Chapter 17: Problem 93
You have 1.50 kg of water at 28.0\(^\circ\)C in an insulated container of negligible mass. You add 0.600 kg of ice that is initially at -22.0\(^\circ\)C. Assume that no heat exchanges with the surroundings. (a) After thermal equilibrium has been reached, has all of the ice melted? (b) If all of the ice has melted, what is the final temperature of the water in the container? If some ice remains, what is the final temperature of the water in the container, and how much ice remains?
Short Answer
Step by step solution
Calculate the energy needed to warm the ice to 0°C
Calculate the energy needed to melt the ice at 0°C
Calculate the energy available from cooling the water to 0°C
Compare energy values to determine if all ice melts
Determine the final temperature and remaining ice mass
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Latent Heat of Fusion
The latent heat of fusion for ice is 334,000 J/kg, meaning each kilogram of ice requires 334,000 joules to fully melt into water. In the given problem, 0.600 kg of ice requires:
- Energy to first reach 0°C from -22°C: calculated separately as it's involved with specific heat capacity.
- Then 200,400 J for melting: \( q_2 = 0.600 \times 334,000 = 200,400 \text{ J} \)
Specific Heat Capacity
In this exercise, two specific heat capacities come into play:
- For ice, it is 2,090 J/kg°C, used for calculating the heat required to bring ice from -22°C to 0°C.
- For water, it is 4,186 J/kg°C, applied for determining the energy release as water cools from 28°C to 0°C.
- \( q_1 = 0.600 \times 2090 \times 22 = 27,588 \text{ J} \)
Thermal Equilibrium
In the problem, thermal equilibrium occurs when the ice and water reach the same temperature — 0°C — because not enough energy is available to melt all the ice.
- The required energy to bring the system to equilibrium conditions is calculated, and excess energy after one component reaches its melting point is used to melt part of the ice.
- Remaining energy is calculated to determine how much ice retains its phase.
Phase Transition
In this exercise's context, we examine the melting of ice into water, a solid-to-liquid transition. The process involves:
- Ice absorbing heat to reach its melting point of 0°C.
- Further absorption of latent heat of fusion for the phase transition at constant temperature.
Phase transitions play a key role in various practical applications, like understanding melting processes in climate systems or designing effective thermal storage solutions.