`10 gm` of ice at `-20^(@)C` is dropped into a calorimeter containing `10 gm` of water at `10^(@)C`, the specific heat of water is twice that of ice. When equilibrium is reached the calorimeter will contain:

Options

A. 20 gm of water
B. 20 gm of ice
C. 10 gm ice and 10 gm of water
D. 5 gm ice and 15 gm water

Answer

To solve the problem step by step, we need to analyze the heat exchange between the ice and the water until thermal equilibrium is reached.

### Step 1: Understand the system
We have:
- 10 g of ice at -20°C
- 10 g of water at 10°C
- The specific heat of water is twice that of ice.

### Step 2: Calculate the heat required to warm the ice to 0°C
The specific heat of ice (let's denote it as \( s \)) is half that of water. Therefore, if the specific heat of water is \( 1 \) cal/g°C, the specific heat of ice will be \( \frac{1}{2} \) cal/g°C.

To calculate the heat \( Q_2 \) required to raise the temperature of the ice from -20°C to 0°C:
\[ Q_2 = m \cdot s \cdot \Delta T \]
Where:
- \( m = 10 \) g (mass of ice)
- \( s = \frac{1}{2} \) cal/g°C (specific heat of ice)
- \( \Delta T = 0 - (-20) = 20 \)°C (change in temperature)

Substituting the values:
\[ Q_2 = 10 \cdot \frac{1}{2} \cdot 20 = 100 \text{ cal} \]

### Step 3: Calculate the heat released by the water as it cools to 0°C
Now, we calculate the heat \( Q_1 \) released by the water when it cools from 10°C to 0°C:
\[ Q_1 = m \cdot s \cdot \Delta T \]
Where:
- \( m = 10 \) g (mass of water)
- \( s = 1 \) cal/g°C (specific heat of water)
- \( \Delta T = 10 - 0 = 10 \)°C (change in temperature)

Substituting the values:
\[ Q_1 = 10 \cdot 1 \cdot 10 = 100 \text{ cal} \]

### Step 4: Analyze the heat exchange
From our calculations:
- Heat gained by ice to reach 0°C: \( Q_2 = 100 \) cal
- Heat lost by water to cool to 0°C: \( Q_1 = 100 \) cal

Since \( Q_1 = Q_2 \), all the heat lost by the water is used to warm the ice to 0°C.

### Step 5: Determine the final state of the system
At this point, the water has cooled to 0°C and the ice has warmed to 0°C. However, there is no heat left to melt the ice, as all the heat exchanged was used to raise the temperature of the ice to 0°C.

Thus, at equilibrium, the calorimeter will contain:
- 10 g of ice at 0°C
- 10 g of water at 0°C

### Final Answer
The calorimeter will contain **10 g of ice and 10 g of water at 0°C**.

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