Entropy from Carnot cycles

The Carnot cycle is formulated directly from the second law of thermodynamics. It is a perfectly reversible, adiabatic cycle consisting of two constant entropy processes and two constant temperature processes. It defines the ultimate efficiency for any process operating between two temperatures. The coefficient of performance (COP) of the reverse Carnot cycle (refrigerator) is expressed as... 

In summary, the Carnot cycle can be used to define the thermodynamic temperature (see Section 2.2b), show that this thermodynamic temperature is an integrating denominator that converts the inexact differential bq into an exact differential of the entropy dS, and show that this thermodynamic temperature is the same as the absolute temperature obtained from the ideal gas. This hypothetical engine is indeed a useful one to consider. 

Carnot efficiency is one of the cornerstones of thermodynamics. This concept was derived by Carnot from the impossibility of a perpetuum mobile of the second kind [ 1]. It was used by Clausius to define the most basic state function of thermodynamics, namely the entropy [2]. The Carnot cycle deals with the extraction, during one full cycle, of an amount of work W from an amount of heat Q, flowing from a hot reservoir (temperature Ti) into a cold reservoir (temperature T2 < T ). The efficiency r] for doing so obeys the following inequality ... 

In the development of the second law and the definition of the entropy function, we use the phenomenological approach as we did for the first law. First, the concept of reversible and irreversible processes is developed. The Carnot cycle is used as an example of a reversible heat engine, and the results obtained from the study of the Carnot cycle are generalized and shown to be the same for all reversible heat engines. The relations obtained permit the definition of a thermodynamic temperature scale. Finally, the entropy function is defined and its properties are discussed. 

The pioneering work in the direction of the second law of thermodynamics is considered to be performed in 1825 by Sadi Carnot investigating the Carnot cycle [51] [40]. Carnot s main theoretical contribution was that he realized that the production of work by a steam engine depends on a flow of heat from a higher temperature to a lower temperature. However, Clausius (1822-1888) was the first that clearly stated the basic ideas of the second law of thermodynamics in 1850 [13] and the mathematical relationship called the inequality of Clausius in 1854 [51]. The word entropy was coined by Clausius in 1854 [51]. 

For any completed reversible cycle, and therefore for the particular case of a Carnot Cycle (which has been studied in the elementary treatment, Chap I), we know from previous consideration that < U = o, i.e the U is once more at its original value Similarly <75 = o, le the entropy of the system is once more at its onginal value when the cycle is complete Since internal energy and entropy depend only on the initial and final states, and these states are, of course, identical for a complete cycle, the entropy and internal energy do not depend on the path followed The expression is, however, not zero, t e there has been a nett gam or loss of external work by the system, and hence is not, zero, there has been a nett addition or subtraction of heat energy to or from the system to balance the work done by or done on the system at some stage or stages of the transformation Let us see what these work and heat terms are in the special case of a Carnot Cycle... 

Figure 4.3-2 The Carnot cycle, (a) Schematic diagram of a Carnot engine. (6) The Carnot cycle on a pressure-volume plot, (c) The Carnot cycle on a temperature-entropy plot. (4) Heat flow into the cycle going from point a to b. (e) Heat flow-into the cycle going from point c to point...

Problem 4.21 A Carnot cycle using steam in a closed system operates between 700 "C and 500 "C. During the isothermal step the steam expands from 20 bar to 10 bar. Perform the energy and entropy balances, calculate the efficiency, and compare to the theoretical value. 

Figure 4.11 The entropy of water. The contours are labeled in bars. The arrows outline a Carnot cycle. Data from program STEAM (Harvey et al. 2000).

This question has been around since Clausius invented the term in 1865, and the answer takes on many forms. Some follow the historical route, from steam engines, to Carnot, Clausius, Thompson, Joule, Rankine, and so on. A particularly lucid, concise account of this history is Purrington (1997). A central feature of this approach is Carnot cycles, as used by Clausius to deduce the existence of the entropy parameter. This approach is rather abstract, and needs some manipulation to be seen to be connected to thermodynamic potentials and chemical reactions. Others emphasize the impossibility of some processes, or the availability of energy, and some have a rather unique viewpoint, such as Reiss (1965), who considers entropy as the degree of constraint. ... 

However, such discussions tend to be rather long and abstract, and exactly how conclusions about irreversible heat transfers in Carnot cycles get transferred to irreversibility in chemical reactions requires even more discussion. This detracts from the task attempted here - the shortest possible, intuitively clear, development of the concepts necessary to use thermodynamics in solving Earth science problems. Readers who want to know more about the historical development of the entropy concept and its deeper meaning must consult the many excellent treatises on this subject. 

Lumsden s Thermodynamics of Alloys develops the thermodynamics of metals from the fundamental laws up to the applications of statistical mechanics. The concept of entropy is approached via randomness rather than via Carnot cycles. Applications of the theory to the correlation of the properties of pure metals are shown and the theories of solution are developed. Solid solutions, and the statistical mechanics of liquids, liquid solutions, and imperfect crystals, are also considered. 

Thus, we obtain the same formula for the efficiency of a heat engine as from the Carnot cycle. Along the way, we realized that when heat flows from hot to cold, the environment wUl gain the exhaust heat 1 1 so that the environment gains enfropy. A profound result of this sort of analysis is that entropy tends to increase in the environment unless there is some other condition and the overall entropy in the universe tends to increase. Especially for biology majors and generally for aU of us. 

The second law of thermodynamics stems from the studies in the 1800 s of heat engines, and in the this period s theories on the motive power of heat. Therefore, the second law is introduced, along with the concept of entropy, through the Carnot cycle for a heat engine operating with an ideal gas. The energy considerations used in the Carnot process are universal and thus they lead to general conditions of equilibrimn for thermodynamic systems. 

Carnot, Nicolas Leonard Sadi (1796-1832)AFrenchphysicistwhobeganhlscareer as a military engineer before turning to scientific research. In 1824 he published a book Reflections on the Motive Power of Fire, which provided for the first lime a general theoretical approach to understanding the conditions under which the efficiency of heat engines could be maximized. The thermodynamic Carnot cycle eventoally led to the concept of entropy. He died aged 36 from cholera. 

In other words, the same way that these imperfections prevent the continuous motion of a body or the conversion of its kinetic energy completely into work, the generation of entropy - resulting also from such physical imperfections - prevents the realization of the Carnot cycle efficiency. 

In a Carnot s cycle, the entropy Qi/Ti is taken from the hot reservoir, and the entropy Q2/T2 is given up to the cold reservoir, and no other entropy change occurs anywhere else. Since these two quantities of entropy are equal and opposite, the entropy. change in the hot reservoir is exactly balanced, or, to use an expression of Clausius, is compensated by an equivalent change in the cold reservoir. Again, in any reversible cycle there is on the whole no production of entropy so that all the changes are compensated. 

Fig. 4-7. Carnot process or reversible cycle. Starting at point A, the cycle consists of an isentropic compression, an isothermal expansion, an isentropic expansion, and an isothermal compression. All steps are reversible. The open arrows symbolize the heat flow in a real technical process where there are temperature drops in the heat transfer to and from the process medium. The entropy flows from the heat source to the heat sink respectively, correspond to the hatched lines in that case.

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