Fick's Law (Schroeder, Thermal Physics Page 47)

In summary, the author uses Fick's law to estimate the time it would take for a dye molecule to diffuse across a glass of water, taking into account the size of the molecule and the volume of the glass. He uses the total number of molecules and the surface at the halfway point to calculate this time. However, some confusion remains as to the specific values and surfaces being considered.
  • #1
torquemada
110
0
This isn't a HW problem - I am just having a hard time following one of his examples.

On page 47 near Eq. 1.71 he says "As a quick example, consider a drop of food coloring added to a glass of water. Imagine that the dye has already spread uniformly through half of the glass. How long would it take to diffuse into the other half? According to Fick's law (Jx= -Ddn/dx; J=net flux, n=particle concentration), I can write very roughly



N/(AΔt)=D(N/V)/Δx (1.71)

where N is the total number of dye molecules, Δx is about 0.1m and V≈A*Δx. I've written the particle flux in terms of the same N to indicate that I want Δt to be the time for approximately all (that is, half) of the molecules to cross from one side of the glass to the other. I don't know how big a molecule of food coloring is, but it can't be too different in size from sucrose so I'll guess D=10^-9 m^2/s. Solving for Δt then gives 10^7 seconds, or almost 4 months."

I'm not following if Δx is the length of the whole glass, or half the glass? Similarly is the volume A*Δx the whole volume or half of the volume?

I am not sure how he got his rough dn/dx? He doesn't really elaborate on that.

The most confusing part though is how he relates it to the left side of the equation.

He says "I've written the particle flux in terms of the same N to indicate that I want Δt to be the time for approximately all (that is, half) of the molecules to cross" - is it all? Or is it half?

Which surface is he evaluating the flux across? I'm assuming the surface at halfway into the glass, but I just want to make sure.

Any elaboration here is greatly appreciated. Thanks
 
Physics news on Phys.org
  • #2
in advance!The Δx he is referring to is half the length of the glass, since he is considering the time it takes for the dye molecules to diffuse from one side of the glass to the other. The volume A*Δx is also half of the total volume as it is the volume of the half of the glass. He is referring to the total number of dye molecules, N, to indicate that he wants Δt to be the time it takes for approximately all, or half, of the molecules to cross from one side of the glass to the other. He is therefore evaluating the flux across the surface at the halfway point of the glass.
 

FAQ: Fick's Law (Schroeder, Thermal Physics Page 47)

1. What is Fick's Law?

Fick's Law is a fundamental equation in physics that describes the diffusion of a substance through a medium. It states that the rate of diffusion is directly proportional to the concentration gradient and the diffusion coefficient of the substance.

2. Who is Fick and how did he discover this law?

Adolph Fick, a German physiologist, first proposed this law in 1855 while studying the diffusion of gases in liquids. He derived the equation by applying the principles of conservation of mass and the second law of thermodynamics.

3. What is the significance of Fick's Law?

Fick's Law is widely used in many fields of science and engineering, including biology, chemistry, and materials science. It helps to understand and predict the movement of particles and molecules in various systems, such as cells, tissues, and industrial processes.

4. Can Fick's Law be applied to both gases and liquids?

Yes, Fick's Law can be applied to both gases and liquids, as long as the diffusion coefficient is known for the specific substance in the given medium. The diffusion coefficient may vary depending on the temperature, pressure, and other factors.

5. Are there any limitations to Fick's Law?

While Fick's Law is a useful and widely applicable equation, it has its limitations. It assumes that the medium is homogeneous and isotropic, meaning that the properties of the medium do not vary with position or direction. In reality, most systems are not perfectly homogeneous, so the accuracy of Fick's Law may decrease in those cases.

Back
Top