Surface energy (S.E.) and surface tension (S.T.)

SURFACE ENERGY. Inside a liquid, a molecule is pulled equally in all directions by its neighbours; but near the surface, a molecule is pulled predominantly downward (only by molecules below it), so the whole surface layer of the liquid is effectively being pulled inward at all times, and the liquid surface naturally tends toward the smallest possible area. To INCREASE the surface area -- by bringing molecules up from the interior to the surface -- work must be done against this inward-pulling force, and the amount of work done is stored as potential energy in the newly created surface: this is why surface molecules always carry greater potential energy than molecules deeper in the liquid. SURFACE ENERGY is defined as this excess energy per unit area of the liquid's free surface, or equivalently the work done in increasing the surface area per unit increase in that area: $\text{S.E.}=\dfrac{\Delta W}{\Delta A}$, expressed in units of J m$^{-2}$ (equivalently N m$^{-1}$). SURFACE TENSION is separately defined as the force per unit length of the liquid's surface, $T=\dfrac{F}{l}$, with SI unit N m$^{-1}$ and dimensional formula $MT^{-2}$. RELATION BETWEEN SURFACE TENSION AND SURFACE ENERGY. Consider a rectangular wire frame ABCD holding a horizontal soap film, with side AB free to slide. The soap film, having TWO free surfaces (front and back), pulls AB inward with a force $F=(2T)l$ due to surface tension. If AB is moved outward by a small distance $\Delta x$ to a new position A'B', the work done against this inward force is $\text{Work}=F\times\Delta x=(2Tl)(\Delta x)$, while the increase in the film's total surface area (counting both faces) is $\Delta A=(2l)(\Delta x)=2l\,\Delta x$. Therefore, $\text{surface energy}=\dfrac{\text{work done}}{\text{increase in surface area}}=\dfrac{2Tl\,\Delta x}{2l\,\Delta x}=T$ -- proving that the surface energy per unit area of a liquid surface is NUMERICALLY EQUAL to its surface tension, which is why the two quantities are so often used interchangeably in practice even though they are conceptually disti …