240 DOC. 20 THEORETICAL ATOMISM
diameter of the smaller molecules
(d) measures a
few tenths of
a
millionth of
a
millimeter,
and that N lies between
1023
and
1024.
Far
more
exact
methods later
[8]
yielded values for N that deviated
by
just
a
little bit over
5
percent from
6.8
.
1023.
In most considerations of the kinetic theory of gases
it is
assumed that the mean
free path is small in comparison with the dimensions of the body confining the gas.
However,
it is
quite possible to realize and
to
treat mathematically cases for which
this assumption
no
longer holds. If the gas pressure
is
one ten-thousandth of
an
atmosphere (pressure of about
0.1
mm mercury), the mean free path
is
already
1
mm.
In
such cases
laws
that have been derived for molecular paths that are vanishingly
small compared with the dimensions of the body are
no
longer valid. For example,
the flow of
gases through
tubes
proceeds
as
if the
gas layer immediately adjacent
to
the wall of the tube
were
sliding
relative
to
the wall of the
tube, indeed, to
a
degree
that
can
be
predicted
on
the basis
of
the
theory. Especially simple
and
interesting
are
the laws for the
case
where the free
path
of the molecules
is
large
relative
to the
pertinent
dimensions of the vessel
walls, e.g.,
relative
to
the diameter of
a
tube
enclosing
the
gas.
In
this
case
laws
apply
that
are
totally
different from those
holding
in
the
usually
considered
case
of
a
path length
that
is
small in
comparison
with the
dimensions of the vessel.
Thus,
for
example,
Knudsen found
theoretically
and
confirmed
experimentally
the
following.
A
gas
container consists of
two
hollow
glass
[9]
spheres
connected
by a
tube
whose
diameter
is
small in
comparison
with the
path
length.
If the
two
hollow
spheres
are
brought
to
different
temperatures
in
such
a
way
that the
falling
off of the
temperature
takes
place along
the
glass tube,
a
higher
pressure
will be established
in
the
warmer
container than
in
the colder
one.
Thus,
the
laws of
hydrostatics do not
hold in these cases!
The methods and results of
the
kinetic theory of gases have also proved fruitful
beyond the domain of the theory of gases.
Van
der Waals added
to
the theory of
gases
by
taking into consideration the volume occupied
by
the molecules and the
[10]
attractive
forces
they
exert
on
each
other;
he
created
a theory
that also
covers-at
least
qualitatively-the
liquid aggregation
state.
Riecke and Drude
developed
a theory
that
explained
the
approximate constancy
of the ratio of the electric
to
the thermal
conductivity
of the
metals,
based
on
the
assumption
that,
in
metals,
freely moving,
electrically charged elementary particles participate
in thermal
agitation
[11] (cf.
Article
20).
The
theory
of
magnetism
also
owes an unexpected upswing
to
the
[12]
kinetic
theory
of
heat.
All of these
things are
mentioned here
only
in
passing.
On the
other
hand,
we
must
still deal with
two
topics
of
great importance, namely
Boltzmann's
general understanding
of the
essence
of irreversible
processes,
and the
insight, gained only recently,
that molecular kinetics
agrees
with
experience only
[13]
within certain limits. These
most
important questions
lead
us
into the thick of the
problems
that
now occupy
theoretical
physicists.
The Case
Where the
Mean Path Is
Not Small in
Comparison
with the
Di-
mensions
of
the Volume
Occupied
by
the Gas.
Applications of
the Kinetic
Theory.