`100g` ice at `0^(@)C` is mixed with `100g` water at `100^(@)C`. The resultant temperature of the mixture is

Options

A. `10^(@)C`
B. `20^(@)C`
C. `30^(@)C`
D. `0^(@)C`

Explanation

To solve the problem of finding the resultant temperature when 100 g of ice at 0°C is mixed with 100 g of water at 100°C, we can follow these steps:

### Step 1: Calculate the heat required to melt the ice
The heat required to melt the ice can be calculated using the formula:
\[ Q_1 = m \cdot L \]
where:
- \( m = 100 \, \text{g} \) (mass of ice)
- \( L = 80 \, \text{cal/g} \) (latent heat of fusion for ice)

Calculating \( Q_1 \):
\[ Q_1 = 100 \, \text{g} \cdot 80 \, \text{cal/g} = 8000 \, \text{cal} \]

### Step 2: Calculate the heat released by the water as it cools down
The heat released by the water when it cools from 100°C to 0°C can be calculated using the formula:
\[ Q_2 = m \cdot s \cdot \Delta T \]
where:
- \( m = 100 \, \text{g} \) (mass of water)
- \( s = 1 \, \text{cal/g°C} \) (specific heat of water)
- \( \Delta T = 100°C - 0°C = 100°C \)

Calculating \( Q_2 \):
\[ Q_2 = 100 \, \text{g} \cdot 1 \, \text{cal/g°C} \cdot 100°C = 10000 \, \text{cal} \]

### Step 3: Calculate the excess heat after melting the ice
After the ice has melted, the total heat available for raising the temperature of the resulting water (from melted ice and the original water) is:
\[ Q_3 = Q_2 - Q_1 \]
Calculating \( Q_3 \):
\[ Q_3 = 10000 \, \text{cal} - 8000 \, \text{cal} = 2000 \, \text{cal} \]

### Step 4: Calculate the final temperature of the mixture
Now, we have 200 g of water (100 g from melted ice + 100 g original water) and 2000 cal of heat to distribute. We can use the formula:
\[ Q_3 = m \cdot s \cdot \Delta T \]
where:
- \( m = 200 \, \text{g} \)
- \( s = 1 \, \text{cal/g°C} \)

Rearranging for \( \Delta T \):
\[ \Delta T = \frac{Q_3}{m \cdot s} \]
Calculating \( \Delta T \):
\[ \Delta T = \frac{2000 \, \text{cal}}{200 \, \text{g} \cdot 1 \, \text{cal/g°C}} = 10°C \]

### Step 5: Determine the final temperature
The initial temperature of the mixture is 0°C (after the ice has melted). Therefore, the final temperature \( T_f \) is:
\[ T_f = 0°C + 10°C = 10°C \]

### Final Answer
The resultant temperature of the mixture is **10°C**.

---