Step 1: Understanding the Concept:
The problem states that the temperature remains constant. According to Boyle's Law, for a fixed mass of gas at a constant temperature, the volume (\(V\)) of the gas is inversely proportional to its pressure (\(P\)).
Mathematically: \(P \propto \frac{1}{V}\) or \(PV = K\) (constant).
This means we can use the equation \(P_1V_1 = P_2V_2\) to find the final volume and subsequently calculate the change in volume.
Step 2: Key Formula or Approach:
1. Initial Pressure (\(P_1\)): At NTP (Normal Temperature and Pressure), the pressure is standard, which is 1 atm.
2. Initial Volume (\(V_1\)): 2.5 dm\(^3\).
3. Final Pressure (\(P_2\)): 1.25 atm.
4. Final Volume (\(V_2\)): To be calculated.
5. Change in Volume (\(\Delta V\)): \(|V_2 - V_1|\).
Step 3: Detailed Explanation:
Applying Boyle's Law:
\[ P_1V_1 = P_2V_2 \]
Substitute the known values:
\[ 1 \text{ atm} \times 2.5 \text{ dm}^3 = 1.25 \text{ atm} \times V_2 \]
Isolating \(V_2\):
\[ V_2 = \frac{1 \times 2.5}{1.25} \]
We can simplify this by noting that \(1.25\) is exactly \(\frac{5}{4}\).
\[ V_2 = \frac{2.5}{1.25} = 2.0 \text{ dm}^3 \]
The question asks for the **change in volume** (\(\Delta V\)):
\[ \Delta V = V_{initial} - V_{final} = 2.5 \text{ dm}^3 - 2.0 \text{ dm}^3 \]
\[ \Delta V = 0.5 \text{ dm}^3 \]
The volume decreases because the pressure was increased from 1 atm to 1.25 atm.
Step 4: Final Answer:
The change in volume of the gas is 0.5 dm\(^3\).
Hence, the correct option is (B).