How Does the Kinetic Theory of Gases Describe Particle Collisions?

[unscientific]

Hi guys, I'm studying the Kinetic Theory of gases from Pauli's book Vol. 3. Here he describes a section on the Mean free path, where the probability of two particles with speed v and v' colliding is described as:

What is f(v,v') here? Is it the velocity distribution function? If so, isn't it simply the fraction of molecules with velocities v and v' ?

 

[Simon Bridge]

$f(\vec v, \vec v')$ appears to be the probability that two molecules with velocities $\vec v$ and $\vec v'$ hit one another - as stated in the accompanying text.

It may be a bit confusing because it is expressed as a small range of velocities.

 

[unscientific]

$f(\vec v, \vec v')$ appears to be the probability that two molecules with velocities $\vec v$ and $\vec v'$ hit one another - as stated in the accompanying text.

It may be a bit confusing because it is expressed as a small range of velocities.

How did they get the exponential expression for f(v²) and f(v'²)?

And it is implied that the probability of both of them colliding is the product: f(v²) f(v'²) dv dv'

Then on its own, what does f(v²)dv and f(v'²)dv' mean?

 

[Simon Bridge]

replace the "f"'s with "p"'s ... you may be able to read it better.

$$p(\vec v^2) = \sqrt{\frac{\alpha^3}{\pi}}e^{-\alpha \vec v^2}$$ ... comes from the distribution of kinetic energies perhaps (as $K\propto v^2$) ...

You may find the following approach easier:
http://physics.bu.edu/~redner/542/refs/reif-chap12.pdf

 

[unscientific]

replace the "f"'s with "p"'s ... you may be able to read it better.

$$p(\vec v^2) = \sqrt{\frac{\alpha^3}{\pi}}e^{-\alpha \vec v^2}$$ ... comes from the distribution of kinetic energies perhaps (as $K\propto v^2$) ...

You may find the following approach easier:
http://physics.bu.edu/~redner/542/refs/reif-chap12.pdf

I have verified that the book meant f(v²) represents the fraction of molecules traveling with velocity v:

Why is the product of f(v) and f(v') the probability that both particles with velocities v and v' collide? Technically the product means fraction of molecules with velocities v and v', implying a molecule having two velocities at the same time --- which doesn't make sense?

 

[Simon Bridge]

Did you read the link I gave you?

 

[unscientific]

Did you read the link I gave you?

Yeah, nothing is said about the relation between probability of collision and distribution of velocities.

Collision time, probability of collision and mean free path was explained, but it didn't relate them to the distribution of velocities?

 

[Simon Bridge]

Is the collision time not related to the relative velocities (w)?
Is the distribution of relative velocities not related to the P(v,v') ?