Nicholas Leonard Sadi Carnot studied heat engines. In England engineers had discovered that heat could be used to expand gases and the expanding gases could be used to drive pistons in cylinders and that the movement of pistons could be used to do everything from pumping water out of flooded tin mines, to moving 'steam trains' along railway tracks. The age of mechanization had arrived.
But Carnot was curious. "How can we know that the steam is used in the most advantageous way possible to produce motive power?" he asked himself in a famous 1824 memoir, and then set out to find the answer.
He was among the first, but not the last, to examine the exchange of heat between hot and cold areas of a system. He considered that 'heat' was a fluid that flowed between those parts of a system that were 'hot' and those parts of a system that were 'cold', much in the same way as water flows from the top of a hill to the bottom of a hill. Like the water, the 'fluid heat' could be made to do useful work (like turning a water wheel, or moving a piston in an engine).
He saw two things about this 'flow' and its conversion into useful work; (1) the available energy to do work depended on the relative difference in temperatures between the 'hot' and the 'cold' part of the system, and (2) the efficiency of the heat engine was the ratio between this available energy and the total energy in the system.
(1) Take the case of water flowing down a hill. If the water falls from a hill 10 meters above sea level into a pond only 4 meters above sea level it has passed through a distance of 6 meters (10 - 4). Assume that this turns a water wheel 100 times (the useful work done). Carnot saw that it did not matter if the hill and water wheel were at sea level or on top of a mountain, what was important was the difference between the two heights (6 meters in this case). This was the available energy to do work.
(2) If all the energy in a load of wood or a pile of coal could be turned into useful work, then the engine would be 100 percent efficient. No engine ever reaches this level of efficiency. No matter how hard engineers try to squeeze every last drop of energy out of the fuel they put into engines, some is always either unavailable or wasted. The ratio of the available energy to the total energy is the maximum efficiency an engine can every reach.
That part of energy which is never available to do work, Rudolf Julius Emanuel Clausius called entropy, a word of uncertain origins. He was able to show that entropy could be measured and expressed in units of calories per degree Celsius.
Carnot realized that in any practical conversion of fuel (wood, coal, oil) into heat and then into mechanical work (turning a steam engine), some part of the stored energy (the potential energy) is always lost of dissipated and does no useful work. This 'lost' energy, unavailable to do work, is the entropy defined by Clausius.
As time went by physicists began to realize that there was more to physics than heat engines, and began to develop a wider appreciation of the idea of entropy. There are now modern versions of the thermodynamic definition of entropy, kinetic definitions of entropy, statistical definitions of entropy, studies of the consequences of the entropy conception, restrictions to this conception, limits to the concept of entropy and even extrascientific applications; enough to make your head spin.
Carnot did not know what he started.