The kinetic theory of the ideal gas envisions the ideal gas as a cloud of
non-interacting point particles. The methods and arguments of kinetic theory
are fundamentally unlike those of Gibbs' ensemble treatment of classical
statistical mechanics. In each element of a Gibbsian ensemble, atoms have fixed
positions and momenta; mechanical variables are evaluated based on atomic
coordinates at a single instant in time. In contrast, kinetic theory relies
on model calculations in which the motions of atoms are computed over some
short time interval 
.
Suppose one traces the trajectories of the atoms of an ideal gas.
During , some atoms collide with a container
wall, while other do not. The total momentum transferred into
the wall by the atom-wall collisions during is 
.
This momentum transfer corresponds to an average force 
; for a wall of area 
the pressure is 
.
Which atoms collide with a wall? Consider V to be an 
cube, with the yz-plane at x = 0 as the wall, as shown in Figure 6-3. To
collide with the wall between the times 0 and , at t = 0 an atom
with x-momentum 
(and velocity 
) must be between
x = 0 and 
. Since atoms must have 
, only
atoms with 
reach the wall. For an ideal gas, the atoms do not
interact with each other, so interatomic collisions do not occur. In a real
gas, collisions between atoms affect which atoms strike each wall.
Figure 6.3: Kinetic model of the ideal gas.
Atoms 1, 2, 3 all have x-component of their velocity 
. Atom 1
collides with the wall during , but atoms 2 and 3 do not. Atom 2 is
too far from the wall ( 
), while atom 3 is headed in the
wrong direction. Atoms 4 and 5 both have 
(with 
) for the x-component of their velocities. While atom 5 is at
some 
, 
still satisfies 
, so atom 5 will reach the wall during .
During , an atom that collides with the wall transfers to it an
impulse . The impulse's magnitude could be obtained by
integrating 
over the atom's trajectory x(t). The
integration, besides being cumbersome, requires complete knowledge of U(x).
Kinetic theory replaces the cumbersome integration with a conservation
argument: Suppose is much larger than the duration of any wall-atom
collision. Each atom has an initial x-momentum . Since an atom is
brought to a stop when it hits the wall, all of must be transferred
into the wall. Furthermore, to maintain equilibrium, the atom cannot just
stick to the wall. If the atoms all stayed on the wall, the supply of gas atoms
within the container would eventually be depleted. A simple assumption,
adequate to maintain equilibrium, is that atoms that strike the wall are
elastically scattered. For collisions with the yz-plane, elastic scattering
takes 
onto 
while leaving 
and 
unchanged.
An elastically-scattered atom having initial momentum therefore
transfers to the wall an impulse 
. [A detailed treatment of atom-
wall collisions and adsorption is a hard problem.]
A fundamental assumption of the kinetic theory of gasses is that a gas atom's momentum is described by the Maxwell-Boltzmann distribution
The kinetic theory ansatz[1] is that W(r, p) is taken to apply at a single time, the evolution of the system to other times being determined by applying the laws of mechanics. Combining all of the above arguments, the average force on a wall due to a single atom is
whencefrom follows
The gas atoms do not interact with each other, so the average force on a wall due to N atoms is just N times the average force from a single atom. The kinetic model thus yields the ideal gas law.
In the kinetic model, U(x) was constrained by several implicit assumptions, notably:
(i) Atoms move ballistically,
(implied by the upper bound of 
) except during short collisions.
(ii) An atom that has collided with a wall then recedes off to infinity (so
that it transfers exactly 
of momentum into the wall, without
experiencing multiple atom-wall collisions.)
(iii) A collision is an identifiable transient phenomenon.
These assumptions are consistent with a purely repulsive short-range U(x), but are not consistent, e. g. with potentials that allow formation of atom-wall bound states (see problem 6-4). These assumptions are completely wrong for atoms in liquids.
A careful reader will note that eq. 6.16 hides several effects.
Suppose an atom strikes the x = 0 plane near 
and (y,z) =
(0,0), its y- and z- momenta both being large and positive. At t = 0, a
simpleminded back-calculation puts such an atom someplace where y, z < 0,
contrary to the requirement that atoms stay inside V. The dashed lines in
Figure 6.5 indicate the notional original location of the atom, starting in a
region which atoms cannot reach, and the atom's
subsequent notional trajectory. If atom-wall collisions are elastic, a proper
back-calculation shows that such an atom must have originated within V, been
scattered by other container walls at some 
, finally
reaching the x = 0 plane via an indirect path.
Figure 6.5 indicates how atom-wall reflections replace atoms coming from
impossible locations (dashed lines) with atoms reflected from allowed
locations(solid lines). Careful analysis shows that this replacement process
has no effect on the pressure of the gas.
Figure 6.4: Atom-wall collisions
