Surface Tension & Viscosity Experiments Surface Tension & Viscosity Experiments Liquids behave as if their surfaces are stretched elastic skins, pulling themselves tight. That pull per unit length is surface tension. It explains why water climbs up a thin glass tube (capillarity) and why insects can walk on water. Inside a moving liquid, layers try to slide past each other; internal friction resists that sliding. That resistance is viscosity. Honey flows slowly because it is highly viscous; water flows quickly as it is less viscous. In the lab, we measure surface tension T by observing how high a liquid rises in a thin capillary tube and measure viscosity η by letting a small sphere fall through a viscous liquid and timing its steady motion. Both methods rely on clean, controlled conditions and careful reading of lengths and times. Small mistakes like using diameter instead of radius, or starting the stopwatch before the sphere reaches terminal velocity, create large errors. Mastering these two staple experiments sharpens your measurement skills: understanding forces at a liquid surface and drag forces in slow, laminar flow. Think of surface tension like a stretched rubber sheet over water, and viscosity like friction between sliding decks of playing cards. Surface tension pulls tight; viscosity resists sliding. remember What we measure and why it works Capillary rise works because the liquid surface curves where it touches the tube wall. If the liquid wets the tube (like water on clean glass), the meniscus is concave and the liquid climbs to reduce surface energy. The upward surface-tension pull balances the weight of the raised column. In Stokes’ method, a small smooth sphere falls through a viscous liquid. Initially it accelerates, but soon viscous drag grows until it balances the net weight (weight minus buoyancy), and the sphere then moves with a constant terminal velocity. Measuring that terminal velocity and using Stokes’ law gives the viscosity. Each method is beautiful because the formula directly emerges from a force balance. Surface tension T of a liquid by capillary rise in a thin clean glass tube. Coefficient of viscosity η of a viscous liquid (e.g., glycerin) using terminal velocity of a small sphere (Stokes’ method). Skill: Reading meniscus correctly, eliminating parallax, handling least count, and identifying steady (terminal) motion. Targets in this experiment Surface tension (T) Force per unit length along a line on the liquid surface, acting tangentially and trying to minimize surface area. SI unit: N/m. Contact angle (θ) Angle inside the liquid between the tangent to the liquid surface and the solid wall at the line of contact. For clean glass–water, θ ≈ 0°; for mercury on glass, θ > 90°. Meniscus Curved liquid surface in a tube. Concave for wetting liquids (e.g., water on glass), convex for non-wetting (e.g., mercury on glass). Rise or depression of a liquid in a thin tube due to surface tension and adhesion with the tube material. Capillary action Viscosity (η) Internal friction of a fluid that opposes relative motion between its layers. SI unit: Pa·s. Viscous drag on a small sphere moving slowly in a viscous fluid: F d = 6 r v . Stokes’ law Constant speed reached by a falling sphere when net force becomes zero: weight = buoyancy + viscous drag. Terminal velocity (v) Dimensionless number indicating flow regime. For a sphere: Re = 2 r v . Stokes’ law holds for Re ≲ 1 (laminar). Reynolds number (Re) The capillary-rise method needs a narrow, clean glass tube vertically dipped in the liquid. The inner radius of the tube must be known accurately (via traveling microscope or pre-calibrated capillary). The rise height is read at the meniscus, with the eye placed tangentially to avoid parallax. For water-like liquids on clean glass, the contact angle is nearly zero so 1 . Any contamination (grease, soap) can change θ and ruin the reading. For mercury, the level is depressed because > 90 and < 0 . Capillary rise relation Rise h in a tube of inner radius r for a liquid of density ( ) and surface tension T with contact angle ( ). Calculates surface tension by relating the measured capillary rise height and inner radius to the liquid's density. Capillary formula applies for a narrow vertical tube, static conditions, wetting angle well-defined, and negligible evaporation/temperature gradients. Use inner radius r (not outer radius), and take appropriate to liquid–glass pair. tip Tube is narrow and vertical; meniscus is part of a sphere with constant curvature near the wall. Liquid is static, isothermal; contact angle θ is well-defined