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 1

Given:

Number of moles of hydrogen gas, n = 3
Initial state: Standard temperature and pressure (STP)
Initial volume = V₁
Final volume = V₂ = V₁ / 2

The walls of the cylinder and the piston are insulated.
Therefore, the process is adiabatic.

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Step 1: Identify the thermodynamic process

Since there is no heat exchange with surroundings:

Q = 0 → Adiabatic process

For an adiabatic process:

P V^γ = constant

For hydrogen (a diatomic gas):

γ = C_p / C_v = 7 / 5

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Step 2: Apply adiabatic relation

P₁ V₁^γ = P₂ V₂^γ

⇒ P₂ / P₁ = (V₁ / V₂)^γ

Since:

V₂ = V₁ / 2

⇒ V₁ / V₂ = 2

Therefore:

P₂ / P₁ = 2^7/5

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Step 3: Evaluate the factor

2^7/5 = 2^1.4 ≈ 2.64

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Final Answer:

When the gas is compressed to half its original volume, the pressure increases by a factor of
2^7/5 ≈ 2.64