Chapter 17: Problem 94
A thirsty nurse cools a 2.00-L bottle of a soft drink (mostly water) by pouring it into a large aluminum mug of mass 0.257 kg and adding 0.120 kg of ice initially at -15.0\(^\circ\)C. If the soft drink and mug are initially at 20.0\(^\circ\)C, what is the final temperature of the system, assuming that no heat is lost?
Short Answer
Step by step solution
Identify Given Information
Calculate Heat Exchange by Ice to 0°C
Calculate Heat Required for Melting Ice
Calculate Heat Lost by Water and Mug
Equate Heat Gained and Lost
Verify the Assumptions
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Heat Transfer
The motto of heat transfer is very much like that of energy: "Energy cannot be created or destroyed, only transferred." This transfer aims at establishing thermal equilibrium, where temperatures level out.
There are three modes of heat transfer: conduction, convection, and radiation. In our context, conduction primarily occurs as the mug, ice, and drink share energy until they reach a uniform temperature.
Understanding these concepts helps explain why heat moves from the soft drink and mug, which lose heat, to the ice, which gains heat.
Specific Heat Capacity
For example, in this exercise, water has a specific heat capacity of 4.18 J/g°C. This means it takes 4.18 Joules of energy to raise the temperature of one gram of water by one degree Celsius.
Similarly, aluminum, with its specific heat capacity of 0.897 J/g°C, requires less energy per gram per degree Celsius change. The differences in specific heat capacities play a crucial part in how quickly or slowly objects warm up or cool down during heat transfer.
- Higher specific heat capacity means a substance stores more heat for a temperature change.
- Metals generally have lower specific heat capacities compared to water, meaning they heat up and cool down more quickly.
Latent Heat of Fusion
In this exercise, the ice at 0°C needs 334 J/g to melt into liquid water. Importantly, even though the ice absorbs a large amount of energy, its temperature does not increase until all ice has melted. This unique aspect is what distinguishes latent heat from specific heat.
- Latent heat is only relevant during phase changes.
- No temperature change occurs during the phase change despite heat absorption.
Melting Ice Process
Once at 0°C, the ice undergoes a phase transformation. To become liquid water, it absorbs latent heat—energy that’s necessary but doesn’t increase temperature. After complete melting, any additional heat changes the temperature of the resulting water.
- Energy is used in two stages: warming to melting point, then phase transition.
- Only after all ice has melted does any temperature rise in liquid water occur.