Chapter 3 The Second Law of Thermodynamics The first law of thermodynamics is the law of energy conservation. The second law has a very different nature; it establishes the rules for the irreversibility of the natural processes. If we bring a pendulum out of its equilibrium position and let it go, its oscillations are ample at the beginning, but gradually decrease in their amplitude and finally stop after a shorter or longer period. The energy, which was initially mechanical energy, has not been lost; rather, it became internal energy of the pendulum and of the surrounding air. The first law does not forbid the inverse process, namely having a pendulum at rest starting oscillations of increasing amplitude, while its temperature and that of the air decrease. Similarly, if we leave a pot of hot coffee on a table, the liquid cools down in time, while the air heats up (not by too much, obviously). We never observe coffee at room temperature heating up while the air cools down. As we shall see, the second law forbids both types of phenomenon. As we already mentioned, the second law was discovered before the first, in the historic period during which engineers were developing thermal engines, namely devices able to perform mechanical work using the heat produced by combustion. In this case, the work of the engineers, theoretical interpretations included, antici- pated that of the physicists. In Sect. 3.1, we state the second law. Both for historical and didactic reasons, we shall give two statements, one attributed to Clausius, one to Lord Kelvin, and then prove their equivalence. All engines operate on cyclic, rather than open, processes. Indeed, an open process can be performed only once, while a cyclic process can continue indefi- nitely. The simplest cycle compatible with the second law exchanges heat with two sources. This is called the Carnot cycle, and we shall study it in the subsequent three sections, together with the fundamental Carnot theorem. In Sect. 3.5, we shall introduce the concept of thermodynamic temperature, which, as we anticipated, allows for extending the scale down to absolute zero. In Sect. 3.6, we shall demonstrate the fundamental Clausius theorem, which leads to the definition of a state function, entropy. This is the function directly © Springer International Publishing Switzerland 2016 A. Bettini, A Course in Classical Physics 2—Fluids and Thermodynamics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-30686-5_3 93