Step 1: Understanding the Question:
The problem requires predicting the change in the volume of an ideal gas when both its pressure and absolute temperature are modified.
We must apply the Ideal Gas Equation, which relates the three state variables: pressure (\(P\)), volume (\(V\)), and temperature (\(T\)).
Step 2: Key Formula or Approach:
The General Gas Equation (Combined Gas Law) is:
\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]
We are given the following relationships between the final and initial states:
\(P_2 = 2P_1\) (Pressure is doubled).
\(T_2 = \frac{T_1}{2}\) (Absolute temperature is halved).
Step 3: Detailed Explanation:
Step A: Rearrange the combined gas law to solve for the final volume \(V_2\):
\[ V_2 = V_1 \cdot \left( \frac{P_1}{P_2} \right) \cdot \left( \frac{T_2}{T_1} \right) \]
Step B: Substitute the given values of \(P_2\) and \(T_2\) into the equation:
\[ V_2 = V_1 \cdot \left( \frac{P_1}{2P_1} \right) \cdot \left( \frac{T_1 / 2}{T_1} \right) \]
Step C: Simplify the expression by canceling the common terms \(P_1\) and \(T_1\):
\[ V_2 = V_1 \cdot \left( \frac{1}{2} \right) \cdot \left( \frac{1}{2} \right) \]
Step D: Perform the final multiplication of the fractions:
\[ V_2 = V_1 \cdot \frac{1}{4} \]
Physically, doubling the pressure forces the molecules closer together (Boyle's Law effect), reducing volume by half. Simultaneously, halving the absolute temperature reduces the kinetic energy and thermal expansion (Charles's Law effect), which also reduces volume by half.
The combined effect of these two independent reductions results in the volume being compressed to one-fourth of its original size.
Step 4: Final Answer:
The final volume of the gas becomes 1/4 of its initial volume.