1 All walls at once. Section 7.1 incorporated only one atom-wall potential. For a cubic volume, there are six potentials, one for the atom of each atom with each wall. Take the atom-wall potentials to be additive. Show the ideal gas law continues to obtain, so long as U is short-ranged. Estimate the correction to P if U has a larger range.
2 The average force. Confirm the steps leading from 6.3 to 6.4.
3 Surface effects. Consider an atom in a long thin square pipe.
What happens if L is not much larger than 
, in the limit
for all 
. Is P larger or smaller
than its ideal-gas law limit? Give a physical interpretation for your answer.
4 A square-well surface potential. Suppose the atom-wall potential has
the form 
, 
; 
. The transitions between values of U are smooth enough that
eq. 6.4 is still integrable. Obtain P as a function of N, V,
T, a. Find the behavior of P as 
. Show 
as 
.
5 Thermodynamics. Derive eqs. 6.12 and 6.13.
6 The thermalizing wall. The kinetic-theory calculation in Section 6-3 assumed that atoms are scattered elastically by collisions with the wall. An alternative assumption is that every atom strikes the wall, sticks briefly, and is re-emitted by the wall with a randomized momentum. The momentum of a re-emitted atom has no relation to the momentum with which the atom struck the wall. However, atoms moving away from the wall must have a Maxwell-Boltzmann momentum distribution. What is the ``randomized velocity distribution'' -- the probability that an atom is emitted with a particular momentum -- for re-emitted atoms? (Hint: not Maxwell-Boltzmann. The answer gives the inverse of the collision rates.) Show that this alternative assumption about atom-wall scattering gives the ideal gas law.
7 The nozzle. Consider a wall containing a long, thin hole. On one
side of the wall is a gas of known N, V, T. Obtain the velocity and momentum
distribution of gas atoms coming out of the hole on the other side of the wall.
[Hint: To pass through the hole, the atom must first reach the wall, a state
of affairs described by eq. 6.15.] This is intrinsically a
three-dimensional problem. Show that a 3-dimensional calculation (in which
the pipe accepts atoms approaching through a fixed range of solid angles) and a
1-dimensional calculation (in which only 
is considered) do not get the
same answer. Why?
8P from kinetic theory. Derive eq. 6.17 from eq. 6.16. Confirm that the average force from N atoms is N times the average force from a single atom.