Surface Tension & Capillarity

The highest-stakes application of this article is in newborn lungs. Alveoli are micrometre-scale wet bubbles; by the curvature–pressure logic previewed above, small wet bubbles need large pressures to stay open. Healthy lungs coat themselves with a pulmonary surfactant that slashes surface tension exactly where alveoli are smallest; premature babies born before the coating develops suffer respiratory distress syndrome, and the therapy — surfactant replacement, delivered since the 1980s — is this chapter administered by nebulizer, credited with cutting RDS mortality several-fold. Elsewhere: inkjet printers fire picolitre drops whose formation and satellite-droplet behaviour are pure surface-tension engineering; lab-on-a-chip diagnostics move blood through capillary channels with no pump at all — Jurin's law as plumbing; water-repellent "lotus-effect" fabrics and self-cleaning glass tune contact angles past 150°; and espresso machines, fountain pens, candle wicks, and paper towels all run on capillarity you can now price with equation (3).

Follow one water molecule. Deep in the bulk, it is bonded to neighbours in every direction — pulled equally all around, net force zero, energetically content. Lift it to the surface and half its social circle vanishes: no neighbours above, only air's occasional molecule. It keeps all its downward and sideways attractions but loses the upward ones. Two consequences: it is pulled inward (which is why surfaces flatten and drops round themselves), and it sits at higher energy than a bulk molecule — creating new surface means promoting molecules into this expensive position. So a liquid surface is a workforce of underpaid molecules, and the liquid economizes: it exposes the fewest molecules it can. A free drop takes the shape with the least surface for its volume — a sphere. A film stretched on a wire frame snaps flat. Two touching droplets merge (less total surface than two). And along any line you draw in the surface, the molecules on one side pull the molecules on the other side with a tangential force proportional to the line's length: the "skin tension" the needle and the pond skater ride on.

One more character enters when a liquid meets a solid wall: the contact angle $\theta$, the angle (measured inside the liquid) between the solid surface and the liquid surface at the line where they meet. It is set by a tug-of-war between cohesion (liquid–liquid attraction) and adhesion (liquid–solid attraction). Water on clean glass: adhesion wins, the water spreads and climbs the wall, $\theta \approx 0°$–$8°$ — a wetting liquid with a concave meniscus. Mercury on glass: cohesion wins overwhelmingly, the mercury balls up and pulls away, $\theta \approx 140°$ — non-wetting, convex meniscus. Water on a waxed jacket: $\theta > 90°$ by design, so rain beads and rolls. The contact angle is the boundary condition that decides which way capillarity goes.

Surface tension $T$ (some books write $S$ or $\gamma$) is defined along a line: if you draw any line of length $L$ in a liquid surface, the surface on one side pulls the surface on the other side with force

where: $T$ — surface tension in N/m, tangent to the surface, perpendicular to the line. Water's $T \approx 0.072$ N/m at room temperature; soap solution ~0.030; mercury a muscular 0.465. These are small forces — 7 grams-weight per metre of line for water — which is why surface tension rules the millimetre world (insects, drops, capillaries) and is irrelevant to ships. The same constant wears an energy costume. Creating new surface area $\Delta A$ promotes molecules to the expensive surface layer, costing work

where: $T$ now reads as surface energy per unit area, J/m². The units agree — $\mathrm{N/m} = \mathrm{J/m^2}$ — and so does the physics: a membrane that costs energy per area is a membrane under tension. Exam questions switch costumes freely: "force on a wire" wants (1), "work to split a drop" wants (2).

Drops, bubbles, and pressure — the preview. A curved surface under tension squeezes what it wraps: the inside of a drop sits at higher pressure than the outside, by $\Delta P = 2T/r$ for a liquid drop and $4T/r$ for a soap bubble (two surfaces again). These results — and the lovely traps hiding in them (smaller bubble, bigger pressure!) — are derived and drilled in their own archetype at the end of this unit; here, carry only the qualitative point: tension + curvature = pressure difference. It is the engine behind capillarity, coming right now.

