On a cold winter day you buy roasted chestnuts from a street...

On a cold winter day you buy roasted chestnuts from a street vendor. You put the change he gives you - coins constituting 9.00 $\mathrm{g}$ of copper at $-12.0^{\circ} \mathrm{C}$ -into the pocket of your down parka. Your pocket already contains 14.0 $\mathrm{g}$ of silver coins at $30.0^{\circ} \mathrm{C}$ . After a short time interval, 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 moment, (a) what is the temperature of the silver coins and $(\mathrm{b})$ at what rate is it changing?

Key Concepts

Conservation of Energy in Thermal Equilibration

This principle states that in an isolated thermal system the total energy remains constant. When two bodies at different temperatures come into contact, the heat lost by the warmer body is exactly equal to the heat gained by the cooler body, until thermal equilibrium is reached. This idea is central for setting up equations that balance the energy exchanged between objects with different masses and specific heat capacities.

Specific Heat Capacity

Specific heat capacity is a material-specific property that indicates the amount of energy required to raise the temperature of a unit mass of a substance by one degree. It plays a crucial role in predicting how different materials respond to heat transfer; substances with higher specific heat capacities change temperature more slowly when absorbing or releasing energy.

Differential Rate of Temperature Change and Heat Transfer Dynamics

When analyzing transient thermal processes, the instantaneous rate of temperature change is important. By applying the differential form of energy conservation, one sets up relationships that relate the time derivatives of the temperatures of different bodies. These relationships incorporate the masses and specific heat capacities of the bodies, and are used to determine how quickly each is gaining or losing heat as they exchange energy.

Transcript

Please give Ace some feedback