Knife blades are often made of hardened carbon steel. The...

Knife blades are often made of hardened carbon steel. The hardening process is a heat treatment in which the blade is first heated to a temperature of $1346^{\circ} \mathrm{F}$ and then cooled down rapidly by immersing it in a bath of water. To achieve the desired hardness, after heating to $1346^{\circ} \mathrm{F}$, a blade needs to be brought to a temperature below $5.00 \cdot 10^{2}{ }^{\circ} \mathrm{F}$. If the blade has a mass of $0.500 \mathrm{~kg}$ and the water is in an open copper container of mass $2.000 \mathrm{~kg}$ and sufficiently large volume, what is the minimum quantity of water that needs to be in the container for this hardening process to be successful? Assume the blade is not in direct mechanical (and thus thermal) contact with the container, and neglect cooling through radiation into the air. Assume no water boils but reaches $100^{\circ} \mathrm{C} .$ The heat capacity of copper around room temperature is $c_{\text {copper }}=386 \mathrm{~J} /(\mathrm{kg} \mathrm{K}) .$ Use the data in the table below for the heat capacity of carbon steel

Key Concepts

Thermal Equilibrium

Thermal equilibrium occurs when two or more objects in thermal contact exchange heat until they reach a common temperature. Understanding this concept is key to analyzing how and when a heated object will cool down by transferring heat to its surroundings, enabling the prediction of final temperatures after energy exchange.

Heat Sink

A heat sink is a medium that absorbs and dissipates heat from a hotter object. Its capacity to remove heat from the system depends largely on its mass and specific heat capacity. This concept is fundamental when designing systems where controlling temperature is vital, as it determines the efficiency of the cooling process.

Specific Heat Capacity

This is a material property that indicates the amount of energy required to raise the temperature of a unit mass of a substance by one degree. It is central to understanding how different materials respond to heating or cooling, and it determines how much heat one material can absorb or release compared to another. This concept is crucial for calculating energy changes in any thermal process.

Conservation of Energy in Heat Transfer

This principle states that energy cannot be created or destroyed, only transferred or transformed. In thermal systems, the heat lost by a hotter object must equal the heat gained by a cooler one, assuming no losses to the environment. This concept allows us to set up energy balance equations to determine critical quantities like the mass of a cooling medium needed for a thermal process.

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