/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 A thin plastic membrane is used ... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 15

A thin plastic membrane is used to separate helium from a gas stream. Under steady-state conditions the concentration of helium in the membrane is known to be \(0.02\) and \(0.005 \mathrm{kmol} / \mathrm{m}^{3}\) at the inner and outer surfaces, respectively. If the membrane is \(1 \mathrm{~mm}\) thick and the binary diffusion coefficient of helium with respect to the plastic is \(10^{-9} \mathrm{~m}^{2} / \mathrm{s}\), what is the diffusive flux?

Short Answer

Expert verified
The diffusive flux of helium through the thin plastic membrane is \(1.5 \times 10^{-5}\,\mathrm{kmol/(m^2 \cdot s)}\).

Step by step solution

01

Identify given data

Here are the given data from the problem: - The concentration of helium in the membrane at the inner surface, \(C_1 = 0.02\,\mathrm{kmol/m^3}\) - The concentration of helium in the membrane at the outer surface, \(C_2 = 0.005\,\mathrm{kmol/m^3}\) - The thickness of the membrane, \(d = 1\,\mathrm{mm}\) - The binary diffusion coefficient of helium with respect to the plastic, \(D = 10^{-9}\,\mathrm{m^2/s}\)
02

Convert units

Let's convert the thickness of the membrane from millimeters to meters: \(d = 1\,\mathrm{mm} \times \frac{1\,\mathrm{m}}{1000\,\mathrm{mm}} = 0.001\,\mathrm{m}\)
03

Calculate the concentration gradient

We need to determine the concentration gradient of the helium across the membrane. The concentration gradient is the change in concentration divided by the thickness of the membrane: \(\frac{\Delta C}{\Delta x} = \frac{C_2 - C_1}{d}\) \(\frac{\Delta C}{\Delta x} = \frac{0.005 - 0.02}{0.001}\) \(\frac{\Delta C}{\Delta x} = -15000\,\mathrm{kmol/m^4}\)
04

Determine the diffusive flux using Fick's law

Now, we'll use Fick's law to find the diffusive flux: \(J = -D \frac{\Delta C}{\Delta x}\) \(J = -\left(10^{-9}\,\mathrm{m^2/s}\right) \left(-15000\,\mathrm{kmol/m^4}\right)\) \(J = 1.5 \times 10^{-5}\,\mathrm{kmol/(m^2 \cdot s)}\) The diffusive flux of helium through the thin plastic membrane is \(1.5 \times 10^{-5}\,\mathrm{kmol/(m^2 \cdot s)}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fick's Law
Fick's Law is the cornerstone of our understanding of diffusion processes. It describes how particles move from an area of higher concentration to an area of lower concentration. Diffusion is driven by the kinetic energy of particles, which causes them to disperse in order to reach a state of equilibrium. Imagine a cup of coffee into which you鈥檝e just added sugar; over time, without stirring, the sugar particles spread out until they are evenly distributed. That鈥檚 diffusion in action.

In mathematical terms, Fick's First Law states that the diffusive flux is proportional to the negative gradient in the concentration of the substance. In practical terms, this can be written as:
\[ J = -D \frac{\Delta C}{\Delta x} \]
where \( J \) is the diffusive flux (the amount of substance that flows through a unit area per unit time), \( D \) is the diffusion coefficient (which quantifies the ease with which a particle moves through a medium), \( \Delta C \) is the change in concentration, and \( \Delta x \) is the distance over which the concentration changes.

Using Fick's Law, we can predict how quickly substances will diffuse across different mediums under various conditions. This has applications in fields ranging from chemical engineering to biology.
Concentration Gradient
A concentration gradient is a fundamental concept in the study of diffusion, and it represents a difference in the concentration of a substance across a space. Think of it as a slope that particles roll down; the steeper the slope, the faster the roll.

In the context of diffusion, particles tend to move down the concentration gradient, from a region of high concentration to a region of low concentration, until the concentration is uniform across the space. This movement is an attempt to balance the uneven distribution of particles.

The concentration gradient is quantified as:
\[ \frac{\Delta C}{\Delta x} \]
where \( \Delta C \) is the change in concentration and \( \Delta x \) is the distance over which the change occurs.

In our exercise scenario, we calculated a steep concentration gradient for helium moving through a plastic membrane, which indicates a significant difference in helium concentration across the membrane and This gradient is what drives the diffusion of helium from one side to the other.
Binary Diffusion Coefficient
The binary diffusion coefficient, symbolized as \( D \), is a crucial parameter in describing how quickly two species will mix or separate in a binary mixture. It encapsulates how the physical properties of the substances and the medium through which they are diffusing affect their movement.

