### The Stirling engine

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:

1. Isothermal expansion from to at temperature .
2. Isochoric lowering of temperature from to at volume .
3. Isothermal contraction from to at temperature .
4. Isochoric increase of temperature from 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.