Dip a clean glass capillary (bore radius $r$) into water. Adhesion curves the water's edge up the glass ($\theta \approx 0$), giving the surface a concave meniscus — approximately a hemisphere of radius $\approx r$. The pressure just below that curved surface is lower than atmospheric (the curved skin holds the difference), so the atmosphere pushes water up the tube until the weight of the raised column repays the pressure deficit. Equilibrium height is Jurin's law:

where: $\theta$ — the contact angle ($\cos\theta \approx 1$ for water on clean glass); $r$ — the tube's inner radius (not diameter, not the meniscus of some other geometry); $\rho$ — the liquid's density. Read the scalings: thinner is higher ($h \propto 1/r$ — halve the bore, double the climb), and the sign of $\cos\theta$ steers everything: mercury's $\theta \approx 140°$ makes $\cos\theta < 0$, so $h < 0$ — mercury is depressed below the outside level, the meniscus convex. That sign flip is a fixture of assertion–reason questions.

1 · Board level — capillary rise (CBSE/ICSE pattern, 3 marks)

A clean glass capillary of inner radius $0.20\ \mathrm{mm}$ is dipped vertically into water ($T = 0.072\ \mathrm{N/m}$, $\rho = 1000\ \mathrm{kg/m^3}$, contact angle $\approx 0°$, $g = 9.8\ \mathrm{m/s^2}$). Find the height of capillary rise.

Step 1. Convert the radius: $r = 0.20\ \mathrm{mm} = 2.0\times10^{-4}\ \mathrm{m}$. (Check the question's word: radius given here; had it said diameter, halve first.)

Step 2. Jurin's law with $\cos 0° = 1$: $h = \dfrac{2T}{\rho g r} = \dfrac{2\times0.072}{1000\times9.8\times2.0\times10^{-4}}$.

Step 3. $h = \dfrac{0.144}{1.96} = \boxed{7.3\times10^{-2}\ \mathrm{m}} \approx 7.3\ \mathrm{cm}$.

Step 4. Sanity: a fifth-of-a-millimetre bore lifts water 7 cm; xylem vessels in plants are finer still — capillarity is one of the mechanisms sap exploits. In a 2 mm bore the rise would be a barely visible 7 mm: the $1/r$ law in action.

2 · Competitive level — work to atomize a drop (JEE-Main/NEET pattern, ~2.5 minutes)

A spherical water drop of radius $R = 1.0\ \mathrm{cm}$ is sprayed into $n = 1000$ identical droplets ($T = 0.072\ \mathrm{N/m}$). Find (a) the radius of each droplet and (b) the work required. (c) Does the water warm or cool slightly as this happens?

Step 1. (a) Volume conservation: $n\cdot\tfrac43\pi r^3 = \tfrac43\pi R^3 \Rightarrow r = R\,n^{-1/3} = 1.0\ \mathrm{cm}/10 = \boxed{1.0\ \mathrm{mm}}$.

Step 2. (b) Surface area jumps from $4\pi R^2$ to $n\cdot4\pi r^2 = 4\pi R^2\, n^{1/3}$: a factor of $n^{1/3} = 10$.

Step 3. Work = $T\,\Delta A = T\,4\pi R^2\,(n^{1/3} - 1) = 0.072\times4\pi\times(10^{-2})^2\times9 = \boxed{8.1\times10^{-4}\ \mathrm{J}}$.

Step 4. (c) You supplied energy that now lives in the enlarged surface; the process absorbs energy, and if done adiabatically the liquid cools slightly. (Merging droplets — the reverse — releases surface energy and warms: that direction powers a whole archetype later in this unit.) Timing note: memorize the skeleton $W = 4\pi R^2 T(n^{1/3}-1)$ — deriving it via volume conservation eats a minute you can spend elsewhere. The paired formula for merging is the same with the sign flipped. Both are JEE-Main regulars.

