# Surface Tension

SURFACE TENSION

The property of a liquid by virtue of which its free surface behaves like a stretched membrane and supports, comparatively heavier objects placed over it is called surface tension.

Surface tension,T=F/L

This property is caused by cohesion of molecules and is responsible for much of the behaviors of liquids.

The free surface of a liquid contracts so that its exposed surface area is a minimum, i.e., it behaves as if it were under tension, somewhat like a stretched elastic membrane. This property is known as surface tension. The surface tension of a liquid varies with temperature as well as dissolved impurities, etc.

When soap is mixed with water, the surface tension of water decreases. Also, the surface tension decreases with increase in temperature.

Surface tension is also seen in the ability of some insects, such as water striders, and even reptiles like basilisk, to run on the water’s surface.

Adhesive Force

The force of attraction acting between the molecules of different substances is called adhesive force, e.g., the force of attracts acting between the molecules of paper and ink ,etc.

Cohesive Force

The force of attraction acting between the molecules of same substances is called cohesive force. e.g., the force of attraction acting between molecules of water, glass, etc.

Cohesive forces and adhesive forces are van der Waals’ forces.

Molecular theory of surface tension

Surface tension has been well- explained by the molecular theory of matter. According to this theory, cohesive forces among liquid molecules are responsible for the phenomenon of surface tension. The molecules well inside the liquid are attracted equally in all directions by the other molecules. The molecules on the surface experience an inward pull.

So, a network is formed against the inward pull, in order to move a molecule to the liquid surface. It results in a greater potential energy on surface molecules. In order to attain minimum potential energy and hence stable equilibrium, the free surface of the liquid tends to have the minimum surface area and thereby it behaves like a stretched membrane.

Surface tension is measured as the force acting normally per unit length on an imaginary line drawn on the free liquid surface at rest. It is represented by the symbol T (or S). It's S.I. The unit is Nm^{-1} and dimensional formula is M^{1}L^{0}T^{-2.}

Surface Energy

If we increase the free surface area of a liquid then work has to be done against the force of surface tension. This work done is stored in liquid as potential energy,

This additional potential energy per unit area of free surface of liquid is called surface energy.

Surface energy (E) = S x &ΔM

where. S = surface tension and ΔA = increase in surface area.

Any Strained body possesses potential energy, which is equal to the work done in bringing it to the present state from its initial unstained state. The surface of liquid is also a strained system and hence the surface of a liquid also has potential energy, which is equal to the work done increasing the surface. This energy per unit area of the surface is called surface energy To derive an expression for surface energy consider a wire frame equipped with a sliding wire AB as shown in figure. A film of soap solution is formed across ABCD of the frame. The side AB is pulled to the left due to surface tension. To keep the wire in position a force F has to be applied to the right. If T is the surface tension and l is the length of AB, then the force due to surface tension over AB is 2lT to the left because the film has two surfaces ( upper and lower) Since the film is in equilibrium F = 2lT

Now, if the wire AB is pulled down, energy will flow from the agent to the film and this energy is stored as potential energy of the surface created just now. Let the wire be pulled slowly through x. Then the work done = energy added to the film from above agent W=Fx = 2lTx Potential energy per unit area ( surface energy) of the film 𝑈 = 2𝑙𝑇𝑥 2𝑙𝑥 = 𝑇 𝑇 = 𝑊 𝑎𝑟𝑒𝑎 Thus surface energy numerically equal to its surface tension It s unit is Joule per square metre ( Jm-2 )

Angle of contact

When the free surface of a liquid comes in contact with a solid, it becomes curved at the point of contact. The angle between the tangent to the liquid surface at the point of contact of the liquid with the solid and the solid surface inside the liquid is called angle of contact. In Fig., QR is the tangent drawn at the point of contact Q. The angle PQR is called the angle of contact. When a liquid has concave meniscus, the angle of contact is acute. When it has a convex meniscus, the angle of contact is obtuse. The angle of contact depends on the nature of liquid and solid in contact. For water and glass, θ lies between 8^{0} and 18^{0} . For pure water and clean glass, it is very small and hence it is taken as zero. The angle of contact of mercury with glass is 138^{0} .

