If there is no heat loss, the heat released by the condensation of x gram of steam at `100^@C` into water at `100^@C` can be used to convert y gram of ice at `0^@C` into water at `100^@C`. Then the ratio of y:x is nearly [Given `L_l = 80 cal//gm` and `L_v= 540 cal//gm`]

Options

A. `1:1`
B. `2:1`
C. `3:1`
D. `4:1`

Answer

To solve the problem, we need to calculate the heat released by the condensation of steam and the heat required to convert ice into water. We will set these two quantities equal to each other since it is stated that there is no heat loss.

### Step-by-Step Solution:

1. **Calculate the heat released by the condensation of steam:**
- The heat released when `x` grams of steam condenses into water at `100°C` can be calculated using the formula:
\[ Q_{\text{steam}} = x \cdot L_v \]
- Where \(L_v\) is the latent heat of vaporization. Given \(L_v = 540 \, \text{cal/g}\), we have:
\[ Q_{\text{steam}} = x \cdot 540 \]

2. **Calculate the heat required to convert ice to water at `100°C`:**
- First, we need to convert `y` grams of ice at `0°C` to `y` grams of water at `0°C`. The heat required for this process is:
\[ Q_1 = y \cdot L_f \]
- Where \(L_f\) is the latent heat of fusion. Given \(L_f = 80 \, \text{cal/g}\), we have:
\[ Q_1 = y \cdot 80 \]

- Next, we need to heat the `y` grams of water from `0°C` to `100°C`. The heat required for this is:
\[ Q_2 = y \cdot S \cdot \Delta T \]
- Where \(S\) is the specific heat of water (1 cal/g°C) and \(\Delta T = 100°C - 0°C = 100°C\):
\[ Q_2 = y \cdot 1 \cdot 100 = 100y \]

3. **Total heat required to convert ice to water at `100°C`:**
- The total heat \(Q_{\text{ice}}\) required is the sum of \(Q_1\) and \(Q_2\):
\[ Q_{\text{ice}} = Q_1 + Q_2 = y \cdot 80 + 100y = 180y \]

4. **Set the heat released equal to the heat required:**
- Since the heat released by the steam equals the heat required to convert the ice, we can write:
\[ Q_{\text{steam}} = Q_{\text{ice}} \]
- Substituting the expressions we derived:
\[ x \cdot 540 = 180y \]

5. **Rearranging the equation to find the ratio \(y:x\):**
- Rearranging gives:
\[ \frac{y}{x} = \frac{540}{180} = 3 \]

6. **Final Ratio:**
- Therefore, the ratio \(y:x\) is:
\[ \frac{y}{x} = 3:1 \]

### Conclusion:
The ratio of \(y\) to \(x\) is nearly \(3:1\).