Surface Tension

The properly of liquid at rest by virtue of which its free surface behaves like a
stretched membrane under tension and tries to occupy minimum surface area
is called surface tension.
l
If F is the force acting on the imaginary line of length l,
𝐹
then Surface tension, T = i.e. F = Tl F
𝑙
Its S.I-unit is N/m and CGS unit is dyne/cm
[ 1 N/m = 1000 dyne/cm]
Dimension of surface tension = [MT-2].
However in practical application, if a thread of length l floats on a liquid
surface, the total length in contact with liquid surface is 2l, since the
force of surface tension acts on both sides of thread and the force of
surface tension,
F = T x 2l
Also, If a ring of negligible thickness and radius r floats on a liquid
surface, then length of ring in contact with liquid surface is 2 (2πr).
So Surface tension force, F = T 2 (2πr) = 4πrT
1. A needle 5cm long can just rest on the surface of the water
without wetting. What is its weight? [T = 0.07N/m] [0.007N]
2. What is the downward force when a circular plate or radius
5cm is taken out of water? [T = 0.072N/m]
3. A rectangular plate of dimensions 6cm by 4cm and thickness 2mm is
placed with its largest face flat on the surface of water. Calculate the
downward force on the plate due to surface tension assuming zero angle of
contact. What is the downward force if the plate is placed vertical so that
it’s longest side just touches the water?
[T = 0.072N/m]
[ 0.014N, 0.00892N]
The surface tension forces act in
all directions, parallel (i.e.
tangential) to the liquid surface.
Therefore, the free surface of a
liquid behaves like an elastic skin
in a 'state of tension.
The surface tension of most
fig: Variation of surface tension with temperature
liquids decreases with increasing
temperature as shown in Fig.
Molecular Theory of Surface Tension
The molecules of a liquid attract each other with a
force of cohesion.
Consider a molecule A of a liquid lying well below the
free surface of a liquid. It has sphere of influence of
radius of the order 10-9 m.
As this molecule is attracted by the neighbouring
molecules lying within the sphere of influence, the
resultant force due to all the molecules on A is zero.

Consider molecule B on the surface of the liquid.

Since there are few molecules of liquid in vapour
state above the free surface, the molecule B
experiences forces of attraction only due to the
molecules lying in the lower half of the sphere of
influence. fig: Molecular theory of surface tension
The resultant of all these forces is downward. So
molecules on surface experiences maximum
downward force.
If we wish to bring a molecule from interior
of the liquid to the surface, some work has to
be done against this force.
This work done on the molecule is stored in it
in the form of potential energy.
For equilibrium, a system must have
minimum potential energy.
So there must be minimum number of
molecules on the liquid surface which is
possible only when the liquid surface is fig: Molecular theory of surface tension
minimum.
That is why the liquid surface contracts like a
stretched elastic membrane.
Some Examples Explaining Surface Tension

(a) Thread on a seep film

(b) Floating needle
(c) When a dry brush is dipped into water, its hair
spread out
(d) Spreading of oil on water surface
Surface Energy
If the surface area of a liquid has to be increased, work has to be done.
The work done is stored in the liquid surface film as its potential
energy.
The potential energy per unit area of the surface film is called surface
energy.
It may also be defined as the amount of work done in increasing the
area of a surface film through unity.
Surface energy
𝑊𝑜𝑟𝑘𝑑𝑜𝑛𝑒 𝑖𝑛 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎
σ=
𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎
Relation between Surface Tension and Surface Energy
Consider a rectangular frame of wire ABCD as
shown in which wire BC is movable.
If we dip the frame in a soap solution, a thin
film is formed which pulls the wire BC towards
left due to surface tension.
If T is surface tension of the film and l is length
of the wire BC, then the force F on BC due to
surface tension is given by,
Fig: Surface Energy
F = T x 2l
Suppose the wire is now moved through a
distance x from BC to B'C' against surface
tension force F so that surface area of the film
increase. In order to increase the film area,
work has to be done against F.
Work done in increasing surface area = F x distance = T(2l)x
where 2lx is increase in surface area.
𝑊𝑜𝑟𝑘𝑑𝑜𝑛𝑒 𝑖𝑛 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑇2𝑙𝑥
Surface energy, σ = = =
𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 2𝑙𝑥
T
Hence surface tension is numerically equal to surface energy.
4. Calculate the workdone in breaking a drop of water of 2mm
diameter into million droplets of same size. [T = 0.072N/m]
[8.9 * 10-5 J]
5. Calculate the workdone in blowing a soap bubble of radius
12 cm if the surface tension of soap solution is 30 dyne /cm.
[0.0108 J]
 Shape of liquid surface

The surface of a liquid is usually curved when it is in contact with a solid.

