On a cold winter day, you buy a hot dog from a street...

On a cold winter day, you buy a hot $\operatorname{dog}$ from a street vendor. Into the pocket of your down parka you put the change he gives you: coins consisting of $9.00 \mathrm{~g}$ of copper at $-12.0^{\circ} \mathrm{C}$. Your pocket already contains $14.0 \mathrm{~g}$ of silver coins at $30.0^{\circ} \mathrm{C}$. A short time later, the temperature of the copper coins is $4.00^{\circ} \mathrm{C}$ and is increasing at a rate of $0.500^{\circ} \mathrm{C} / \mathrm{s}$. At this time (a) what is the temperature of the silver coins, and (b) at what rate is it changing? (Neglect energy transferred to the surroundings.)

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Key Concepts

Energy Balance in Transient Heat Transfer

In problems involving time-varying temperatures, the energy balance is applied in differential form. This means equating the rate of change of thermal energy in one body to that in another, which is critical for determining unknown temperatures or rates of temperature change when multiple substances interact.

Rate of Temperature Change

The rate at which a substance's temperature changes is linked to the heat transfer rate and its specific heat capacity and mass. Expressed as dT/dt, it provides a dynamic view of how quickly a system is heating up or cooling down.

Conservation of Energy

This principle states that energy cannot be created or destroyed in an isolated system. In thermal problems, it implies that the heat gained by one part of a system must equal the heat lost by another, allowing for the analysis of heat exchange between substances.

Specific Heat Capacity

Specific heat capacity is the amount of energy required to raise the temperature of a unit mass of a material by one degree. It is essential for relating changes in thermal energy to changes in temperature for different substances.

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