QCC Notes
CLASS 11 · PHYSICSJEE MAIN × NEETहिंदी
§ 9.5NCERT Class 11 · Physics · Chapter 9

Surface Tension, Surface Energy, Excess Pressure & Capillarity

QCC Notes · Class 11 Physics NCERT-aligned Updated 26 Sep 2026 9 min read

A free liquid surface exhibits a distinctive tendency to contract to the minimum possible surface area, behaving as if covered by a stretched elastic membrane under isotropic tension. This interfacial phenomenon is known as surface tension. It governs the spherical shape of rain droplets, the walking of water striders on ponds, and the ascent of sap through capillary xylem vessels.

1. Molecular Theory of Surface Tension

Intermolecular interactions involve two primary forces: cohesive forces (attraction between like liquid molecules) and adhesive forces (attraction between liquid and solid/gas boundary molecules). The maximum distance over which attractive intermolecular forces operate effectively is called the molecular range (c ≈ 10-9 m). An imaginary sphere of this radius surrounding a molecule is its sphere of molecular influence.

   Free Surface: ~~~~~~~~~~~~~~ Molecule B (On Surface) ~~~~~~~~~~~~~~
                                    |
                               (   |  / )  Half-sphere in vapor:
                                ( --o-- )   Net downward cohesive pull!
                                 ( /| )    Excess Surface Energy U = T * Delta A
                                    |
   Bulk Interior:               (  | / )
                                ( --o-- )   Molecule A (In Deep Bulk)
                                ( / |  )   Surrounded symmetrically:
                                            Net cohesive force = 0
Fig 9.5A: Comparison of cohesive forces acting on a deep bulk molecule versus an interface surface molecule.
  • Molecule in Bulk Liquid: Attracted equally in all directions by surrounding molecules. Resultant cohesive force is zero.
  • Molecule in Surface Film (Thickness ≈ 10-9 m): Possesses fewer liquid molecules above it. Consequently, it experiences a net downward unbalanced inward pull toward the bulk interior.
  • Surface Energy: To bring a molecule from the interior to the surface against this inward attractive pull, work must be done. This work is stored as potential energy called surface energy. The state of minimum potential energy dictates that a liquid contracts to minimize its surface area.

2. Quantitative Definition & Surface Energy Relationship

The surface tension (T or S) of a liquid is defined as the tangential force acting per unit length along an imaginary line drawn on the free liquid surface:

T = F/L
  • SI Unit: N/m or J/m2.
  • Dimensional Formula: [T] = [M1 L0 T-2] (identical dimensions to force constant k).

Relation between Surface Tension and Surface Energy

Consider a U-shaped wire frame holding a liquid film of width L with a movable sliding wire. Because a liquid film has two free surfaces (top and bottom), the surface tension force pulling the slider inward is F = 2 T L. To displace the wire outward by distance Δ x, the work done is:

W = F · Δ x = (2 T L)Δ x = T (2 L Δ x) = T · Δ A
Work Done in Changing Surface Area:
W = T · Δ A
Hence, surface tension is numerically and dimensionally equal to the surface energy per unit increase in surface area (T = W / Δ A).

Work Done in Splitting / Coalescing Drops

  • Splitting 1 Large Drop of Radius R into n Identical Droplets of Radius r: By volume conservation, 4/3π R3 = n (4/3π r3) ⇒ r = R / n1/3.
    Δ A = n(4π r2) - 4π R2 = 4π R2 (n1/3 - 1)
    W = T · Δ A = 4π T R2 (n1/3 - 1)
    Because energy is absorbed, the temperature of the liquid droplets drops: Δ θ = 3T/(ρ s)(1/r - 1/R).
  • Blowing a Soap Bubble from Radius R1 to R2 (2 surfaces!):
    W = T · Δ Atotal = T · 2 ( 4π R22 - 4π R12 ) = 8π T (R22 - R12)

3. Excess Pressure inside Drops and Bubbles

Due to surface tension, a curved liquid surface exerts a net inward compressive force toward the concave side. For mechanical equilibrium, the pressure on the concave side must exceed the pressure on the convex side by an excess pressure Δ P:

Interface Geometry Number of Free Surfaces Excess Pressure (Δ P = Pin - Pout) Formula Context
Spherical Liquid Droplet 1 surface (liquid-air)
Δ P = 2T/R
 Raindrop, mercury droplet in air
Spherical Cavity (Air Bubble in Liquid) 1 surface (air-liquid)
Δ P = 2T/R
 Air bubble inside an aquarium or boiling water
Hollow Soap Bubble 2 surfaces (inner & outer)
Δ P = 4T/R
 Thin liquid soap film in air
Cylindrical Liquid Jet 1 cylindrical surface
Δ P = T/R
 Liquid emerging from circular nozzle

4. Angle of Contact & Capillary Ascent (Jurin's Law)

Angle of Contact (θc)

The angle of contact is defined as the angle made by the tangent to the liquid surface at the point of contact with the solid wall inside the liquid:

  • Acute (θc < 90°): Adhesive forces exceed cohesive forces (Fa > Fc/√(2)). The liquid wets the solid, meniscus is concave, and the liquid ascends in a capillary tube (e.g. water on glass, θc ≈ 0°).
  • Obtuse (θc > 90°): Cohesive forces dominate (Fc/√(2) > Fa). The liquid does not wet the solid, meniscus is convex, and the liquid depresses in a capillary tube (e.g. mercury on glass, θc ≈ 135°).

