The pressure of a gas filled in a closed jar increases 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 a gas, which is derived from the ideal gas law. Here are the steps to find the initial temperature of the gas:

### Step 1: Understand the relationship
From the ideal gas law, we know that for a fixed amount of gas at constant volume, pressure (P) is directly proportional to temperature (T). This can be expressed as:
\[ P \propto T \]
This means that if the temperature changes, the pressure will also change proportionally.

### Step 2: Set up the equation
We can express the change in pressure and temperature using the formula:
\[ \frac{dP}{P} = k \cdot \frac{dT}{T} \]
where \( k \) is a constant that depends on the proportionality. In this case, we can consider \( k = 1 \) for simplicity since we are looking for a direct relationship.

### Step 3: Substitute the given values
We are given that the pressure increases by 0.2%. This can be expressed as:
\[ \frac{dP}{P} = \frac{0.2}{100} = 0.002 \]
We are also given that the temperature increases by \( dT = 2°C \). We need to convert this into Kelvin for our calculations, but since the change in Celsius is the same as the change in Kelvin, we can use:
\[ dT = 2 \]

### Step 4: Write the equation with the known values
Now substituting the known values into the equation:
\[ 0.002 = \frac{2}{T} \]
where \( T \) is the initial temperature in Kelvin.

### Step 5: Solve for T
Rearranging the equation to solve for \( T \):
\[ T = \frac{2}{0.002} \]
\[ T = 1000 \text{ K} \]

### Step 6: Convert to Celsius
To convert the temperature from Kelvin to Celsius, we use the formula:
\[ T(°C) = T(K) - 273.15 \]
So,
\[ T(°C) = 1000 - 273.15 = 726.85 \text{ °C} \]
We can round this to:
\[ T(°C) \approx 727 \text{ °C} \]

### Final Answer
The initial temperature of the gas is approximately **727 °C**.

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