A 0.075-kg ice cube at 0.0^∘ C is dropped into a Styrofoam cup holding 0.33 kg of water at 14^∘

step 1 High approval here it is given. Ice of mass 0 .075 KG at initial temperature 0 degrees Celsius W

A 0.075-kg ice cube at 0.0^∘ C is dropped into a Styrofoam...

A $0.075-\mathrm{kg}$ ice cube at $0.0^{\circ} \mathrm{C}$ is dropped into a Styrofoam cup holding $0.33 \mathrm{kg}$ of water at $14^{\circ} \mathrm{C} .$ (a) Find the final temperature of the system and the amount of ice (if any) remaining. Assume the cup and the surroundings can be ignored. (b) Find the initial temperature of the water that would be enough to just barely $y$ melt all of the ice.

Key Concepts

Conservation of Energy

This principle states that energy cannot be created or destroyed, only transferred. In thermal problems, this means that the total heat lost by warmer substances must equal the total heat gained by cooler substances in an isolated system. Applying this concept allows us to relate the loss of heat from the water to the gain of heat in the ice, including both its temperature change and any phase change.

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. This concept is crucial when dealing with ice melting, as it explains how energy is consumed to overcome molecular bonds rather than increasing the temperature of the substance.

Specific Heat Capacity

Specific heat capacity is the amount of energy required to raise the temperature of one kilogram of a substance by one degree Celsius. This concept is used to calculate how much energy is needed to change the temperature of a given mass of water or ice, helping to determine the energy exchange during heating or cooling processes.

Thermal Equilibrium

Thermal equilibrium is the state in which all parts of a system reach the same temperature, meaning there is no net heat flow between them. In problems involving mixing substances at different temperatures, the end condition is usually thermal equilibrium, and it is found by ensuring that the energy gained by one component is equal to the energy lost by the other.

Phase Change Calculations

Phase change calculations involve accounting for the energy required to change the state of a substance (e.g., melting, freezing) without a temperature change. This requires using the latent heat values alongside the specific heat capacities to fully account for energy exchanges when a substance undergoes a phase transition, such as ice melting into water.

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