capillary rise

A liquid of density $\rho$ and surface tension $\sigma$ rises in a capillary of inner radius $r$ to a height$$\begin{array}{c}h=\frac{2\sigma \cos \theta }{\rho gr}\end{array}$$where $\theta$ is the contact angle made by the liquid meniscus with the capillary’s surface.

The liquid rises due to the forces of adhesion, cohesion, and surface tension. If adhesive force (liquid-capillary) is more than the cohesive force (liquid-liquid) then liquid rises as in case of water rise in a glass capillary. In this case, the contact angle is less than 90 deg and the meniscus is concave. If adhesive force is less than the cohesive force then liquid depresses as in case of mercury in a glass capillary. In this case, the contact angle is greater than 90 deg and the meniscus is convex.

The formula for capillary rise can be derived by balancing forces on the liquid column. The weight of the liquid ($\pi r^{2}h\rho g$) is balanced by the upward force due to surface tension ($2\pi r\sigma \cos \theta$). This formula can also be derived using pressure balance.

The capillary rise experiment is used to measure the surface tension of a liquid.