Find the final temperature and composition of the mixtue of 1 kg of ice at `0^(@)C` and 1.5 kg of water at `45^(@)C`. Given that specific heat of water is 4200 J/kg and latent heat of fusion of ice is `3.36xx10^(5)J//kg`

Answer

To solve the problem of finding the final temperature and composition of the mixture of 1 kg of ice at 0°C and 1.5 kg of water at 45°C, we can follow these steps:

### Step 1: Understand the Initial Conditions
- We have 1 kg of ice at 0°C.
- We have 1.5 kg of water at 45°C.
- The specific heat of water (s) = 4200 J/kg°C.
- The latent heat of fusion of ice (L) = 3.36 x 10^5 J/kg.

### Step 2: Assume the Final Temperature
Assume the final temperature (T_final) of the mixture is 0°C. This is a reasonable assumption since the ice is at 0°C and the water is at a higher temperature.

### Step 3: Calculate the Heat Absorbed by Ice
If x kg of ice melts, the heat absorbed by the ice can be calculated using the formula:
\[ Q_{absorbed} = x \cdot L \]
Where:
- \( Q_{absorbed} \) is the heat absorbed by the ice.
- \( L \) is the latent heat of fusion.

### Step 4: Calculate the Heat Released by Water
The heat released by the water as it cools down from 45°C to 0°C can be calculated using:
\[ Q_{released} = m_{water} \cdot s \cdot (T_{initial} - T_{final}) \]
Where:
- \( m_{water} = 1.5 \) kg (mass of water).
- \( s = 4200 \) J/kg°C (specific heat of water).
- \( T_{initial} = 45°C \) and \( T_{final} = 0°C \).

Calculating this gives:
\[ Q_{released} = 1.5 \cdot 4200 \cdot (45 - 0) \]
\[ Q_{released} = 1.5 \cdot 4200 \cdot 45 \]
\[ Q_{released} = 283500 \, \text{J} \]

### Step 5: Set Up the Equation
Since the heat absorbed by the ice must equal the heat released by the water:
\[ x \cdot L = Q_{released} \]
Substituting the values we have:
\[ x \cdot (3.36 \times 10^5) = 283500 \]

### Step 6: Solve for x
Now we can solve for x:
\[ x = \frac{283500}{3.36 \times 10^5} \]
Calculating this gives:
\[ x \approx 0.843 \, \text{kg} \]

### Step 7: Determine Remaining Ice and Water
- Remaining ice = Initial ice - melted ice = \( 1 - 0.843 = 0.157 \, \text{kg} \)
- Total water after melting = Initial water + melted ice = \( 1.5 + 0.843 = 2.343 \, \text{kg} \)

### Step 8: Final Results
- Final temperature of the mixture = 0°C
- Composition of the mixture:
- Ice remaining = 0.157 kg
- Water = 2.343 kg

### Final Answer
The final temperature is 0°C, and the composition of the mixture is 0.157 kg of ice and 2.343 kg of water.

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