One kilogram of ice at `0^(@)C` is mixed with one kilogram of water at `80^(@)C`. The final temperature of the mixture is
(Take : specific heat of water `=4200 J kg^(-1) K^(-1)`, latent heat of ice `= 336 kJ//kg^(-1)`)

Options

A. `40^(@)C`
B. `60^(@)C`
C. `0^(@)C`
D. `50^(@)C`

Explanation

To solve the problem of mixing 1 kg of ice at \(0^\circ C\) with 1 kg of water at \(80^\circ C\), we need to determine the final temperature of the mixture. We will use the principles of heat transfer and the concepts of latent heat and specific heat.

### Step-by-step Solution:

1. **Identify the Heat Required to Melt the Ice**:
- The latent heat of ice is given as \(336 \, \text{kJ/kg}\).
- For 1 kg of ice, the heat required to melt it completely (Q1) is:
\[ Q_1 = \text{mass} \times \text{latent heat} = 1 \, \text{kg} \times 336 \, \text{kJ/kg} = 336 \, \text{kJ} \]

2. **Calculate the Heat Released by the Water**:
- The specific heat of water is given as \(4200 \, \text{J/kg} \cdot \text{K}\) or \(4.2 \, \text{kJ/kg} \cdot \text{K}\).
- The maximum heat that can be released by 1 kg of water when it cools from \(80^\circ C\) to \(0^\circ C\) (Q2) is:
\[ Q_2 = \text{mass} \times \text{specific heat} \times \Delta T = 1 \, \text{kg} \times 4.2 \, \text{kJ/kg} \cdot \text{K} \times (80 - 0) \, \text{K} \]
\[ Q_2 = 1 \times 4.2 \times 80 = 336 \, \text{kJ} \]

3. **Compare the Heat Required and Released**:
- We find that \(Q_1 = 336 \, \text{kJ}\) (heat required to melt the ice) and \(Q_2 = 336 \, \text{kJ}\) (heat released by the water).
- Since the heat released by the water is equal to the heat required to melt the ice, the entire 1 kg of ice will melt, and the water will cool down to \(0^\circ C\).

4. **Determine the Final Temperature**:
- After the ice has completely melted, the final temperature of the mixture will be \(0^\circ C\) because all the heat from the water has been used to melt the ice, and there is no excess heat to raise the temperature above \(0^\circ C\).

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