# A Brief History of Entropy pt. 2 – Talk About the Revolutions

Around 1750 Europe and especially England saw the unfolding of the so-called first industrial revolution. Engineers learned how to build mechanical machines powered by steam, a thing that in the Ancient Greece was seen just as a recreational pastime. Now those steam engines were used to make clothes, turn mills and possibly to move people. But one problem remained: their efficiencies were unsatisfactory. Unfortunately for the English, this will be solved by a young frog-eater.

### Paris, France, 1783

In this year the illuminated French revolutionary Lazare Carnot published Essai sur les machines en général. He proposed for the first time in history the idea that in a mechanical apparatus shocks and frictions dissipated a sizable fraction of usable work. However, Lazare preferred politics over science. Too bad: he was a main character during the French Revolution, but after Napoleon crowned himself emperor on 2 December 1804 he decided to leave his public life, he retired and eventually he died in exile in Prussia. He left us a famous theorem and gave his nephew Sadi a first glimpse of entropy.

Sadi Carnot was a young and brilliant military engineer who wrote in 1824 a booklet entitled Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. This work is now considered a cornerstone of classical thermodynamics and physics. In this book, Sadi explained his theory of heat engines: power is generated by the falling of the caloric fluid from a hot to a cold body. The  Carnot cycle, the Carnot heat engineCarnot’s theorem and thermodynamic efficiency are all explained in detail for the first time. Probably, his major contribution to Science is the discovery of his theorem: the ultimate efficiency of a thermal engine is given by the Carnot’s cycle and depends only on the ratio of the temperatures of the two reservoirs used to operate the machine:

$\eta = 1 - \frac{T_{L}}{T_{H}}$

Unfortunately for him, this book passed almost unnoticed for some 10 years, until a mining engineer (Emile Clapeyron) re-discovered it in 1834.

### Berlin, Prussia, 1850

Rudolf Clausius, a German professor, re-re-discovered Sadi Carnot’s work reading a paper written by the French engineer Clapeyron. Clausius stated in his paper called Über die bewegende Kraft der Wärme that there was a contradiction between the Carnot’s theorem and the conservation of energy. Before anything, he classified heat in three categories:

1. Heat used to rise the temperature of a body
2. Heat used to produce “exterior” work (typically exerted by a piston)
3. Heat used to increase the “interior” energy of a body

After stating the first law of thermodynamics in the modern fashion, Clausius went further proving that in a transformation the passage of a quantity of heat Q from a temperature T1 to a temperature T2 the “equivalence-value” (now called entropy) has a value Q(1/T2 – 1/T1). According to Carnot’s theorem, in a cycle this value must be equal or less than zero. The equivalence is obtained if the cycle is made of reversible transformations. In formulae:

$\int\frac{\delta Q}{T} \leq 0$

Later on, he called this quantity entropy (Greek: “inner transformation”) in analogy to the word energy (Greek: “inner work”). Clausius said nothing about the origin of the symbol S. Apparently, it stands for Sadi (Carnot) or perhaps for “State” of the system.