Surface Tension & Capillarity Surface Tension & Capillarity Surface Tension Phenomena Phenomenon Driving Force Formula Contact Angle Role Common Example Capillary rise climbs with 2T , while bubbles double the pressure trouble with 4T/R . Capillary Action Resultant of Adhesion and Surface Tension h = 2T r g Rise if < 90 , Fall if > 90 Water rising in a glass tube Excess Pressure (Liquid Drop) Surface tension force towards the center P ex = 2T R Determines the curvature of the liquid interface Formation of spherical raindrops Excess Pressure (Soap Bubble) Surface tension force from two film surfaces P ex = 4T R Requires very small for film stability Blowing soap bubbles in air Meniscus Shape Relative strength of Cohesion vs Adhesion F net = F a + F c Concave for < 90 , Convex for > 90 Curved surface of water vs mercury Coalescence of Drops Minimization of total surface area/energy E = 4 R 2 T (1 - n 1/3 ) Reduces surface energy via contact merging Small mercury droplets forming one large drop Splitting of a Drop External work done against surface tension W = 4 R 2 T (n 1/3 - 1) Increases total surface area A Atomizer or spray nozzle operation Floating Needle Vertical component of surface tension film mg = 2Tl Requires to adjust for weight balance Greased needle resting on water surface Detergent Action Reduction of surface tension by surfactants T = T pure - T sol Decreases to improve wetting Washing clothes with soap solution surface tension phenomena Imagine the liquid surface isn't just a random boundary, but a stretched, elastic skin. Deep inside a cup of water, every molecule is surrounded by friends pulling on it from all directions, so it feels no net force. At the surface there is no crowd above, so surface molecules feel a net inward pull. This imbalance tightens the surface molecules together like a drumhead and makes the surface try to shrink to the smallest possible area. That is why droplets tend to be spherical and why tiny insects can walk on water. The competition in all such situations is between cohesive forces inside the liquid that favor a small area and external factors like gravity or adhesion to a solid that may stretch or curve the surface. Understanding this balance explains surface energy, contact angle, capillary rise, and the extra pressure that exists inside tiny drops and soap bubbles. remember Real-world picture: a crowded dance floor seen from above. People in the middle are jostled equally from all sides, but those at the boundary are pulled only inward by the crowd. The boundary line becomes tight and resists intrusion like a stretched rope. A liquid surface behaves in the same spirit. Cohesive forces Attractive forces between molecules of the same liquid that pull surface molecules inward and make the surface contract. Adhesive forces Attraction between molecules of a liquid and those of another material (like glass). Adhesion competes with cohesion and helps set the meniscus shape and the contact angle. Surface tension (T or σ) Force per unit length acting along a line drawn on the liquid surface, tangent to the surface and perpendicular to the line. SI unit: N/m. Surface energy Work required to create unit area of new surface at constant temperature. Numerically equal to surface tension for a pure liquid at fixed temperature. Unit: J/m². Angle of contact (θ) Angle measured inside the liquid between the solid surface and the tangent to the liquid surface at the line of contact. Wetting vs non-wetting If θ < 90°, the liquid wets the solid (water on clean glass, concave meniscus). If θ > 90°, it does not wet (mercury on glass, convex meniscus). Rise or fall of a liquid in a narrow tube (capillary) due to surface tension and contact angle effects. Capillarity Meniscus The curved liquid surface near a solid boundary. Concave for wetting liquids and convex for non-wetting liquids. Two equally powerful viewpoints are used in problems. The force picture treats the surface like a tight string on the perimeter that can pull with a force T per unit length. The energy picture counts the work needed to create extra surface area: to stretch a surface by A at fixed temperature needs work T , A . For static shapes, both viewpoints agree and provide quick routes to results such as the extra pressure inside a small drop or the height of liquid in a capillary. Definition of surface tension Force per unit length acting tangentially to the surface and perpendicular to the chosen line. Surface energy relation Work per unit area required to create new surface at fixed temperature. Units: J/m 2 = N/m . Determines the energy cost required to stretch or change the area of a liquid-gas or liquid-liquid interface. Because enlarging a surface costs energy, systems spontaneously try to reduce surface area. A free droplet tends toward a sphere, the shape with the minimum surface area for a given volume. Soap bubbles have two free surfaces (inner and outer), so any change in radius changes