Question:
A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume ?
Answer:
The cylinder is completely insulated from its surroundings. As a result, no heat is exchanged between the system (cylinder) and its surroundings. Thus, the process is adiabatic.
Initial pressure inside the cylinder = P1
Final pressure inside the cylinder = P2
Initial volume inside the cylinder = V1
Final volume inside the cylinder = V2
Ratio of specific heats, γ = 1.4
For an adiabatic process, we have:
P1V1γ = P2V2γ
The final volume is compressed to half of its initial volume.
∴ V2 = V1/2
P1V1γ = P2(V1/2)γ
P2/P1 = V1γ / (V1/2)γ
= 2γ = 21.4 = 2.639
Hence, the pressure increases by a factor of 2.639.
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