The pressure of a gas filled in a closed vessel increase by 0.4% when temperature is increased by `1^(@) C`. Find the initial temperature of the gas.

Options

A. `250 K`
B. `250^(@)C`
C. `2500 K`
D. `25^(@)C`

Explanation

To find the initial temperature of the gas in a closed vessel where the pressure increases by 0.4% when the temperature is raised by 1°C, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between pressure and temperature**:
According to the ideal gas law, for a closed vessel where volume (V) and the number of moles (n) of gas are constant, pressure (P) is directly proportional to temperature (T). This can be expressed as:
\[ \frac{P_1}{P_2} = \frac{T_1}{T_2} \]

2. **Define the initial and final conditions**:
- Let the initial pressure be \( P_1 = P \).
- The pressure increases by 0.4%, so the final pressure \( P_2 \) can be expressed as:
\[ P_2 = P + 0.004P = 1.004P \]
- Let the initial temperature be \( T_1 = T \) (in Kelvin).
- The final temperature after an increase of 1°C is:
\[ T_2 = T + 1 \]

3. **Set up the equation using the proportionality**:
Substituting the values into the proportionality equation:
\[ \frac{P}{1.004P} = \frac{T}{T + 1} \]
This simplifies to:
\[ \frac{1}{1.004} = \frac{T}{T + 1} \]

4. **Cross-multiply to eliminate the fraction**:
Cross-multiplying gives:
\[ T + 1 = 1.004T \]

5. **Rearrange the equation**:
Rearranging the equation:
\[ 1 = 1.004T - T \]
\[ 1 = 0.004T \]

6. **Solve for T**:
Dividing both sides by 0.004:
\[ T = \frac{1}{0.004} = 250 \text{ K} \]

Thus, the initial temperature of the gas is **250 K**.