540 g of ice at `0^(@)C` is mixed with 540 g of water at `80^(@)C`. The final temperature of the mixture is

Options

A. `0^(@)C`
B. `40^(@)C`
C. `80^(@)C`
D. less than `0^(@)C`

Explanation

To solve the problem of mixing 540 g of ice at 0°C with 540 g of water at 80°C, we will use the principle of conservation of energy, which states that the heat lost by the water will be equal to the heat gained by the ice until thermal equilibrium is reached.

### Step-by-step Solution:

1. **Identify the masses and temperatures:**
- Mass of ice, \( m_1 = 540 \, \text{g} \)
- Initial temperature of ice, \( T_1 = 0^\circ C \)
- Mass of water, \( m_2 = 540 \, \text{g} \)
- Initial temperature of water, \( T_2 = 80^\circ C \)

2. **Define the final temperature:**
- Let the final temperature of the mixture be \( T_f \).

3. **Calculate the heat gained by the ice:**
- The ice will first absorb heat to melt into water at 0°C, and then it will absorb heat to increase its temperature to \( T_f \).
- The heat gained by the ice can be expressed as:
\[ Q_{\text{ice}} = m_1 \cdot L_f + m_1 \cdot c \cdot (T_f - T_1) \]
- Where:
- \( L_f = 80 \, \text{cal/g} \) (latent heat of fusion)
- \( c = 1 \, \text{cal/g°C} \) (specific heat of water)

Substituting the values:
\[ Q_{\text{ice}} = 540 \cdot 80 + 540 \cdot 1 \cdot (T_f - 0) \]
\[ Q_{\text{ice}} = 43200 + 540 T_f \]

4. **Calculate the heat lost by the water:**
- The water will lose heat as it cools down from 80°C to \( T_f \):
\[ Q_{\text{water}} = m_2 \cdot c \cdot (T_2 - T_f) \]
Substituting the values:
\[ Q_{\text{water}} = 540 \cdot 1 \cdot (80 - T_f) \]
\[ Q_{\text{water}} = 540 (80 - T_f) \]

5. **Set the heat gained equal to the heat lost:**
\[ Q_{\text{ice}} = Q_{\text{water}} \]
Therefore:
\[ 43200 + 540 T_f = 540 (80 - T_f) \]

6. **Simplify the equation:**
Expanding the right side:
\[ 43200 + 540 T_f = 43200 - 540 T_f \]
Rearranging gives:
\[ 540 T_f + 540 T_f = 43200 - 43200 \]
\[ 1080 T_f = 0 \]
\[ T_f = 0^\circ C \]

7. **Conclusion:**
The final temperature of the mixture is \( 0^\circ C \).

### Final Answer:
The final temperature of the mixture is \( 0^\circ C \).