Whenever a spring is stretched we say some work is done on it and it is stored in the form of Potential energy known as Elastic potential energy. We can say if the body is in its undeformed state there will be no potential energy.

Similarly when we talk about free surfaces of the liquid, as we know the free surfaces of liquids are considered to be stretched membrane, then the surface will store some potential energy due to stretched surface and since it’s only at the surface of the liquid is termed as Surface energy or Surface free energy.

## Surface Free Energy

Surface energy is the potential energy of the liquid molecules, that will help the molecules to remain at the surface of the liquid.

All the molecules at the surface try to reach the bottom layer, which will decrease the surface energy. Surface energy is the amount of work done to increase the surface area of the liquid surface.

Mathematically, the surface energy is the work done per unit area of the liquid surface.

⇒ Surface energy = \[\frac{Workdone}{Area}\] Joules/m\[^{2}\]

The SI unit of surface energy is Joules/m2 or Newton/meter(N/m).

If the surface is less then the liquid surface will exert high surface energy (Ex: metals, Oxides, Ceramics). Similarly, if the surface area is more then the surface will exert low surface energy (Ex: Plastics, Rubbers, etc..).

The materials having low surface energy are classified as low surface energy materials.

### Relation Between Surface Tension and Surface Energy

Let us derive a relation between the surface tension and surface energy. Now, consider a rectangular wireframe with a slider attached to it as shown in the figure below. The rectangular wireframe is dipped in the soap water that will result in soap film due to the surface tension.

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Due to surface tension, the soap film will exert force on the slider. Then the net force on the slider due to surface tension is given by:

⇒ F = 2Tl ………..(1)

Where,

T - Surface Tension of the soap film

l - Length of the slider

(The multiplicative factor 2 is used, as there are two surfaces of the slider, and the force exerted on both surfaces slider due to surface tension.)

For slowly moving the slider outwards, the external force acting on it will be,

⇒ F\[_{ext}\] = 2Tl……….(2)

Where,

T - Surface Tension of the soap film

l - Length of the slider

Now, let us displace the slider by Δx distance outwards, then there will be some work done on it.

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Whatever work is done to move the slider will be stored in the form of its potential energy. Therefore, the work done in displacing the slider is given by:

⇒ dW = F\[_{ext}\] . Δx

⇒ ΔU = dW = 2Tl . Δx ………(3)

Where,

ΔU - Increase in the surface energy

On rearranging the above expression,

⇒ ΔU = T . ΔS ………….(4)

Where,

ΔS - Increase in surface area of the film

Therefore, the relation between surface tension and surface energy is given by the following expression:

⇒ T = \[\frac{\Delta U}{\Delta S}\] ……….(5)

Equation (5) gives the relation between surface tension and surface energy.

### Example:

1) A Film of Water is Formed Between Two Parallel Wires Each of Length 20cm and the Separation Between them is 1cm. Calculate the Work Required to Increase their Surface Area by 2mm. (Surface Tension of Water is 72 х 10⁻³ N/m)

Solution:

Before starting the solution of the given example, let us first visualize the given data.

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Given that,

The length of the wire = 20cm

The separation between the wires = 1cm = 10⁻²m

The displacement due to the external force ∆x = 2mm = 2 х 10⁻³m

The surface tension of water = T = 72 х 10⁻³ N/m

Now, we know that the relation between surface tension and surface energy (work done),

⇒ ∆U = T.∆S …..(1)

Where,

⇒ ∆S = 2l∆x (Increase in surface area of the film)

⇒ ∆S = 2 х 20 х 2 х 10⁻³ = 80 х 10⁻³m² …….(2)

Substituting equation (2) in (1),

⇒ ∆U = (72 х 10⁻³) . (80 х 10⁻³) = 5.76 х 10⁻³ J

⇒ ∆U = 5.76 mJ

Therefore, 5.76mJ of work is done to increase the surface area by 2mm.

### Did You Know

The surface tension of every material is different, thus the surface energy of the materials will also be varying from one material to another. The surface energy of liquid will be different from that of the surface energy of metals. The surface energy of water will be depending upon the angle of contact.

The surface energy of few materials is as listed below:

1: The Dimensional Formula of Surface Energy?

Ans: The dimensional formula for surface energy is MT⁻².

2: What is Specific Surface Energy?

Ans: Specific surface energy is the increase in the free energy when surface area increased by unit area.