Molecular origin of surface tension
A liquid holds together because its molecules attract one another. Consider a molecule deep inside the liquid: it is surrounded on all sides by neighbours, so the cohesive attractions pull it equally in every direction and the net force on it is zero. A molecule sitting at the surface is different. Only its lower hemisphere is filled with liquid; above it lies vapour or air with far fewer molecules. The downward and sideways pulls no longer cancel, and the molecule experiences a net inward cohesive force directed into the bulk.
Because work must be done against this inward pull to bring a molecule up to the surface, every surface molecule carries extra potential energy compared with one in the interior — NCERT estimates it as roughly half the energy needed to remove the molecule from the liquid entirely, i.e. about half the heat of vaporisation. A liquid therefore behaves so as to keep its surface area as small as the surroundings permit, because minimum area means minimum surface energy. This tendency to contract is exactly what we call surface tension.
Surface tension: force per length, energy per area
The macroscopic definition follows directly. Imagine drawing any line of length $L$ on the liquid surface. The molecules on one side pull the line one way, those on the other pull it back; the surface is in tension just like a stretched sheet. Surface tension $S$ is the force acting per unit length of such a line, directed perpendicular to the line and lying in the plane of the surface:
$$ S = \frac{F}{L} $$
Its SI unit is the newton per metre ($\text{N m}^{-1}$). But there is a second, equivalent way to read $S$. Because creating new surface costs energy, $S$ also equals the surface energy stored per unit area of the interface, with unit joule per square metre ($\text{J m}^{-2}$). Dimensionally the two units coincide, $\text{J m}^{-2} = \text{N m}\,/\,\text{m}^2 = \text{N m}^{-1}$, so the same number describes both pictures. More precisely, $S$ is the energy of the interface between two media and depends on both — a liquid–air value differs from a liquid–solid value.
| Reading of $S$ | Definition | Unit | Use it when… |
|---|---|---|---|
| Force per unit length | $S = F/L$, force in the plane of the surface, ⟂ to a line of length $L$ | $\text{N m}^{-1}$ | A wire, film boundary, ring or plate is being pulled by the surface |
| Surface energy per unit area | $S = W/\Delta A$, work to create extra area $\Delta A$ at constant temperature | $\text{J m}^{-2}$ | You must find work/energy to form a drop, film or bubble |
Surface energy and the work to grow a surface
NCERT derives $S$ from a soap film stretched on a U-shaped frame closed by a sliding bar of length $l$ (Figure 2). Pull the bar out by a small distance $d$ against the surface's inward force $F$. The work done is $W = Fd$, and by energy conservation this is stored as new surface energy. The crucial detail is that a film has two liquid surfaces — front and back — so when the bar moves $d$, the total new area created is $2\,l\,d$. If $S$ is the energy per unit area,
$$ S \,(2\,l\,d) = F\,d \qquad\Longrightarrow\qquad S = \frac{F}{2l} $$
For a single surface (one interface, not a film), the work to create extra area $\Delta A$ is simply
$$ W = S\,\Delta A. $$
This is the equation behind every "energy required to form a drop/bubble" problem. To form a liquid drop of radius $R$ from scattered liquid, the new area is $4\pi R^2$ (one surface) so $W = S\cdot 4\pi R^2$. To blow a soap bubble of radius $R$, the film has two surfaces, so $W = S\cdot 2(4\pi R^2) = 8\pi R^2 S$. The factor of two reappears.
The same curved-surface pressure that governs bubbles drives liquid up narrow tubes — see capillary action for the rise formula $h = 2S\cos\theta / (\rho g r)$.
Angle of contact: wetting vs non-wetting
Where a liquid surface meets a solid, the surface curves. The angle of contact $\theta$ is the angle between the tangent to the liquid surface at the point of contact and the solid surface, measured inside the liquid. Its value is fixed by the balance of three interfacial tensions at the contact line — liquid–air $S_{la}$, solid–air $S_{sa}$, and solid–liquid $S_{sl}$ — through the relation $S_{la}\cos\theta + S_{sl} = S_{sa}$.
