100 g of ice (latent heat 80 cal/g, at `0^(@)C`) is mixed with 100 g of water (specific heat `1" cal"//g-""^(@)C`) at `80^(@)C`. The final temperature of the mixture will be :-

Options

A. `0^(@)C`
B. `40^(@)C`
C. `80^(@)C`
D. `lt0^(@)C`

Answer

To find the final temperature of the mixture when 100 g of ice at 0°C is mixed with 100 g of water at 80°C, we can use the principle of conservation of energy. The heat lost by the water will be equal to the heat gained by the ice.

### Step-by-Step Solution:

1. **Identify the given data:**
- Mass of ice, \( m_{\text{ice}} = 100 \, \text{g} \)
- Latent heat of fusion of ice, \( L = 80 \, \text{cal/g} \)
- Mass of water, \( m_{\text{water}} = 100 \, \text{g} \)
- Specific heat of water, \( s = 1 \, \text{cal/g°C} \)
- Initial temperature of water, \( T_{\text{water}} = 80°C \)
- Initial temperature of ice, \( T_{\text{ice}} = 0°C \)

2. **Calculate the heat gained by the ice:**
The ice will first absorb heat to melt into water at 0°C, and then it will warm up if there is any heat left. The heat gained by the ice can be expressed as:
\[ Q_{\text{ice}} = m_{\text{ice}} \cdot L + m_{\text{ice}} \cdot s \cdot (T_f - T_{\text{ice}}) \]
where \( T_f \) is the final temperature of the mixture.

Substituting the values:
\[ Q_{\text{ice}} = 100 \cdot 80 + 100 \cdot 1 \cdot (T_f - 0) \]
\[ Q_{\text{ice}} = 8000 + 100T_f \]

3. **Calculate the heat lost by the water:**
The water will lose heat as it cools down to the final temperature \( T_f \):
\[ Q_{\text{water}} = m_{\text{water}} \cdot s \cdot (T_{\text{water}} - T_f) \]
Substituting the values:
\[ Q_{\text{water}} = 100 \cdot 1 \cdot (80 - T_f) \]
\[ Q_{\text{water}} = 8000 - 100T_f \]

4. **Set the heat gained by the ice equal to the heat lost by the water:**
\[ Q_{\text{ice}} = Q_{\text{water}} \]
\[ 8000 + 100T_f = 8000 - 100T_f \]

5. **Solve for \( T_f \):**
Rearranging the equation:
\[ 100T_f + 100T_f = 8000 - 8000 \]
\[ 200T_f = 0 \]
\[ T_f = 0°C \]

### Final Answer:
The final temperature of the mixture will be \( 0°C \).