The heat evolved for the rise of water when one end of the...

The heat evolved for the rise of water when one end of the capillary tube of radius $r$ is immersed vertically into water, is: (Assume surface tension $=T$ and density of water to be $\rho$ ) (a) $\frac{2 \pi T}{\rho g}$ (b) $\frac{\pi T^{2}}{\rho g}$ (c) $\frac{2 \pi T^{2}}{\rho g}$ (d) none of these

The heat evolved for the rise of water when one end of the capillary tube of radius $r$ is immersed vertically into water, is: (Assume surface tension $=T$ and density of water to be $\rho$ )
(a) $\frac{2 \pi T}{\rho g}$
(b) $\frac{\pi T^{2}}{\rho g}$
(c) $\frac{2 \pi T^{2}}{\rho g}$
(d) none of these

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Key Concepts

Capillary Action

Capillary action is the phenomenon where a fluid rises or falls in a narrow tube due to the interplay between adhesive forces (between the fluid and the tube material) and cohesive forces (within the fluid itself). This effect is fundamental in explaining how liquids can move against gravity in small channels and is governed by the balance between surface tension and gravitational forces.

Surface Tension

Surface tension is the property of a liquid that causes its surface to behave like a stretched elastic membrane. It is quantified as the energy per unit length or force per unit line and arises from the cohesive forces between liquid molecules. In capillary phenomena, surface tension is key to the formation of the meniscus and the subsequent lifting of the liquid within the tube.

Gravitational Potential Energy

Gravitational potential energy in the context of capillary rise refers to the energy required to raise a column of liquid against the force of gravity. As the liquid rises in a capillary tube, its center of mass is elevated, and the work done or energy change is calculated based on the volume of the liquid raised, the gravitational acceleration, and the height through which the center of mass is lifted. This energy is balanced by the change in surface energy due to capillarity.

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