An expression of excess pressure inside a liquid drop:

To find the magnitude of excess pressure in a liquid, we consider a liquid drop of finite radius. If P_{i }and P_{0} be the inside and outside pressure in a liquid drop then, the excess pressure is P

P = P_{i} - P_{0} in which pressure is always more than

Let's take a liquid through of center O and radius R in which the inside pressure is P_{i} and outside pressure is P_{0}. Due to the difference in pressure or excess pressure, the size of liquid trough increases through the distance OR then, the work done in including the area or distance by DR is

dw=force∗distancedw=force∗distance

=P∗4πR2∗dR=P∗4πR2∗dR ------- (i)

But we also know that the surface energy is,

σ=ΔWΔAσ=ΔWΔA

or, ΔW=σ∗ΔAΔW=σ∗ΔA

=T∗ΔA=T∗ΔA --------- (iii)

Where, σ=Tσ=T

The work done in equation (ii) and (iii) is same, so

T∗ΔA=P∗4πR2∗dRT∗ΔA=P∗4πR2∗dR ------------ (iv)

But,

ΔA=A2−A1ΔA=A2−A1

=4π (R+dR)^{2}−4πR^{2}

=4π (R+dR)^{2}−4πR^{}

=4π[R^{2}+2RdR+ (dR)^{2}−R^{2}]

=4π[R^{2}+2RdR+(dR)^{2}−R^{2}]

=4π[2RdR+(dR)2]=4π[2RdR+(dR)^{2}]

Since dR is small, (dR)^{2} will be very small and we can neglect it. So, neglecting (dR)^{2} we get

ΔA

=4π∗2RdRΔA

=4π∗2RdR ---------- (v)

Putting the value of ΔA in equation (iv) we get

T∗4π∗2RdR=P∗4πR2∗dRT∗4π∗2RdR=P∗4πR2∗dR

Or, 2T=P∗R2T=P∗R

or, P=2TRP=2T/R

or, Pi−P_{o}=2T/R ----------(vi)

*The phenomenon of rise or fall of liquids in capillary tube is known as capillarity.*

Surface tension by capillary rise method

Let us consider a capillary tube of uniform bore dipped vertically in a beaker containing water. Due to surface tension, water rises to a height h in the capillary tube as shown in above Fig.. The surface tension T of the water acts inwards and the reaction of the tube R outwards. R is equal to T in magnitude but opposite in direction. This reaction R can be resolved into two rectangular components.

(i) Horizontal component R sin θ acting radially outwards

(ii) (ii) Vertical component R cos θ acting upwards.

The horizontal component acting all along the circumference of the tube cancel each other whereas the vertical component balances the weight of water column in the tube. Total upward force = R cos θ × circumference of the tube

F = 2πr R cos θ or F = 2πr T cos θ ...(1) [∵ R = T ]

This upward force is responsible for the capillary rise. As the water column is in equilibrium, this force acting upwards is equal to weight of the water column acting downwards.

(i.e) F = W ...(2)

Now, volume of water in the tube is assumed to be made up of (i) a cylindrical water column of height h and (ii) water in the meniscus above the plane

CD. Volume of cylindrical water column = πr2h Volume of water in the meniscus = (Volume of cylinder of height r and radius r) – (Volume of hemisphere)

∴Volume of water in the meniscus= 𝜋𝑟 ^{2} × 𝑟 − 2 3 𝜋𝑟 ^{3} = 1 3 𝜋𝑟 ^{3}

∴Total volume of water in the tube 𝜋𝑟 2ℎ + 1 3 𝜋𝑟 3 = 𝜋𝑟 2 (ℎ + 𝑟 ^{3} ) If ρ is the density of water, then weight of water in the tube is 𝑊 = 𝜋𝑟 2 (ℎ + 𝑟 3 ) 𝜌𝑔 ..(3)

Substituting (1) and (3) in (2), 𝜋𝑟 2 (ℎ + 𝑟 3 ) 𝜌𝑔 = 2𝜋𝑟𝑇𝑐𝑜𝑠𝜃 𝑇 = 𝜋𝑟 2 (ℎ + 𝑟 3 ) 𝜌𝑔 2𝜋𝑟𝑐𝑜𝑠𝜃 Since r is very small, r/3 can be neglected compared to h.

𝑇 = ℎ𝑟𝜌𝑔 /2𝑐𝑜𝑠𝜃

For water θ is very small cosθ =1

𝑇 = ℎ𝑟𝜌𝑔/ 2