Surface Tension: theory | Physics Circle

CONCEPT

Molecular Theory & Factors

In one line
Surface molecules feel a net inward pull, so the surface behaves like a stretched membrane trying to shrink.

INTERMOLECULAR

3.1.1 Intermolecular forces

Cohesive	Adhesive
same substance	different substances
two drops merge into one	water wets glass

Their origin is electromagnetic. They do not obey the inverse square law.

SURFACE TENSION

3.1.2 What is surface tension?

Force on either side of a line on the surface

T = FL

Force per unit length on either side of an imaginary line, along the surface and at right angles to the line.
Unit: N m⁻¹, dyne cm⁻¹. Dimensions: [MT⁻²].
A scalar, independent of the area of the surface.
A small drop is spherical: least area for a given volume.

MOLECULES

3.1.3 Molecular theory

Each circle is a sphere of influence

Molecular range: the largest distance of attraction, about 10⁻⁹ m.
Sphere of influence: sphere of that radius around a molecule.
A, B (deep): pulled equally on all sides, net force zero.
C (in the film): net downward force.
D (on the surface): maximum downward force.
Surface molecules have maximum potential energy, so the surface tends to shrink.

TEMPERATURE

3.1.4 Effect of temperature

T_t = T₀(1 − αt)

T falls as temperature rises. Exception: molten copper and cadmium.
At the critical temperature, T = 0.
Hot soup tastes better.

IMPURITIES

3.1.5 Effect of impurities

Added	T
highly soluble (NaCl)	increases
sparingly soluble (phenol)	decreases
insoluble, soap, detergent	decreases

USES

3.1.6 Everyday uses

Soap lowers T of water: clothes clean easily, spraying is easier.
Water does not wet grease: the angle of contact is obtuse.

Practise PYQs on this topicMolecular Theory & Factors · 15 questions with solutions

CONCEPT

Force due to Surface Tension

In one line
Force = T × total length of contact with the liquid. Count both sides wherever the liquid touches both.

RULE

3.2.1 Force needed to lift a body

F = T × (length of contact) + W

Body	Force
needle or wire, length l	2lT + W
thin ring, radius r	4πrT + W
hollow disc, r₁ and r₂	2π(r₁ + r₂)T + W
circular plate, radius r	2πrT + W
square frame, side l	8lT + W
square plate, side l	4lT + W

Lifting a ring out of the liquid

FLOATING

3.2.2 Floating needle, wire or coin

Vertical parts of T balance the weight

A greased needle floats on water without breaking the surface.
Wire of length L: πr²Ldg = 2LT.

r_max = √(2Tπdg)

Coin of radius r, thickness t: πr²tρg = 2πrT.

r_max = 2Tρgt

Disc with surface bent at θ to its edge: Weight = upthrust + 2πrT cos θ.

FILM

3.2.3 Soap film on a frame

A film has two surfaces.
Slider of length l supporting W: W = 2lT.
A thread loop on a film turns circular when the film inside it is pierced.

TUBE

3.2.4 Drop from a tube

W_drop = 2πrT

PLATES

3.2.5 Water film between two plates

Plates are hard to pull apart

F = 2ATt = 2A²TV

t is the film thickness; volume V = At.

HALF DROP

3.2.6 Drop half immersed

Weight = upthrust + surface tension force

43πr³ρg = 23πr³dg + 2πrT

r² = 3T(2ρ − d)g

Practise PYQs on this topicForce due to Surface Tension · 22 questions with solutions

CONCEPT

Surface Energy & Work Done

In one line
Surface tension is also the energy per unit area of a surface. New surface costs work.

ENERGY

3.3.1 Surface energy

Bringing a molecule to the surface needs work, stored as potential energy.
Potential energy per unit area of surface is surface energy.
Unit: J m⁻². Dimensions: [MT⁻²].

FILM

3.3.2 Work on a soap film

Sliding wire pulled through x

Force on the wire: F = T × 2l.
Moved through x: area grows by 2lx on the two sides together.

W = T × ΔA (film: ΔA = 2 × change in area)

T = WΔA

So surface tension is work done per unit area.
Film expanded from A₁ to A₂: W = 2T(A₂ − A₁).
Two wires of length l, gap raised by d: W = 2Tld.
At constant T, energy is proportional to area.

