The lower end of a capillary tube of radius r is placed...

The lower end of a capillary tube of radius $r$ is placed vertically in water. Then with the rise of water in the capillary, heat evolved is (a) $+\frac{\pi^{2} r^{2} h^{2}}{J} d g$ (b) $+\frac{\pi r^{2} h^{2} d g}{2 J}$ (c) $-\frac{\pi r^{2} h^{2} d g}{2 J}$ (d) $-\frac{\pi r^{2} h^{2} d g}{J}$

The lower end of a capillary tube of radius $r$ is placed vertically in water. Then with the rise of water in the capillary, heat evolved is
(a) $+\frac{\pi^{2} r^{2} h^{2}}{J} d g$
(b) $+\frac{\pi r^{2} h^{2} d g}{2 J}$
(c) $-\frac{\pi r^{2} h^{2} d g}{2 J}$
(d) $-\frac{\pi r^{2} h^{2} d g}{J}$

Key Concepts

Capillary Action

Capillary action refers to the ability of a liquid to flow in narrow spaces without the assistance of external forces, and sometimes even in opposition to gravity. This phenomenon occurs due to the combination of adhesive forces between the liquid and the tube wall, and cohesive forces within the liquid, often resulting in a rise or fall of the liquid in the confined space. In problems involving capillarity, the balance between surface tension forces and gravitational forces is central to understanding the equilibrium height of the liquid column.

Surface Energy Changes

In capillary phenomena, surface energy changes play a key role. When a liquid rises in a capillary tube, the areas of contact between the liquid and the solid increase, which corresponds to a change in the interfacial energy. These changes in surface energies must be accounted for in the energy balance of the system, and they are often related to the work done by or against surface tension forces during the process.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by a body due to its position in a gravitational field. In the context of capillary rise, as the liquid ascends against gravity, its gravitational potential energy increases. Balancing this increase with the decrease in surface energy (or the work done by surface tension) is essential for determining the net energy changes in the system, including any heat evolved or absorbed.

Energy Conservation in Capillary Phenomena

The principle of energy conservation is fundamental when analyzing capillary phenomena. The energy supplied or released during the liquid rise is linked to changes in both gravitational potential energy and surface energy. The difference in these energy contributions must be accounted for in the form of mechanical work, and any imbalance manifests as heat evolved or absorbed. This thermodynamic perspective provides insight into the underlying processes that occur as the liquid moves within the capillary tube.

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