---
title: "Derive surface tension and capillarity"
subject: "Fluid Mechanics"
chapter: 1
key: "semester_3/fluid_mechanics:surface-tension-capillarity-derivation"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/surface-tension-capillarity-derivation
site: BCT Engineering
---

# Derive surface tension and capillarity

Fluid Mechanics · Chapter 1: Fluid and its physical properties

- Asked in: 2074 Asoj, 2 marks
- Video: [Deriving the Capillary Action Formula](https://www.youtube.com/watch?v=PwoKPvFW8NI) — 1Mjourney
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/surface-tension-capillarity-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-1

## Question

Derive an expression for surface tension and capillarity.

## Short answer

Surface tension σ causes excess pressure inside a droplet: Δp = 4σ/d (soap bubble: 8σ/d). Capillary rise balances surface tension with weight: h = (4σ cosθ)/(ρ g d). For water-glass θ≈0° gives rise; mercury θ≈130° gives depression.

## Answer

#### Surface tension
**Surface tension** $\sigma$ is the tensile force per unit length acting along the free surface of a liquid (N/m), due to cohesion between molecules.

**Liquid droplet of diameter $d$:** the pressure force over the cut section balances the surface-tension force round its circumference:

$$
\Delta p\,\frac{\pi d^{2}}{4}=\sigma\,\pi d\ \Rightarrow\ \Delta p=\frac{4\sigma}{d}
$$

For a soap bubble (two surfaces), $\Delta p=\dfrac{8\sigma}{d}$.

#### Capillarity
In a vertical tube of diameter $d$ dipped in a liquid with contact angle $\theta$, the upward component of the surface-tension force round the circumference supports the weight of the raised column of height $h$:

$$
\sigma\cos\theta\,(\pi d)=\rho g\,\frac{\pi d^{2}}{4}\,h
$$

$$
h=\frac{4\sigma\cos\theta}{\rho g d}
$$

For water in glass $\theta\approx0^\circ$ (rise); for mercury $\theta\approx130^\circ$ so $h$ is negative (depression).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For a liquid droplet of diameter d, what is the excess pressure?
   A. 2 sigma / d
   B. 8 sigma / d
   C. 4 sigma / d
   D. sigma / d
   Answer: C. Delta p (pi d^2/4) = sigma pi d gives Delta p = 4 sigma/d. (Key point: Droplet pressure)

2. Why is the soap bubble excess pressure 8 sigma/d?
   A. It has two surfaces, doubling the tension force
   B. It is larger than a droplet
   C. It contains air
   D. It has no surface tension
   Answer: A. A soap film has inner and outer surfaces, so the surface-tension force is doubled. (Key point: Soap bubble)

3. Which balance gives h = 4 sigma cos(theta)/(rho g d)?
   A. Surface tension pull x circumference balanced against column weight
   B. Pressure force equals buoyancy
   C. Viscous force equals weight
   D. Surface tension equals pressure at the top
   Answer: A. sigma cos(theta) pi d = rho g (pi d^2/4) h. (Key point: Capillary rise balance)

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