BLOWING

3.3.3 Blowing a drop or bubble

Body	Surfaces	Work, r₁ to r₂
drop	one	4πT(r₂² − r₁²)
soap bubble	two	8πT(r₂² − r₁²)

New bubble: W = 8πr²T

In terms of diameter: W = 2πT(D² − d²).
W ∝ r² ∝ V^2/3.
Hotter solution has smaller T, so the work is a little less.

Practise PYQs on this topicSurface Energy & Work Done · 30 questions with solutions

CONCEPT

Splitting & Merging of Drops

In one line
Volume stays the same, area changes. Splitting needs energy; merging releases it.

SPLITTING

3.4.1 Splitting a drop

One drop becomes n equal drops

43πR³ = n43πr³ ⇒ R³ = nr³ ⇒ R = n^1/3 r

Work = T × (final area − initial area). All these forms are equal:

W = 4πT(nr² − R²)

W = 4πR²T(n^1/3 − 1)

W = 4πTr²n^2/3(n^1/3 − 1)

W = 4πTR³(1r − 1R)

Quick count: 8 drops: n^1/3 = 2. 27: 3. 64: 4. 125: 5. 512: 8. 729: 9. 1000: 10.

RATIO

3.4.2 Ratio of surface energies

E of n small dropsE of big drop = n^1/3

Sprayed into n drops: the energy becomes n^1/3 times.
One big drop to one small drop: n^2/3 : 1.
Spraying without outside work cools the liquid.

MERGING

3.4.3 Merging into a big drop

Area falls, so energy is released.

E = 4πr²T(n − n^2/3) = 4πR²T(n^1/3 − 1)

If surface energy of each drop is E, released energy = E(n − n^2/3).
In terms of volume V: E = 3VT(1r − 1R).

USE

3.4.4 Where the energy goes

Used as	Result
heat in the big drop	Δθ = 3TJSd(1r − 1R)
kinetic energy	v = √(6Td(1r − 1R))

J = mechanical equivalent of heat, S = specific heat, d = density.

Practise PYQs on this topicSplitting & Merging of Drops · 54 questions with solutions

CONCEPT

Excess Pressure: Drops & Bubbles

In one line
Pressure on the concave side of a curved surface is larger. The smaller the radius, the larger the excess.

RULES

3.5.1 Excess pressure in different cases

Drop

Bubble in air

Bubble in liquid

Bubble at depth h

Surface	ΔP
plane	0
drop, or bubble in a liquid	2TR
soap bubble in air	4TR
bubble at depth h in liquid	2TR + hρg
cylindrical	TR
unequal radii, one surface	T(1R₁ + 1R₂)

A soap bubble has two surfaces, so it has twice the excess of a drop.
Absolute pressure inside: P = P₀ + ΔP.

RATIOS

3.5.2 Quick ratios

ΔP ∝ 1r ⇒ r₁r₂ = ΔP₂ΔP₁

V ∝ r³ and m ∝ r³: three times the excess gives 1 : 27 in volume.
Blow air in: r grows, so the pressure inside falls.
n drops merged: R = n^1/3r, so ΔP becomes 1n^1/3 times.

CONNECTED

3.5.3 Two connected bubbles

Air flows from the small bubble to the large one

Smaller bubble has higher pressure.
Air flows until the small bubble shrinks to a droplet.

INSIDE

3.5.4 Bubble inside a bubble

P₂ − P₀ = 4Tr₁ + 4Tr₂

Equivalent: 1r = 1r₁ + 1r₂

MERGING

3.5.5 Two bubbles combine

Isothermal: PV adds up

P_aV_a + P_bV_b = P_cV_c

T = P₀(c³ − a³ − b³)4(a² + b² − c²)

In vacuum: c² = a² + b²

Common surface of two bubbles in contact has radius below.

r = abb − a

Mercury drops merging (no air inside) just keep their volume: R³ = R₁³ + R₂³.

JEE · NEET

Corner

Double bubble

Interface bends towards the larger bubble

Inner surface: excess pressure runs from the concave to the convex side.

1r = 1r₁ − 1r₂ ⇒ r = r₁r₂r₂ − r₁

Charged soap bubble

Charging pushes the surface outward, so the bubble grows.

