Steam at `100^(@)C` enters a radiator and leaves as water ( at `80^(@)C )`. Take the heat of vaporization to be 540 cal`//`g .Of the total energy given off as heat, what percent arises from the cooling of the water?

Options

A. 1
B. 0.54
C. 0.26
D. 0.036

Answer

To solve the problem, we need to calculate the total heat energy released when steam condenses to water and then cools down from 100°C to 80°C. We will then determine what percentage of this total energy comes from the cooling of the water.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Heat of vaporization of steam, \( L = 540 \, \text{cal/g} \)
- Initial temperature of steam, \( T_1 = 100^\circ C \)
- Final temperature of water, \( T_2 = 80^\circ C \)
- Temperature change, \( \Delta T = T_1 - T_2 = 100 - 80 = 20^\circ C \)

2. **Calculate the Heat Released During Condensation**:
- Let the mass of steam be \( m \) grams.
- The heat released when steam condenses to water is given by:
\[ Q_{\text{condensation}} = m \cdot L = m \cdot 540 \, \text{cal} \]

3. **Calculate the Heat Released During Cooling**:
- The specific heat of water, \( c = 1 \, \text{cal/g}^\circ C \).
- The heat released when the water cools from 100°C to 80°C is given by:
\[ Q_{\text{cooling}} = m \cdot c \cdot \Delta T = m \cdot 1 \cdot 20 = 20m \, \text{cal} \]

4. **Calculate the Total Heat Released**:
- The total heat released as the steam condenses and then cools is:
\[ Q_{\text{total}} = Q_{\text{condensation}} + Q_{\text{cooling}} = 540m + 20m = 560m \, \text{cal} \]

5. **Calculate the Percentage of Heat from Cooling**:
- The percentage of heat released due to cooling of the water is given by:
\[ \text{Percentage} = \left( \frac{Q_{\text{cooling}}}{Q_{\text{total}}} \right) \times 100 = \left( \frac{20m}{560m} \right) \times 100 \]
- Simplifying this gives:
\[ \text{Percentage} = \left( \frac{20}{560} \right) \times 100 = \left( \frac{1}{28} \right) \times 100 \approx 3.57\% \]

6. **Final Result**:
- Therefore, the percentage of the total energy given off as heat that arises from the cooling of the water is approximately **3.57%**.