The pressure of a gas filled in a closed jar increased by 0.2%. When temperature is increased by `2^@ C`. Find the initial temperature of the gas.

Explanation

To solve the problem, we will use the relationship between pressure and temperature for an ideal gas, which is given by the ideal gas law. The pressure of a gas is directly proportional to its absolute temperature when the volume and the number of moles of gas are constant.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The relationship between pressure (P) and temperature (T) for a fixed volume of gas is given by:
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
where \(P_1\) and \(T_1\) are the initial pressure and temperature, and \(P_2\) and \(T_2\) are the final pressure and temperature.

2. **Identify the Changes**:
We know that the pressure increases by 0.2%, which can be expressed as:
\[ P_2 = P_1 + 0.002 P_1 = 1.002 P_1 \]
The temperature increases by \(2^\circ C\), which is equivalent to an increase of \(2 K\) in absolute terms.

3. **Express the Final Temperature**:
If the initial temperature is \(T_1\), then the final temperature \(T_2\) can be expressed as:
\[ T_2 = T_1 + 2 \]

4. **Substitute into the Ideal Gas Law**:
Substitute \(P_2\) and \(T_2\) into the ideal gas law relationship:
\[ \frac{P_1}{T_1} = \frac{1.002 P_1}{T_1 + 2} \]

5. **Cancel \(P_1\)**:
Since \(P_1\) is common on both sides, we can cancel it out:
\[ \frac{1}{T_1} = \frac{1.002}{T_1 + 2} \]

6. **Cross Multiply**:
Cross multiplying gives:
\[ T_1 + 2 = 1.002 T_1 \]

7. **Rearranging the Equation**:
Rearranging the equation:
\[ 1.002 T_1 - T_1 = 2 \]
\[ 0.002 T_1 = 2 \]

8. **Solve for \(T_1\)**:
Dividing both sides by \(0.002\):
\[ T_1 = \frac{2}{0.002} = 1000 \text{ K} \]

9. **Convert to Celsius**:
To convert the temperature from Kelvin to Celsius:
\[ T_1 = 1000 - 273 = 727^\circ C \]

### Final Answer:
The initial temperature of the gas is \(727^\circ C\).