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Question #42523

Difficulty: 8/10 | 🟢 Free

The Question

A spherical ice nanoparticle of radius $R$ is in phase equilibrium with surrounding liquid water at atmospheric pressure $P_0$ . The surface tension of the ice-water interface is $\gamma$ . Due to surface reconstruction, the density of ice at the boundary layer (where melting occurs) is $\rho_s$ , which differs from the bulk ice density $\rho_i$ . The density of bulk water is $\rho_w$ . The latent heat of fusion per unit mass is $L$ , and the normal melting point at $P_0$ is $T_0$ . Assuming the temperature shift $\Delta T \ll T_0$ , which of the following expressions correctly represents the shift in the melting point $\Delta T$ of the nanoparticle?

Options

A.
$-\frac{2\gamma T_0}{\rho_s L R}$
B.
$\frac{2\gamma T_0 (\rho_s - \rho_w)}{\rho_s \rho_w L R}$
C.
$-\frac{2\gamma T_0}{\rho_i L R}$
D.
$\frac{2\gamma T_0 (\rho_i - \rho_w)}{\rho_i \rho_w L R}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Latent Heat and Volume Change|Excess Pressure (Laplace Law)|Approximations (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the precise application of phase equilibrium thermodynamics (Gibbs-Thomson effect) under curved interfaces.

A common trap is blindly applying the standard Clausius-Clapeyron equation ( $dT/dP = T\Delta V/L$ ), which incorrectly assumes both phases experience the Laplace pressure, leading to the erroneous inclusion of water's density.

Furthermore, it tests the understanding that phase transitions occur at the boundary, meaning the local surface density ( $\rho_s$ ), not the bulk density ( $\rho_i$ ), dictates the chemical potential shift.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Latent Heat and Volume Change
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free