Math Problem Statement
if you have 31 moles of ice at -20 C and 26.6 moles of liquid water at 100 C, what is the final temperature
Solution
To find the final equilibrium temperature when combining ice at and liquid water at , we need to consider energy transfer. Specifically, we'll apply the principle of conservation of energy, where the heat gained by the ice must equal the heat lost by the water.
Given:
- Mass of ice: 31 moles
- Initial temperature of ice:
- Mass of water: 26.6 moles
- Initial temperature of water:
Constants:
- Heat capacity of ice ():
- Heat capacity of water ():
- Heat of fusion of ice ():
- Molar mass of water:
To find the final temperature, we'll go through the following steps:
- Calculate the energy required to warm the ice from to .
- Calculate the energy required to melt the ice at .
- Consider the cooling of the water from to a lower temperature.
- Determine if the melted ice and water reach thermal equilibrium.
Let's proceed with these calculations.
1. Heat required to warm the ice to :
The formula for heating ice: where:
- is the mass of ice in grams.
- is the specific heat capacity of ice.
- is the temperature change.
Converting moles of ice to grams:
2. Heat required to melt the ice:
The formula for melting ice:
3. Heat released by cooling the water from to :
The formula for cooling water: Converting moles of water to grams: The heat released by cooling water to :
Energy Balance Equation
For energy conservation: Substituting the known values:
Final Answer:
The final temperature of the system will be , as the temperature cannot go below freezing due to phase change considerations, implying that not all the ice melts, and the system stabilizes with ice and water at .
Would you like a detailed explanation of these steps or have any questions?
Related Questions:
- What happens to the energy exchange if more ice is added to the system?
- How would the final temperature change if the initial temperature of the ice were different?
- What is the specific role of the heat of fusion in this scenario?
- Can you explain how specific heat values differ between ice and water?
- How does the conservation of energy apply to phase changes?
Tip: Always consider the phase change temperature as a key factor when dealing with energy transfer problems involving ice and water.
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Math Problem Analysis
Mathematical Concepts
Thermodynamics
Energy Conservation
Phase Change
Formulas
q = m * c * ΔT (heat transfer formula)
q = m * ΔH_f (heat of fusion calculation)
Energy balance equation: q_1 + q_2 = q_3
Theorems
Conservation of Energy
Suitable Grade Level
Grades 11-12
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