A gas is heated through `2^(@)C` in a closed vessel. Its pressure is increased by 0.4%. The initial temperature of the gas is

Options

A. `250^(@)C`
B. `100^(@)C`
C. `-75^(@)C`
D. `-23^(@)C`

Answer

To solve the problem, we will use the relationship between pressure and temperature for an ideal gas, which states that pressure is directly proportional to temperature when volume and the number of moles of gas are constant.

### Step-by-Step Solution:

1. **Understand the relationship**: According to the ideal gas law, for a fixed amount of gas at constant volume, the pressure (P) is directly proportional to the absolute temperature (T). This can be expressed as:
\[ P \propto T \]
Therefore, we can write:
\[ \frac{\Delta P}{P} = \frac{\Delta T}{T} \]
where \(\Delta P\) is the change in pressure, \(\Delta T\) is the change in temperature, and \(T\) is the initial temperature.

2. **Identify the given values**:
- The change in pressure \(\Delta P\) is given as 0.4%. This can be expressed as:
\[ \Delta P = 0.004P \]
- The change in temperature \(\Delta T\) is given as 2°C.

3. **Set up the equation**: Substitute the known values into the proportionality equation:
\[ \frac{0.004P}{P} = \frac{2}{T} \]
Simplifying this gives:
\[ 0.004 = \frac{2}{T} \]

4. **Solve for T**: Rearranging the equation to solve for \(T\):
\[ T = \frac{2}{0.004} \]
Calculating this gives:
\[ T = 500 \text{ K} \]

5. **Convert to Celsius**: Since the question asks for the temperature in degrees Celsius, we need to convert from Kelvin to Celsius using the formula:
\[ T(°C) = T(K) - 273 \]
Thus,
\[ T(°C) = 500 - 273 = 227°C \]

6. **Check the options**: The options provided were:
- 250°C
- 100°C
- -75°C
- -23°C

Since 227°C is not listed in the options, we can conclude that there may be an error in the options provided.

### Final Answer:
The initial temperature of the gas is **227°C**.