But since M is constant,
dM = Sd[sigma] + [sigma]dS = 0. (14)
Eliminating dS from equations (13) and (14), and dividing by S, we
find
de
T = e - [sigma]--------, (15)
d[sigma]
In this expression [sigma] denotes the mass of unit of area of the
film, and e the energy of unit of area.
If we take the axis of z normal to either surface of the film, the
radius of curvature of which we suppose to be very great compared with
its thickness c, and if [rho] is the density, and [chi] the energy of
unit of mass at depth z, then
_
/ c
[sigma] = | [rho] dz, (16)
_/0
and
_
/ c
e = | [chi] [rho] dz. (17)
_/0
Both [rho] and [chi] are functions of z, the value of which remains
the same when z - c is substituted for z. If the thickness of the film
is greater than 2 [epsilon], there will be a stratum of thickness c -
2 [epsilon] in the middle of the film, within which the values of
[rho] and [chi] will be [rho]0 and [chi]0. In the two strata on either
side of this the law, according to which [rho] and [chi] depend on the
depth, will be the same as in a liquid mass of large dimensions. Hence
in this case
_
/ [epsilon]
[sigma] = (c - 2[epsilon]) [rho]0 + 2 | [rho]d[nu], (18)
_/0
_
/ [epsilon]
e = (c - 2[epsilon]) [chi]0[rho]0 + 2 | [chi][rho]d[nu], (19)
_/ 0
d[sigma] de de
-------- = [rho]0, -- = [chi]0[rho]0, .: -------- = [chi]0,
dc dc d[sigma]
_ _
/ [epsilon] / [epsilon]
T = 2 | [chi][rho]d[nu] - 2[chi]0 | [rho]d[nu] =
_/0 _/ 0
_
/ [epsilon]
2 | ([chi] - [chi]0)[rho]d[nu]. (20)
_/0
Hence the tension of a thick film is equal to the sum of the tensions
of its two surfaces as already calculated (equation 7). On the
hypothesis of uniform density we shall find that this is true for
films whose thickness exceeds [epsilon].
The symbol [chi] is defined as the energy of unit of mass of the
substance. A knowledge of the absolute value of this energy is not
required, since in every expression in which it occurs it is under the
form [chi] - [chi]0, that is to say, the difference between the
energy in two different states. The only cases, however, in which we
have experimental values of this quantity are when the substance is
either liquid and surrounded by similar liquid, or gaseous and
surrounded by similar gas. It is impossible to make direct
measurements of the properties of particles of the substance within
the insensible distance [epsilon] of the bounding surface.
When a liquid is in thermal and dynamical equilibrium with its vapour,
then if [rho]' and [chi]' are the values of [rho] and [chi] for the
vapour, and [rho]0 and [chi]0 those for the liquid,
[chi]' - [chi]0 = JL - p(1/[rho]' - 1/[rho]0), (21)
where J is the dynamical equivalent of heat, L is the latent heat of
unit of mass of the vapour, and p is the pressure. At points in the
liquid very near its surface it is probable that [chi] is greater than
[chi]0, and at points in the gas very near the surface of the liquid
it is probable that [chi] is less than [chi]', but this has not as yet
been ascertained experimentally. We shall therefore endeavour to apply
to this subject the methods used in Thermodynamics, and where these
fail us we shall have recourse to the hypotheses of molecular physics.
We have next to determine the value of [chi] in terms of the action
between one particle and another. Let us suppose that the force
between two particles m and m' at the distance f is