Count the Surfaces, Then Multiply Tension by Area

A film (a soap film on a frame, a liquid film on a slider) has two free surfaces, so the force it exerts on a slider of length l is 2 T l and the work to enlarge it by ΔA is 2 T ΔA. A film holding a weight of 1.5 x 10⁻² N on a 60 cm slider has 2 T x 0.6 = 1.5 x 10⁻², so T = 0.0125 N/m. A soap film of surface tension 3 x 10⁻² N/m stretched from 5 x 5 cm² to 6 x 5 cm² has ΔA = 5 x 10⁻⁴ m², so the work is 2 x 3 x 10⁻² x 5 x 10⁻⁴ = 30 x 10⁻⁶ J.

A drop has one free surface, so its surface energy is 4π R² T. When n drops of radius r merge into one drop of radius R, volume is conserved: n r³ = R³. Two drops of radius R that merge give a drop of radius 2^(1/3) R, so the ratio of total surface energy before and after is 2 : 2^(2/3) = 2^(1/3) : 1. For n drops of radius r merging into a drop of radius R and volume V, the energy released is 3 V T (1/r - 1/R). A drop of diameter D that splits into 27 equal drops has each radius one third of the original, so the area triples and the energy supplied is 2π T D². Merging releases energy; splitting needs energy.

A soap bubble has two surfaces, inside and outside, so its surface energy is 2 x 4π R² T. When the volume of a bubble increases by 700%, the volume becomes 8 times the original, so the radius doubles and the area becomes 4 times: the work is 2 T x 4π (4R² - R²) = 24π R² T.

Insects and spiders walk on water because the surface behaves like a stretched elastic membrane, and the surface tension supports their weight. Detergents help remove grease because they lower the oil-water surface tension and hold the dirt suspended in the water as small droplets, so a statement that detergent raises the surface tension is wrong, and raising the viscosity is not the reason.