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Difficulty: MediumPneumatics and Gas Laws

A pneumatic lifting system uses compressed air to operate a main lifting cylinder. Initially, the cylinder contains 2.0 m32.0 \text{ m}^3 of air at an absolute pressure of 100 kPa100 \text{ kPa}. To prepare for a heavy lift, the system's compressor forces the piston inward, reducing the trapped air's volume to 0.5 m30.5 \text{ m}^3. Assuming the temperature of the air remains constant during this process, what is the new absolute pressure inside the cylinder?

  1. A
    100 kPa100 \text{ kPa}
  2. B
    25 kPa25 \text{ kPa}
  3. 400 kPa400 \text{ kPa}Answer
  4. D
    200 kPa200 \text{ kPa}

Answer

The new absolute pressure is 400 kPa400 \text{ kPa}.
According to Boyle's Law, the pressure of a fixed mass of gas is inversely proportional to its volume, provided the temperature remains constant. The equation is P1V1=P2V2P_1V_1 = P_2V_2. By substituting the given values (100 kPa×2.0 m3=P2×0.5 m3100 \text{ kPa} \times 2.0 \text{ m}^3 = P_2 \times 0.5 \text{ m}^3), we find that reducing the volume to one-quarter of its original size forces the pressure to quadruple, resulting in a final pressure of 400 kPa400 \text{ kPa}.

Step-by-Step Solution

1
Identify the relevant gas law for a situation with changing pressure and volume at a constant temperature.
Boyle's Law (P1V1=P2V2P_1V_1 = P_2V_2) applies to this scenario.
The problem specifies a trapped gas undergoing volume compression while maintaining a constant temperature.
2
Extract the known initial and final values from the problem statement.
Initial pressure (P1P_1) = 100 kPa100 \text{ kPa}. Initial volume (V1V_1) = 2.0 m32.0 \text{ m}^3. Final volume (V2V_2) = 0.5 m30.5 \text{ m}^3.
Organizing known variables is necessary before attempting to solve for the unknown final pressure (P2P_2).
3
Substitute the known values into Boyle's Law equation and solve for the unknown final pressure.
100×2.0=P2×0.5100 \times 2.0 = P_2 \times 0.5. This simplifies to 200=0.5P2200 = 0.5P_2, which yields P2=400 kPaP_2 = 400 \text{ kPa}.
Calculating the algebraic solution provides the final absolute pressure required by the question.

Key Concept

Boyle's Law (Inverse Relationship of Pressure and Volume)
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Topics

Pneumatics and Gas Laws

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