We will calculate the efficiency of a different type of cyclic reversible thermodynamic process called the Stirling cycle. We also have at hand demonstration of such a Stirling engine which we will test after having gone through the calculation.
The Stirling cycle has the following steps:
to 
at temperature 
.
to 
at volume .
to at temperature .
to
We first calculate the amount of work done. Using the result previously
derived for the isothermal process we can immediately write
As opposed to the Carnot cycle where heat only flows to the system in
isothermal processes heat flows to the working gas of a Stirling engine
both during the isochoric heating process and during the isothermal
expansion. For these processes we can write
We can then derive the efficiency of the engine for converting heat into
mechanical work:
where we defined 
as the efficiency of the corresponding
Carnot cycle and we used the equipartition theorem to
introduce the number of degrees of freedom of the
working substance (Note that a previous version of this document had a misprint in this formula
which was repeated on the blackboard in class).
Thus we clearly see that the Stirling cycle is less
efficient than the Carnot cycle. The reduction in efficiency
results from the isochoric process where heat goes in but no work
gets done. The efficiency of the Stirling engine is greater
the fewer degrees of freedom per atom in the working substance,
so a machine based on a Mon-atomic gas would work better than one based on
a more complicated molecular gas. We also see that the larger the ratio of
volumes involved the close the Stirling process approximates the Carnot
cycle.
After understanding the mathematical description of the Stirling cycle we explain and test an actual Stirling engine.