Ice starts forming in a lake with water at 0^∘ C, when the atmospheric temperature is -10^∘ C. I

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Ice starts forming in a lake with water at 0^∘ C, when the...

Ice starts forming in a lake with water at $0^{\circ} \mathrm{C}$, when the atmospheric temperature is $-10^{\circ} \mathrm{C}$. If time taken for $1 \mathrm{~cm}$ of ice to be formed is $7 \mathrm{~h}$, the time taken for the thickness of ice to increase from $1 \mathrm{~cm}$ to $2 \mathrm{~cm}$ is (a) $7 h$ (b) less than $7 h$ (c) more than $7 \mathrm{~h}$ but less than $14 \mathrm{~h}$ (d) more than $14 \mathrm{~h}$

Ice starts forming in a lake with water at $0^{\circ} \mathrm{C}$, when the atmospheric temperature is $-10^{\circ} \mathrm{C}$. If time taken for $1 \mathrm{~cm}$ of ice to be formed is $7 \mathrm{~h}$, the time taken for the thickness of ice to increase from $1 \mathrm{~cm}$ to $2 \mathrm{~cm}$ is
(a) $7 h$
(b) less than $7 h$
(c) more than $7 \mathrm{~h}$ but less than $14 \mathrm{~h}$
(d) more than $14 \mathrm{~h}$

Key Concepts

Phase Change and Latent Heat

When a substance undergoes a change of state (for example, from liquid to solid), it releases or absorbs energy known as latent heat. This concept is central in understanding freezing processes because the removal of latent heat governs how quickly the phase change occurs, independent of temperature change in the substance itself.

Thermal Conduction

Thermal conduction is the process by which heat energy is transmitted through a material due to a temperature gradient. It is governed by Fourier’s law, which states that the heat flux is proportional to the temperature gradient and the material’s thermal conductivity. This underlying mechanism is critical in analyzing how heat is transferred from a warmer region to a colder one during processes such as freezing.

Insulation Effects and Thermal Resistance

As a solid layer forms during the freezing process, it acts as an insulator that resists the flow of heat from the warmer substance to the colder environment. This increase in thermal resistance means that as the layer thickens, it takes progressively longer for heat to be removed from the underlying material, thereby slowing down the rate of phase change.

Nonlinear Time Dependence in Freezing

Due to the insulating effect of the already formed solid layer, the time required for further freezing is not linearly proportional to the increase in thickness. Instead, heat conduction through the growing layer typically results in a nonlinear relationship, often quadratic in nature, reflecting that additional growth in the layer requires disproportionately more time as the layer thickens.

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