---
title: "Surface tension and capillarity"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:surface-tension-capillarity"
source: https://www.bctengineer.com/q/semester_6/hydro_power/surface-tension-capillarity
site: BCT Engineering
---

# Surface tension and capillarity

Hydro Power · Chapter 1: Introduction to Basic Fluid Mechanics and Hydraulics

- Asked in: 2073 Bhadra, 4 marks
- Video: [Surface Tension of Water, Capillary Action, Cohesive and Adhesive Forces - Work & Potential Energy](https://www.youtube.com/watch?v=WBk7mGb7TKw) — The Organic Chemistry Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/surface-tension-capillarity
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

Discuss surface tension and capillarity.

## Short answer

Surface tension is the tensile force per unit length on a liquid surface, making it behave like a stretched membrane. Capillarity is the rise or fall in a thin tube due to surface tension, with height h = 4σ cosθ/(ρ g d); water rises, mercury falls.

## Answer

#### Surface tension
**Surface tension** ($\sigma$) is the tensile force per unit length acting along the free surface of a liquid, so that the surface behaves like a stretched membrane. Unit: N/m.
- Cause: a molecule inside the liquid is pulled equally in all directions by cohesion, but a molecule at the surface is pulled only inwards. This creates a net inward pull and a "skin" at the surface.
- Examples: water drops are spherical, small insects walk on water, a needle can float.
- Pressure inside a droplet: $p = \dfrac{4\sigma}{d}$; inside a soap bubble (two surfaces): $p = \dfrac{8\sigma}{d}$; inside a liquid jet: $p = \dfrac{2\sigma}{d}$.
- $\sigma$ decreases with increase in temperature. For water at 20 °C, $\sigma \approx 0.073$ N/m.

#### Capillarity
**Capillarity** is the rise or fall of a liquid in a small-diameter tube dipped in it, caused by surface tension together with adhesion and cohesion.
- If adhesion > cohesion (water and glass), the liquid wets the tube, the meniscus is concave and the liquid **rises**.
- If cohesion > adhesion (mercury and glass), the meniscus is convex and the liquid **falls** (capillary depression).

Equating the upward surface tension force to the weight of the raised column:
$$
\sigma \pi d \cos\theta = \rho g \frac{\pi d^2}{4} h \;\Rightarrow\; h = \frac{4\sigma\cos\theta}{\rho g d}
$$
where $\theta$ is the contact angle ($\approx 0^\circ$ for water–glass, $\approx 130^\circ$ for mercury–glass). The rise is inversely proportional to tube diameter, so manometer and piezometer tubes are made wider than about 6 mm to keep capillary error small.

## Check yourself

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

1. Surface tension arises because a molecule at the free surface of a liquid:
   A. Is pulled equally in all directions
   B. Is pulled only inwards by cohesion, giving a net inward pull
   C. Is repelled by air molecules
   D. Has higher temperature
   Answer: B. Surface molecules lack neighbours above, so a net inward cohesive pull forms a skin. (Key point: Cause of surface tension)

2. For the same diameter d, the excess pressure inside a soap bubble compared with a liquid droplet is:
   A. Half (2 sigma/d)
   B. Equal (4 sigma/d)
   C. Double (8 sigma/d)
   D. Four times (16 sigma/d)
   Answer: C. A droplet has p = 4 sigma/d while a soap bubble has two surfaces, p = 8 sigma/d. (Key point: Pressure in droplet and bubble)

3. The capillary rise h = 4 sigma cos(theta)/(rho g d) shows that rise is:
   A. Directly proportional to tube diameter
   B. Inversely proportional to tube diameter
   C. Independent of tube diameter
   D. Proportional to d^2
   Answer: B. h varies as 1/d, so narrow tubes give larger rise; manometer tubes are made wide to reduce error. (Key point: Capillary rise)

4. Mercury in a glass tube shows capillary depression because:
   A. Adhesion > cohesion and the meniscus is concave
   B. Cohesion > adhesion and the meniscus is convex
   C. Its contact angle is about 0 degrees
   D. Surface tension of mercury is zero
   Answer: B. Cohesion exceeds adhesion for mercury-glass (theta about 130 degrees), giving a convex meniscus and fall. (Key point: Rise or fall)

## Related in Hydro Power

- [Definitions of basic fluid properties](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Afluid-property-definitions) — Hydro Power, chapter 1, asked 3×

## Related in other subjects

- [Derive surface tension and capillarity](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Asurface-tension-capillarity-derivation) — Fluid Mechanics, chapter 1