Carnot cycle
Carnot cycle[edit]
The concept of entropy arose from Rudolf Clausius's study of the Carnot cycle.^{[11]} In a Carnot cycle, heat Q_{H} is absorbed isothermally at temperature T_{H} from a 'hot' reservoir and given up isothermally as heat Q_{C} to a 'cold' reservoir at T_{C}. According to Carnot's principle, work can only be produced by the system when there is a temperature difference, and the work should be some function of the difference in temperature and the heat absorbed (Q_{H}). Carnot did not distinguish between Q_{H} and Q_{C}, since he was using the incorrect hypothesis that caloric theory was valid, and hence heat was conserved (the incorrect assumption that Q_{H} and Q_{C} were equal) when, in fact, Q_{H} is greater than Q_{C}.^{[12]}^{[13]} Through the efforts of Clausius and Kelvin, it is now known that the maximum work that a heat engine can produce is the product of the Carnot efficiency and the heat absorbed from the hot reservoir:

$W=\left({\frac {T_{\text{H}}T_{\text{C}}}{T_{\text{H}}}}\right)Q_{\text{H}}=\left(1{\frac {T_{\text{C}}}{T_{\text{H}}}}\right)Q_{\text{H}}$
(1)
To derive the Carnot efficiency, which is 1 − T_{C}/T_{H} (a number less than one), Kelvin had to evaluate the ratio of the work output to the heat absorbed during the isothermal expansion with the help of the Carnot–Clapeyron equation, which contained an unknown function called the Carnot function. The possibility that the Carnot function could be the temperature as measured from a zero temperature, was suggested by Joule in a letter to Kelvin. This allowed Kelvin to establish his absolute temperature scale.^{[14]} It is also known that the work produced by the system is the difference between the heat absorbed from the hot reservoir and the heat given up to the cold reservoir:

$W=Q_{\text{H}}Q_{\text{C}}$
(2)
Since the latter is valid over the entire cycle, this gave Clausius the hint that at each stage of the cycle, work and heat would not be equal, but rather their difference would be a state function that would vanish upon completion of the cycle. The state function was called the internal energy and it became the first law of thermodynamics.^{[15]}
Now equating (1) and (2) gives
 ${\frac {Q_{\text{H}}}{T_{\text{H}}}}{\frac {Q_{\text{C}}}{T_{\text{C}}}}=0$
or
 ${\frac {Q_{\text{H}}}{T_{\text{H}}}}={\frac {Q_{\text{C}}}{T_{\text{C}}}}$
This implies that there is a function of state that is conserved over a complete cycle of the Carnot cycle. Clausius called this state function entropy. One can see that entropy was discovered through mathematics rather than through laboratory results. It is a mathematical construct and has no easy physical analogy. This makes the concept somewhat obscure or abstract, akin to how the concept of energy arose.
Clausius then asked what would happen if there should be less work produced by the system than that predicted by Carnot's principle. The righthand side of the first equation would be the upper bound of the work output by the system, which would now be converted into an inequality
 $W<\left(1{\frac {T_{\text{C}}}{T_{\text{H}}}}\right)Q_{\text{H}}$
When the second equation is used to express the work as a difference in heats, we get
 $Q_{\text{H}}Q_{\text{C}}<\left(1{\frac {T_{\text{C}}}{T_{\text{H}}}}\right)Q_{\text{H}}$
 or
 $Q_{\text{C}}>{\frac {T_{\text{C}}}{T_{\text{H}}}}Q_{\text{H}}$
So more heat is given up to the cold reservoir than in the Carnot cycle. If we denote the entropies by S_{i} = Q_{i}/T_{i} for the two states, then the above inequality can be written as a decrease in the entropy
 $S_{\text{H}}S_{\text{C}}<0$
 or
 $S_{\text{H}}<S_{\text{C}}$
The entropy that leaves the system is greater than the entropy that enters the system, implying that some irreversible process prevents the cycle from producing the maximum amount of work predicted by the Carnot equation.
The Carnot cycle and efficiency are useful because they define the upper bound of the possible work output and the efficiency of any classical thermodynamic system. Other cycles, such as the Otto cycle, Diesel cycle and Brayton cycle, can be analyzed from the standpoint of the Carnot cycle. Any machine or process that converts heat to work and is claimed to produce an efficiency greater than the Carnot efficiency is not viable because it violates the second law of thermodynamics. For very small numbers of particles in the system, statistical thermodynamics must be used. The efficiency of devices such as photovoltaic cells requires an analysis from the standpoint of quantum mechanics.