Surface tension and surface energy

A skin on the surface

A steel needle laid gently on water floats; an insect walks on a pond. The surface of a liquid behaves like a stretched skin.

The reason is in the molecules. One deep inside is pulled equally on every side. One at the surface is pulled only sideways and inward, so the surface molecules have extra energy, and the liquid keeps its surface as small as it can. That is why drops are round.

Surface tension and surface energy

Surface tension $S$ is the force per unit length with which the surface pulls across any line drawn on it, in $\text{N/m}$. Water $0.072\ \text{N/m}$, soap solution about $0.03\ \text{N/m}$, mercury $0.465\ \text{N/m}$.

It is also the surface energy: the work needed to make each square metre of new surface. Stretching a soap film by an area $\Delta A$ takes

because a film has two surfaces, front and back.

Pressure inside drops and bubbles

A curved surface squeezes what is inside. The pressure inside is higher than outside by

• a drop, or an air bubble inside a liquid (one surface): $\Delta p = \dfrac{2S}{r}$;
• a soap bubble in air (two surfaces): $\Delta p = \dfrac{4S}{r}$.

You should find that small bubbles have the highest pressure, which is why the first puff into a bubble pipe is the hardest.

Watch out. Count the surfaces: one for a drop or an air bubble in water, two for a soap bubble.

A worked example

How much work does it take to blow a soap bubble of radius $5.0\ \text{cm}$, with $S = 0.025\ \text{N/m}$, and what is the excess pressure inside?

1. New surface. Two surfaces of $4\pi r^2$: $8\pi \times (0.050\ \text{m})^2 = 6.28 \times 10^{-2}\ \text{m}^2$.
2. Work. $0.025\ \text{N/m} \times 6.28 \times 10^{-2}\ \text{m}^2 = 1.57 \times 10^{-3}\ \text{J}$.
3. Pressure. $\dfrac{4 \times 0.025\ \text{N/m}}{0.050\ \text{m}} = 2.0\ \text{Pa}$.

Practise with new numbers

Try it in Python

The program works out the soap bubble of the example.

Remember these

• $S$: force per unit length, or energy per unit area, in $\text{N/m}$.
• A film has two surfaces: $W = 2S\,\Delta A$.
• Drop: $\dfrac{2S}{r}$; soap bubble: $\dfrac{4S}{r}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why are small drops of liquid round?

2. A film's area grows by $30\ \text{cm}^2$, with $S = 0.040\ \text{N/m}$. How much work is done?

3. What is the excess pressure inside a mercury drop of radius $2.0\ \text{mm}$?

4. Two soap bubbles are joined by a tube. Which way does the air flow?