notesonly.in

Suppose we have 1 mole of air (γ = 1.4) in a sealed piston. Initially the pressure is 1 atm, the volume is 1.0 L, and the temperature is 300 K. The gas is compressed adiabatically to a final volume of 0.5 L. Find the final pressure and temperature. Step 1: Use the adiabatic relation P V^γ = constant. P1 V1^γ = P2 V2^γ (1 atm)(1.0 L)^1.4 = P2(0.5 L)^1.4 Step 2: Solve for P2. P2 = 1 atm × (1.0/0.5)^1.4 = 1 atm × (2)^1.4 ≈ 1 atm × 2.64 ≈ 2.64 atm. Step 3: Use the temperature–volume relation T V^(γ−1) = constant. T1 V1^(γ−1) = T2 V2^(γ−1) 300 K × (1.0 L)^(0.4) = T2 × (0.5 L)^(0.4) Step 4: Solve for T2. T2 = 300 K × (1.0/0.5)^0.4 = 300 K × (2)^0.4 ≈ 300 K × 1.32 ≈ 396 K. Answer: After adiabatic compression to 0.5 L, the pressure rises to about 2.6 atm and the temperature rises to about 396 K.