Thermodynamic Cycle with Air · USAPhO 2010 #A3
Physics · thermodynamics
Problem
A cylindrical pipe contains a movable piston that traps 2.00 mols of air. Originally, the air is at one atmosphere of pressure, a volume V₀, and at a temperature of T₀ = 298 K. First (process A) the air in the cylinder is compressed at constant temperature to a volume of 1/4 V₀. Then (process B) the air is allowed to expand adiabatically to a volume of V = 15.0 L. After this (process C) this piston is withdrawn allowing the gas to expand to the original volume V₀ while maintaining a constant temperature. Finally (process D) while maintaining a fixed volume, the gas is allowed to return to the original temperature T₀. Assume air is a diatomic ideal gas, no air flows into, or out of, the pipe at any time, and that the temperature outside the remains constant always. Possibly useful information: Cp = 7R/2, Cv = 5R/2, 1 atm = 1.01×10⁵ Pa. a. Draw a P-V diagram of the whole process. b. How much work is done on the trapped air during process A? c. What is the temperature of the air at the end of process B?Intuitive explanation
This problem traces a gas through four linked processes—each with different constraints (constant , adiabatic, constant again, constant )—forming a closed cycle on a - diagram. The key is recognizing that the ideal gas law and the first law of thermodynamics connect pressure, volume, and temperature at each stage, so you can track the state variables through each process. For work calculations, remember that takes different forms depending on whether temperature, pressure, or volume is held fixed.
Intuitive explanation
This problem traces a gas through four linked processes—each with different constraints (constant , adiabatic, constant again, constant )—forming a closed cycle on a - diagram. The key is recognizing that the ideal gas law and the first law of thermodynamics connect pressure, volume, and temperature at each stage, so you can track the state variables through each process. For work calculations, remember that takes different forms depending on whether temperature, pressure, or volume is held fixed.