Cup A contains 100 grams of water at 0 C and cup B contains 200 grams of water at 50 C.
The contents of the two cups are mixed together in an insulated container (no heat can leak in or out).
What is the final temperature of the water in the container?
- 50 C
- Lower than 0 C
- Higher than 50 C
- 25 C
- Between 0 C and 25 C
- Between 25 C and 50 C
The Correct Answer and Explanation is :
To determine the final temperature of the water when the contents of the two cups are mixed, we apply the principle of conservation of energy. No heat is lost to the surroundings, so the heat lost by the warmer water will be equal to the heat gained by the cooler water.
Given Data:
- Cup A: ( m_A = 100 ) g of water at ( T_A = 0^\circ C )
- Cup B: ( m_B = 200 ) g of water at ( T_B = 50^\circ C )
- Specific heat capacity of water: ( c = 4.18 ) J/g°C
Step 1: Set Up the Energy Conservation Equation
The heat lost by the hot water is:
[
q_{\text{lost}} = m_B c (T_B – T_f)
]
The heat gained by the cold water is:
[
q_{\text{gained}} = m_A c (T_f – T_A)
]
Since the system is insulated:
[
q_{\text{lost}} + q_{\text{gained}} = 0
]
Substituting values:
[
200 \times 4.18 \times (50 – T_f) = 100 \times 4.18 \times (T_f – 0)
]
Canceling out ( 4.18 ) from both sides:
[
200 (50 – T_f) = 100 T_f
]
Expanding:
[
10000 – 200T_f = 100T_f
]
Solving for ( T_f ):
[
10000 = 300T_f
]
[
T_f = \frac{10000}{300} = 33.3^\circ C
]
Step 2: Select the Correct Answer
Since ( 33.3^\circ C ) is between 25°C and 50°C, the correct answer is:
Option 6: Between 25°C and 50°C.
Explanation:
- The hotter water loses heat, and the cooler water gains heat until thermal equilibrium is reached.
- The final temperature depends on the mass and initial temperatures of both samples.
- Because the mass of the warm water (200 g) is greater than that of the cold water (100 g), the final temperature is closer to 50°C than to 0°C.
- The temperature does not go below 0°C since there is no phase change involved.
I will now generate an image to visually represent this heat transfer process.
The image visually represents the heat transfer process, showing the initial temperatures of the two cups, the mixing in an insulated container, and the final equilibrium temperature of approximately 33.3°C. Let me know if you need any modifications or further explanations!