Surface tension phenomena — capillarity, drop shape, excess pressure explained

Question

Explain surface tension and its three main effects: (a) capillary rise, (b) shape of liquid drops, and (c) excess pressure inside a soap bubble. Water rises 5 cm in a glass capillary tube. If the surface tension of water is 0.073 N/m, find the radius of the tube.

(JEE Main + NEET + CBSE 11 pattern)

Solution — Step by Step

Surface tension ( $T$ ) is the force per unit length along the surface of a liquid, acting tangentially. It arises because surface molecules are pulled inward by intermolecular forces, creating a “stretched membrane” effect.

Three key manifestations:

Capillarity — liquid rises or falls in narrow tubes
Drop shape — small drops are spherical (surface tension minimises surface area)
Excess pressure — pressure inside a curved surface is higher than outside

h = \frac{2T\cos\theta}{\rho g r}

For water on glass: $\theta \approx 0°$ (wetting), so $\cos\theta = 1$ .

Given: $h = 0.05$ m, $T = 0.073$ N/m, $\rho = 1000$ kg/m $^3$ , $g = 9.8$ m/s $^2$ .

r = \frac{2T\cos\theta}{\rho g h} = \frac{2 \times 0.073 \times 1}{1000 \times 9.8 \times 0.05}

r = \frac{0.146}{490} = 2.98 \times 10^{-4} \text{ m} \approx \mathbf{0.3 \text{ mm}}

Inside a liquid drop: $\Delta P = \frac{2T}{R}$

Inside an air bubble in liquid: $\Delta P = \frac{2T}{R}$

Inside a soap bubble (two surfaces): $\Delta P = \frac{4T}{R}$

A soap bubble has two surfaces (inner and outer), so the excess pressure is doubled compared to a liquid drop.

flowchart TD
    A["Surface Tension Effects"] --> B["Capillary Rise"]
    A --> C["Spherical Drop Shape"]
    A --> D["Excess Pressure"]
    B --> E["h = 2T cosθ / ρgr"]
    B --> F["Wetting: rises (water-glass)"]
    B --> G["Non-wetting: falls (mercury-glass)"]
    D --> H["Liquid drop: ΔP = 2T/R"]
    D --> I["Soap bubble: ΔP = 4T/R"]
    C --> J["Minimum surface area for given volume = sphere"]

Why This Works

Capillary rise occurs because the adhesive force between water and glass is stronger than the cohesive force between water molecules. The liquid climbs the wall, creating a curved meniscus. Surface tension along this meniscus pulls the liquid upward until the weight of the liquid column balances the upward pull.

The excess pressure inside a curved surface exists because the surface tension acts along the curved surface, creating a net inward force. To maintain equilibrium, the pressure inside must exceed the outside pressure. Smaller radius means more curvature, so higher excess pressure — this is why smaller bubbles have higher internal pressure.

Alternative Method — Energy Approach

The capillary rise can also be derived using energy minimisation. The system minimises total energy (surface energy + gravitational PE). When the liquid rises by $h$ , it reduces surface energy (liquid-glass interface replaces air-glass interface) but gains gravitational PE. The equilibrium height balances these two energies.

For JEE, remember that if two soap bubbles of radii $R_1$ and $R_2$ merge, the combined bubble has radius given by: $\frac{4T}{R} = \frac{4T}{R_1}$ or $\frac{4T}{R_2}$ — actually, if they combine completely, use conservation of gas molecules (Boyle’s law) along with excess pressure to find the new radius.

Common Mistake

Students use $\Delta P = 2T/R$ for a soap bubble. A soap bubble has TWO surfaces (inner and outer), so the correct formula is $\Delta P = 4T/R$ . A liquid drop and an air bubble inside liquid each have only ONE surface, so they use $2T/R$ . To remember: soap bubble = double the excess pressure of a liquid drop of the same size.

Question

Solution — Step by Step

Why This Works

Alternative Method — Energy Approach

Common Mistake

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