50 g of ice at `0^@C` is mixed with 50 g of water at `80^@C` . The final temperature of the mixture is (latent heat of fusion of ice `=80 cal //g , s_(w) = 1 cal //g ^@C)`

Options

A. `0^@C`
B. `40^@C`
C. `60^@C`
D. less than `0^@C`

Answer

To solve the problem of finding the final temperature of the mixture of ice and water, we can use the principle of conservation of energy. The heat lost by the warm water will be equal to the heat gained by the ice.

### Step-by-step Solution:

1. **Identify the components:**
- Mass of ice, \( m_i = 50 \, \text{g} \)
- Initial temperature of ice, \( T_i = 0^\circ C \)
- Mass of water, \( m_w = 50 \, \text{g} \)
- Initial temperature of water, \( T_w = 80^\circ C \)
- Latent heat of fusion of ice, \( L = 80 \, \text{cal/g} \)
- Specific heat of water, \( s_w = 1 \, \text{cal/g}^\circ C \)

2. **Calculate the heat gained by the ice:**
- The ice will first melt and then the resulting water will warm up.
- Heat gained by ice during melting:
\[ Q_{\text{melt}} = m_i \cdot L = 50 \, \text{g} \cdot 80 \, \text{cal/g} = 4000 \, \text{cal} \]
- After melting, the temperature of the melted ice (now water) will increase to the final temperature \( T_f \):
\[ Q_{\text{heat}} = m_i \cdot s_w \cdot (T_f - T_i) = 50 \, \text{g} \cdot 1 \, \text{cal/g}^\circ C \cdot (T_f - 0) \]
\[ Q_{\text{heat}} = 50 \, T_f \, \text{cal} \]

3. **Calculate the heat lost by the warm water:**
- The warm water will cool down to the final temperature \( T_f \):
\[ Q_{\text{lost}} = m_w \cdot s_w \cdot (T_w - T_f) = 50 \, \text{g} \cdot 1 \, \text{cal/g}^\circ C \cdot (80 - T_f) \]
\[ Q_{\text{lost}} = 50 \cdot (80 - T_f) \, \text{cal} \]

4. **Set up the energy conservation equation:**
- The heat gained by the ice (both melting and warming) is equal to the heat lost by the water:
\[ Q_{\text{melt}} + Q_{\text{heat}} = Q_{\text{lost}} \]
\[ 4000 + 50 T_f = 50 (80 - T_f) \]

5. **Solve for \( T_f \):**
- Expand the equation:
\[ 4000 + 50 T_f = 4000 - 50 T_f \]
- Combine like terms:
\[ 50 T_f + 50 T_f = 4000 - 4000 \]
\[ 100 T_f = 0 \]
\[ T_f = 0^\circ C \]

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