(III) Approximately how long should it take 11.0 kg of ice at 0^∘ C to melt when it is placed in

step 1 For this problem, since we're dealing with heat conduction, the equation for heat conducted thro

(III) Approximately how long should it take 11.0 kg of ice...

(III) Approximately how long should it take 11.0 $\mathrm{kg}$ of ice at $0^{\circ} \mathrm{C}$ to melt when it is placed in a carefully sealed Styrofoam ice chest of dimensions $25 \mathrm{cm} \times 35 \mathrm{cm} \times 55 \mathrm{cm}$ whose walls are 1.5 $\mathrm{cm}$ thick? Assume that the conductivity of Styrofoam is double that of air and that the outside temperature is $32^{\circ} \mathrm{C} .$

Key Concepts

Thermal Resistance and Insulation

Thermal resistance quantifies a material’s opposition to heat flow, and it depends on factors such as the material’s thermal conductivity and its thickness. Insulating materials, like Styrofoam, have high thermal resistance, meaning they slow down the rate of heat transfer. Understanding the thermal resistance is key in determining how effectively the insulation reduces the heat entering or leaving a system, thereby influencing the rate at which processes like melting occur.

Energy Balance in Phase Changes

The concept of energy balance involves equating the heat entering a system with the energy required for a phase change, alongside any losses or gains. In the context of melting, the heat conducted through the walls of an insulated container must eventually match the energy needed to convert ice into water. This balance provides a framework to understand and calculate the timing of phase transitions under given thermal conditions.

Fourier's Law of Thermal Conduction

This principle governs how heat energy moves through a material when there is a temperature difference. It states that the heat transfer rate is proportional to the thermal conductivity of the material, the surface area through which the heat flows, and the temperature difference, and inversely proportional to the thickness of the material. In problems like these, Fourier’s law is fundamental in calculating how quickly heat can penetrate insulating barriers, such as the walls of an ice chest.

Latent Heat of Fusion

Latent heat of fusion is the amount of energy required to change a substance from a solid to a liquid at its melting point without changing its temperature. When analyzing the melting of ice, this concept is essential because it determines the total energy needed to melt a given mass of ice once the temperature threshold is met. This energy requirement directly impacts the time scale of the phase change process.

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