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Chapter 14: Problem 179

The basic equation describing the diffusion of one medium through another stationary medium is (a) \(j_{A}=-C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) (b) \(j_{A}=-D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) (c) \(j_{A}=-k \frac{d\left(C_{A} / C\right)}{d x}\) (d) \(j_{A}=-k \frac{d T}{d x}\) (e) none of them

Short Answer

Expert verified
a) \(j_{A} = -C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) b) \(j_{A} = -D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) c) \(j_{A} = -k \frac{d\left(C_{A} / C\right)}{d x}\) d) \(j_{A} = -k \frac{d T}{d x}\) e) None of them Answer: a) \(j_{A} = -C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\)

Step by step solution

01

Option (a)

\(j_{A} = -C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) In this equation, \(j_{A}\), the flux, is directly proportional to the negative gradient of the molar ratio. It takes into consideration the gradient with respect to the position '\(x\)' and a proportionality constant, \(D_{A B}\). This option seems to represent Fick's first law appropriately.
02

Option (b)

\(j_{A} = -D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) This expression is similar to option (a) but without the term 'C', which represents a concentration. Fick's law states that the flux depends on the total concentration, so this option is incorrect.
03

Option (c)

\(j_{A} = -k \frac{d\left(C_{A} / C\right)}{d x}\) This option has the same format as option (b) but uses the proportionality constant 'k' instead of the diffusion coefficient \(D_{A B}\). Since Fick's law specifically mentions the diffusion coefficient, this option is incorrect.
04

Option (d)

\(j_{A} = -k \frac{d T}{d x}\) In this option, the gradient is of temperature with respect to position '\(x\)', rather than concentration or molar ratio. This does not represent Fick's first law of diffusion, which is based on concentration gradients.
05

Option (e)

None of them Since we have already analyzed each option, and option (a) seems to represent Fick's first law correctly, there is no need to consider this option. In conclusion, the correct equation describing the diffusion of one medium through another stationary medium is given by:
06

Answer

Option (a) \(j_{A} = -C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\)

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Most popular questions from this chapter

14-45 Consider a rubber membrane separating carbon dioxide gas that is maintained on one side at \(2 \mathrm{~atm}\) and on the opposite at \(1 \mathrm{~atm}\). If the temperature is constant at \(25^{\circ} \mathrm{C}\), determine (a) the molar densities of carbon dioxide in the rubber membrane on both sides and \((b)\) the molar densities of carbon dioxide outside the rubber membrane on both sides.

A glass of milk left on top of a counter in the kitchen at \(15^{\circ} \mathrm{C}, 88 \mathrm{kPa}\), and 50 percent relative humidity is tightly sealed by a sheet of \(0.009-\mathrm{mm}\)-thick aluminum foil whose permeance is \(2.9 \times 10^{-12} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{Pa}\). The inner diameter of the glass is \(12 \mathrm{~cm}\). Assuming the air in the glass to be saturated at all times, determine how much the level of the milk in the glass will recede in \(12 \mathrm{~h}\). Answer: \(0.0011 \mathrm{~mm}\)

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