`M g` of ice at `0^@C` is mixed with `M g` of water at `10^@ c`. The final temperature is

Options

A. `8^@ C`
B. `6^@C`
C. `4^@ C`
D. `0^@ C`

Explanation

To solve the problem of mixing M grams of ice at 0°C with M grams of water at 10°C, we need to analyze the heat transfer between the ice and the water.

### Step-by-step Solution:

1. **Identify the Heat Required to Melt Ice:**
- The ice at 0°C needs to be converted into water at 0°C. The latent heat of fusion of ice is 80 calories per gram.
- Therefore, the heat required to convert M grams of ice into water is:
\[ Q_{\text{melt}} = M \times 80 \text{ calories} \]

2. **Identify the Heat Released by Water:**
- The water at 10°C will cool down to 0°C. The specific heat of water is 1 calorie per gram per degree Celsius.
- The heat released when M grams of water cools from 10°C to 0°C is:
\[ Q_{\text{cool}} = M \times 10 \times 1 = M \times 10 \text{ calories} \]

3. **Compare the Heat Transfer:**
- The heat required to melt the ice (Q_melt) is:
\[ Q_{\text{melt}} = M \times 80 \text{ calories} \]
- The heat released by the water (Q_cool) is:
\[ Q_{\text{cool}} = M \times 10 \text{ calories} \]

4. **Determine the Final Temperature:**
- Since \( Q_{\text{melt}} > Q_{\text{cool}} \):
\[ M \times 80 > M \times 10 \]
- This means that the heat released by the water is not enough to completely melt the ice. Therefore, the final temperature of the mixture will be 0°C, as the ice will absorb all the heat available from the water until it reaches 0°C.

5. **Conclusion:**
- The final temperature of the mixture when M grams of ice at 0°C is mixed with M grams of water at 10°C is:
\[ \text{Final Temperature} = 0°C \]