Surface Tension and Capillarity
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Direct answer
Surface tension S is force per unit length on a liquid surface (also energy per unit area, both numerically equal in N/m), roughly 0.073 N/m for water and 0.48 N/m for mercury at room temperature, and it exists because surface molecules are pulled inward, so the surface contracts like stretched film. Consequences with formulas: excess pressure inside a liquid drop is 2S/r, inside a soap bubble 4S/r (two surfaces), and a soap film on a loop is pierced by 2S per contact line. Capillary rise follows Jurin's law, h = 2 S cos(theta)/(r rho g), positive for water in glass (theta near 0 degrees) and negative — capillary depression — for mercury in glass (theta about 140 degrees).
What you must remember
- Two equal definitions: S = F/l = dW/dA; water at 20 degrees is about 0.073 N/m and falls linearly with temperature to zero near the boiling point.
- Excess pressure ladder: liquid drop 2S/r; soap bubble 4S/r; air bubble in liquid 2S/r; cylindrical jet or tube S/r — the denominator is the radius of curvature, the multiplier counts the surfaces.
- Angle of contact: water–glass nearly 0 degrees (wetting), mercury–glass about 140 degrees (non-wetting); detergents reduce both S and theta, which is how they clean.
- Capillary rise (Jurin's law): h = 2 S cos(theta)/(r rho g); halving the tube radius doubles the rise; for water with theta = 0, h r = 2 S/(rho g) is constant along the same liquid.
- Energy of a film: blowing a soap bubble of radius r needs surface energy 8 pi r^2 S (two surfaces); splitting one drop into n drops conserves volume, shrinks each radius by n^(1/3), and multiplies total surface energy by n^(1/3).
- Needle on water: floats by surface tension although denser than water — the maximum weight supportable is about 2 S l for a wire of length l before the film breaks.
- Temperature and impurity: surface tension falls with temperature and with surface-active impurities like detergent; dissolved salts raise it slightly.
- Pattern note: Main asks the excess-pressure and capillary-rise numericals; Advanced asks energy accounting in drop coalescence and splitting.
One drop split into eight
Take a water drop of radius 3 mm and spray it into 8 identical droplets. Volume conservation fixes the new radius: 8 × (4/3)pi r^3 = (4/3)pi R^3, so r = R/2 = 1.5 mm. Surface energy before is 4 pi R^2 S; after, 8 × 4 pi (R/2)^2 S = 2 × 4 pi R^2 S. The energy has doubled, and the difference is the work the spraying hand supplied. The general law for n droplets: radii shrink by n^(1/3), surface area grows by n^(1/3), and the work done is 4 pi R^2 S (n^(1/3) − 1).
The capillary formula rewards the same clean bookkeeping. A tube of radius 0.5 mm dipped in water (S = 0.073, theta ~ 0) rises h = 2 × 0.073/(0.0005 × 1000 × 9.8) = 0.0298 m, about 3 cm; halve the radius and 6 cm appears.
Where students slip
The 2S/r versus 4S/r distinction drops marks every session: a soap bubble has an inner and an outer surface, a liquid drop has one — candidates who count surfaces by intuition instead of by inspection of the film misapply it. Second, cos(theta) is signed: for mercury in glass it is negative and capillary depression follows; quoting a positive rise from the formula without the angle's sign is the error. Third, in energy questions students divide area change by volume change casually; the work done against surface tension is S × change in area, full stop. Finally, radius versus diameter in Jurin's law — the r is the capillary's inner radius, and using the bore diameter halves every answer.
Frequently asked questions
Why does a soap bubble have double the excess pressure of a drop?
A bubble carries two free surfaces, inner and outer, each contributing 2S/r, giving 4S/r total; a drop exposes only one surface.
What determines whether a liquid rises or falls in a capillary?
The contact angle: h = 2 S cos(theta)/(r rho g) is positive for wetting liquids (water in glass) and negative for non-wetting (mercury in glass, theta about 140 degrees).
What happens when several small droplets coalesce into one big drop?
Total surface area and surface energy decrease, and the energy released slightly warms the drop — radius grows by n^(1/3) of the small radius.
How does a detergent help in cleaning?
It lowers both S and the contact angle, letting water spread over and penetrate the fabric — two effects working jointly.
Why can a steel needle float on water although steel is denser?
The water surface, behaving like a stretched membrane, exerts surface tension along the contact line sufficient to balance the needle's weight until the film is penetrated.