The diffusion coefficient depends on factors such as:
  • The size of the diffusing particles (smaller particles typically diffuse faster)
  • The nature of the medium (particles may diffuse more easily through a gas than a viscous liquid)
  • The temperature of the environment (higher temperatures generally increase the diffusion rate due to increased kinetic energy)
  • The presence of barriers or channels within the medium
For gases in particular, the diffusion coefficient is influenced by the pressure and temperature of the system according to a relationship known as Graham's law.

In our textbook problem, a binary diffusion coefficient of \( 10^{-9} \mathrm{m^2/s} \) for helium in a plastic material was given, signifying that helium atoms difffuse through the plastic at a rate determined by this coefficient. Understanding the diffusion coefficient is essential for predicting the speed and extent of diffusion in a given system.

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Most popular questions from this chapter

As an employee of the Los Angeles Air Quality Commission, you have been asked to develop a model for computing the distribution of \(\mathrm{NO}_{2}\) in the atmosphere. The molar flux of \(\mathrm{NO}_{2}\) at ground level, \(N_{\mathrm{A}, 0}^{N}\), is presumed known. This flux is attributed to automobile and smoke stack emissions. It is also known that the concentration of \(\mathrm{NO}_{2}\) at a distance well above ground level is zero and that \(\mathrm{NO}_{2}\) reacts chemically in the atmosphere. In particular, \(\mathrm{NO}_{2}\) reacts with unburned hydrocarbons (in a process that is activated by sunlight) to produce PAN (peroxyacetylnitrate), the final product of photochemical smog. The reaction is first order, and the local rate at which it occurs may be expressed as \(\dot{N}_{\mathrm{A}}=-k_{1} C_{\mathrm{A}}\). (a) Assuming steady-state conditions and a stagnant atmosphere, obtain an expression for the vertical distribution \(C_{\mathrm{A}}(x)\) of the molar concentration of \(\mathrm{NO}_{2}\) in the atmosphere. (b) If an \(\mathrm{NO}_{2}\) partial pressure of \(p_{\mathrm{A}}=2 \times 10^{-6}\) bar is sufficient to cause pulmonary damage, what is the value of the ground level molar flux for which you would issue a smog alert? You may assume an isothermal atmosphere at \(T=300 \mathrm{~K}\), a reaction coefficient of \(k_{1}=0.03 \mathrm{~s}^{-1}\), and an \(\mathrm{NO}_{2}\)-air diffusion coefficient of \(D_{\mathrm{AB}}=0.15 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}\).

Hydrogen gas is used in a process to manufacture a sheet material of \(6-\mathrm{mm}\) thickness. At the end of the process, \(\mathrm{H}_{2}\) remains in solution in the material with a uniform concentration of \(320 \mathrm{kmol} / \mathrm{m}^{3}\). To remove \(\mathrm{H}_{2}\) from the material, both surfaces of the sheet are exposed to an airstream at \(500 \mathrm{~K}\) and a total pressure of \(3 \mathrm{~atm}\). Due to contamination, the hydrogen partial pressure is \(0.1 \mathrm{~atm}\) in the airstream, which provides a convection mass transfer coefficient of \(1.5 \mathrm{~m} / \mathrm{h}\). The mass diffusivity and solubility of hydrogen (A) in the sheet material (B) are \(D_{\mathrm{AB}}=2.6 \times 10^{-8} \mathrm{~m}^{2} / \mathrm{s}\) and \(S_{\mathrm{AB}}=160 \mathrm{kmol} / \mathrm{m}^{3} \cdot\) atm, respectively. (a) If the sheet material is left exposed to the airstream for a long time, determine the final content of hydrogen in the material \(\left(\mathrm{kg} / \mathrm{m}^{3}\right)\). (b) Identify and evaluate the parameter that can be used to determine whether the transient mass diffusion process in the sheet can be assumed to be characterized by a uniform concentration at any time during the process. Hint: This situation is analogous to that used to determine the validity of the lumped-capacitance method for a transient heat transfer analysis. (c) Determine the time required to reduce the hydrogen mass density at the center of the sheet to twice the limiting value calculated in part (a).

