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?
The pressure of the gas increases by a factor of approximately 2.64.
Explanation
This question involves an adiabatic process. The context clearly states that the walls of the cylinder are made of a heat insulator and the piston is insulated by having a pile of sand on it. This means no heat exchange occurs with the surroundings (Q = 0), which is the defining characteristic of an adiabatic process. For such a process, the relation PV^γ = constant holds, where γ (gamma) is the adiabatic index. For hydrogen, a diatomic gas, γ = 1.4. When the gas is compressed to half its original volume, we can calculate the pressure ratio using this relation.
Solution Steps
Step 1: Identify the process - Since the walls are heat insulators and the piston is insulated, no heat exchange occurs. This is an adiabatic process.
Step 2: Apply the adiabatic relation - For an adiabatic process: P₁V₁^γ = P₂V₂^γ, where γ is the adiabatic index.
Step 3: Determine γ for hydrogen - Hydrogen is a diatomic gas, so γ = Cₚ/Cᵥ = 1.4 (or 7/5).
Step 4: Apply the compression condition - Volume is compressed to half: V₂ = V₁/2
Step 5: Calculate the pressure ratio: P₁V₁^γ = P₂V₂^γ P₁V₁^(1.4) = P₂(V₁/2)^(1.4) P₂/P₁ = (V₁/(V₁/2))^(1.4) = 2^(1.4) 2^(1.4) 2.64