If 10 g of ice is added to 40 g of water at `15^(@)C`, then the temperature of the mixture is (specific heat of water `= 4.2 xx 10^(3) j kg^(-1) K^(-1)`, Latent heat of fusion of ice `= 3.36 xx 10^(5) j kg^(-1)`)

Options

A. `15^(@)C`
B. `12^(@)C`
C. `10^(2)C`
D. `0^(@)C`

Explanation

To solve the problem of finding the final temperature of the mixture when 10 g of ice is added to 40 g of water at 15°C, we will follow these steps:

### Step 1: Identify the given data
- Mass of ice, \( m_{ice} = 10 \, \text{g} = 10 \times 10^{-3} \, \text{kg} \)
- Mass of water, \( m_{water} = 40 \, \text{g} = 40 \times 10^{-3} \, \text{kg} \)
- Initial temperature of water, \( T_{water} = 15 \, \text{°C} \)
- Initial temperature of ice, \( T_{ice} = 0 \, \text{°C} \)
- Specific heat of water, \( c_{water} = 4.2 \times 10^{3} \, \text{J/kg/K} \)
- Latent heat of fusion of ice, \( L_f = 3.36 \times 10^{5} \, \text{J/kg} \)

### Step 2: Calculate the heat required to melt the ice
The heat required to convert the ice at 0°C to water at 0°C can be calculated using the formula:
\[ Q_{melt} = L_f \times m_{ice} \]
Substituting the values:
\[ Q_{melt} = 3.36 \times 10^{5} \, \text{J/kg} \times 10 \times 10^{-3} \, \text{kg} = 3360 \, \text{J} \]

### Step 3: Calculate the heat released by the water as it cools down
The heat released by the water when it cools from 15°C to 0°C is given by:
\[ Q_{cool} = m_{water} \times c_{water} \times \Delta T \]
Where \( \Delta T = T_{initial} - T_{final} = 15 - 0 = 15 \, \text{°C} \).
Substituting the values:
\[ Q_{cool} = 40 \times 10^{-3} \, \text{kg} \times 4.2 \times 10^{3} \, \text{J/kg/K} \times 15 \, \text{K} \]
Calculating this:
\[ Q_{cool} = 40 \times 10^{-3} \times 4.2 \times 10^{3} \times 15 = 2520 \, \text{J} \]

### Step 4: Compare the heat required and the heat released
- Heat required to melt the ice: \( Q_{melt} = 3360 \, \text{J} \)
- Heat released by the water: \( Q_{cool} = 2520 \, \text{J} \)

Since \( Q_{melt} > Q_{cool} \), the water cannot provide enough heat to melt all the ice. Therefore, some of the ice will remain unmelted.

### Step 5: Determine the final state of the mixture
Since the water can only cool down to 0°C and not provide enough heat to melt all the ice, the final temperature of the mixture will be:
\[ T_{final} = 0 \, \text{°C} \]

### Conclusion
The final temperature of the mixture when 10 g of ice is added to 40 g of water at 15°C is \( 0 \, \text{°C} \).

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