two areas simultaneously, doubling both the energy and the surface-tension force compared to a single-surface drop. Laplace pressure for a single-surface spherical interface. Excess pressure in a spherical liquid drop Excess pressure in a soap bubble Two surfaces in a bubble double the surface-tension force. Determines the pressure difference across any curved fluid interface based on the surface tension and its principal radii of curvature. Height of rise (or fall, if <0 ) in a narrow tube of radius r . Capillary rise (Jurin's law) tip Sign of decides rise or fall: for water in clean glass, 0 , so 1 and the liquid rises. For mercury on glass, > 90 , is negative and the level falls. h = 2 g r Static equilibrium; meniscus forms a small spherical section. Uniform, constant contact angle θ along the tube wall. Tube radius r is small (capillary scale); viscous forces irrelevant in final static height. Capillary rise: h = 2 g r Only the vertical component of surface tension supports the liquid column. Density times volume times g. Equilibrium of vertical forces. Jurin's law for rise (or fall if <0 ). Static height of a liquid column in a narrow tube. Directly proportional to surface tension and , inversely proportional to tube radius and fluid weight density g . P ex,drop = 2T R , P ex,soap bubble = 4T R Work done by inside pressure in expanding the sphere. Differentials for a sphere; a soap bubble has two surfaces so dA bubble = 2 8 R , dR . Energy cost to create extra surface area. For a drop (one surface). Laplace pressure for a single-surface drop (or an air bubble in a liquid). Two surfaces in a soap bubble double the pressure difference. Excess pressure: drop (2T/R) and soap bubble (4T/R) Spherical interface of radius R. Quasi-static change of radius by dR, keeping temperature constant. Pressure inside higher than outside by P ex. Angle of contact is set by a competition: cohesion tries to keep molecules together, while adhesion tries to pull the liquid towards the solid. A clean glass surface attracts water strongly, so the water surface dips near the wall (concave meniscus) and is acute. Mercury has much stronger cohesion than adhesion to glass, so the surface bulges upward (convex meniscus) and is obtuse. In capillarity, only the vertical component matters for supporting or depressing the column. Classic trap: in a soap bubble the excess pressure is 4T/R , not 2T/R . Two surfaces double both the surface energy change and the net inward force. neet-alert Factors that affect surface tension Temperature: increases in temperature reduce T (thermal agitation weakens cohesion). For many liquids approximately linear over small ranges. Impurities: highly soluble salts in water generally increase T ; surfactants like soap or detergents strongly decrease T . Surface contamination (grease, dust): usually lowers T and changes the effective contact angle. Dissolved gases: can slightly modify T and nucleate bubbles. Insect standing on water with visible dimples and labels showing surface tension and hydrophobic hairs. Water strider supported by surface tension. Dimples around the hydrophobic legs show the depressed meniscus; the tight surface balances the insect’s weight. Schematic sphere showing water molecules with arrows indicating balanced forces inside and inward pull at the surface. Molecular view of a droplet: bulk molecules feel balanced pulls; surface molecules feel a net inward pull, creating the elastic skin of surface tension. Isolated water droplet with reflections, photographed at very high shutter speed. High‑speed photo of a nearly perfect spherical water droplet in air. The sphere minimizes surface area for a given volume due to surface tension. Liquid Surface tension σ (N/m) Typical contact angle on clean glass Remarks Reference values at about 20 °C (typical textbook numbers). Water (pure) 0.0728 ≈ 0° Strongly wetting; concave meniscus; rises in glass. Soap solution 0.025–0.040 Small Surfactants reduce σ; bubbles stabilize easily. Mercury 0.485 > 90° (≈ 140°) Non-wetting; convex meniscus; level falls in glass. Ethanol 0.022 Small Wetting; σ much lower than water. System Number of free surfaces Excess pressure P ex Comment Liquid drop in air (or air bubble in liquid) 2T/R Single spherical interface Soap bubble in air 4T/R Inner and outer surfaces Energy language gives quick estimates. Work needed to split or stretch surfaces equals T , A . If a big drop (radius R ) breaks into n equal smaller drops (radius r ), volume conservation gives R 3 =nr 3 . Initial area is 4 R 2 and final area is n 4 r 2 = 4 n r 2 . The energy absorbed is W = T(4 n r 2 - 4 R 2 ) = 4 T R 2 (n 1/3 -1) . Such relations help decide whether mechanical agitation will create sprays or foam, and why surfactants that reduce T make bubble formation easier. Capillary formula limits: it assumes a narrow tube, a uniform contact angle, a spherical meniscus, and static equilibrium. For wide tubes ( r large), rise h becomes