500g of ice at `0°C` is mixed with 1g steam at `100°C`. The final temperature of the mixture is

Options

A. `0°C`
B. `50°C`
C. `40°C`
D. `100°C`

Answer

To solve the problem of mixing 500 g of ice at 0°C with 1 g of steam at 100°C, we need to analyze the heat exchange between the two substances.

### Step-by-Step Solution:

1. **Identify the mass and initial temperatures**:
- Mass of ice (m_ice) = 500 g
- Initial temperature of ice (T_ice) = 0°C
- Mass of steam (m_steam) = 1 g
- Initial temperature of steam (T_steam) = 100°C

2. **Calculate the heat required to melt the ice**:
- The latent heat of fusion of ice (L_f) = 80 cal/g
- Heat required to melt the ice (Q_ice) = m_ice × L_f
\[ Q_{ice} = 500 \, \text{g} \times 80 \, \text{cal/g} = 40000 \, \text{cal} \]

3. **Calculate the heat released by the steam when it condenses to water**:
- The latent heat of vaporization of steam (L_v) = 540 cal/g
- Heat released when steam condenses (Q_steam_condense) = m_steam × L_v
\[ Q_{steam\_condense} = 1 \, \text{g} \times 540 \, \text{cal/g} = 540 \, \text{cal} \]

4. **Calculate the heat released by the condensed steam as it cools down to 0°C**:
- After condensing, the steam is now water at 100°C. We need to cool this water to 0°C.
- The specific heat of water (c) = 1 cal/g°C
- Heat released by the water as it cools (Q_steam_cool) = m_steam × c × ΔT
\[ Q_{steam\_cool} = 1 \, \text{g} \times 1 \, \text{cal/g°C} \times (100°C - 0°C) = 100 \, \text{cal} \]

5. **Total heat released by the steam**:
\[ Q_{total\_steam} = Q_{steam\_condense} + Q_{steam\_cool} = 540 \, \text{cal} + 100 \, \text{cal} = 640 \, \text{cal} \]

6. **Compare heat gained by ice and heat released by steam**:
- Heat required to melt the ice = 40000 cal
- Total heat released by steam = 640 cal

Since the heat released by the steam (640 cal) is less than the heat required to melt the ice (40000 cal), the ice will not completely melt.

7. **Final temperature of the mixture**:
- Since the ice does not completely melt, the final temperature of the mixture remains at 0°C.

### Conclusion:
The final temperature of the mixture is **0°C**.