Finding average angle in kinetic theory of gases

In summary, the author is trying to determine the average angle calculation between particles that have hit a plane boundary. They are unsure of how to do it and need help from someone else. They also include a velocity function which just cancels out.
  • #1
Kevin Willis
25
3

Homework Statement



"Show that particles hitting a plane boundary have traveled a distance 2λ/3 perpendicular to the plane since their last collision, on average."

Homework Equations


(Root mean path squared) <x> = 2^(.5)λ

λ = ( 2^(.5) * n * sigma )^(-1)

ANSWER:
upload_2015-2-17_11-49-17.png

The Attempt at a Solution


[/B]
I already knew I needed <x cos(theta)>.
The book tells me that I can split x and cos(theta) apart by this rule:
"
upload_2015-2-17_11-51-35.png
"

My issue is that I do not understand how to calculate the average angle. I know the boundaries are 0 < theta < pi/2.
I would expect the average angle to be pi/4.

upload_2015-2-17_12-2-58.png


(NOTE: I measured the angle from the plane and should have shown measurement from the z axis down to satisfy the cos(theta).

So I figured that if I can just determine the average angle, I could plug it into <x> * cos(theta) and I would be done. However, that does not seem to work.

Upon looking at the answer, I am clueless of what they are doing in the angle integral:
upload_2015-2-17_12-5-39.png
. Why is sin(theta) here?

I am also unsure why they are taking the mean probability distributions
upload_2015-2-17_12-7-27.png
and dividing them by the integral of the probability functions
upload_2015-2-17_12-8-29.png
.And lastly, I don't understand why they included the velocity function, even though it just cancels out anyways.

Thanks.
 
Last edited:
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  • #2
Kevin Willis said:
I do not understand how to calculate the average angle.
But you don't want the average angle - you want the average of cos(θ).
Despite the diagram, I believe θ here is the angle to the normal. Consider a hemisphere centred on the impact point on the wall. Particles come equally likely from each small area element on the hemisphere. For the region between θ and θ+dθ from the normal, what area is that? What integral can you write down for the average of cos(θ)?
 
  • #3
upload_2015-2-20_14-54-1.png

This should be the average of cos(θ).
 

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  • #4
Kevin Willis said:
View attachment 79367
This should be the average of cos(θ).
Not sure we're in agreement over which angle is theta. Since you are interested in perpendicular distance and are taking that as cos(theta), I believe theta is the angle to the normal. But in your diagram you mark an angle to the plane, and the above calculation also seems to be taking it as the angle to the plane.
 
  • #5
I do mean angle from the normal. In the OP I wrote "(NOTE: I measured the angle from the plane and should have shown measurement from the z axis down to satisfy the cos(theta)." directly under the diagram.

So, cos(theta) here is measuring perpendicular distance. Parallel would be sin(theta). I have submitted the correction below to clear any misunderstandings.
upload_2015-2-20_15-49-30.png
 
  • #6
Kevin Willis said:
I do mean angle from the normal. In the OP I wrote "(NOTE: I measured the angle from the plane and should have shown measurement from the z axis down to satisfy the cos(theta)." directly under the diagram.

So, cos(theta) here is measuring perpendicular distance. Parallel would be sin(theta). I have submitted the correction below to clear any misunderstandings.View attachment 79369
Ok, then I disagree with your integral for the average value of cos theta. How do you arrive at that?
 
  • #7
Kevin Willis said:

Homework Statement



I am also unsure why they are taking the mean probability distributions View attachment 79220and dividing them by the integral of the probability functions View attachment 79221.

Oh, that part I know. It's normalization. Depending on how you define that probability density its integration over all domain might not be 1, so you want to make sure the value you're calculating gets normalized.
 

1. What is the average angle in kinetic theory of gases?

The average angle in kinetic theory of gases refers to the average direction of motion of gas particles in a given system. It is a measure of the average kinetic energy and velocity of the gas particles.

2. How is the average angle calculated in kinetic theory of gases?

The average angle is calculated by taking the sum of the angles of motion of all gas particles in a system and dividing it by the total number of particles. This calculation is based on the assumption that gas particles move in random directions.

3. Why is the average angle important in kinetic theory of gases?

The average angle is important in kinetic theory of gases because it helps us understand the behavior and properties of gases. It is used to calculate important factors such as pressure, temperature, and volume of a gas.

4. How does the average angle affect the behavior of gases?

The average angle affects the behavior of gases by influencing their speed and direction of motion. A higher average angle results in faster and more chaotic movement of gas particles, while a lower average angle results in slower and more orderly movement.

5. Can the average angle change in a gas system?

Yes, the average angle can change in a gas system. It depends on various factors such as temperature, pressure, and volume. A change in these factors can alter the average angle and consequently affect the behavior of the gas particles.

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