and uniform along the circumference. Pressure inside the liquid is hydrostatic: p = p 0 + g z . Weight of the liquid in the column equals the upward vertical component of surface-tension force at equilibrium. Capillary rise: h = 2T g r Only the vertical component T contributes to lifting the column. Approximating the column as a cylinder of height h and radius r (meniscus volume is small). h = 2T g r Surface tension of water T = 0.073 ( N/m ) at room temperature Density of water ( = 1000 , kg/m 3 ) Contact angle ( 0 ) so = 1 Inner radius of tube r = 0.50 mm g = 9.8 ( m/s 2 ) Use h = 2T g r with r in metres. easy Rise h in the tube. Capillary rise of water in a thin glass tube. Reading the meniscus correctly is crucial. For concave menisci (water on clean glass), read the lowest point of the curve aligned with the scale using a microscope or a magnifier. Avoid parallax by keeping the eye’s line of sight perpendicular to the scale. Ensure the tube is clean: rinse with chromic acid or alcohol-acetone mix, then with distilled water, and flame-dry if allowed. A contaminated wall changes θ and thus , directly biasing T. Non-wetting liquids (like mercury in glass) have > 90 so < 0 , leading to a depression instead of a rise. The same formula predicts negative h. For all liquids, the liquid level always rises in a capillary. neet-alert Do not use tube diameter in the capillary formula. r is the INNER RADIUS. Using diameter doubles the denominator and halves the calculated h or T. 1 / radius (1/r) 1/m Plot of h vs 1/r is a straight line through origin; slope gives surface tension. custom Linear relation: h ∝ 1/r for fixed T, ρ, g, and θ. Capillary rise (h) 1/r 2 T cos θ / (ρ g) × (1/r) Slope = 2T cosθ / (ρ g) control control dependent 2D PLOT Capillary rise vs 1/radius h = slope invr invr slope 2T·cosθ/(ρg) Indicative values; actual lab values vary with temperature and purity. Liquid Surface tension T (N/m) at ~20–25°C Contact angle on clean glass Water (distilled) 0.072–0.073 ≈ 0° (good wetting) Glycerin 0.063 ≈ 0° Soap solution (typical) 0.025–0.035 ≈ 0° Mercury 0.46–0.49 > 90° (non-wetting; depression) If you measure rise h to determine T, invert the capillary formula: T = h , g r 2 . Estimate uncertainty as T/T h/h + r/r + / + g/g + ( )/( ) . Since g and ρ are often known well, the dominant errors come from h, r, and θ. Minimizing meniscus reading error and calibrating r precisely are the fastest ways to reduce uncertainty. Use measured h and known r, ρ, g, θ to compute T. Surface tension from rise Use this equation to determine the surface tension by measuring the height of the liquid column in the capillary tube. Surface tension T. Find surface tension from capillary rise data. Use T = h , g r 2 with all quantities in SI. medium Measured rise h = 28.0 mm Inner radius r = 0.40 mm Density ( = 1000 , kg/m 3 ), g = 9.8 ( m/s 2 ) Clean glass–water: = 1 N/m Viscosity by Stokes’ method (terminal velocity) Drop a small, smooth, dense sphere (e.g., steel or glass ball bearing) into a tall jar filled with a viscous liquid (like glycerin). After a short transient, the sphere falls at constant speed. Mark two points a known distance apart on the jar and time the sphere between them after it has reached steady motion. Use the measured terminal velocity v and the densities of the sphere (ρ) and liquid (σ) to compute η via Stokes’ law. Viscous drag on a sphere of radius r moving at speed v in a viscous liquid. Stokes’ drag This formula is valid for calculating the viscous drag force on a small, spherical object moving slowly through a fluid under laminar flow c Viscosity from terminal velocity Derived from force balance at terminal velocity: weight = buoyancy + viscous drag. = 2 r 2 ( - ) g 9 v = 2 r 2 ( - ) g 9 v Laminar, steady flow around a small sphere (Re ≲ 1). Sphere is rigid, smooth, and far from walls (container of effectively infinite extent). Fluid is Newtonian, incompressible, and homogeneous; no slip at the surface. Terminal velocity reached: acceleration ≈ 0. Conditions for Stokes’ law: small sphere, Re ≲ 1 (laminar), start timing only after the sphere has moved several radii and the speed is steady, and keep the sphere well away from the walls and bottom. Control temperature: η decreases with temperature. tip Even for dense spheres, buoyancy B = (4/3) r 3 g is often 10–20% of weight, and it exactly appears in the derivation. Ignoring it overestimates η. Buoyancy is negligible for dense spheres, so it can be ignored in the viscosity formula. Terminal velocity v m/s Plot of v vs r 2 is a straight line through origin; slope yields η. For fixed