A 0.30 -m-thick sheet of ice covers a lake. The air...

A 0.30 -m-thick sheet of ice covers a lake. The air temperature at the ice surface is $-15^{\circ}$ C. In five minutes, the ice thickens by a small amount. Assume that no heat flows from the ground below into the water and that the added ice is very thin compared to $0.30 \mathrm{~m}$. Find the number of millimeters by which the ice thickens.

A 0.30 -m-thick sheet of ice covers a lake. The air temperature at the ice surface is $-15^{\circ}$
C. In five minutes, the ice thickens by a small amount. Assume that no heat flows from the ground below into the water and that the added ice is very thin compared to $0.30 \mathrm{~m}$. Find the number of millimeters by which the ice thickens.

Key Concepts

Heat Conduction

Heat conduction is the process by which thermal energy is transferred through a material from regions of higher temperature to regions of lower temperature. In the context of the problem, the temperature difference between the water below and the cold air above drives the conduction of heat through the ice, which is fundamental to understanding how the rate of ice thickening is determined.

Thermal Conductivity

Thermal conductivity is a material property that indicates how efficiently heat is transmitted through a substance. For ice, this property determines how quickly heat is conducted from the water to the colder air, thereby influencing the rate at which the ice can grow in thickness.

Latent Heat of Fusion

The latent heat of fusion is the amount of energy that must be removed from a substance to change its phase from liquid to solid without changing its temperature. In phase change problems like ice formation, this concept is crucial because the energy removed by conduction is used to freeze a layer of water, and the amount of ice formed is directly related to the latent heat released during the phase change.

Energy Conservation in Phase Change

Energy conservation in phase change problems involves equating the rate of thermal energy loss by conduction through a material to the energy needed to change the phase of a substance (in this case, water into ice). This principle allows us to compute the incremental growth of the ice layer by linking the conductive heat loss to the latent heat required for freezing.

Transcript

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