Equilibrium Temp: Mixing Water & Ice (Solved!)
Hey guys! Ever wondered what happens when you mix hot water and ice? It's not just about getting a lukewarm drink; there's some cool (pun intended!) physics involved. Let's dive into a classic problem: figuring out the final temperature when we throw some ice into hot water. We'll break it down step by step, so you can become a pro at solving these types of problems.
The Problem: Hot Water Meets Ice
Here's the scenario: We've got 200 cm³ of water sitting pretty at 40°C in a super-insulated container (no heat escaping here!). Then, we introduce three ice cubes, each weighing 20 grams and chilling at -20°C. The big question is: What's the final temperature of the water once everything settles down? Let's tackle this physics puzzle together.
Key Concepts: Heat Transfer and Equilibrium
Before we jump into calculations, let's refresh some key concepts. The magic behind this problem is heat transfer. Heat always flows from a warmer object to a cooler one. In our case, the hot water will try to warm up the ice, and the ice will try to cool down the water. This exchange continues until they reach a thermal equilibrium, meaning they both have the same temperature. No more heat flows when they're in sync.
Another crucial idea is the concept of specific heat capacity. This is basically how much energy it takes to raise the temperature of 1 gram of a substance by 1 degree Celsius. Water has a relatively high specific heat capacity, which means it takes a good amount of energy to change its temperature. Ice and liquid water have different specific heat capacities, and we'll need to keep that in mind.
Finally, we can't forget about the latent heat of fusion. This is the energy needed to change a substance from a solid to a liquid (or vice versa) without changing its temperature. In our problem, the ice needs to absorb enough heat to melt into water before it can start warming up to the final temperature.
Breaking Down the Process: A Roadmap to the Solution
To solve this, we'll need to consider all the heat exchanges happening in the system. It might seem complex, but we can break it down into manageable steps:
- Heating the Ice: First, the ice at -20°C needs to warm up to its melting point (0°C). We'll calculate the heat required for this using the specific heat capacity of ice.
- Melting the Ice: Once the ice reaches 0°C, it needs to melt into liquid water at 0°C. This requires energy in the form of latent heat of fusion. We'll calculate this heat separately.
- Heating the Melted Ice: Now we have liquid water at 0°C. This water will warm up to the final equilibrium temperature. We'll use the specific heat capacity of water for this calculation.
- Cooling the Water: The initial hot water at 40°C will cool down to the final equilibrium temperature. Again, we'll use the specific heat capacity of water for this step.
- Heat Balance: The principle of heat balance is our guiding light. The total heat lost by the hot water must equal the total heat gained by the ice (in its various stages). This will give us an equation we can solve for the final equilibrium temperature.
Step-by-Step Calculation: Let's Get Numerical!
Alright, let's crunch some numbers! We'll use the following information:
- Volume of water: 200 cm³ (which is approximately 200 grams since the density of water is about 1 g/cm³)
- Initial temperature of water: 40°C
- Mass of each ice cube: 20 g
- Number of ice cubes: 3
- Total mass of ice: 3 * 20 g = 60 g
- Initial temperature of ice: -20°C
- Specific heat capacity of ice (c_ice): approximately 2.1 J/g°C
- Specific heat capacity of water (c_water): approximately 4.186 J/g°C
- Latent heat of fusion of ice (L_f): approximately 334 J/g
Step 1: Heating the Ice
The heat (Q₁) required to raise the temperature of the ice from -20°C to 0°C is given by:
Q₁ = m_ice * c_ice * ΔT
Where:
- m_ice is the mass of the ice (60 g)
- c_ice is the specific heat capacity of ice (2.1 J/g°C)
- ΔT is the change in temperature (0°C - (-20°C) = 20°C)
Q₁ = 60 g * 2.1 J/g°C * 20°C = 2520 J
Step 2: Melting the Ice
The heat (Q₂) required to melt the ice at 0°C is given by:
Q₂ = m_ice * L_f
Where:
- m_ice is the mass of the ice (60 g)
- L_f is the latent heat of fusion (334 J/g)
Q₂ = 60 g * 334 J/g = 20040 J
Step 3: Heating the Melted Ice
Let's say the final equilibrium temperature is T_f. The heat (Q₃) required to raise the temperature of the melted ice (now water) from 0°C to T_f is:
Q₃ = m_ice * c_water * (T_f - 0°C)
Q₃ = 60 g * 4.186 J/g°C * T_f
Q₃ = 251.16 * T_f J
Step 4: Cooling the Water
The heat (Q₄) lost by the original water as it cools from 40°C to T_f is:
Q₄ = m_water * c_water * (40°C - T_f)
Where:
- m_water is the mass of the water (200 g)
- c_water is the specific heat capacity of water (4.186 J/g°C)
Q₄ = 200 g * 4.186 J/g°C * (40°C - T_f)
Q₄ = 837.2 * (40°C - T_f) J
Step 5: Heat Balance
Now for the grand finale! The heat gained by the ice must equal the heat lost by the water:
Q₁ + Q₂ + Q₃ = Q₄
Plugging in our values:
2520 J + 20040 J + 251.16 * T_f J = 837.2 * (40°C - T_f) J
Simplifying the equation:
22560 + 251.16 * T_f = 33488 - 837.2 * T_f
Combining the T_f terms:
1088.36 * T_f = 10928
Solving for T_f:
T_f = 10928 / 1088.36
T_f ≈ 10.04 °C
The Answer: Equilibrium at Around 10°C
So, after all that calculation, we find that the final equilibrium temperature is approximately 10°C. That's option A in our multiple-choice list!
Why This Matters: Real-World Applications
Understanding heat transfer and equilibrium isn't just about acing physics exams. It's everywhere in the real world! Think about:
- Cooking: Getting the right temperature for cooking food involves heat transfer principles. Whether you're boiling water, baking a cake, or searing a steak, understanding how heat moves is key.
- Climate Control: Heating and cooling systems in our homes and buildings rely on heat transfer. Insulation, ventilation, and the efficiency of our appliances are all related to these concepts.
- Engineering: From designing engines to building bridges, engineers need a solid grasp of heat transfer to ensure safety and efficiency.
- Weather: The Earth's climate is driven by heat transfer from the sun, and understanding these processes helps us predict weather patterns and climate change.
Practice Makes Perfect: More Problems to Try
Want to become a true heat transfer master? Try solving similar problems with different values. Change the amount of water, the initial temperatures, or the mass of the ice. See how the final temperature changes! You can also explore more complex scenarios, like adding multiple substances or considering heat loss to the environment (though that makes the math a bit trickier!).
The more you practice, the more comfortable you'll become with these concepts. And who knows, maybe you'll even impress your friends with your newfound physics skills!
Conclusion: Physics is Awesome!
So, there you have it! We've successfully tackled a heat transfer problem and found the equilibrium temperature. By breaking it down step-by-step and understanding the underlying concepts, we can solve even the trickiest physics puzzles. Keep exploring, keep questioning, and remember that physics is all around us!