The sign of $(S_{sl} - S_{sa})$ decides everything. When $S_{sl} > S_{la}$ the liquid is attracted weakly to the solid and strongly to itself; $\theta$ is obtuse, the meniscus is convex, and the liquid does not wet the solid — mercury on glass is the textbook case. When $S_{sl} < S_{la}$ the liquid clings to the solid; $\theta$ is acute, the meniscus is concave, and the liquid wets the solid — water on clean glass behaves this way.
This is also why soaps, detergents and dyes work. They are wetting agents: dissolved in water they lower $S_{la}$ and shrink the angle of contact so the liquid penetrates fabric and dirt. Waterproofing agents do the opposite — they raise the contact angle so water beads off the fibres.
| Property | Wetting liquid (e.g. water–glass) | Non-wetting liquid (e.g. mercury–glass) |
|---|---|---|
| Angle of contact $\theta$ | Acute, $\theta < 90^\circ$ | Obtuse, $\theta > 90^\circ$ |
| Meniscus shape | Concave (curves upward at walls) | Convex (bulges upward at centre) |
| Interfacial energies | $S_{sl} < S_{la}$ | $S_{sl} > S_{la}$ |
| Behaviour in a capillary | Liquid rises | Liquid is depressed |
| Adhesion vs cohesion | Adhesion (to solid) dominates | Cohesion (within liquid) dominates |
Excess pressure in drops, bubbles and cavities
Because the surface is in tension, a curved liquid surface squeezes the fluid on its concave side: the pressure just inside is greater than just outside. Take a spherical drop of radius $R$ in equilibrium and let its radius grow by $\Delta R$. The extra surface energy of the single liquid–air surface is $S\,(8\pi R\,\Delta R)$, and this is paid for by the work done by the pressure difference $(P_i - P_o)$ acting over the expanding surface, $(P_i - P_o)\,4\pi R^2\,\Delta R$. Equating the two,
$$ P_i - P_o = \frac{2S}{R} \quad\text{(liquid drop, one surface)}. $$
An air bubble (cavity) inside a liquid has the same single liquid–gas interface, so its excess pressure is also $\dfrac{2S}{R}$. A soap bubble is the exception: it is a thin film with air on both sides, so it has two liquid–air surfaces. Each surface contributes $2S/R$, doubling the result:
$$ P_i - P_o = \frac{4S}{R} \quad\text{(soap bubble, two surfaces)}. $$
Factors that change surface tension
Surface tension is an intrinsic property of the interface and does not depend on the area of the surface, but it responds to several physical conditions.
- Temperature. $S$ decreases as temperature rises. Greater thermal motion increases molecular separation, weakening the net inward cohesion; at the critical temperature $S$ falls to zero. NCERT notes that, like viscosity, surface tension usually falls with temperature.
- Impurities and surfactants. Soluble surfactants — soaps, detergents — lower $S$ sharply, which is what makes them effective wetting and cleaning agents. Highly soluble impurities such as common salt can raise $S$ slightly.
- Nature of the two media. $S$ is the energy of the interface between two materials, so a liquid–air value differs from a liquid–solid or liquid–liquid value.
- Contamination of the surface. A film of oil or grease on water markedly lowers its surface tension, which is why camphor specks dance on a clean water surface but stop once oil is present.
Surface tension also depends on the curvature it produces, not the other way round: a smaller radius means a larger excess pressure ($\propto 1/R$). This is why two soap bubbles of unequal size, when joined, see the smaller one empty into the larger — the smaller bubble holds the higher internal pressure.
Surface tension in one breath
- Surface molecules feel a net inward cohesive pull, giving the surface extra energy and a tendency to minimise area.
- $S = F/L$ (force per length, $\text{N m}^{-1}$) $= W/\Delta A$ (energy per area, $\text{J m}^{-2}$) — same number, two pictures.
- Work to create new area: $W = S\,\Delta A$. Drop (one surface) $4\pi R^2 S$; soap bubble (two surfaces) $8\pi R^2 S$.
- Angle of contact: acute → wetting, concave meniscus (water–glass); obtuse → non-wetting, convex meniscus (mercury–glass).
- Excess pressure: liquid drop $2S/R$, air cavity $2S/R$, soap bubble $4S/R$. Count the interfaces.
- $S$ decreases with temperature, drops with surfactants/impurities, and is independent of surface area.