If m is the minimum amount of steam of `100^(@)C` required to melt 12 gm ice completely, then find the value of 10cm.

Explanation

To find the minimum amount of steam required to melt 12 grams of ice completely, we will use the principle of conservation of energy, where the heat released by the steam will be equal to the heat absorbed by the ice.

### Step-by-Step Solution:

1. **Identify the Heat Released by Steam**:
The heat released when steam condenses into water can be calculated using the formula:
\[ Q_1 = m \cdot L_v \]
where \( m \) is the mass of the steam and \( L_v \) is the latent heat of vaporization of water. The latent heat of vaporization of water is approximately \( 540 \, \text{cal/g} \).

2. **Identify the Heat Required to Melt Ice**:
The heat required to melt ice can be calculated using the formula:
\[ Q_2 = m_{ice} \cdot L_f \]
where \( m_{ice} \) is the mass of the ice (12 grams) and \( L_f \) is the latent heat of fusion of ice. The latent heat of fusion of ice is approximately \( 80 \, \text{cal/g} \).

Therefore, for 12 grams of ice:
\[ Q_2 = 12 \, \text{g} \cdot 80 \, \text{cal/g} = 960 \, \text{cal} \]

3. **Set Up the Equation**:
According to the principle of conservation of energy:
\[ Q_1 = Q_2 \]
Substituting the expressions for \( Q_1 \) and \( Q_2 \):
\[ m \cdot L_v = 960 \, \text{cal} \]

4. **Substitute the Values**:
Substitute \( L_v = 540 \, \text{cal/g} \):
\[ m \cdot 540 = 960 \]

5. **Solve for \( m \)**:
Rearranging the equation to find \( m \):
\[ m = \frac{960}{540} \approx 1.78 \, \text{g} \]

### Final Answer:
The minimum amount of steam required to melt 12 grams of ice completely is approximately \( 1.78 \, \text{grams} \).

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