Summary

General Principles

Heat Engines

Devices that transform heat into work. They require two energy reservoirs at different temperatures.

The figure illustrates the energy transfer diagram of a heat engine.

Thermal efficiency

η = \frac{W_{out}}{Q_{H}} = \frac{what you get}{what you pay}

Second-law limit:

η \leq 1 - \frac{T_{C}}{T_{H}}

Refrigerators

Devices that use work to transfer heat from a colder object to a hotter object.

The figure illustrates the energy transfer diagram of a refrigerator.

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Coefficient of performance

K = \frac{Q_{C}}{W_{in}} = \frac{what you get}{what you pay}

Second-law limit:

K \leq \frac{T_{C}}{T_{H} - T_{C}}

Important Concepts

A perfectly reversible engine (a Carnot engine) can be operated as either a heat engine or a refrigerator between the same two energy reservoirs by reversing the cycle and with no other changes.

A Carnot heat engine has the maximum possible thermal efficiency of any heat engine operating between $T_{H}$ and $T_{C}$ because $Δ S_{universe} = 0$ :

$η_{Carnot} = 1 - \frac{T_{C}}{T_{H}}$
A Carnot refrigerator has the maximum possible coefficient of performance of any refrigerator operating between $T_{H}$ and $T_{C} :$

$K_{Carnot} = \frac{T_{C}}{T_{H} - T_{C}}$

The Carnot cycle for a gas engine consists of two isothermal processes and two adiabatic processes.

The figure illustrates the Pressure versus Volume graph of the Carnot engine. The graph represents a cyclic process.

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An energy reservoir is a part of the environment so large in comparison to the system that its temperature doesn’t change as the system extracts heat energy from or exhausts heat energy to the reservoir. All heat engines and refrigerators operate between two energy reservoirs at different temperatures $T_{H}$ and $T_{C}$ .

The work $W_{s}$ done by the system has the opposite sign to the work done on the system.

W_{s} = area under p V curve

The figure illustrates a pressure versus volume graph.

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Applications

To analyze a heat engine or refrigerator:

Model Identify each process in the cycle.

Visualize Draw the $p V$ diagram of the cycle.

Solve There are several steps:

Determine p, V, and T at the beginning and end of each process.
Calculate $Δ E_{th}, W_{s}$ , and Q for each process.
Determine $W_{in}$ or $W_{out}, Q_{H}$ , and $Q_{C}$ .
Calculate $η = W_{out} / Q_{H}$ or $K = Q_{C} / W_{in}$ .

Review Verify $(Δ E_{th})_{net} = 0$ . Check signs.