η, (ρ − σ), and g, v scales linearly with r 2 . custom r 2 m 2 control dependent control r 2 2 r 2 (ρ − σ) g / (9 η) Slope = 2(ρ − σ)g / (9η) 2D PLOT Terminal velocity vs r² v = slope rsq rsq slope 2(ρ−σ)g/(9η) Viscosity η. Viscosity of glycerin using terminal velocity. Use = 2 r 2 ( - ) g 9 v . medium Sphere radius r = 1.0 mm = 1.0 10 -3 , m Sphere density = 7800 , kg/m 3 (steel) Glycerin density = 1260 , kg/m 3 Measured terminal speed v = 1.2 10 -2 , m/s g = 9.8 , m/s 2 Pa·s Start timing only after the sphere reaches terminal velocity. If you time too early, v is smaller, and η comes out too large. Drop the sphere well above the first mark and discard the initial transient zone. neet-alert Finite-container ‘wall effects’ increase the drag above 6 r v . A rough correction for moderate r/R is F d,wall 6 r v ,(1 + 2.4 ,r/R) , where R is the jar radius. This increases the inferred η by roughly the same factor. Better: use a wide jar so r/R ≪ 1 and you can ignore corrections within your experimental uncertainty. r = 1.0 10 -3 , m , R = 1.0 10 -2 , m (so r/R = 0.10) v(measured) = 1.20 10 -2 , m/s = 7800 , kg/m 3 , = 1260 , kg/m 3 Assume F d = 6 r v ,(1+2.4 ,r/R) g = 9.8 , m/s 2 Pa·s and dimensionless Replace v in denominator effectively by v/(1+2.4 r/R), or multiply η by (1+2.4 r/R). Then compute Re = 2 r v / using corrected η. Stokes’ law with corrections and Reynolds number check hard Corrected η and Reynolds number Re. Hard check: Wall correction and laminarity test. WBV balance At terminal fall: W = B + V. Weight down equals Buoyancy + Viscous drag up. Clean the capillary thoroughly; fix it vertically in a stand with a transparent beaker below. Dip the lower end into the liquid and wait for the level to stabilize. Focus a traveling microscope at the meniscus; align cross-wire tangential to the lowest point (for concave). Measure the rise h from the external reservoir level; measure inner radius r separately. Repeat for multiple tubes or remeasure to average; compute T via T = h , g r 2 . Capillary rise: key steps Fill a tall jar with viscous liquid; mark two horizontal lines separated by a known distance s. Drop a small sphere gently along the axis; allow it to pass a few centimeters to reach terminal speed. Start the stopwatch when it crosses the first mark; stop at the second. Repeat and average. Compute terminal velocity v = s/t; calculate η using the known r, ρ, σ, and g. Check Re; if Re ≲ 1 and r/R ≪ 1, Stokes’ assumptions hold. Stokes’ method: key steps h (capillary rise) Traveling microscope / scale 0.01 cm (typ.) Parallax at meniscus Eye level, focus cross-wire at tangent r (inner radius) Microscope / calibration 10 µm (typ.) Using outer diameter Calibrate or use bore gauge t (fall time) Stopwatch 0.1 s (typ.) Starting too soon Discard transient zone s (mark spacing) Scale on jar 1 mm (typ.) Misalignment Use a vertical reference ρ, σ (densities) From handbook Wrong temperature Use correct-temperature tables Quantity Measured by Least count / precision Common error Mitigation Viscosity from terminal velocity of a sphere in a viscous fluid under Stokes’ law. High-yield; check laminarity (Re ≲ 1) and wall effects. Temperature matters: surface tension usually decreases slightly with temperature; viscosity often decreases strongly. Note the lab temperature and, if available, apply tabulated temperature corrections. remember Uncertainty tips: Average several timings to reduce random error; compute standard deviation to estimate timing uncertainty. For capillarity, measure r at multiple orientations and average to reduce ellipticity bias. When reporting T or η, include uncertainty: e.g., = (1.20 0.06) , Pa s , explaining how you propagated errors. Sign conventions and units: Always convert mm to m before substitution. Write densities in kg/m 3 and g in m/s 2 . For surface tension, N/m is standard; for viscosity, Pa·s is standard (1 Pa·s = 1 N·s/ m 2 ). Keep two significant figures in final results unless your data justify more. Edge cases to recognize: Very wide tubes make h tiny—hard to resolve; very narrow tubes can trap air bubbles and roughness dominates. In viscosity, very small spheres give tiny v (timing dominated by reaction time), while very large spheres may increase Re and break Stokes’ regime; also they may induce wall effects if r is not ≪ R. neet-alert Do not confuse distance in the n-th second with total distance in kinematics when estimating terminal velocity onset. Terminal velocity is the speed after transients die out; it is not the average speed from release. Troubleshooting capillary: If the meniscus keeps drifting, check