374 Problems €4 Solutions on Thermddynam'cs 8' Statistical Mechanics(b) a fast molecule?Solution:
( wzs c 0 nsin)Consider the motion of a molecule A. Only those molecules whose
centers separate from the center of A by distances smaller than or equal to
the effective diameter of the molecule can collide with A. We can imagine
a cylinder, whose axis coincides with the orbit of the center of A, with a
radius equal the effective diameter d of the molecule. Then all the molecules
whose centers are in the cylinder will collide with A. The cross section of
this cylinder is u = ad2.
In the time interval t, the path length of A is tLt (iz is the average
relative speed), which corresponds to a volume act of the cylinder. The
number of collisions A suffers with other molecules is naGt (n is the number
density). The frequency of collisions is therefore
- V V
Hence the mean free path is A = : = ---=. From the Maxwell distribution,
z nuu
we can show that U = v'k. Thus
- V V
A=- (^1) - - 1
don fland2 '
A more precise calculation from the Maxwell distribution gives the mean
free path of a molecule whose speed is V = pz as
m
-1 A=- x2
nd2 '
where $(z) = zexp(-z2) + (2z2 + 1) exp(-y2)dy.
LX
(a) For a slow molecule, iT ---* 0, or z -+ 0,
Am---- X - mv
find2 nd2dm '
(b) For a fast molecule, V -+ 03, or z -+ 03,
-1
Am-.
and2