5 g of ice at `0^@C` and 20 g of water at `45^@C` are mixed. The temperature of the mixture will be

Options

A. `10^@C`
B. `20^@C`
C. `30^@C`
D. `40^@C`

Explanation

To solve the problem of finding the final temperature of the mixture when 5 g of ice at 0°C is mixed with 20 g of water at 45°C, 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 as it melts and warms up.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Mass of ice, \( m_{ice} = 5 \, \text{g} \)
- Initial temperature of ice, \( T_{ice} = 0 \, \text{°C} \)
- Mass of water, \( m_{water} = 20 \, \text{g} \)
- Initial temperature of water, \( T_{water} = 45 \, \text{°C} \)
- Latent heat of fusion of ice, \( L_f = 80 \, \text{cal/g} \)
- Specific heat of water, \( c_{water} = 1 \, \text{cal/g°C} \)

2. **Assume Final Temperature:**
- Let the final temperature of the mixture be \( T \) (in °C).

3. **Calculate Heat Gained by Ice:**
- The ice will first absorb heat to melt and then the resulting water will absorb heat to increase its temperature to \( T \).
- Heat gained by the ice:
\[ Q_{ice} = m_{ice} \cdot L_f + m_{ice} \cdot c_{water} \cdot (T - T_{ice}) \]
\[ Q_{ice} = 5 \, \text{g} \cdot 80 \, \text{cal/g} + 5 \, \text{g} \cdot 1 \, \text{cal/g°C} \cdot (T - 0) \]
\[ Q_{ice} = 400 \, \text{cal} + 5T \, \text{cal} \]

4. **Calculate Heat Lost by Water:**
- The warm water will lose heat as it cools down to \( T \).
- Heat lost by the water:
\[ Q_{water} = m_{water} \cdot c_{water} \cdot (T_{water} - T) \]
\[ Q_{water} = 20 \, \text{g} \cdot 1 \, \text{cal/g°C} \cdot (45 - T) \]
\[ Q_{water} = 20(45 - T) \, \text{cal} \]

5. **Set Heat Gained Equal to Heat Lost:**
- According to the principle of calorimetry:
\[ Q_{ice} = Q_{water} \]
\[ 400 + 5T = 20(45 - T) \]

6. **Solve the Equation:**
- Expand and rearrange the equation:
\[ 400 + 5T = 900 - 20T \]
\[ 5T + 20T = 900 - 400 \]
\[ 25T = 500 \]
\[ T = \frac{500}{25} = 20 \, \text{°C} \]

7. **Final Answer:**
- The final temperature of the mixture is \( T = 20 \, \text{°C} \).