1) Concept & Molecular Picture

Idea: Molecules at the surface experience a net inward cohesive pull, making the surface behave like a stretched membrane.

• Cohesion = attraction between molecules of the same liquid.
• Adhesion = attraction between liquid and a different surface (e.g., glass).
• Consequences: spherical droplets, meniscus formation, capillarity, soap bubbles, insects walking on water.

2) Definition & Units

Surface tension (also called surface force per unit length) is defined as

$$ T \equiv \frac{F}{L} $$

• SI unit: \( \mathrm{N\,m^{-1}} \)
• CGS unit: \( \mathrm{dyne\,cm^{-1}} \)

Surface energy: Work required to increase the surface area by unit amount. In SI, numerical value of surface energy per unit area equals \(T\) (J m\(^{-2}\) ↔ N m\(^{-1}\)).

3) Excess Pressure (Laplace law)

These follow from mechanical equilibrium of a curved surface under tension.

4) Capillarity & Angle of Contact

Capillary rise/fall in a tube of radius \(r\):

$$ h = \frac{2T\cos\theta}{\rho g r} $$

• \(\theta\): angle of contact (acute for wetting liquids like water on glass → rise; obtuse for non-wetting like mercury on glass → fall).
• \(\rho\): density of liquid, \(g\): acceleration due to gravity.

Meniscus: Concave when adhesion \(>\) cohesion (\(\theta<90^\circ\)); convex when cohesion \(>\) adhesion (\(\theta>90^\circ\)).

5) Temperature & Impurities

• \(T\) decreases with temperature. Empirically: $$ T(T_{\text{abs}}) \approx T_0 \big(1 – k\,T_{\text{abs}}\big), \quad k>0. $$ \(T \to 0\) near the critical temperature.
• Surface-active agents (soaps/detergents) reduce \(T\) and enhance wetting/cleaning.
• Gas above liquid (air vs another immiscible liquid) also affects the measured \(T\).

6) Work & Energy at Surfaces

To create new area \( \Delta A \) at constant \(T\):

$$ W = T\,\Delta A, \qquad \text{so} \quad \frac{dW}{dA} = T. $$

Interpretation: \(T\) is the surface free energy per unit area (isothermal, reversible addition of area).

7) Typical Surface Tension Values (at ~20–25 °C)

Values are indicative for classroom use; exact values depend on temperature and purity.

8) Illustrative Examples

Ex. 1 — Excess pressure in soap bubble

For a bubble of radius \( r = 1.0\,\text{mm} \) with \( T = 0.030\,\mathrm{N\,m^{-1}} \):

$$ \Delta P = \frac{4T}{r} = \frac{4\times 0.030}{1.0\times 10^{-3}} = 120\,\text{Pa}. $$

Ex. 2 — Capillary rise of water

\( r = 0.50\,\text{mm},\; T = 0.072\,\mathrm{N\,m^{-1}},\; \rho = 1000\,\mathrm{kg\,m^{-3}},\; \theta \approx 0^\circ \):

$$ h = \frac{2T\cos\theta}{\rho g r} = \frac{2 \times 0.072 \times 1}{1000 \times 9.8 \times 0.5\times 10^{-3}} \approx 0.029\,\text{m} \;=\; 2.9\,\text{cm}. $$

9) Quick Checks

1. State the SI unit of surface tension and surface energy per unit area.
   Ans: Both numerically \( \mathrm{N\,m^{-1}} \) (and \( \mathrm{J\,m^{-2}} \) for surface energy).
2. Why does mercury form a convex meniscus in glass?
   Ans: Cohesion \( \gt \) adhesion ⇒ \( \theta > 90^\circ \).
3. Show that \( h \propto \dfrac{1}{r} \) for a wetting liquid in a capillary.
   Ans: From \( h=\dfrac{2T\cos\theta}{\rho g r} \) with \(T,\theta,\rho,g\) fixed.

10) Common Applications

• Cleaning action of soaps/detergents (reduced \(T\) improves wetting).
• Capillary action in plant xylem; wicks in lamps and pens.
• Drop formation, emulsions/foams stabilization with surfactants.
• Coating & printing processes (wetting, spread, leveling depend on \(T\) and \(\theta\)).