Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Phyaica 389

good the vacuum ought to be if the beam is to remain collimated for at
least 1 meter. (1 torr = ImmRg).
(UC, Berkeley)
Solution:
(a) The Maxwell distribution is given by

There are nAv, fdv,dvydv, atoms in the velocity interval v - v + dv that
escape through the area A of the hole. The number of atoms that pass
through the second hole (the collimator C) is

m 312 1 2kT A

=nA (27rkT) - 5 (m) dz

With A = 10 x 0.1 = 1.0 mm2 = lo-' m2,

d = 10 cm = 1.0 x 10-'m ,

p = 6 x lop3 torr = 0.80N/m2 ,

T = 550 K

we have 4 = 6 x 10l1 s-l.

(b) D(v)dv = Cv3e- *" dv, where C is the normalizing factor given

Hence

(c) Assume that the vacuum region is at room temperature T = 300 K.

Since the mean free path is 1 = 1 m, we have

kT 1.38 x x 300

= 0.414 Pa

P-z= 1 x 10-20

= 3 x 10-~ torr ,