to reach 8.97 × 10−8 gmole/cm3 at a position of 0.3 cm from the organic-aqueous interface?
b. What is the TCE concentration in the aqueous phase after infinite time (t → ∞)?
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25.6 We are interested in the diffusion of CO2 gas out of a randomly porous adsorbent material slab of 2.0 cm thickness, as shown in the figure below. Initially, the gas space inside the porous material contains 10% mole CO2 (A) and 90 mole% air (B). The process is maintained at 25ºC and total system pressure of 1.0 atm. At these conditions, the binary molecular diffusion coefficient of CO2 in air is 0.161 cm2/s, but the effective diffusion coefficient of CO2 within the porous medium is only 0.010 cm2/s. Fresh air containing no CO2 blows over the surface of the slab so that the convective mass-transfer coefficient for CO2 transfer (kc) is 0.0025 cm/s. How long will it take for the CO2 concentration of the gas space inside the porous material to be reduced to only 2.0 mole% CO2 at a depth of 1.6 cm from the exposed surface of the slab?...
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25.7 Consider the porous slab shown in the figure below. Very tiny pores of 20 A diameter run through the 2.0 cm slab in parallel array. This device will ultimately serve as a drug delivery vehicle for a drug that is soluble in ethanol. As part of the device development, we are interested in diffusion aspects of this unit with respect to ethanol and water containing no drug. The pores are initially filled with liquid ethanol. Ethanol (molecular weight 46 g/gmole) has an approximate molecular diameter of 4 Å, and water (molecular weight 18 g/gmole) has an approximate molecular diameter of 3 Å. The viscosity of liquid ethanol is 0.85 cP at 313 K. This ethanol-filled porous slab is placed in a large, well-mixed vat of liquid water at 313 K. Water diffuses into the ethanol-filled pores of the slab. After 10 min of contact time, what is the concentration of water 2.0mm(0.2 cm) into the slab? It can be assumed that water does not penetrate very far into the pores of the slab, and that the surrounding liquid is essentially pure water at all times....
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25.8 One step in the processing of cucumbers to pickles is the pickling process itsel
f. In one method of making pickles, young cucumbers with no waxy skins are soaked in a NaCl solution overnight. To initiate the pickling process, acetic acid is added to the salt solution to make the pickling brine. The acetic acid acts as a preservative, and the salt solution keeps the cucumber from swelling as the acetic acid diffuses into the cucumber. When a certain concentration of acetic acid inside the cucumber is achieved, the cucumber is considered “pickled” and will remain safe and tasty to eat for a long time. In the present pickling process, the temperature is 80ºC, and the pickling brine solution contains an acetic acid concentration of 0.900 kgmole/m3. The cucumbers are 12.0 cm in length and 2.5 cm in diameter. The amount of cucumbers relative to the amount of pickling brine solution is small so that the bulk liquid-phase concentration of acetic acid remains essentially constant during the pickling process. Initially, the cucumbers do not contain any acetic acid, and are maintained at 80ºC. End effects can be neglected. It can also be assumed that the diffusion coefficient of acetic acid into the pickle approximates the diffusion coefficient of acetic acid in water. The diffusion coefficient of acetic acid in water at 20ºC (not 80ºC) is 1.21 × 10−5 cm2/s.
a. How long will it take (in hours) for the center of the pickle to reach a concentration of 0.864 kgmole/m3? For part (a), consider that the liquid is very well mixed so that external convective mass-transfer resistances can be neglected.
b. Refer to part (a). What is the new required time if convective mass transfer around the pickle is now such that kc = 1.94 × 10−5 cm/s?
