4.3 Liquid State: Surface Tension, Kelvin Equation, Capillary Action, Viscosity and Liquid Crystals

4.3.1 Surface tension and surface energy ★★★

• A molecule inside the liquid is pulled equally on all sides; one at the surface feels a net inward pull. Bringing molecules to the surface costs work, so the liquid shrinks its area. γ is the work to make unit new area (surface Gibbs energy) or the force per unit length along the surface.SN
• Stronger intermolecular forces give larger γ: Hg 0.485, water 0.072, ethanol 0.022 N m⁻¹ at 20–25 °C. Ethanol (H-bonded) > dimethyl ether: the two are isomers with similar dispersion forces, but only ethanol forms O–H···O bonds (b.p. 78 °C against −25 °C) CSE 2024 Q3(b)(i) 5m.SN
• γ is intensive (energy per area), like the dielectric constant; energy and enthalpy are extensive Geo Pre 2024 Q49. γ falls as T rises and vanishes at the critical temperature (Add-on 4.3.8).
• Breaking up a drop costs γΔσ at constant volume: , so . A drop of radius 0.4 cm split into 125 drops: r = 0.08 cm, Δσ = 10.053 − 2.011 = 8.042 cm², ΔE = 72 × 8.042 = 579 erg = 5.79 × 10⁻⁵ J CSE 2021 Q3(a) 10m.SN

4.3.2 Curved surfaces: the Laplace equation ★★★

• Derivation: let a drop grow from r to r + dr. Work against the pressure difference Δp·4πr²dr equals the surface work γ·8πrdr, so Δp = 2γ/r. Pressure is always higher on the concave side.SN
• Water drop r = 150 nm: Δp = 2(0.072)/1.50 × 10⁻⁷ = 9.6 × 10⁵ Pa (9.5 atm) CSE 2024 Q3(b)(ii) 5m.SN
• Maximum bubble pressure: a bubble blown at the tip of a capillary in a liquid has one surface; the pressure is largest when it is a hemisphere of the tube radius. Δp = 2.16 torr = 288.0 Pa, γ = 0.072 N m⁻¹: r = 2γ/Δp = 5.0 × 10⁻⁴ m, diameter 0.10 cm CSE 2023 Q4(c) 10m.SN
• Plain hydrostatics is asked in the same unit: a 15 cm straw at 45° lifts water a vertical h = 15 sin 45° = 10.6 cm, so Δp = ρgh = 1000 × 9.81 × 0.1061 = 1.04 × 10³ Pa = 7.8 torr (only the vertical height counts) CSE 2023 Q4(b) 10m.

4.3.3 Wetting, contact angle and capillary action ★★★

• Contact angle θ: angle inside the liquid between the solid and the liquid surface at the contact line. Young’s equation balances the three tensions along the solid. θ < 90°: the liquid wets (adhesion beats cohesion, water on clean glass, θ ≈ 0); θ > 90°: non-wetting (mercury on glass, about 140°).SN
• From Young, : for the same solid, the liquid with the larger own surface tension has the larger θ. Liquid A (γ seven times B) has the higher contact angle; with gravity switched off A gathers into one nearly spherical drop touching the glass at a small patch, while B (wetting) creeps over the walls and coats the beaker CSE 2023 Q1(e)(ii) 5m.SN
• Capillary rise: the vertical pull 2πrγ cos θ round the rim holds up the column πr²hρg, so h = 2γ cos θ/ρgr. Water in a tube of radius 0.1 mm rises 14.7 cm; mercury (θ ≈ 140°) is depressed. Method of measuring γ: capillary rise (θ ≈ 0).SN
• Same tube: h_B/h_A = (γ_B/γ_A)(ρ_A/ρ_B). A has half the γ and twice the density of B; h_A = 2.0 cm, so h_B = 2.0 × 2 × 2 = 8.0 cm CSE 2022 Q4(c) 10m.SN
• Other methods of measuring γ: drop weight (stalagmometer, γ₁/γ₂ = m₁/m₂ for equal drop sizes), maximum bubble pressure (4.3.2), du Noüy ring (force to pull a ring off the surface).

4.3.4 The Kelvin equation ★★★

1. Derivation: the liquid in a drop is under extra pressure Δp = 2γ/r (4.3.2). Raising the pressure on a liquid by Δp raises its chemical potential by V_mΔp; at equilibrium the vapour must match it: RT ln(p/p*) = V_m·2γ/r.SN
2. Water at 25 °C (γ 0.072 N m⁻¹, V_m 18.07 cm³ mol⁻¹): p/p* = 2.86 at r = 1 nm, 1.11 at 10 nm, 1.01 at 100 nm, 1.001 at 1 μm (Fig. 4.3.3).SN

• Consequences: small drops evaporate and large ones grow (Ostwald ripening); vapour can be supersaturated without condensing until dust or ions give large nuclei (cloud seeding, Wilson cloud chamber); a concave meniscus in a narrow pore has a lower vapour pressure, so vapour condenses in pores below p* (capillary condensation).SN
• Superheating: a vapour bubble starts at molecular size. Inside a tiny cavity the vapour pressure is below p*, while the pressure needed to hold the bubble open is p_ext + 2γ/r (2.2 atm for r = 1 μm in boiling water). At the normal b.p. p* only equals p_ext, so the bubble collapses; the liquid must overshoot the b.p. before boiling starts, then boils violently (bumping). Boiling chips supply ready-made cavities CSE 2025 Q1(e) 5m.SN