Clausius Formulation of the Second Law

Clausius proceeded to demonstrate the power of entropy to express the deep consequences of the second law. We begin by introducing the inequality of Clausius, which complements Carnot s theorem (4.25) for the irreversible case. 

For this purpose, consider a change of state A — B, which can be achieved by either a reversible path (with heat qmy and work wrev) or an irreversible path (with heat girrev and work wirrevX as shown schematically below  

As shown in Section 3.2, the reversible path yields maximum useful work  

From the first law, the internal energy change A /(A — B) must be independent of the path between states  

The sense of this inequality is preserved if we divide both sides by T ( 0)  

A JJrev — 7rev H- Wrev — Af/irrev(A B) — irrev H- Wjrrev 

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The general inequality (4.48) leads to the famous Clausius formulation of the second law ... 

This is in fact the Carnot-Clausius formulation of the Second Law [94-96] we stress here especially that the inverse implication of (1.18) is not valid (e.g., on-off cycle of electrical heating has q > 0 but q+ = 0). 

Assuming that the inner shell is a finite system with an intrinsic quantum property to expand, the process will be spontaneous . We have to do with the case where the heat passes spontaneously from a colder body to the hotter one , which is in a clear contradiction with the second law of thermodynamics in the original Clausius wording. Such a straightforward argument is rather naive (however, not too rare in the literature) and does not stand the confrontation even with a more advanced Clausius formulation of the second law of thermodynamics claiming ... 

The full significance of these observations could not be appreciated in advance of the formulation of the second law of thermodynamics by Lord Kelvin and Clausius in the early 1850 s. In a paper published in 1857 that was probably the first to treat the thermodynamics of elastic deformation, Kelvin showed that the quantity of heat Q absorbed during the (reversible) elastic deformation of any body is related in the following manner to the change with temperature in the work — TFei required to produce the deformation ... 

The term entropy, which literally means a change within, was first used in 1851 by Rudolf Clausius, one of the formulators of the second law of thermodynamics. A rigorous quantitative definition of entropy involves statistical and probability considerations. However, its nature can be illustrated qualitatively by three simple examples, each demonstrating one aspect of entropy. The key descriptors of entropy are randomness and disorder, manifested in different ways. 

The above definitions reflect the Clausius view of the origin of entropy at the beginning of the twentieth century a reformulation of thermodynamics by -> Born and Caratheodory showed firstly that the formulation of the second law of - thermodynamics requires a consideration of the heat and work relationships of at least two bodies, as implicitly discussed above, and that entropy arises in this formulation from the search for an integrating factor for the overall change in heat, dq when the simultaneous changes in two bodies are considered. The Born-Caratheodory formulation then leads naturally to the restriction that only certain changes of state are possible under adiabatic conditions. 

Several famous scientists have contributed to certain aspects of the second law of thermodynamics, among them are Carnot, Joule, Kelvin, Clausius, Planck, and Boltzmann. Various formulations of the second law have been created. This process still continues. Subsequently we will mainly focus on the formulation of Clausius. 

The formulation of the second law according to Clausius in the version from 1865 consists essentially of two statements. 

Other formulations of the second law have appeared in science. Some of these originate even earlier than the formulation of Clausius. 

This Statement was later refined by Planck. On first glance, the formulation of the second law of Clausius and the formulation of Kelvin and Planck do not seem to have much in common. However, it is shown in elementary texts that both formulations are equivalent. 

There are several formulations of the second law of thermodynamics.The so-called Clausius statement says that in spontaneous processes heat cannot fiow from a lower-temperature body to a higher-temperature body. The Thomson (Lord Kelvin) statement says that heat cannot be completely converted into work. 

The task of finding a mathematical formulation of the second law of thermodynamics was accomplished by Sadi Carnot and Clausius on the traditional, macroscopic side, and, again, by Boltzmann from a molecular, statistical perspective. What is sought is a state function that quantitatively describes the degree of dispersion in a chemical system. This is entropy, and its symbol is S. It must increase in any irreversible process. 

Rudolf Julius Emanuel Clausius, German physicist, K6slin (now Koszalin) 2.1.1822, fBonn 24.8.1888 one of the developers of the mechanical theory of heat his achievements encompass the formulation of the second law and the introduction of the entropy concept. 

The work of Carnot, published in 1824, and later the work of Clausius (1850) and Kelvin (1851), advanced the formulation of the properties of entropy and temperature and the second law. Clausius introduced the word entropy in 1865. The first law expresses the qualitative equivalence of heat and work as well as the conservation of energy. The second law is a qualitative statement on the accessibility of energy and the direction of progress of real processes. For example, the efficiency of a reversible engine is a function of temperature only, and efficiency cannot exceed unity. These statements are the results of the first and second laws, and can be used to define an absolute scale of temperature that is independent of ary material properties used to measure it. A quantitative description of the second law emerges by determining entropy and entropy production in irreversible processes. 

Prove that the Clausius statement is a necessary consequence of the second law, as formulated in this chapter. 

Then (Sq/T) becomes a state function called entropy and T the absolute temperature. As a state function, entropy is path-independent. Eqn (1.25) is a mathematical statement of the second law of thermodynamics. The introduction of the integrating factor for 8q causes the thermal energy to be split into an extensive factor S and an intensive factor T. Clausius defined the entropy with the integrating factor of the inverse of absolute temperature in T 8q) = dS. Similarly, integrating factor 1/P in IP 6W) = dV leads to exact differential dV, which is formulated by Clapeyron in 1834. Introducing Eqn (1.25) into the first law of thermodynamics dU =8q + yields the combined first and second laws of thermodynamics... 

The original proposal of Clausius for the Second Law of Thermodynamics was to formulate the fact that heat flows from a position of higher temperature to a lower one (Yamamoto 1987). In this sequence it is natural to introduce the condition of positive definiteness for the heat flux as the second part of the Second Law of Thermodynamics. 

Around 1850, however, the German physicist Rudolf Clausius and the English physicist William Thomson (later Lord Kelvin) independently showed that the concept of energy conservation implied that the work capacity of heat included the actual conversion of heat into work. Clausius and Thomson each independently formulated the limits for energy conversion processes of the second law. In 1865, Clausius postulated the fundamental principle of the constant increase of entropy. 

As formulated by Clausius, the Second Law introduces another state function, entropy (symbol S). Since entropy is a state function, its nmnerical value is determined solely by the state of the system and not by how that state is reached. Nevertheless, the prescription to calculate entropy depends on one particular path. According to the mathematical statement of the second law ... 

It is desirable to find some common measure (preferably a quantitative measure) of the tendency to change and of the direction in which change can occur. In the 1850s, Clausius and Kelvin independently formulated the second law of thermodynamics, and Clausius invented the term entropy S (from the Greek word TpoTT-rj, which means transformation), to provide a measure of the transformational content or the capacity for change. In this chapter, we will develop the properties of this function and its relationship to the direction and extent of natural processes as expressed in the second law of thermodynamics. 

The second law of thermodynamics (Clausius formulation) In isolated systems, spontaneous changes are always accompanied by a net increase in entropy. 

Rational thermodynamics is formulated based on the following hypotheses (i) absolute temperature and entropy are not limited to near-equilibrium situations, (ii) it is assumed that systems have memories, their behavior at a given instant of time is determined by the history of the variables, and (iii) the second law of thermodynamics is expressed in mathematical terms by means of the Clausius-Duhem inequality. The balance equations were combined with the Clausius-Duhem inequality by means of arbitrary source terms, or by an approach based on Lagrange multipliers. 

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