If pressure of a gas contained in a closed vessel is increased by `0.4%` when heated by `1^(@)C`, the initial temperature must be

Options

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

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 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature in Kelvin

### Step 2: Set Up Initial Conditions
Let the initial pressure be \( P \), the initial temperature be \( T \) (in Kelvin), and the volume be \( V \). The equation for the initial state is:
\[ PV = nRT \]

### Step 3: Set Up Final Conditions
When the temperature is increased by \( 1^\circ C \), the new temperature becomes \( T + 1 \) (in Kelvin). The pressure increases by \( 0.4\% \), so the new pressure is:
\[ P + 0.004P = 1.004P \]
The equation for the final state is:
\[ (1.004P)V = nR(T + 1) \]

### Step 4: Divide the Two Equations
Now, we can divide the final state equation by the initial state equation:
\[ \frac{(1.004P)V}{PV} = \frac{nR(T + 1)}{nRT} \]
This simplifies to:
\[ 1.004 = \frac{T + 1}{T} \]

### Step 5: Solve for Temperature
Cross-multiplying gives:
\[ 1.004T = T + 1 \]
Rearranging the equation, we find:
\[ 1.004T - T = 1 \implies 0.004T = 1 \implies T = \frac{1}{0.004} = 250 \text{ K} \]

### Step 6: Convert to Celsius (if needed)
To convert Kelvin to Celsius, we use the formula:
\[ T_{C} = T_{K} - 273.15 \]
However, since the question asks for the initial temperature in Kelvin, we can leave it as \( 250 \text{ K} \).

### Final Answer
The initial temperature of the gas must be \( 250 \text{ K} \).

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