Derivation of Capillary Ascent Formula

When a narrow glass tube of internal bore radius r is immersed vertically into a wetting liquid of density ρ and surface tensionT:

                                 |  |
                                 |  |  Capillary Bore Radius r
                                 |  |  Meniscus Radius R = r / cos(theta)
                                 |()|  Concave Meniscus
                                 |  |  
                                 |  |  Upward Surface Tension Force:
                                 |  |  Fup = T * (2 * pi * r) * cos(theta)
                                 |  |  
                                 |  |  Balances Weight of Column:
                                 |  |  W = m * g = (pi * r2 * h) * rho * g
                          ~~~~~~~+--+~~~~~~~ Free Liquid Surface Level
Fig 9.5B: Balance between the vertical component of surface tension around the circumference and the weight of the elevated liquid column.

Upward vertical pulling force around the contact perimeter: Fup = (2π r) T cos θc.

Downward weight of the elevated liquid column: W = (π r2 h)ρ g.

Equating forces (Fup = W):

2π r T cos θc = π r2 h ρ g
Ascent Formula:
h = (2T cos θc)/(r ρ g) = 2T/(R ρ g)
where R = r / cos θc is the radius of curvature of the spherical liquid meniscus.

Jurin's Law & Insufficient Tube Length

  • Jurin's Law: For a given liquid-solid pair, h · r = constant. Narrower capillary tubes produce greater heights of liquid rise.
  • Capillary of Insufficient Length (l < h): If a tube of length l < h is dipped into water, water rises to the very top lip and stops. Water never overflows! The meniscus simply flattens out, increasing its radius of curvature from R to R' such that:
    l · R' = h · R ⇒ R' = R (h/l)

JEE & NEET Solved Practice Problems

Problem 1: Calculate the work done in blowing a soap bubble of radius 5 cm from a soap solution of surface tension T = 0.03 N/m. Take π = 3.142.
Solution:
A soap bubble has two free surfaces (inner and outer).
Initial radius R1 = 0, final radius R2 = 0.05 m.
Total increase in surface area:
Δ A = 2 × [4π R22 - 0] = 8π R22

Work done:
W = T · Δ A = 8π T R22 = 8 × 3.142 × 0.03 × (0.05)2

W = 8 × 3.142 × 0.03 × 0.0025 = 1.885 × 10-3 J = 1.885 mJ
Problem 2: A clean glass capillary tube of internal diameter 0.4 mm is dipped vertically into a beaker containing water (T = 0.07 N/m, ρ = 1000 kg/m3, θc = 0°). (a) Find the height to which water rises in the tube. (b) If the capillary tube is pushed down so that only 3 cm of tube projects above the water surface, find the new radius of curvature of the meniscus. Take g = 9.8 m/s2.
Solution:
(a) Internal radius r = d/2 = 0.2 mm = 2 × 10-4 m.
Theoretical capillary rise:
h = (2T cos θc)/(r ρ g) = (2 × 0.07 × cos 0°)/((2 × 10-4) × 1000 × 9.8) = 0.14/1.96 = 0.0714 m = 7.14 cm

(b) With available tube length l = 3 cm = 0.03 m < h, water rises to the top and flattens out without overflowing.
Since h R = l R', and initially R = r / cos 0° = 0.2 mm:
R' = R (h/l) = 0.2 mm × (7.14/3.0) ≈ 0.476 mm

Frequently Asked Questions

Q1. Why do small liquid drops assume a perfectly spherical shape in the absence of gravity?
Surface tension imparts extra potential energy (surface energy U = T * A) to molecules residing on a liquid's boundary. To achieve stable mechanical equilibrium, a liquid minimizes its surface potential energy by minimizing its surface area for a given volume. For any fixed volume, a sphere mathematically possesses the minimum possible surface area, so surface tension pulls the drop into a spherical shape.
Q2. Why is the excess pressure inside a soap bubble double that inside a liquid droplet of the same radius?
A liquid droplet has only one liquid-gas interface separating inside from outside (Delta P = 2T / R). A hollow soap bubble consists of a thin spherical liquid film possessing two separate free liquid surfaces (an inner air-liquid surface and an outer liquid-air surface). Therefore, surface tension acts across both surfaces simultaneously, yielding double the excess pressure: Delta P = 4T / R.
Q3. What happens if a capillary tube of insufficient height is dipped into water? Does water overflow?
No, water will never overflow! If the capillary tube height h' is less than the theoretical rise height h, water rises to the top edge and stops. The curved meniscus automatically flattens, increasing its radius of curvature R' such that the product h' * R' = h * R = constant. It forms a flatter meniscus without spilling a single drop.
Your progress

Saved on this device only — no account, no sign-in.