very small and the meniscus shape deviates from spherical; the simple formula then loses accuracy. tip Hyperbola: h falls inversely with r; positive for wetting liquids (θ < 90°) and negative for non-wetting (θ > 90°). Tube radius r Capillary height h cm Jurin’s law signature: h ∝ 1/r for fixed σ, θ, ρ, g. custom sigma control control theta dependent Very narrow tube → large rise 0.25 mm 6 cm 1.0 mm Four times radius → one-fourth height 1.5 cm By definition, the angle of contact is measured inside the liquid between the tangent to the liquid surface and the solid surface. Angle of contact is measured in the air, above the surface. Capillary rise depends on the tube height or on how fast the tube is inserted. The final static height is independent of insertion speed and tube length (as long as the tube is tall enough). It depends only on , , r , , and g . Excess pressure is the same for a droplet and a soap bubble of the same radius. A soap bubble has two surfaces, so P ex =4T/R , which is double the value for a single-surface droplet. “WaGla Wets, Hg Humps.” Water on glass wets (θ small → concave meniscus, rise). Mercury humps up (θ obtuse → convex meniscus, fall). Remember: rise ∝ . neet-alert Do not plug diameter into Jurin’s formula. The variable r is the inner radius of the tube. Using diameter doubles r and halves the rise, leading to a factor-of-2 error. cm easy r = 0.50 mm = 5.0 10 -4 , m = 0.0728 , N/m =0 =1 =1000 , kg/m 3 g=9.8 , m/s 2 A clean glass capillary of radius 0.50 mm is dipped in pure water at 20 °C ( =0.0728 , N/m , =1000 , kg/m 3 ). Assume =0 and g=9.8 , m/s 2 . Find the height of water rise. Use Jurin’s law h= 2 g r . Find the excess pressure inside (a) a spherical water drop of radius 1.0 , mm and (b) a soap bubble of the same radius. Take T water =0.073 , N/m and T soap =0.030 , N/m . R = 1.0 , mm = 1.0 10 -3 , m For a drop: P ex =2T/R For a soap bubble: P ex =4T/R medium Pa P ex for the two cases Initial sphere radius R = 2.0 , mm Number of small drops n = 8 Surface tension T = 0.073 , N/m Work W required hard A large drop of water of radius R=2.0 , mm breaks into n=8 equal small droplets. If T=0.073 , N/m , calculate the work required for this process (assume no change in temperature). Energy increase equals T times increase in surface area. With R 3 =nr 3 , r = R/n 1/3 and A = 4 (n r 2 - R 2 ) = 4 R 2 (n 1/3 -1) . r A = 0.30 mm, r B = 0.90 mm h ∝ 1/r for fixed σ, θ, ρ, g Two identical glass capillaries are used with water at 20 °C ( =0.0728 , N/m , =1000 , kg/m 3 , g=9.8 , m/s 2 ). Tube A has radius 0.30 , mm and tube B has radius 0.90 , mm . Compute the ratio h A/h B and the numerical heights (assume =0 ). cm medium h A / h B and numerical values Measuring surface tension by capillary rise is straightforward. Clean the glass thoroughly so that the contact angle is reproducible and close to its tabulated value. Measure the inner radius of the tube accurately (using traveling microscope or by mercury thread method) and record the steady height difference between the liquid in the tube and the reservoir. Use = g r h 2 . Because h varies inversely with r , even a small error in radius creates a large error in ; careful calibration of r is crucial. Interfacial phenomena in biology and medicine often hinge on the same physics. Alveoli in the lungs are lined with a surfactant that reduces surface tension; otherwise the large Laplace pressure for tiny radii would make small alveoli collapse into larger ones. Capillary action helps transport water in plants through very fine xylem vessels. In the laboratory, microfluidic devices use capillarity for self-filling channels without pumps. Dimensional check: [ ]= N/m = kg ,s -2 , and h= g r has units kg ,s -2 kg ,m -2 ,s -2 m = m . If your computed h is not in meters, re-check r (often mistaken as diameter). remember Beyond simple tubes, the same idea explains wicking in porous materials, rise in paper towels, and the curvature of menisci in thin gaps. In complex geometries, the curvature of a surface at each point sets the pressure jump: P = ( 1 R 1 + 1 R 2 ) where R 1 and R 2 are principal radii of curvature (Young–Laplace equation). For a sphere both radii equal R , giving the familiar formulas above. Temperature dependence is usually monotonic: decreases with T and vanishes at the critical temperature where liquid and vapor phases become indistinguishable. Practically, even a few degrees rise can change capillary heights noticeably, which is why experiments should record temperature and, if needed, correct using tabulated data. Procedure sketch: measuring σ from capillary rise Clean the capillary thoroughly using detergent, distilled water, and finally alcohol; dry without touching the inner wall. Measure inner diameter with a microscope or by forming a mercury thread of known length and mass to infer radius from volume. Immerse the tube vertically in the liquid and wait for the height to stabilize. Measure the vertical height