Thermal Energy Analysis: Melted Ice and Warm Water Fusion Heat Calculation

In this article, we will explore the process of mixing ice and warm water, focusing on the fusion heat involved. The objective is to determine the amount of heat required to melt the ice and bring the resulting mixture to a final temperature. We will apply the principle of the conservation of energy, illustrating how the heat lost by the warm water is equal to the heat gained by the melting ice.

Given Parameters and Constants

Mass of ice: 0.5 kg Initial temperature of ice: 0 °C Mass of water: 0.30 kg Initial temperature of water: 50 °C Final temperature: 6.7 °C Specific heat of water: 4184 J/kg·°C Latent heat of fusion of ice: 334000 J/kg

Calculations and Steps

Heat Lost by the Warm Water

The heat lost by the warm water can be calculated using the formula:

Qlost mwater × cwater × (Twater - Tf)

Substituting the given values:

Qlost 0.30 kg × 4184 J/kg·°C × (50 °C - 6.7 °C)

Performing the calculation:

Qlost 0.30 × 4184 × 43.3 ≈ 5434.56 J

Heat Gained by the Ice

The heat gained by the ice involves two stages:

Heat to melt the ice at 0 °C. Heat to warm the melted ice water from 0 °C to 6.7 °C.

The total heat gained is given by:

Qgained mice × Lf mice × cwater × (Tf - Tice)

Substituting the given values:

Qgained 0.5 kg × 334000 J/kg 0.5 kg × 4184 J/kg·°C × (6.7 °C - 0 °C)

Performing the calculation:

Qgained 167000 J 0.5 × 4184 × 6.7 ≈ 167000 J 14028.4 J ≈ 181028.4 J

Conclusion

Setting the heat lost equal to the heat gained:

Qlost Qgained

5434.56 J 181028.4 J

We observe that the heat gained by the ice is much greater than the heat lost by the water. Therefore, all the heat from the warm water is used to melt the ice and increase the temperature of the melted ice to the final temperature.

Thus, the fusion heat involved in this process is approximately:

167000 J

This detailed analysis confirms the principle of the conservation of energy in the context of heat transfer between ice and warm water. Understanding this process is crucial in fields such as thermal engineering, chemical processes, and environmental studies.