# Surface tension equals the free energy of the liquid surface – prove it

**Surface tension (S) equals the free energy (E) of the liquid surface (Temperature remaining constant)**.

Let’s see how we can prove that mathematically using an experimental setup.

## how to prove that Surface tension equals the free energy of the liquid surface?

Figure 1 above shows a frame XPQX1 over which wire RS can slide without any friction.

**Step 1**

Form a liquid film of soap solution and place the arrangement in a vertical plane.

Wire RS starts moving vertically **upwards**! This is due to surface tension.

As wire RS move towards the side PQ area of film decreases. This means there is a vertically upward force, F, due to Surface Tension (S.T.) on RS.

**Step 2**

Suspend a small weight w from RS so that RS is in equilibrium.

Obviously, F = w.

Experiments show F is directly proportional to the length of RS i.e.,* l*.

We write, the net force on RS due to surface tension as **F = 2 S l**, where S is a constant depending on the nature of the film. The factor of 2 in the above equation is due to the presence of

*two free surfaces*.

Had there been only ONE free surface then, F = S*l*

**S is known as the surface tension of the liquid surface.**

**Definition**: *Surface tension is numerically equal to force per length acting along the liquid surface (i.e. a tangential force) at right angles to any arbitrary line on the liquid surface.*

**SI unit of Surface tension is Nm ^{–1} and its dimensions are MT^{–2}.**

**Step 3**

In Fig. (b), wire RS is pulled down very slowly (so that surface film does not break) to position R’S’ by an infinitesimally small distance Δ*l*.

The work done, *dw*, against the force of Surface Tension on RS = F× Δ*l* = 2S*l* × Δ*l*.

Let *E be the free energy per unit area of the liquid surface*.

The work done *dw* is used to increase the free energy of the liquid surface.

**Now, Increase in free energy = Additional area of the film created × E**

= 2×area RR ‘SS’×E

= 2× l × Δ*l* ×E

**Step 4**

**From the law of conservation of energy**

**2S l × Δl = 2× l × Δl ×E **

or, **S = E**

**In other words; when the temperature remains constant, surface tension (S) equals the free energy (E) per unit area of the liquid surface.**

**Summary**

- Surface tension is numerically equal to force per length acting along the liquid surface (i.e. a tangential force) at right angles to any arbitrary line on the liquid surface.
- SI unit of Surface tension is Nm
^{–1}and its dimensions are MT^{–2}. - When the temperature remains constant, surface tension (S) equals the free energy (E) per unit area of the liquid surface.