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Show that the coefficient of surface tension T is given by T=rhPg/2 where P, h...

Show that the coefficient of surface tension T is given by T=rhPg/2 where P, h and r are density, height and radius of the system respectively.?

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The properties of a liquid surface give very good
evidence for the existence of molecules and tell
us quite a lot about the behaviour of these
molecules.
The free surface of a liquid can show many
interesting properties due to a phenomenon
known as surface tension. Surface tension
explains why liquid drops are spherical (in the
absence of a gravitational field), why water rises
up a capillary tube, why the insects called pond-
skaters can walk on water, why your waterproof
tent will not let in rain, why small pieces of
camphor behave erratically when dropped on
water and why the bristles of a paint brush cling
together when it is lifted out of water. The
motion of ripples on a water surface is also
governed by surface tension.
Surface tension is known to be due to
intermolecular attractions in the liquid surface
and these forces produce a skin effect on the
surface. The forces between individual pairs of
molecules are very small, and so in a definition
of the surface tension we consider the effect of
a large number of molecules in a line in the
surface.
Consider a line of unit length drawn in the
surface of the liquid and think of the forces
acting on the molecules in that line. Clearly the
forces will act in all directions in the surface but
we will consider only those components of force
acting at right angles to the line (see Figure 1).
The force on the whole line is the sum of all the
forces on the individual molecules. Notice that
any given molecule is in equilibrium due to equal
and opposite forces acting on it.
The size of the force of surface tension in a
liquid surface is governed by a property called
the coefficient of surface tension for that liquid.
The coefficient of surface tension of the liquid is
defined as follows:
The coefficient of surface tension of a liquid is
the force acting in the surface of the liquid at
right angles to a line of unit length in the surface
of the liquid.
The units for surface tension are therefore
newtons per metre (Nm -1 ) and the dimensions
are ML 0T -2 . The following table gives the
surface tensions of some common liquids.
Liquid Surface tension
(Nm -1x10 3)
Liquid Surface tension
(Nm -1x10 3)
Methylated
spirits
22.6 Ethyl
alcohol
22.3
Glycerol 63.4 Turpentine 27
Water 72.7 Bromine 41.5
Benzene 28.9 Acetone 23.7
Mercury 472
Olive oil 32
The surface tension of a liquid decreases with
increase in temperature and vanishes at the
critical temperature. All values given in the table
are for temperatures of 20 o C, except that for
gold which is for 1130o C.
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