One kg of ice at `0^(@)C` is mixed with 1 kg of water at `10^(@)C`. The resulting temperature will be

Options

A. between `0^(@)C` and `10^(@)C`
B. `0^(@)C`
C. less than `0^(@)C`
D. greater than `0^(@)C`

Explanation

To solve the problem of mixing 1 kg of ice at 0°C with 1 kg of water at 10°C, we will follow these steps:

### Step 1: Understand the scenario
We have:
- 1 kg of ice at 0°C
- 1 kg of water at 10°C

When these two substances are mixed, the heat from the water will be used to melt some of the ice.

### Step 2: Calculate the heat required to melt the ice
The heat required to convert ice at 0°C to water at 0°C is given by the formula:
\[ Q = m \cdot L_f \]
where:
- \( Q \) is the heat required,
- \( m \) is the mass of the ice,
- \( L_f \) is the latent heat of fusion of ice (approximately 80 cal/g).

Since we have 1 kg of ice:
- Convert kg to grams: \( 1 \, \text{kg} = 1000 \, \text{g} \)
- Thus, the heat required to melt the ice is:
\[ Q = 1000 \, \text{g} \cdot 80 \, \text{cal/g} = 80000 \, \text{cal} \]

### Step 3: Calculate the heat released by the water
The heat released by the water as it cools down from 10°C to 0°C is given by:
\[ Q = m \cdot c \cdot \Delta T \]
where:
- \( m \) is the mass of the water,
- \( c \) is the specific heat capacity of water (approximately 1 cal/g°C),
- \( \Delta T \) is the change in temperature.

For 1 kg of water:
- Convert kg to grams: \( 1 \, \text{kg} = 1000 \, \text{g} \)
- The heat released by the water is:
\[ Q = 1000 \, \text{g} \cdot 1 \, \text{cal/g°C} \cdot (10 - 0) \, \text{°C} = 10000 \, \text{cal} \]

### Step 4: Compare the heat required and the heat released
- Heat required to melt the ice: \( 80000 \, \text{cal} \)
- Heat released by the water: \( 10000 \, \text{cal} \)

Since the heat released by the water (10000 cal) is less than the heat required to melt all the ice (80000 cal), not all the ice will melt.

### Step 5: Determine the final state of the mixture
Since the water can only provide 10000 cal, it will only partially melt the ice. The remaining ice will still be at 0°C, and the water will also be at 0°C after losing its heat.

### Conclusion
The resulting temperature of the mixture will be 0°C, with some ice remaining.

### Final Answer
The resulting temperature will be **0°C**.

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