How Does Doubling the Temperature Affect the Mean Free Path in a Gas?

[Sarah0001]

Homework Statement

"The mean free path is independent of temperature at constant volume."

Relevant Equations

mean free path inversely proportional to density (=mass of gas/volume)
it is at constant volume density of gas remains same.
Assumption, all N particles in Volume V move with avg speed (rms speed) from avg KE for a given temperature T.

Say Temperature of a gas doubles, I do not understand how the average distance between particles (mean free path) is unaffected if they are traveling √2 times as fast in a fixed volume V. Root 2 as a factor of increase because T*2 --> KE*2 --> V*2 --> Vrms*√2
Is it because relative to one another the if both particles are moving at the same rate then, the distance between them is fixed, regardless of the rate at which distance is covered.

 

[haruspex]

how the average distance between particles (mean free path) is unaffected if they are traveling √2 times as fast

Imagine watching a video of them. If you speed ot up or slow it down the particles will move exactly the same distance between collisions.