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. `2.5:1`

Explanation

To solve the problem, we need to equate the heat lost by the condensation of steam to the heat gained by the melting of ice and heating the resulting water. Let's break it down step by step.

### Step 1: Identify the heat loss from steam condensation
When steam condenses into water at 100°C, it releases heat. The heat released by the condensation of \( x \) grams of steam can be calculated using the latent heat of vaporization (\( L_v \)):
\[ \text{Heat lost} = x \cdot L_v = x \cdot 540 \, \text{cal} \]

### Step 2: Identify the heat gain by ice
The heat gained by \( y \) grams of ice at 0°C to convert it into water at 100°C consists of two parts:
1. The heat required to melt the ice at 0°C into water at 0°C.
2. The heat required to raise the temperature of the resulting water from 0°C to 100°C.

The heat required to melt the ice is given by:
\[ \text{Heat to melt ice} = y \cdot L_l = y \cdot 80 \, \text{cal} \]

The heat required to raise the temperature of the water from 0°C to 100°C is given by:
\[ \text{Heat to raise temperature} = y \cdot s \cdot \Delta T = y \cdot 1 \cdot (100 - 0) = y \cdot 100 \, \text{cal} \]

### Step 3: Combine the heat gain expressions
Now, we can express the total heat gained by the ice:
\[ \text{Total heat gained} = \text{Heat to melt ice} + \text{Heat to raise temperature} = y \cdot 80 + y \cdot 100 = y \cdot (80 + 100) = y \cdot 180 \, \text{cal} \]

### Step 4: Set heat lost equal to heat gained
According to the principle of conservation of energy (no heat loss), we can set the heat lost equal to the heat gained:
\[ x \cdot 540 = y \cdot 180 \]

### Step 5: Solve for the ratio \( \frac{y}{x} \)
Rearranging the equation gives us:
\[ \frac{y}{x} = \frac{540}{180} = 3 \]

### Conclusion
Thus, the ratio \( \frac{y}{x} \) is nearly \( 3:1 \).