When a gas filled in a closed vessel is heated through `2^(@)C`, its pressure increases by 0.4%. What is the initial temperature of gas ?

Explanation

To solve the problem, we will use the ideal gas law and the relationship between pressure, volume, and temperature. Here are the steps to find the initial temperature of the gas:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a closed vessel filled with gas. When the gas is heated by 2°C, its pressure increases by 0.4%. We need to find the initial temperature of the gas.

2. **Identify the Variables**:
- Let the initial pressure be \( P_1 = P \).
- The final pressure after heating is \( P_2 = P + 0.004P = 1.004P \) (since 0.4% of \( P \) is \( 0.004P \)).
- Let the initial temperature be \( T_1 = T \) (in Kelvin).
- The final temperature after heating is \( T_2 = T + 2 \) (since we are heating by 2°C).

3. **Use the Ideal Gas Law**: Since the volume is constant (closed vessel), we can use the relationship:
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]

4. **Substitute the Known Values**:
\[ \frac{P}{T} = \frac{1.004P}{T + 2} \]

5. **Cancel \( P \) from Both Sides** (as long as \( P \neq 0 \)):
\[ \frac{1}{T} = \frac{1.004}{T + 2} \]

6. **Cross-Multiply**:
\[ 1 \cdot (T + 2) = 1.004 \cdot T \]
This simplifies to:
\[ T + 2 = 1.004T \]

7. **Rearrange the Equation**:
\[ 2 = 1.004T - T \]
\[ 2 = 0.004T \]

8. **Solve for \( T \)**:
\[ T = \frac{2}{0.004} = 500 \text{ K} \]

### Final Answer:
The initial temperature of the gas is **500 Kelvin**.

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