4
PHENOMENA OF
CAPILLARITY
P
=
P
-
OD
(SO2
1
2
p
If
we now
also
assume
that the
density of
the liquid is
constant
up
to
its
surface,
which
is
made
plausible
by
the fact that the
energy
of
the
surface
is
independent
of
temperature,
then
we
are
able
to
calculate
the
potential
energy per
unit
volume
in
the interior
of
the liquid,
and
that
per
unit surface.
I.e., if
we
put
i
2
•+00
£=-00
r+ao
+00
dxdydz.tp
y~-oo
J
Z=-oo
A
x2+y2+z2
=
K
,
then the potential
energy
per
unit
volume
is
(SC )2
P
=
P
-
K
%-
oo
Vz
If
we
imagine
a
liquid
of
volume
V
and
surface
S,
we
obtain
by
integration
(SC )2
P
=
P
-
K 01
OD
V
(Sc
)2
v
-
r
-J-
.o
,
where
the
constant K'
denotes
rX]=
1
x'
=0
rV-1
y'=0
rZ'
=0
Z]=-oo
^
£=-oo
£=oo
y=oo
y=-oo
2^=oo
^=0
dx.dy.dz.dx]
.dy]
.dz
A
(£-£'
)2+(y-y]
)'2+(z-z]
)2
Since
nothing
is
known
about
Q, we
naturally
do not get
any
relationship between
K
and
K'.
One
should
keep
in
mind, to begin
with,
that
we
cannot
know
whether
or
not
the molecule
of the
liquid
contains the n-fold
mass
of the
gas
molecule,
but it follows
from
our
derivation
that
this
does not
change
our
expression
for the potential
energy
of the liquid.
Based
on
the
assumptions
we
have
just
made, we
obtain the
following
expression
for the potential
energy
of
the
surface:
[9]
[10]