**How-To:**

The diagram at left depicts the Pressure-Volume diagram of a
heat engine.
A heat engine is a mechanism in which heat energy
from a hot reservoir is tranfered to an ideal gas, which then,
through a series of ideal gas processes,
converts part of that energy into mechanical work and releases the rest back as
thermal energy into a cold reservoir. All ideal gas processes follow
the equations:

*PV = nRT*, and
*ΔE*_{sys} = Q_{on sys} + W_{on sys}
= 1.5nRΔT, where R is the ideal gas constant (8.31), n is the
moles of ideal gas, and T is the temperature of the system.
The efficiency of a heat engine what percentage of the heat
added to the gas is converted to mechanical energy.
*(N = -W*_{total}/Q_{total})

To compute the engine's efficiency, find the work and heat
transferred at each step, then sum these numbers over all steps and
take the quotient.

The heat engine shown at left starts in the lower right and moves
clockwise, and consists of any of the following ideal gas processes:

Process |
Constant Quantity |
Q_{surr on sys} (J) |
W_{surr on sys} (J) |
ΔE (J) |

**Adiabatic** |
P*V^{γ} (γ=1.67) |
0 |
1.5nRΔT |
1.5nRΔT |

**Isothemal** |
T |
nRT*ln(V_{f}/V_{i}) |
-nRT*ln(V_{f}/V_{i}) |
0 |

**Isobaric** |
P |
2.5nRΔT |
-nRΔT |
1.5nRΔT |

**Isochoric** |
V |
1.5nRΔT |
0 |
1.5nRΔT |

**Solution:**

The heat transfer to the system, work done on the system, change in
energy of the system, and change in temperature of the system between
each point on the PV diagram, starting in the lower right, is shown
in the table below:

Process |
Q_{surr on sys} (J) |
W_{surr on sys} (J) |
ΔE (J) |
T_{i} (K) |
T_{f} (K) |

12 |
-4.69e+03 |
1.88e+03 |
-2.81e+03 |
7.92e+02 |
5.95e+02 |

23 |
0.00e+00 |
1.21e+04 |
1.21e+04 |
5.95e+02 |
1.44e+03 |

34 |
9.84e+03 |
-9.84e+03 |
0.00e+00 |
1.44e+03 |
1.44e+03 |

41 |
0.00e+00 |
-9.30e+03 |
-9.30e+03 |
1.44e+03 |
7.92e+02 |

Efficiency: 52.34%