A gas is filled in a container of volume V at `121^@ C`. To what temperature should it be heated in order that `(1)/(4)` th of the gas may escape out of the vessel?

Explanation

To solve the problem, we need to find the temperature to which the gas should be heated so that one-fourth of it escapes from the container. We will use the ideal gas law and the relationship between the number of moles and temperature.

### Step-by-Step Solution:

1. **Convert the Initial Temperature to Kelvin:**
The initial temperature is given as \(121^\circ C\). To convert this to Kelvin, we use the formula:
\[ T_1 = 121 + 273 = 394 \, K \]

2. **Determine the Initial Number of Moles:**
Let the initial number of moles of the gas be \(N\).

3. **Calculate the Remaining Moles After Escape:**
If one-fourth of the gas escapes, then three-fourths of the gas remains. Therefore, the remaining number of moles is:
\[ N_2 = N - \frac{1}{4}N = \frac{3}{4}N \]

4. **Set Up the Ideal Gas Law Relationship:**
According to the ideal gas law, we have:
\[ P_1 V = N_1 R T_1 \quad \text{and} \quad P_2 V = N_2 R T_2 \]
Since the volume \(V\) and the gas constant \(R\) remain constant, we can write:
\[ \frac{N_1 T_1}{N_2 T_2} = 1 \]

5. **Substituting the Values:**
Substituting \(N_1 = N\), \(N_2 = \frac{3}{4}N\), and \(T_1 = 394 \, K\):
\[ \frac{N \cdot 394}{\frac{3}{4}N \cdot T_2} = 1 \]
Simplifying this gives:
\[ \frac{394}{\frac{3}{4} T_2} = 1 \]

6. **Solving for \(T_2\):**
Rearranging the equation to solve for \(T_2\):
\[ T_2 = \frac{394 \cdot \frac{3}{4}}{1} = 394 \cdot \frac{4}{3} \]
Calculating this gives:
\[ T_2 = \frac{1576}{3} \approx 525.33 \, K \]

7. **Convert \(T_2\) Back to Celsius:**
To convert the final temperature back to Celsius:
\[ T_2 = 525.33 - 273 \approx 252.33^\circ C \]

### Final Answer:
The temperature to which the gas should be heated is approximately \(252.33^\circ C\).

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