In the history of physics, the concept of entropy developed in response to the observation that a certain amount of functional energy released from combustion reactions is always lost to dissipation or friction and is thus not transformed into useful work. Early heat-powered engines such as Thomas Savery's (1698), the Newcomen engine (1712) and Nicolas-Joseph Cugnot's steam tricycle (1769) were inefficient, converting about 0.5% of the input energy into useful work output. Over the next two centuries, physicists investigated this puzzle of lost energy; the result was the concept of entropy. In the early 1850s, Rudolf Clausius set forth the concept of the thermodynamic system and posited the argument that in any irreversible process a small amount of heat energy δQ is incrementally dissipated across the system boundary. Clausius continued to develop his ideas of lost energy, and coined the term entropy. Since the mid-20th century, the concept of entropy has found application in the field of information theory, describing an analogous loss of data in information transmission systems.
Classical thermodynamic views In 1803, mathematician Lazare Carnot published a work entitled Fundamental Principles of Equilibrium and Movement. This work includes a discussion on the efficiency of fundamental machines, i.e. pulleys and inclined planes. Carnot saw through all the details of the mechanisms to develop a general discussion on the conservation of mechanical energy. Over the next three decades, Carnot's theorem was taken as a statement that in any machine the accelerations and shocks of the moving parts all represent losses of moment of activity, i.e. the useful work done. From this Carnot drew the inference that perpetual motion was impossible. This loss of moment of activity was the first-ever rudimentary statement of the second law of thermodynamics and the concept of 'transformation-energy' or entropy, i.e. energy lost to dissipation and friction. Carnot died in exile in 1823. During the following year his son Sadi Carnot, having graduated from the École Polytechnique training school for engineers, but now living on half-pay with his brother Hippolyte in a small apartment in Paris, wrote Reflections on the Motive Power of Fire. In this book, Sadi visualized an ideal engine in which any heat (i.e., caloric) converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Building on his father's work, Sadi postulated the concept that "some caloric is always lost" in the conversion into work, even in his idealized reversible heat engine, which excluded frictional losses and other losses due to the imperfections of any real machine. He also discovered that this idealized efficiency was dependent only on the temperatures of the heat reservoirs between which the engine was working, and not on the types of working fluids. Any real heat engine could not realize the Carnot cycle's reversibility, and was condemned to be even less efficient. This loss of usable caloric was a precursory form of the increase in entropy as we now know it. Though formulated in terms of caloric, rather than entropy, this was an early insight into the second law of thermodynamics.
Clausius' definitions
1854 definition
In his 1854 memoir, Clausius first develops the concepts of interior work, i.e. that "which the atoms of the body exert upon each other", and exterior work, i.e. that "which arise from foreign influences [to] which the body may be exposed", which may act on a working body of fluid or gas, typically functioning to work a piston. He then discusses the three categories into which heat Q may be divided:
Heat employed in increasing the heat actually existing in the body. Heat employed in producing the interior work. Heat employed in producing the exterior work. Building on this logic, and following a mathematical presentation of the first fundamental theorem, Clausius then presented the first-ever mathematical formulation of entropy, although at this point in the development of his theories he called it "equivalence-value", perhaps referring to the concept of the mechanical equivalent of heat which was developing at the time rather than entropy, a term which was to come into use later. He stated:
the second fundamental theorem in the mechanical theory of heat may thus be enunciated: If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat Q from work at the temperature T, has the equivalence-value:
Q T {\displaystyle {\frac {Q}{T}}}
and the passage of the quantity of heat Q from the temperature T1 to the temperature T2, has the equivalence-value:
Q ( 1 T 2 − 1 T 1 ) {\displaystyle Q\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right)}
wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected. In modern terminology, that is, the terminology introduced by Clausius himself in 1865, we think of this equivalence-value as "entropy", symbolized by S. Thus, using the above description, we can calculate the entropy change ΔS for the passage of the quantity of heat Q from the temperature T1, through the "working body" of fluid, which was typically a body of steam, to the temperature T2 as shown below:
If we make the assignment:
S = Q T {\displaystyle S={\frac {Q}{T}}}
Then, the entropy change or "equivalence-value" for this transformation is:
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