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Question #42572

Difficulty: 8/10 | 🟢 Free

The Question

A capillary tube of radius $r$ is partially immersed in a liquid of density $\rho$ and surface tension $\gamma$ . The liquid rises to form a concave meniscus with a contact angle of $0^\circ$ . The entire system is in thermodynamic equilibrium at a constant temperature $T$ . Let $\Delta s_0$ be the specific entropy of vaporization (per unit mass) of the liquid at a flat surface at temperature $T$ . Assuming the vapor behaves as an ideal gas and the liquid is perfectly incompressible, which of the following expressions represents the specific entropy of vaporization, $\Delta s$ , at the curved meniscus?

Options

A.
$\Delta s_0 + \frac{2\gamma}{\rho r T}$
B.
$\Delta s_0 - \frac{2\gamma}{\rho r T}$
C.
$\Delta s_0 + \frac{\gamma}{\rho r T}$
D.
$\Delta s_0 - \frac{\gamma}{\rho r T}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Entropy and Reversibility|Latent Heat and Volume Change|Capillary Rise Mechanics

Why This Question Is Tricky

Common Mistakes

This question targets the intersection of capillary mechanics and thermodynamics, specifically the Kelvin equation and phase equilibrium.

Students often treat capillary rise as a purely mechanical phenomenon (balancing hydrostatic pressure and surface tension) and ignore the thermodynamic coupling of the meniscus curvature to the liquid's chemical potential and entropy.

The trap is assuming that the entropy or enthalpy of vaporization remains constant regardless of surface curvature.

By requiring the balance of Gibbs free energy ( $\Delta g = 0$ ) and recognizing that the liquid's pressure drop ( $\Delta P = -2\gamma/r$ ) decreases its enthalpy, students must deduce that the specific entropy of vaporization must increase to maintain equilibrium.

This prevents the 'perpetual motion' error typical in novice capillary models and tests a deep understanding of Maxwell relations applied to phase boundaries.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Entropy and Reversibility
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free