Derive surface tension and capillarity

Surface tension

Surface tension $\sigma$ is the tensile force per unit length acting along the free surface of a liquid (N/m), due to cohesion between molecules.

Liquid droplet of diameter $d$: the pressure force over the cut section balances the surface-tension force round its circumference:

$$\Delta p\,\frac{\pi d^{2}}{4}=\sigma\,\pi d\ \Rightarrow\ \Delta p=\frac{4\sigma}{d}$$

For a soap bubble (two surfaces), $\Delta p=\dfrac{8\sigma}{d}$.

Capillarity

In a vertical tube of diameter $d$ dipped in a liquid with contact angle $\theta$, the upward component of the surface-tension force round the circumference supports the weight of the raised column of height $h$:

$$\sigma\cos\theta\,(\pi d)=\rho g\,\frac{\pi d^{2}}{4}\,h$$ $$h=\frac{4\sigma\cos\theta}{\rho g d}$$

For water in glass $\theta\approx0^\circ$ (rise); for mercury $\theta\approx130^\circ$ so $h$ is negative (depression).