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The Carnot cycle

The most efficient such engine possible is described by the Carnot cycle, named after the French engineer Sadi Carnot who invented it. We do not describe the equipment that makes this process run, just steps which the gas goes through in one cycle:

  1. Isothermal expansion at temperature tex2html_wrap_inline253 with heat transfer tex2html_wrap_inline255
  2. Adiabatic cooling to temperature tex2html_wrap_inline257
  3. Isothermal compression at temperature tex2html_wrap_inline257 with heat transfer tex2html_wrap_inline261
  4. Adiabatic heating to temperature tex2html_wrap_inline253
The first law of thermodynamics implies that
because the cyclic nature of the process requires that there be no change in the internal energy upon completing one cycle. The efficiency of the process is
We could now consider details of the process such as what volume changes and what temperatures are involved in order to determine how these factors affect the overall efficiency of the process. An easier way and one which provides some insight is to use the entropy state function. Clearly the entropy of the thermodynamic system consisting of the gas which goes through the Carnot cycle does not change through one cycle because S is a state function. The total entropy change is readily calculated as
Introducing this simple result in Eq. 11 yields
As is argued in the book one can show that no thermodynamic process involving reservoirs at temperatures tex2html_wrap_inline253 and tex2html_wrap_inline257 can have an efficiency greater than that of the corresponding Carnot cycle. Thus the maximum achievable efficiency of any cyclic thermodynamic process is limited simply by the ratio of temperatures in the corresponding low and high temperature reservoirs.

Collin Broholm
Mon Dec 8 01:33:45 EST 1997