100 g of ice at `0^(@)C` is mixed with 100 of water `80^(@)C`. The final temperature of the mixture will be

Options

A. `0^(@)C`
B. `20^(@)C`
C. `40^(@)C`
D. `60^(@)C`

Answer

To solve the problem of mixing 100 g of ice at 0°C with 100 g of water at 80°C, we can follow these steps:

### Step 1: Identify the heat released by the water.
The water at 80°C will cool down to 0°C, releasing heat in the process. We can calculate the heat released (Q_water) using the formula:
\[ Q_{\text{water}} = m \cdot c \cdot \Delta T \]
Where:
- \( m = 100 \, \text{g} \) (mass of water)
- \( c = 1 \, \text{cal/g°C} \) (specific heat capacity of water)
- \( \Delta T = (0 - 80) = -80 \, \text{°C} \)

Calculating:
\[ Q_{\text{water}} = 100 \cdot 1 \cdot (0 - 80) = 100 \cdot 1 \cdot (-80) = -8000 \, \text{cal} \]
The negative sign indicates that heat is being released.

### Step 2: Identify the heat required to melt the ice.
The ice at 0°C will absorb heat to convert into water at 0°C. The heat absorbed (Q_ice) can be calculated using the formula:
\[ Q_{\text{ice}} = m \cdot L \]
Where:
- \( m = 100 \, \text{g} \) (mass of ice)
- \( L = 80 \, \text{cal/g} \) (latent heat of fusion of ice)

Calculating:
\[ Q_{\text{ice}} = 100 \cdot 80 = 8000 \, \text{cal} \]

### Step 3: Compare the heat released by water and the heat absorbed by ice.
From the calculations:
- Heat released by water: \( Q_{\text{water}} = -8000 \, \text{cal} \)
- Heat absorbed by ice: \( Q_{\text{ice}} = 8000 \, \text{cal} \)

### Step 4: Apply the principle of conservation of energy.
According to the principle of conservation of energy, the heat lost by the water will be equal to the heat gained by the ice:
\[ Q_{\text{water}} + Q_{\text{ice}} = 0 \]
This means:
\[ -8000 + 8000 = 0 \]
The heat released by the water is completely absorbed by the ice.

### Step 5: Determine the final temperature of the mixture.
Since all the ice has melted into water at 0°C and the water has cooled down to 0°C, the final temperature of the mixture will be:
\[ \text{Final Temperature} = 0 \, \text{°C} \]

### Conclusion:
The final temperature of the mixture is 0°C.

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