A pond of water at 0^∘ C is covered with a layer of ice 4.00...

A pond of water at $0^{\circ} \mathrm{C}$ is covered with a layer of ice $4.00 \mathrm{cm}$ thick. If the air temperature stays constant at $-10.0^{\circ} \mathrm{C},$ what time interval is required for the ice thickness to increase to $8.00 \mathrm{cm} ?$ Suggestion: Use Equation 20.16 in the form $$\frac{d Q}{d t}=k A \frac{\Delta T}{x}$$ and note that the incremental energy $d Q$ extracted from the water through the thickness $x$ of ice is the amount required to freeze a thickness $d x$ of ice. That is, $d Q=$ Led dx, where $\rho$ is the density of the ice, $A$ is the area, and $L_{f}$ is the latent heat of fusion.

Key Concepts

Fourier's Law of Heat Conduction

Fourier's law describes the rate of heat transfer through a material due to a temperature gradient. It states that the heat flux is proportional to the negative gradient of temperatures and the materials' thermal conductivity. This principle is key when analyzing how thermal energy is conducted through the ice layer from the water at 0°C to the colder air, emphasizing the role of the ice thickness as a barrier to heat flow.

Latent Heat of Fusion

Latent heat of fusion is the energy released or absorbed when a substance changes phase between solid and liquid states at a constant temperature. In the context of ice growth, it represents the amount of energy that must be removed from the water to convert it into ice. This concept is critical for linking the energy transfer due to conduction with the incremental increase in ice thickness.

Temperature Gradient

The temperature gradient is the difference in temperature over a certain distance, which drives the rate of heat transfer in conductive processes. In this scenario, the gradient exists between the warm water (0°C) and the cold air (-10°C), and it decreases as the ice thickens. Understanding this gradient is essential for determining how quickly energy is extracted from the water to facilitate further freezing.

Differential Analysis in Phase Change

The differential analysis involves establishing a relationship between an infinitesimal amount of heat removed (or energy transferred) and an infinitesimal change in ice thickness. By equating the rate of heat conduction to the energy requirement for freezing a small layer of water (taking into account the latent heat and density), one can derive a differential equation whose integration yields the time required for a specified increase in ice thickness.

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