What would be the final temperature of the mixture when `1kg` of ice at `-10^(@)C` is mixed with `4.4kg` of water of `30^(@)C`? (Specific heat capacity of ice `=2100Jkg^(-1)K^(-1))`

Answer

To find the final temperature of the mixture when `1 kg` of ice at `-10°C` is mixed with `4.4 kg` of water at `30°C`, we can follow these steps:

### Step 1: Identify the heat transfer
When ice is mixed with water, heat will flow from the warmer water to the colder ice until thermal equilibrium is reached. We need to consider two cases: (1) some of the ice melts, and (2) all of the ice melts.

### Step 2: Calculate the heat required to warm the ice to 0°C
The heat required to raise the temperature of the ice from `-10°C` to `0°C` can be calculated using the formula:
\[ Q_1 = m_{ice} \cdot c_{ice} \cdot \Delta T \]
Where:
- \( m_{ice} = 1 \, \text{kg} \)
- \( c_{ice} = 2100 \, \text{J/kg°C} \)
- \( \Delta T = 0 - (-10) = 10°C \)

Substituting the values:
\[ Q_1 = 1 \cdot 2100 \cdot 10 = 21000 \, \text{J} \]

### Step 3: Calculate the heat required to melt the ice
Next, we need to calculate the heat required to melt the ice at `0°C` into water at `0°C`:
\[ Q_2 = m_{ice} \cdot L_f \]
Where:
- \( L_f = 336000 \, \text{J/kg} \) (latent heat of fusion)

Substituting the values:
\[ Q_2 = 1 \cdot 336000 = 336000 \, \text{J} \]

### Step 4: Calculate the heat released by the water
Now, we calculate the heat released by the water as it cools down from `30°C` to the final temperature \( T \):
\[ Q_{water} = m_{water} \cdot c_{water} \cdot (T_{initial} - T) \]
Where:
- \( m_{water} = 4.4 \, \text{kg} \)
- \( c_{water} = 4200 \, \text{J/kg°C} \)
- \( T_{initial} = 30°C \)

Substituting the values:
\[ Q_{water} = 4.4 \cdot 4200 \cdot (30 - T) \]

### Step 5: Set up the heat balance equation
The heat gained by the ice (both warming and melting) must equal the heat lost by the water:
\[ Q_1 + Q_2 = Q_{water} \]
Substituting the values we calculated:
\[ 21000 + 336000 = 4.4 \cdot 4200 \cdot (30 - T) \]

### Step 6: Simplify the equation
First, calculate the right side:
\[ 4.4 \cdot 4200 = 18480 \]
So the equation becomes:
\[ 357000 = 18480 \cdot (30 - T) \]

### Step 7: Solve for \( T \)
Expanding the equation:
\[ 357000 = 554400 - 18480T \]
Rearranging gives:
\[ 18480T = 554400 - 357000 \]
\[ 18480T = 197400 \]
Now, divide by `18480`:
\[ T = \frac{197400}{18480} \approx 10.67°C \]

### Step 8: Conclusion
The final temperature of the mixture when `1 kg` of ice at `-10°C` is mixed with `4.4 kg` of water at `30°C` is approximately **10.67°C**.

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