3 · Competitive level — lifting a ring off water (JEE-Main pattern, ~2 minutes)

A thin wire ring of radius $5.0\ \mathrm{cm}$ and weight $0.030\ \mathrm{N}$ rests flat on a water surface. Find the minimum vertical force needed to pull it free ($T = 0.072\ \mathrm{N/m}$; the water wets the wire).

Step 1. As the ring lifts, it drags up a collar of water attached along both the inner and outer edges of the wire — two contact circles, each of circumference $2\pi r$ (thin wire: both ≈ same $r$).

Step 2. Surface-tension pull-down at breakaway: $F_T = T \times 2(2\pi r) = 0.072\times2\times2\pi\times0.05 = 4.5\times10^{-2}\ \mathrm{N}$.

Step 3. Total minimum pull: $F = W + F_T = 0.030 + 0.045 = \boxed{7.5\times10^{-2}\ \mathrm{N}}$.

Step 4. Note the structure: the tension term ($0.045$ N) exceeds the ring's weight — at centimetre scales the skin outmuscles gravity, which is the entire theme of this chapter. Timing note: the examiner's coin-flip is the contact-line count — a ring or a needle pulls TWO lines of water (factor 2); a vertical plate's perimeter counts once around. Draw one cross-section, count the liquid junctions, then compute. Ninety seconds with the picture; a guaranteed error without it.

The factor of 2 census. Film on a slider: 2 surfaces → $2TL$. Soap bubble: 2 surfaces → $4T/r$. Water drop or air bubble inside water: 1 surface → $2T/r$. Ring/needle pulled off water: 2 contact lines. Miscounting the 2 is this chapter's signature error — take the census before substituting.

Radius vs diameter in Jurin's law. $h = 2T\cos\theta/\rho g r$ uses the tube's inner radius. Questions love giving the diameter.

Units of $T$. N/m, equivalently J/m² — not N/m² (that's pressure/stress). An option quoting surface tension in pascals is an auto-eliminate.

Contact angle direction. Measured inside the liquid, from the solid wall to the surface tangent. Wetting: $\theta < 90°$, concave meniscus, rise. Non-wetting: $\theta > 90°$, convex, depression. Mercury numericals arrive with $\theta \approx 140°$ and $\cos\theta \approx -0.77$: keep the minus sign and report a depression.

Short-tube "overflow". If the tube's length above the liquid is less than the predicted $h$, water rises to the top and the meniscus reduces its curvature until the balance closes — it never fountains out. Energy conservation is not negotiable; this is a stock assertion–reason item.

Temperature and impurities. Heating lowers $T$ (near-linear over exam ranges); soluble surfactants lower it dramatically; highly soluble salts raise it slightly. "Hot soapy water cleans best" is this trap's real-world mnemonic.

Small means surface-ruled. Surface forces scale with length; weights scale with volume. Below a few millimetres, tension wins (striders, drops, capillaries); above, gravity wins (lakes are flat). If your capillary-rise answer is in metres for a millimetre bore, re-check powers of ten.

Which $g$. CBSE 9.8; JEE/NEET usually 10. Jurin's $h$ scales as $1/g$, so the choice shifts answers by 2% — match the paper's declaration.

Texts. NCERT Physics Class 11, Ch. 9 §9.6 · HC Verma Vol. 1, Ch. 14 · Resnick–Halliday–Walker Ch. 14. The excess-pressure machinery ($2T/r$, $4T/r$, joined bubbles) is derived and drilled in the Bubbles, Drops & Excess Pressure archetype.

Provenance. All results standard. A verifier should recompute: the slider's $2TLx = T(2Lx)$ agreement; example 1's $h = 0.144/1.96 = 7.3$ cm; example 2's $r = 1$ mm, $\Delta A$ factor 10, $W = 8.1\times10^{-4}$ J; example 3's $F_T = 0.045$ N and total 0.075 N.

Prerequisites. Fluids at Rest: Pressure & Pascal's Law. Leads to. Bubbles, Drops & Excess Pressure (archetype) · Drops Merging & Terminal Velocity (archetype).