Time for a lake to freeze over. When the air temperature is below 0^∘ C, the water at the surfac

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Time for a lake to freeze over. When the air temperature is...

Time for a lake to freeze over. When the air temperature is below $0^{\circ} \mathrm{C},$ the water at the surface of a lake freezes to form a sheet of ice. If the upper surface of an ice sheet 25.0 $\mathrm{cm}$ thick is at $-10.0^{\circ} \mathrm{C}$ and the bottom surface is at $0.00^{\circ} \mathrm{C},$ calculate the time it will take to add 2.0 $\mathrm{mm}$ to the thickness of this sheet.

Key Concepts

Thermal Conduction

Thermal conduction is the process by which heat energy is transmitted through a material due to a temperature gradient. In problems involving freezing, the rate at which heat is lost through the ice depends on the material's thermal conductivity and the temperature difference across the ice layer.

Fourier's Law

Fourier's Law of heat conduction quantitatively relates the heat flux (the rate of heat transfer per unit area) to the negative gradient of temperature and the thermal conductivity of the material. This law is used to estimate the rate of energy loss through the ice by considering the temperature difference between the top and bottom surfaces.

Latent Heat of Fusion

The latent heat of fusion is the amount of energy that must be removed from a substance in order to change its phase from liquid to solid without changing its temperature. For freezing water, this concept is crucial as it determines the energy required to convert a given mass of water into ice.

Energy Balance in Phase Change

The energy balance in phase change problems involves equating the rate of heat removal by conduction (calculated via Fourier's Law) to the energy that must be removed to freeze a given mass of water (determined by the latent heat of fusion). This balance allows one to calculate the time required for a specified thickness of ice to form.

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