Chapter 17: Problem 97
In a container of negligible mass, 0.0400 kg of steam at 100\(^\circ\)C and atmospheric pressure is added to 0.200 kg of water at 50.0\(^\circ\)C. (a) If no heat is lost to the surroundings, what is the final temperature of the system? (b) At the final temperature, how many kilograms are there of steam and how many of liquid water?
Short Answer
Step by step solution
Write the Heat Exchange Equation
Calculate Heat Released by Steam
Calculate Heat Released When Cooled to Final Temperature
Calculate Heat Absorbed by Initial Water
Equate Heat Released and Absorbed
Determine the Final Quantity of Steam and Water
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Heat Transfer
- **Direction of Heat Flow**: Heat naturally flows from a hotter object to a cooler one. Here, this means that the steam (hotter) will transfer heat to the cooler water.
- **Mechanism**: Heat can be transferred through different methods such as conduction, convection, and radiation. But in this exercise, conduction is the main method because heat is transferred directly between the steam and the water.
- **No Loss Assumption**: It’s assumed that no heat is lost to the surroundings. This is an idealized situation meaning all the heat stays within the system and none escapes into the air or container.
Latent Heat
- **Latent Heat of Vaporization**: For water, the latent heat of vaporization is crucial. It’s the heat necessary to convert water in gaseous state (steam) to liquid.
- **Calculation**: The latent heat of vaporization for water is 2260 kJ/kg. In this exercise, it tells us how much energy is released when steam changes into liquid water.
- **Energy Release**: When steam condenses to water, it releases this latent energy, which is then used to heat the remaining water solution.
Phase Change
- **From Gas to Liquid**: The initial state of steam is gas. When heat is released during condensation, the steam transforms into liquid water.
- **Impact on Temperature**: During this change, the temperature of the steam doesn’t fall until it has turned completely into water. This is because latent heat is involved, maintaining the 100°C temperature until completion of the phase shift.
- **Final Outcome**: After the condensation process completes, the new liquid formed by the condensed steam mixes with the original water, affecting the overall temperature of the system.
Equilibrium Temperature
- **Reaching Equilibrium**: As heat is exchanged between the steam and the water, their initial temperature differences reduce until they balance out.
- **Calculation**: Using the formula: \( Q_{released} = Q_{absorbed} \), where the energy released by the steam is set equal to the energy absorbed by the water, an equilibrium temperature is found.
- **Final Measurement**: The equilibrium temperature tells us how warm the total amount of water becomes after all energy exchanges and the phase change completes. This is achieved without loss of heat to the environment as per the problem's conditions.