Surface Tension Phenomena — Capillarity, Drop Shape, Excess Pressure Explained

Question

How does surface tension explain capillarity, the shape of liquid drops, and excess pressure inside bubbles and drops?

Solution — Step by Step

Surface tension ( $T$ or $\gamma$ ) is the force per unit length acting along the surface of a liquid, pulling the surface taut like a stretched membrane.

T = \frac{F}{l}

SI unit: N/m. For water at 20 degrees C: $T \approx 0.073$ N/m.

Why does it exist? Molecules at the surface have unbalanced intermolecular forces — they are pulled inward by neighbours below but have no neighbours above. This creates a net inward force that minimises the surface area. That is why free drops are spherical — a sphere has the least surface area for a given volume.

Liquid drop (one surface):

\Delta P = \frac{2T}{R}

Soap bubble (two surfaces — inner and outer):

\Delta P = \frac{4T}{R}

The pressure inside is always greater than outside. Smaller radius = higher excess pressure. This is why small bubbles shrink and big bubbles grow when connected (the air flows from higher pressure to lower pressure).

When a narrow tube is dipped in a liquid:

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

where $\theta$ = contact angle, $r$ = tube radius, $\rho$ = liquid density.

Water in glass ( $\theta < 90°$ , $\cos\theta > 0$ ): meniscus is concave, liquid rises
Mercury in glass ( $\theta > 90°$ , $\cos\theta < 0$ ): meniscus is convex, liquid falls

graph TD
    A[Surface Tension Effects] --> B[Drop Shape]
    A --> C[Capillarity]
    A --> D[Excess Pressure]
    B --> B1[Free drops are spherical - minimum surface area]
    B --> B2[On surface: shape depends on contact angle]
    C --> C1["Rise: theta < 90 degrees - water on glass"]
    C --> C2["Fall: theta > 90 degrees - mercury on glass"]
    C --> C3["Height inversely proportional to tube radius"]
    D --> D1["Drop: Delta P = 2T/R"]
    D --> D2["Bubble: Delta P = 4T/R"]
    D --> D3[Smaller radius = higher pressure]

Why This Works

All three phenomena — drop shape, capillarity, and excess pressure — come from the same physics: surface tension tries to minimise surface area. A sphere minimises area (drop shape). A curved meniscus in a tube creates a pressure difference that drives capillary rise. The excess pressure inside a curved surface follows from the balance between surface tension force and pressure force.

JEE Main and NEET both ask capillary rise calculations. The formula $h = \frac{2T\cos\theta}{\rho g r}$ is a must-know. Common variations: “If the tube radius is halved, what happens to height?” (doubles). “If the tube is tilted, does the height change?” (vertical height stays the same, but liquid travels further along the tube).

Alternative Method

For excess pressure, use the energy argument: to expand a drop of radius $R$ by $dR$ , the work done against surface tension ( $T \cdot 8\pi R\, dR$ ) equals the work done by excess pressure ( $\Delta P \cdot 4\pi R^2\, dR$ ). Equating gives $\Delta P = 2T/R$ .

For a soap bubble (two surfaces), the surface area term doubles, giving $\Delta P = 4T/R$ .

Common Mistake

Students use $\Delta P = 2T/R$ for soap bubbles instead of $4T/R$ . A soap bubble has TWO liquid surfaces (inner and outer), each contributing $2T/R$ to the excess pressure. A liquid drop has only ONE surface. Confusing these gives an answer that is off by a factor of 2. Always ask: is it a drop (one surface) or a bubble (two surfaces)?

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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