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Chapter 14: Problem 46

A tube has a length of 0.015 \(\mathrm{m}\) and a cross-sectional area of \(7.0 \times 10^{-4} \mathrm{m}^{2} .\) The tube is filled with a solution of sucrose in water. The diffusion constant of sucrose in water is \(5.0 \times 10^{-10} \mathrm{m}^{2 / \mathrm{s}}\) . A difference in concentration of \(3.0 \times 10^{-3} \mathrm{kg} / \mathrm{m}^{3}\) is maintained between the ends of the tube. How much time is required for \(8.0 \times 10^{-13} \mathrm{kg}\) of sucrose to be transported through the tube?

Short Answer

Expert verified
Approximately 1.14 seconds.

Step by step solution

01

Understand Fick's First Law

Fick's first law of diffusion can be expressed as \( J = -D \frac{dC}{dx} \), where \( J \) is the diffusion flux, \( D \) is the diffusion constant, \( \frac{dC}{dx} \) is the concentration gradient. The negative sign indicates the diffusion occurs from high to low concentration.
02

Set Up Fick's First Law

Reformulate Fick's law to solve for \( J \). Here, \( J = \frac{Q}{At} \), where \( Q \) is the amount of sucrose transported, \( A \) is the cross-sectional area, and \( t \) is the time. Set \( \frac{Q}{At} = D \frac{\Delta C}{L} \) where \( \Delta C \) is the concentration difference and \( L \) is the length of the tube.
03

Plug in the Known Values

Substitute the given values into the equation: \( \frac{8.0 \times 10^{-13}}{7.0 \times 10^{-4} \cdot t} = 5.0 \times 10^{-10} \frac{3.0 \times 10^{-3}}{0.015} \).
04

Simplify and Solve for Time

First, calculate \( D \frac{\Delta C}{L} = 5.0 \times 10^{-10} \times \frac{3.0 \times 10^{-3}}{0.015} = 10^{-10} \). Then solve \( \frac{8.0 \times 10^{-13}}{7.0 \times 10^{-4} \cdot t} = 10^{-10} \). Rearrange to find \( t = \frac{8.0 \times 10^{-13}}{7.0 \times 10^{-4} \times 10^{-10}} \).
05

Calculate the Time Required

Perform the calculation: \( t = \frac{8.0 \times 10^{-13}}{7.0 \times 10^{-14}} = \frac{8}{7} \times 10^{-13+14} = \frac{8}{7} \times 10^{1} = \approx 1.14 \). The required time is approximately 1.14 seconds.

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Key Concepts

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

Diffusion Flux
Diffusion flux is a central concept within Fick's First Law of diffusion. It represents the amount of substance that moves through a unit area per unit time due to concentration differences. Imagine it like a river of molecules, where the flux tells you how many molecules flow across a "cross-section" of the river every second. For Fick's law, this is represented by the symbol \( J \).
The formula for diffusion flux is \( J = -D \frac{dC}{dx} \), where:
  • \( J \) is the diffusion flux measured in units like kg/m虏路s.
  • \( D \) is the diffusion constant.
  • \( \frac{dC}{dx} \) is the concentration gradient.
The negative sign in Fick's law indicates that diffusion occurs spontaneously from high to low concentration. This aligns with the tendency of particles to move towards equilibrium.
Concentration Gradient
The concentration gradient is a measure of how concentration changes over a distance. It tells us how steep or gentle the concentration difference is between two points. In essence, it's the difference per unit length, and it drives the diffusion process.
In Fick's First Law, the concentration gradient is represented as \( \frac{dC}{dx} \). For the tube problem, this gradient helps determine how quickly sucrose will diffuse from one end to the other.
When the concentration gradient is large, particles move more quickly from regions of high concentration to low concentration. Think of it like a steep hill, where a ball rolls down faster than on a flat surface.
Diffusion Constant
The diffusion constant, represented by \( D \), is a measure of how easily a substance can diffuse through a medium. It depends on the type of particles diffusing and the medium they鈥檙e moving through.
The diffusion constant has units of area over time (e.g., m虏/s), reflecting how fast the diffusion occurs. In practical terms, a higher diffusion constant means the substance diffuses more easily and quickly.
For the sucrose in water example, the diffusion constant of \( 5.0 \times 10^{-10} \) m虏/s indicates a moderate rate of diffusion, suitable for liquids, where particles encounter resistance but still move over time.

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Most popular questions from this chapter

A clown at a birthday party has brought along a helium cylinder, with which he intends to fill balloons. When full, each balloon contains 0.034 \(\mathrm{m}^{3}\) of helium at an absolute pressure of \(1.2 \times 10^{5} \mathrm{Pa}\) . The cylinder contains helium at an absolute pressure of \(1.6 \times 10^{7} \mathrm{Pa}\) and has a volume of 0.0031 \(\mathrm{m}^{3}\) . The temperature of the helium in the tank and in the balloons is the same and remains constant. What is the maximum number of balloons that can be filled?

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