Thermodynamic Cycles Class Exercises
AP Physics
For a heat engine to be useful it must be able to operate continuously. Practically, this means that it has to follow a cycle of
processes that returns the system to its original state so it can start over again. We will continue to assume that all processes
are reversible, even though no actual heat engine works that way. We will continue to use a gas filled piston as our standard
heat engine.
1. Sketch the P-V graph for a gas-filled piston going through each of the following cycles:
a) isothermal expansion, isobaric compression,
c) adiabatic expansion, isometric cooling,
isometric heating
adiabatic compression, isometric heating
b) isobaric expansion, adiabatic expansion,
isothermal compression, isometric heating
d) isothermal expansion, adiabatic expansion,
isothermal compression, adiabatic compression
For the remainder of the problems, we will be following thermodynamic cycles. In each problem the
challenge will be the same:
a) Sketch a P-V diagram for the cycle and label the pressure volume and temperature of each state
b) Find the work, heat and change in internal energy for each process
c) Find the net work, net heat and net change in internal energy for
d) Find the ideal efficiency of the engine
2. Let’s imagine one of the simplest heat engine cycles possible. A piston full of 0.000298 kg of helium
gas [cv = 3128 J/(kg K) goes through these stages: isobaric compression, isometric heating, isobaric
expansion, isometric cooling.
The gas initially occupies a volume of 0.004 m3 at a pressure of 200000 Pa and a temperature of 1292 K.
i) During the isobaric compression, the gas gives off 1504 J of energy as heat. At the end it occupies
0.001 m3 of space and has cooled to 323 K
ii) During the isometric heating, the gas’s temperature rises to 807.5 K and the pressure climbs to
500000 Pa
iii) During the isobaric expansion, the gas returns to a volume of 0.004 m3 as the temperature climbs to
3230 K
iv) After the isometric cooling, the gas loses 1807.5 J of internal energy and returns to its original state.
3. Could you make the cycle go backwards?
4. Imagine a heat engine that goes through the following cycle: an isobaric expansion, and isometric
cooling and an isothermal compression. The system is a piston filled with 0.0056 kg of hydrogen gas
[cv = 10217 J/(kg K)], which starts out occupying a volume of 0.02 m3 at a pressure of 430000 Pa and a
temperature of 100 °C.
i) During the isobaric expansion, the gas absorbs 59841 J of energy as heat while the volume grows to
0.06 m3 and the temperature rises to 846.5 °C.
ii) During the isometric cooling the pressure drops to143,333 Pa as the temperature returns to 100 °C.
iii) During the isothermal compression, the gas returns to its original state as 9448 J of work are done
on it.
5. The Stirling cycle is a thermodynamic cycle that follows the following steps: isometric cooling,
isothermal compression, isometric heating, isothermal expansion. We will investigate a Stirling engine
that uses 5.058 x 10-5 kg of nitrogen [cv = 742.86 J/(kg K)] which starts at 150,000 Pa of pressure and
600 K of temperature enclosed in a piston that provides 6 x 10-5 m3 of volume..
i) During the isometric cooling, the gas loses 3.757 J of energy in the form of heat while its pressure
drops to 125,000 Pa.
ii) It requires 13.438 J of work to isothermally compress the gas to a volume of 1 x 10-5 m3.
iii) The system’s pressure rises to 750,000 Pa as it isometrically heats.
iv) The system absorbs 16.126 J of energy in the form of heat as it isothermally returns to its original
state
6. The Brayton/Joule cycle models the performance of gas turbines by following a path that consists of
an adiabatic compression, an isobaric expansion, an adiabatic expansion and an isobaric compression.
One turbine uses 0.174 kg of helium [cv = 3128.3 J/(kg K)] which initially occupies 1 m3 at atmospheric
pressure and 7 °C.
i) The gas heats up to 347 °C as 186,000 J of work are done to adiabatically compress it to 0.3 m3.
ii) The gas’s internal energy goes up by 676,020 J as it expands isobarically until it is 0.9 m3.
iii) The gas loses 557400 J of internal energy as it expands adiabatically back to atmospheric pressure
iv) The gas is compressed isobarically until it return to its original conditions.
7. A diesel engine injects the fuel as droplets instead of vapor, which slows down the burning so it
follows this cycle: isobaric expansion, adiabatic expansion, isometric cooling, adiabatic
compression. A large Diesel engine is made from a piston filled with 20.2 kg of nitrogen gas [cv =
742.9 J/(kg K)]. The gas starts out occupying a volume of 2.4 m3 at a pressure of 2,141,610.681 Pa
and a temperature of 856.65 K.
i) The gas expands isobarically to 6.0 m3
ii) The gas does 7,780,000 J of work as it expands adiabatically to a volume of 12.0 m3
iii) The gas cools isometrically
iv) It gains 6,100,000 J of internal energy as adiabatically expands back to its original state.
8. Just to be clever, you want to design a heat engine that uses each basic thermodynamic process: an
adiabatic expansion, and isobaric compression, isothermal compression and isometric heating. You
build a small engine from a piston filled with 5.94 x 10-6 kg of argon [cv = 313.35 J/(kg K)]. The gas
starts out occupying a volume of 3.0 x 10-6 m3 at a pressure of 245,164 Pa and a temperature of 595.4 K.
i) The gas loses 0.318 J of internal energy expanding adiabatically as the pressure drops to 105,000 Pa
ii) The system undergoes an isobaric compression
iii) The gas has 0.1208 J of work done on it as it compresses isothermally
iv) During the isometric heating, the gas raises its temperature by 255.4 K to return to the original state.