A gas is filled in a vessel of volume V at `35^(@)C`. To what temperature should it be heated in order that half of the gas may escape out?

Explanation

To solve the problem of determining the temperature to which a gas must be heated in order for half of it to escape from a vessel, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

2. **Initial Conditions**:
We have a gas at an initial temperature \( T_1 = 35^\circ C \). To use the ideal gas law, we need to convert this temperature to Kelvin:
\[ T_1 = 35 + 273 = 308 \, K \]

3. **Final Conditions**:
We want to find the temperature \( T_2 \) when half of the gas escapes. If the initial number of moles is \( n \), after half escapes, the final number of moles \( n_2 \) will be:
\[ n_2 = \frac{n}{2} \]

4. **Using the Relationship of Moles and Temperature**:
Since the product \( nT \) is constant, we can write:
\[ n_1 T_1 = n_2 T_2 \]
Substituting the known values:
\[ n \cdot 308 = \left(\frac{n}{2}\right) T_2 \]

5. **Simplifying the Equation**:
We can cancel \( n \) from both sides (assuming \( n \neq 0 \)):
\[ 308 = \frac{1}{2} T_2 \]
Multiplying both sides by 2 gives:
\[ T_2 = 2 \cdot 308 = 616 \, K \]

6. **Convert Kelvin to Celsius**:
To find the final temperature in degrees Celsius, we convert from Kelvin:
\[ T_2 = 616 - 273 = 343^\circ C \]

### Final Answer:
The temperature to which the gas should be heated in order for half of it to escape is:
\[ \boxed{343^\circ C} \]