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?
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Answer 1

Explanation:

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

Answer 2

Explanation:

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.