Problems and Solutions on Thermodynamics and Statistical Mechanics

3 70 Problems €4 Solutions on Thermodynamics d Statistical Mechanics

Solution:

An apparatus that could be used is shown in Fig. 2.39. Atomic beam

enters into the cylinder R of diameter D after it passes through slits S1 and

S2. The cylinder R rotates about its axis with angular velocity w. Suppose

an atom arrives at point p' on the cylinder, $' = s. The time taken for

the atom to travel from S2 to p' is t = -, where v is its velocity.

D

V

During this time, the cylinder has rotated through an angle 0 = wt.

Thus

8 = D -. # 1 --^1 D2w

2 2v

m

2

The energy of the atom is therefore E = -v2 = mD4w2/8s2. Hence there is
a one-to-one correspondence between s and E. By measuring the thickness
distribution of the atomic deposition on the cylinder we can determine the
energy distribution of the atoms.

2180

Write the Maxwell distribution, P(v,, uY, v~), for the velocities of mole-

cules of mass M in a gas at pressure p and temperature 2'. (If you have

forgotten the normalization constant, derive it from the Gaussian integral,

J-00

When a clean solid surface is exposed to this gas it begins to absorb

molecules at a rate W (moIecules/s.cm2).

A molecule has absorption probability 0 for a normal velocity compo-

nent less than a threshold VT, and absorption probability 1 for a normal

velocity greater than vr. Derive an expression for W.

Solution:

( was co nsin)

The Maxwell distribution of velocity is given by

We take the z-axis normal to the solid surface. Then the distribution of

the component v, of velocity is