Question #6b4cb

1 Answer
Truong-Son N.
Dec 11, 2017

The mean-free path #l# for a particular particle following the Maxwell-Boltzmann distribution is given by:

#l = 1/(sqrt2 rho sigma)#,

where #rho = N/V# is the number of particles per unit volume, and #sigma = pi(2r)^2 = pid^2# is the collision cross section. Note that #d = 2r# is the diameter of the particle, and #r# is its radius.

You can think of #rho# as the number density. Next, for ideal gases, the ideal gas law applies:

#PV = Nk_BT#

And so, #N/V = P/(k_BT)# and:

#bar|ul(" "l = 1/sqrt2 (k_BT)/(pid^2P)" ")|#

And now we can see that #color(blue)(l prop T)# and #color(blue)(l prop 1/P)#.

This says that at higher temperatures, particles generally travel farther before colliding with something else. It should also make sense that at lower pressures, they can again travel farther before hitting something else.

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