for temperature gradients or air currents. If the rise differs between repeats, inspect for residual grease or for incomplete wetting. If readings are systematically low, re-check r and make sure you are not including wall thickness. Troubleshooting viscosity: If timing varies widely, increase the distance s between marks. If the sphere hugs the wall, use a wider jar or guide the release along the axis. If v increases with distance, you started timing too early; wait longer before the first mark or raise it. Data processing pattern: Record 5–10 time measurements for a given sphere and spacing s; compute each v = s/t, then the mean v and its uncertainty (standard error). If you have several sphere radii, plot v vs r 2 and fit a straight line. From the slope m = 2( - )g/(9 ) , solve for η using all data at once—this reduces random error compared to single-pair calculations. If v = m r 2 from a linear fit, invert to get η. Slope method for viscosity Dimensional sanity checks help avoid formula slips. In h = 2T g r , units are (N/m) / [(kg/ m 3 )(m/ s 2 )(m)] = (N/m) / (N/ m 2 ) = m, good. In = 2 r 2 ( - ) g 9 v , units are ( m 2 )(kg/ m 3 )(m/ s 2 ) / (m/s) = (kg/(m·s)), which is Pa·s. If your computed unit doesn’t reduce correctly, you likely used diameter instead of radius or missed a conversion. Very heavy spheres may increase Re or deform the flow; they also reach higher v, shrinking timing precision advantage. Choose radii to keep Re small and v measurable without violating Stokes’ regime. Using a heavier sphere always improves accuracy. Reporting style: Summarize with a clear statement: “At 25 C , the surface tension of the given liquid by capillary rise is T = (0.060 0.005) , N/m .” and “The viscosity of glycerin at 28 C by Stokes’ method is = (1.20 0.06) , Pa s ; Re = 0.03 .” Include key conditions like temperature, contact angle assumption, and whether wall effects were negligible. Practical contact angle: For water on truly clean glass, θ is very small but rarely exactly zero; if your calculated T is consistently below standard values, suspect θ > 0°, contamination, or that you read the meniscus at the wrong point. For mercury, ensure you read the crest of the convex meniscus and consider the depression from the external liquid level. Meniscus geometry: For better precision, some procedures add a small correction for the meniscus volume (the raised column is not a perfect cylinder). At the NEET-UG level, this correction is usually negligible if r is small and readings are careful. tip Why v vs r 2 is powerful: Measuring several r values spreads data across a wider range of v, making linear regression robust against stopwatch jitter. Also, if the line doesn’t pass near the origin, it hints at systematic issues—wall drag, mis-timed starts, or density mismatch errors. Choice of sphere material: Steel spheres are dense and available in precise sizes; glass spheres reduce density mismatch for very viscous liquids and can lengthen the time window for accurate timing. Always verify sphericity; pitted or non-spherical particles produce erratic drag. Cleaning protocol checklist: Rinse the capillary with detergent, with distilled water, with alcohol/acetone, then with distilled water again; dry without leaving lint. For the viscosity jar, remove previous residues and degas the liquid if bubbles persist. Bubbles on the sphere surface reduce effective density and alter drag. Cross-check with density difference: If (ρ − σ) is small, terminal velocity will be small and sensitive to timing errors. Consider using denser spheres or a liquid with lower σ to improve (ρ − σ) while keeping Re small. Alternatively, increase s (mark separation) to improve time resolution. Beyond NEET: For higher accuracy, one can fit the transient approach to terminal velocity using the time constant = 2 r 2 sphere /(9 ) and verify when steady state is reached. However, in school labs, we rely on a long settling distance before the first timing mark and the Re check. Quick glossary Surface tension Pull per unit length on a liquid surface; minimizes surface area. Angle between liquid surface and wall within the liquid; sets meniscus shape. Contact angle Capillary rise (h) Height a liquid climbs or depresses in a thin tube due to T and θ. Internal friction of a fluid; resistance to flow. Viscosity Terminal velocity Constant speed when net force on a falling sphere becomes zero. Dimensionless indicator of flow regime; small Re implies laminar flow. Reynolds number Re Upthrust equal to the weight of displaced fluid. Buoyancy Stokes’ law Drag on a sphere at low Re: F d = 6 r v .