Classical Thermodynamics : Lecture 14 - The Clausius inequality

In this lecture I first provide a proof of the Clausius inequality, a restriction that all cyclic processes must obey. While the proof shows that this inequality is a consequence of the second law of thermodynamics, I then go on to show that it, in its turn implies the second law. Both the Kelvin-Planck statement of the second law and the Clausius statement of the second law follow simply from this. Not only that, it is easy to prove the Carnot theorems from the inequality. Indeed, we demonstrate that it is possible to use the inequality to establish a stronger result - the generalized Carnot theorem. This states that no engine can be more efficient than a Carnot engine that runs between the maximum and minimum temperatures attained during its cycle. The real importance of the Clausius inequality is that it leads to the concept of entropy. We end this lecture by indicating how the concept of entropy enters into thermodynamics.