Carnot Cycle on Ideal Gas

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The Carnot cycle is an idealization for a heat engine operating reversibly between two reservoirs at temperatures and . The working substance is assumed to be one mole of an ideal gas with heat-capacity ratio . (For a monatomic ideal gas, has its maximum value at .) The four steps of the cycle are most commonly plotted on a pressure-volume diagram, shown on the left, with alternate isotherms (red curves ofconstant temperature) and adiabatics or isentropics (blue curves of constant entropy). A simple alternative representation is therefore a rectangle on a temperature-entropy diagram, shown as an inset.

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An accurately-drawn pV diagram of a Carnot cycle approximates a narrow crescent, in contrast to the more familiar pictures shown in many physical chemistry texts that have aspect ratios around 1.

A schematic diagram of an idealized heat engine is shown on the right. In each cycle of the engine, a quantity of heat is withdrawn from the hot reservoir at temperature . The fraction is rejected to the cold reservoir at temperature , with the difference converted into work. Numerical values of , and in kJ are shown. Since the heat is essentially wasted, the efficiency of the heat engine is expressed as the ratio . The efficiency of a Carnot cycle depends only on the temperatures of the two reservoirs: , where and are measured on the absolute temperature scale (in K). The efficiency is always less than 1. To get , the cold reservoir would have to be at , absolute zero, which is unattainable. The area enclosed by either the pV or TS curves equals the work produced per cycle.

The sliders enable you to select , the temperatures and , and the volumes and in the upper isothermal step.

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Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshot 1: A heat engine operating between the freezing and boiling points of water has an efficiency of only about 27%. Steam engines' efficiencies are in this approximate range.

Snapshot 2: 50% efficiency for

Snapshot 3: 90% efficiency for

Carnot cycles are discussed in all physical chemistry texts. See, for example, P. W. Atkins, Physical Chemistry, San Francisco: W. H. Freeman, 2006 or S. M. Blinder, Advanced Physical Chemistry, New York: Macmillan, 1969.



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