Derive the relation between surface tension and surface energy per unit area.

Surface tension and surface energy per unit area are closely related concepts. Let's derive the relation between them.

Surface Tension:
Surface tension is defined as the force acting per unit length along the boundary of a liquid surface, perpendicular to the line drawn in the liquid. It is denoted by the symbol "γ" and has units of force per unit length (N/m or J/m²).

Surface Energy:
Surface energy, also known as interfacial energy, is the amount of energy required to increase the surface area of a liquid by a unit amount. It is denoted by the symbol "E" and has units of energy per unit area (J/m²).

Derivation:
Consider a liquid droplet of radius "r." When the droplet is in equilibrium, the force acting inward due to surface tension is balanced by the pressure difference across the curved surface. The force due to surface tension is given by the equation:

Force due to surface tension = 2πrγ

The pressure difference across the curved surface is given by the equation:

Pressure difference = 2ΔP = Pinside - Poutside

Where Pinside is the pressure inside the droplet and Poutside is the pressure outside the droplet.

The work done to increase the surface area of the droplet by a small amount, dA, is given by:

dW = Force due to surface tension × dA

dW = (2πrγ) × dA

Since surface energy, E, is defined as the work done per unit area, we can write:

dW = E × dA

E × dA = (2πrγ) × dA

E = 2πrγ

The above equation gives the relation between surface energy per unit area (E) and surface tension (γ) for a liquid droplet of radius "r."

Note: This derivation assumes the droplet to be spherical and in equilibrium. The same relation can be derived for other shapes and interfaces as well, considering the appropriate geometry and equilibrium conditions.