Referring to Problem 14.34, a more representative model of respiration in a spherical organism is one for which oxygen consumption is governed by a firstorder reaction of the form \(\dot{N}_{\mathrm{A}}=-k_{1} C_{\mathrm{A}}\). (a) If a molar concentration of \(C_{\mathrm{A}}\left(r_{o}\right)=C_{\mathrm{A}, o}\) is maintained at the surface of the organism, obtain an expression for the radial distribution of oxygen, \(C_{\mathrm{A}}(r)\), within the organism. Hint: To simplify solution of the species diffusion equation, invoke the transformation \(y \equiv r C_{\mathrm{A}}\). (b) Obtain an expression for the rate of oxygen consumption within the organism. (c) Consider an organism of radius \(r_{o}=0.10 \mathrm{~mm}\) and a diffusion coefficient of \(D_{\mathrm{AB}}=10^{-8} \mathrm{~m}^{2} / \mathrm{s}\). If \(C_{\mathrm{A}, o}=5 \times 10^{-5} \mathrm{kmol} / \mathrm{m}^{3}\) and \(k_{1}=20 \mathrm{~s}^{-1}\), estimate the corresponding value of the molar concentration at the center of the organism. What is the rate of oxygen consumption by the organism?

A 100-mm-long, hollow iron cylinder is exposed to a \(1000^{\circ} \mathrm{C}\) carburizing gas (a mixture of \(\mathrm{CO}\) and \(\mathrm{CO}_{2}\) ) at its inner and outer surfaces of radii \(4.30\) and \(5.70 \mathrm{~mm}\), respectively. Consider steady-state conditions for which carbon diffuses from the inner surface of the iron wall to the outer surface and the total transport amounts to \(3.6 \times 10^{-3} \mathrm{~kg}\) of carbon over \(100 \mathrm{~h}\). The variation of the carbon composition (weight \(\%\) carbon) with radius is tabulated for selected radii. $$ \begin{array}{lllllllll} r(\mathrm{~mm}) & 4.49 & 4.66 & 4.79 & 4.91 & 5.16 & 5.27 & 5.40 & 5.53 \\ \text { Wt.C }(\%) & 1.42 & 1.32 & 1.20 & 1.09 & 0.82 & 0.65 & 0.46 & 0.28 \end{array} $$ (a) Beginning with Fick's law and the assumption of a constant diffusion coefficient, \(D_{\mathrm{C}-\mathrm{Fe}}\), show that \(d \rho_{\mathrm{C}} / d(\ln r)\) is a constant. Sketch the carbon mass density, \(\rho_{\mathrm{C}}(r)\), as a function of \(\ln r\) for such a diffusion process. (b) The foregoing table corresponds to measured distributions of the carbon mass density. Is \(D_{\mathrm{C}-\mathrm{Fe}}\) constant for this diffusion process? If not, does \(D_{\mathrm{C}-\mathrm{Fe}}\) increase or decrease with an increasing carbon concentration? (c) Using the experimental data, calculate and tabulate \(D_{\mathrm{C}-\mathrm{Fe}}\) for selected carbon compositions.

A solar pond operates on the principle that heat losses from a shallow layer of water, which acts as a solar absorber, may be minimized by establishing a stable vertical salinity gradient in the water. In practice such a condition may be achieved by applying a layer of pure salt to the bottom and adding an overlying layer of pure water. The salt enters into solution at the bottom and is transferred through the water layer by diffusion, thereby establishing salt-stratified conditions. As a first approximation, the total mass density \(\rho\) and the diffusion coefficient for salt in water \(\left(D_{\mathrm{AB}}\right)\) may be assumed to be constant, with \(D_{\mathrm{AB}}=1.2 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). (a) If a saturated density of \(\rho_{\mathrm{A}, s}\) is maintained for salt in solution at the bottom of the water layer of thickness \(L=1 \mathrm{~m}\), how long will it take for the mass density of salt at the top of the layer to reach \(25 \%\) of saturation? (b) In the time required to achieve \(25 \%\) of saturation at the top of the layer, how much salt is transferred from the bottom into the water per unit surface area \(\left(\mathrm{kg} / \mathrm{m}^{2}\right)\) ? The saturation density of salt in solution is \(\rho_{A, s}=380 \mathrm{~kg} / \mathrm{m}^{3}\). (c) If the bottom is depleted of salt at the time that the salt density reaches \(25 \%\) of saturation at the top, what is the final (steady-state) density of the salt at the bottom? What is the final density of the salt at the top?
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