difference h between the meniscus inside the tube and the free surface in the beaker. Note the contact angle θ (for water on clean glass, assume θ≈0° unless contamination is suspected). Compute = g r h 2 and estimate uncertainties, especially from r. When mercury falls in a glass capillary, many students forget to put a negative sign for . The formula itself gives a negative h (meaning fall). Report the magnitude of fall as |h| with a note that the level is depressed. neet-alert Why does a soap film pull more strongly than a single interface? Because the film has two sides in contact with air. When you draw a rectangular wire frame with a movable slider and dip it into soap solution, pulling the slider increases the area of both surfaces. The horizontal pull equals 2T times the length of the moveable edge, not T . Force on a soap-film slider L is the length of the slider inside the film; the factor 2 accounts for the two surfaces. In estimating whether a light needle can float on water, compare the downward weight to the maximum upward force supplied by surface tension at the contact line: approximately F max (total wetting perimeter) for 0 . If the water is contaminated or warm, is reduced and the needle sinks. Contact angle can depend on surface preparation. A greasy or waxed glass surface becomes effectively non-wetting for water, making large and h small or negative. This is why wash basins treated with hydrophobic coatings repel water drops which bead up rather than spread. Dimensional reasoning is a good rescue tool. In h= 2 g r , twice or half of any one parameter scales h accordingly. Proportionality questions are common in NEET: expect options that test whether you recognize h , h , and h 1/( g r) . Boundary cases to sanity-check answers If r 0 , h in the formula, but in reality the tube cannot be thinner than molecular scales; viscosity and evaporation intervene. If =90 , =0 and h=0 (no rise or fall). As R for a droplet/bubble, P ex 0 so large containers have negligible curvature pressure. At very high temperatures close to critical, 0 and capillary action disappears. Numerical shortcuts: convert millimeters to meters early to avoid unit slips; keep g as 9.8 m/s 2 for two-significant-figure answers; for quick estimates, r 2 h gives the column volume and g the weight per unit volume. For bubbles/drops, memorizing P 1/R avoids re-derivations in the exam hall. Surface-tension driven flows (Marangoni effect) occur if varies along a surface (for example, due to temperature or concentration gradients). While dynamic flows are beyond NEET scope, remember that our static formulas apply only after the system stops moving and the contact line settles. Worked-ratio practice: if a capillary with radius r shows rise h , another with radius 3r will show rise h/3 . If the second tube is coated to make =60 while the first has =0 , the ratio becomes h 1/h 2 = ( 0 )/( 60 ) (r 2/r 1) = (1/0.5) 3=6 . Such combined changes are common in multiple-choice questions. Practical corrections: in very narrow tubes, the meniscus is truly curved, so the elevation of the bottom of the meniscus is slightly higher than the center by a fraction of r . Experimental protocols specify whether h is measured to the top of the meniscus or its center; be consistent. When using P ex =2T/R for an air bubble in a liquid, ensure that T is the surface tension of the liquid–air interface, not of the liquid–liquid interface. If the bubble is in a soap solution, use the solution’s T. tip Surface tension is anisotropic only if the interface itself is anisotropic (e.g., liquid crystals). For ordinary isotropic liquids in NEET problems, T is the same in all directions along the surface, so the pull along any line is simply T per unit length regardless of orientation. Is your h positive for wetting and negative for non-wetting? Did you use radius, not diameter, in capillary problems? For soap films, did you include the factor of 2 for two surfaces? Are units consistent (N/m for σ, m for r, kg/m³ for ρ)? Do results make sense in boundary limits (r→0, R→∞)? Quick checkpoints before final answers Force per unit length along a line on a liquid surface; equals surface energy per unit area. T, Surface tension (T or σ) Work stored in creating surface area: W=T , A . E s Surface energy Angle measured inside the liquid between the solid surface and the tangent to the liquid surface. Angle of contact Meniscus Curved surface near a boundary; concave for wetting, convex for non-wetting. Capillary rise Static height change in a narrow tube: h= 2 g r . Pressure inside a curved interface minus outside pressure. Drop: 2T/R ; soap bubble: 4T/R . Excess pressure Cohesion/Adhesion Attraction between like molecules / between different materials. Wetting Case with <90 ; liquid spreads on the solid. >90 ; liquid beads up and avoids the solid surface. Non-wetting General curvature-pressure relation: P = (1/R 1 + 1/R 2) . Young–Laplace equation Recap glossary