The curved surface of the liquid is called meniscus.
However, the shape of the meniscus is determined by the relative strength
of cohesive and adhesive forces acting on the molecules.
6. Calculate the workdone in increasing the radius of soap
bubble in air from 1cm to 2cm. The surface tension of soap
bubble is 30 dyne / cm. [2262.85 erg]
Fig. (a) shows water in contact with a vertical glass plate. Consider a molecule A on the
liquid surface near the glass.
The resultant adhesive force P on A due to attraction of glass molecules is perpendicular
to the surface of glass. The resultant cohesive force Q on A due to attraction of the
neighbouring liquid molecules is directed towards the interior of the liquid.
Whether P is more or Q is more depends on the liquid concerned and on the solid with
which it is in contact.
In this case
P > Q so their resultant R is directed outward from water as a result the liquid tends to
stick to the wall and forms concave meniscus as shown in Fig.(b).
Fig.(c) shows mercury in contact with glass. Here Q > P so their
resultant R is directed towards the interior of the mercury.
As a result the mercury surface is pulled away from the glass wall and
forms convex meniscus as shown in Fig (d).
 Angle of Contact and Capillarity
The eagle θ which the tangent to the liquid
surface at the point of contact makes with the
solid surface inside the liquid is called angle of
contact or capillary angle.
The angle of contact is acute (i.e. less than 90o) in
case of the liquids which wet the walls of the
container as shown in Water Fig.(a)
It is obtuse (i.e. greater than 90°) for the liquids
which do not wet the walls of the container as
shown in Fig.(b).
The angle of contact of pure water and clean glass
is zero.
Fig: Angle of Contact
The angle of contact for ordinary water and glass,
is about 8° while for mercury and glass, it is 1400.
 Capillarity

Fig: Capillarity

The tube of very fine bore is known as capillary tube.

When a glass capillary tube opened at both ends is dipped vertically in water,
the water in the tube will rise above the level of water in the vessel as shown
in fig(a).
In case of mercury, the liquid is depressed in the tube below the level of
mercury in the vessel as shown in fig(b).
This rise or fall of a liquid in a tube of very fine bore is called capillarity or
capillary action.
Some examples of capillarity are:
(i) Oil rises in cotton wicks of lamps through the small
capillaries between the threads.
(ii) A blotting paper absorbs ink by capillary action
(iii) The tip of the nib of a pen is split to provide capillary
action for rise of ink.
(iv) We use towels to dry our body after taking bath.
Measurement of Surface Tension by Capillary Rise Method

Consider a capillary tube of radius r open

at both ends and dipped into a liquid
(e.g., water) which has a concave
meniscus as shown in Fig.
Let θ be the angle of contact, h be the
height of liquid rises, ρ be the density of
the liquid and T be the surface tension of
liquid.
The surface tension forces causes the
liquid to exert a downward directed
force T on the walls of the tube.
This force T acts along the tangent at the
point of contact A.
From Newton's third law of motion, the
tube exerts an equal and opposite Fig: Capillary rise method
reaction 'R'.
The reaction (R = T) can be resolved into two
rectangular components so that
T sinθ and T cosθ are obtained along
horizontal and vertical respectively.
The horizontal components cancel to each
other whereas the vertical components are
added which pulls the liquid upward.
The component T cosθ acts along the whole
circumference of the meniscus.
Fig: Capillary rise method
Total upward force = T cos θ x 2πr …… Eqn(1)
Volume of the liquid in the tube above the
free surface of liquid is given by,
V=

Fig: Capillary rise method