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Heat transfer is the process by which thermal energy moves from one object or substance to another. It occurs whenever there is a temperature difference and continues until the objects reach thermal equilibrium, meaning they are at the same temperature.

There are three main modes of heat transfer:

• Conduction, which occurs in solids where heat is transferred through direct contact.
• Convection, which occurs in fluids (liquids or gases) where heat is carried through the fluid's movement.
• Radiation, where heat is transferred through electromagnetic waves without requiring a medium.

In the exercise, heat transfer occurs when the ice cube is placed in the hot water. The heat moves from the warmer water to the colder ice until they reach the same temperature. Calculations, such as the one in the solution, allow us to find the amount of heat transferred and the new equilibrium temperature.

Specific heat capacity is a property of a material that indicates how much heat energy is required to raise the temperature of a unit mass of the substance by one degree. It is usually denoted by the symbol 'c' and is measured in units like joules per kilogram per kelvin (J/kg·K) or kilojoules per kilogram per Kelvin (kJ/kg·K).

In the context of the exercise, water has a specific heat capacity of \(4.18 kJ/kg.K\). This means that to increase the temperature of one kilogram of water by one degree Celsius, \(4.18 kJ\) of heat is required. The high specific heat capacity of water compared to most other substances explains why water is effective in moderating temperature changes and why it takes a significant amount of heat to change its temperature.

The heat of fusion, often symbolized as \(L_f\), is the amount of heat energy required to change a unit mass of a substance from solid to liquid state without changing its temperature. For water, the heat of fusion is approximately \(334 kJ/kg\).

This is a crucial concept in understanding the melting of ice in our exercise. The 100-gram ice cube needs a certain amount of heat to transition from solid ice to liquid water at the same temperature (\(0^\circ C\)). This phase change requires a substantial amount of energy due to the breaking of hydrogen bonds within the ice structure. The provided solution shows how this heat of fusion is used to calculate the total heat required to melt the ice, and that is a fundamental step in determining the final equilibrium temperature of the combined water and melted ice system.