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25.9 Spherical polymer beads of 3.0mm .0.3 cm. diameter contain residual solvent from the polymer-casting process. Initially, the bead contains 0.20 wt% of residual solvent uniformly distributed within the polymer. The residual solvent will be removed from the bead material by drying the beads in a fluidized bed of air. This sink for mass transfer will cause the solvent molecules inside the bead to transfer to the surface of the bead. The air flow through the fluidized bed is very high so that convective mass-transfer resistances are not present and the effective concentration of emitted solvent vapor into the bulk flowing air is equal to zero. At the process conditions of the fluidized bed, the effective diffusion coefficient of residual solvent molecules in the polymer material is 4.0 × 10−7 cm2/s.
a. How long will it take (in hours) for the solvent in the center of the bead to reach 0.002 wt%?
b. How long will it take (in s) for the solvent to reach 0.18 wt% composition at a depth of 0.1 mm from the surface of the bead?
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25.10 A small spherical bead is used as a controlled drug-release capsule in the gastrointestinal system (i.e., your stomach). In this particular case, a 0.10-cm diameter bead has a uniform initial concentration of 0.20 mmole/L of the drug griseofulvin (species A). The diffusion coefficient of griseofulvin within the bead material is 1.5 × 10−7 cm2/s. Upon release from the bead, the drug is immediately consumed so that the surface concentration is essentially zero.
a. Using the concentration-time charts, determine the time it will take for concentration of the griseofulvin in the center of the bead to reach 10% of its initial value.
b. Using the infinite series analytical solution for cA(r,t), determine the time it will take for concentration of the griseofulvin in the center of the bead to reach 10% of its initial value, and then plot out the concentration profile from r = 0 (center) to r = R (surface). Carry out your calculations on a computer spreadsheet.
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25.11 Consider a rectangular-shaped gel tablet of thickness 0:125 cm and width 0:50 cm. The edges of the gel tablet are sealed so that diffusion only occurs along the thickness of the tablet. The initial concentration of the drug Dramamine (species A) in the gel is 64 mg=cm3, to provide a total drug dosage of 2:0 mg. The concentration of A at the exposed surface of the tablet is maintained at zero. What is the total amount (mA) of drug released after 1.0 and 2.0 hours, respectively? Hint: Use the infinite series analytical solution for mA(t).
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25.12 As society searches for technical solutions to global warming, one approach to sequester carbon-dioxide rich greenhouse gases is to capture the CO2 within an adsorbent material at high pressure. An experiment designed to evaluate a candidate adsorbent material is presented in the figure below, which consists of a 10 cm × 10 cm slab with 100 cm2 of exposed surface. A gas mixture of 10 mole% CO2 (species A) and 90 mole% N2 (species B) at 15.0 atm total system pressure and temperature of 200ºC is contacted with a microporous material designed to selectively adsorb the CO2 from the gas mixture. The partitioning of CO2 gas within the adsorbent is described by...where QA is the amount of CO2 adsorbed per unit volume of porous adsorbent (mol CO2/cm3 adsorbent), cA is the concentration of CO2 in the gas phase within the porous adsorbent (mol CO2/cm3 gas pore space), and K′ is the apparent CO2 adsorption constant (cm3 gas/cm3 adsorbent). The experiment is conducted as an unsteady-state process where the total amount of CO2 captured within the adsorbent is measured after a given time. Initially, there is no CO2 adsorbed on the solid or in the pore space, and the process is considered dilute with respect to CO2 in the gas phase. The diffusion process is modeled as semi-infinite sink for CO2, with CO2 flux given by...cAs is the molar concentration of CO2 at the outer surface of the slab. In the above equation, ... is the apparent diffusion coefficient of CO2 within the adsorbent material under conditions where the solute adsorbs onto the surface of the pores, which includes the following terms: apparent CO2 adsorption constant (K′), the void fraction of the porous adsorbent (ε), and the effective diffusion coefficient of CO2 within the solid under non-CO2 adsorbing conditions (DAe). In the present system, K′ = 1776.2 cm3/cm3, ε= 0.60, and DAe = 0.018 cm2/s. After a total contact time of 60 min, what was the total mass of CO2 captured by the slab? Consider that the slab acts as a semi-infinite sink for CO2, and that convective mass-transfer resistances are eliminated so that cAs = cA∞....
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