Deriving the diffusion constant D

[Rukas Kang]

Homework Statement

Prove that the diffusion constant D of an ideal gas can be expressed as D=1/3λv(v bar) where λ is the mean free pass of the gas molecule. And v bar is the average speed of the gas molecules obtained from the kinetic theory of gases.(Use the picture to setup the net flux change of gas molecules flowing from a negative to positive z direction.)

Homework Equations

Use the Taylor expansion of the form
f (x) = a0f(0)+ a1(df/dx)(x=0)x +...+ an(d^nf/dx^n)(x=0)x^n
for calculating the concentration c(z) on z axis at z = ±λ with the initial condition c(z=0)=c0 . When doing so, use the first two terms only of the expansion and use a1 = 2/3 for this particular system. You may need to use the molecular flux Φ (i.e. # of molecules crossing a unit area per a unit time) of the gas.

The Attempt at a Solution

I've tried to solve as deriving the the coefficient of viscosity, but I couldn't. ;(

 

[anathema]

Thank you for your post. I would be happy to assist you in proving the expression for the diffusion constant D of an ideal gas.

First, let's define some variables for clarity:
- D: diffusion constant
- λ: mean free path of gas molecules
- v bar: average speed of gas molecules
- c(z): concentration of gas molecules at a given position z
- c0: initial concentration of gas molecules at z=0
- Φ: molecular flux of gas molecules (i.e. number of molecules crossing a unit area per unit time)

To begin, let's consider the net flux of gas molecules across a unit area in the z direction, as shown in the picture provided. We can express this as:

Φ = Φ(z+λ) - Φ(z)

Where Φ(z+λ) is the flux at z+λ and Φ(z) is the flux at z.

Next, we can use the Taylor expansion provided in the forum post to calculate the concentration c(z) at z=±λ. We can write this as:

c(z+λ) = c(z) + a1(dc/dz)(z)λ + O(λ^2)

Where a1=2/3, as given in the forum post. Note that we are using the first two terms of the expansion, as instructed.

Now, let's substitute this expression into our net flux equation:

Φ = [c(z) + a1(dc/dz)(z)λ + O(λ^2)][c(z+λ) - c(z)]

We can simplify this further by using the definition of Φ and rearranging the terms:

Φ = [c(z) + a1(dc/dz)(z)λ + O(λ^2)]/λ

Next, we can use the kinetic theory of gases to express the concentration c(z) in terms of the average speed v bar of the gas molecules. According to the kinetic theory, the concentration c(z) is directly proportional to the average speed v bar, and we can write this as:

c(z) = c0(v bar)

Where c0 is the initial concentration at z=0.

Substituting this into our net flux equation, we get:

Φ = [c0(v bar) + a1(dc/dz)(z)λ + O(λ^2)]/λ

Now, we can use the