Practise PYQs on this topicExcess Pressure: Drops & Bubbles · 60 questions with solutions

CONCEPT

Angle of Contact & Capillarity

In one line
The liquid rises in a narrow tube if it wets the tube, and the narrower the tube, the higher it rises.

ANGLE

3.6.1 Angle of contact

Angle measured inside the liquid

θ	Meniscus	Wets?	Level
below 90°	concave	yes	rises
90°	plane	no	no change
above 90°	convex	no	falls

Angle between the tangents to the liquid and to the solid, inside the liquid.
θ = 0° pure water and glass; 90° water and silver; 138° mercury and glass.
It depends on the pair of liquid and solid, not on the inclination of the wall.
It decreases on heating.
Wax coating raises θ: this is water proofing.

Three common pairs

CAPILLARITY

3.6.2 Capillary rise

Liquid column balances the pressure drop

Pressure just below the meniscus is lower by 2TR.
R = rcos θ.

hρg = 2TR

h = 2T cos θrρg

Examples: ink in blotting paper, oil in a wick, towel soaking water.
With the meniscus included: h + r3.

JURIN

3.6.3 Jurin’s law

hr = constant

Area a times: radius √a times.
For different liquids: h ∝ T cos θρ.
Same h with same T, r: cos θ ∝ ρ.
h does not depend on the shape of the tube.

Same meniscus radius, same height

SHORT

3.6.4 Tube shorter than h

No overflow: the meniscus flattens

hr = Lr′

Liquid reaches the top and stays; it never overflows.
Apparent angle: cos θ′ = Lh.

TILTED

3.6.5 Tilted tube

Vertical height stays the same

l = hcos α

COLUMN

3.6.6 Mass and force of the column

Upward force = T cos θ × 2πr = weight of the column.
m = πr²hρ. As hr is constant, m ∝ r.

GRAVITY

3.6.7 Changing g

h ∝ 1g.
Moon: 6 times. Mine: g′ = g(1 − dR).
Lift going up: g + a, h falls. Going down: g − a, h rises.
Free fall or satellite: g = 0, water fills the tube to its top.

U-TUBE

3.6.8 U-tube of unequal bores

Narrow limb stands higher

h = 2T cos θρg(1r₁ − 1r₂)

JEE · NEET

Corner

Shape of drops

Three surface tensions meet at the edge

T_SL + T_LA cos θ = T_SA ⇒ cos θ = T_SA − T_SLT_LA

T_SA > T_SL: θ acute (water on glass).
T_SA < T_SL: θ obtuse (mercury on glass, water on lotus leaf).
T_SL + T_LA cos θ < T_SA: the liquid spreads.

Force behind the meniscus

F_c < √2 F_a: concave

F_c = √2 F_a: plane

F_c > √2 F_a: convex

F_a acts along the normal to the wall, F_c at 45° to the vertical

tan α = F_c√2 F_a − F_c

The free surface stands at right angles to the resultant.

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EXAM-DAY CARD

Just see this before the exam

T = FL

definition

W = T ΔA

energy

W = 4πR²T(n^1/3 − 1)

splitting

ΔP = 4TR

bubble

h = 2T cos θrρg

capillary

hr = constant

Jurin

FORMULA SHEET

Whole chapter on one page

Basics

T = FL = WΔA

T_t = T₀(1 − αt)

[T] = MT⁻²

Force

ring 4πrT, plate 2πrT

frame 8lT, wire 2lT

plates: F = 2ATt

Work

film: 2T ΔA

drop: 4πT(r₂² − r₁²)

bubble: 8πT(r₂² − r₁²)

Drops

R = n^1/3 r

W = 4πR²T(n^1/3 − 1)

E_smallE_big = n^1/3

Excess pressure

drop 2TR, bubble 4TR

c² = a² + b² (vacuum)

r = abb − a

Capillary

h = 2T cos θrρg

hr = constant, h ∝ 1g

m ∝ r, l = hcos α

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

Film or bubble with one surface

Drop excess pressure as 4TR

Area halved means h doubled

Short tube overflows

CORRECT ✓

Film and bubble: two surfaces

Drop 2TR, bubble 4TR

h ∝ 1r, and area ∝ r²

Rises to the top, no overflow