CARATHEODORY THEOREM THERMODYNAMICS PDF

The second law of thermodynamics states that the total entropy of an isolated system can never .. Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the Carnot’s theorem () is a principle that limits the maximum efficiency for any possible engine. this energy, as stated by the first law of thermodynamics, is so widely applied that the law introduced by Caratheodory [6], which asserts that arbitrarily close to any .. At last we are in a position to state the definitive theorem concerning the. PDF | During a recent round of teaching a course on Thermodynamics and alluded to in many books, is seldom presented with the theorems needed for its.

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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibriumor is undergoing a fictive reversible process. In all processes that occur, including spontaneous processes[1] the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense.

The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past. Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanicsclassical or quantumexplains the microscopic origin of the law.

The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnotwho in showed that there is an upper limit to the efficiency of conversion of heat to work, in a heat engine. The first law of thermodynamics provides the basic definition of internal energyassociated with all thermodynamic systemsand states the rule of conservation of energy.

For example, heat always flows spontaneously from hotter to colder bodies, and never the reverse, unless external work is performed on the system. The explanation of the phenomena was given in terms of entropy. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality [9].

This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature T and the temperature of the surroundings T surr.

The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics — the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.

The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. The second law allows [ how? These statements cast the law in general physical terms citing the impossibility of certain processes.

The Clausius and the Kelvin statements have been shown to be equivalent.

The historical origin of the second law of thermodynamics was in Carnot’s principle. It refers to a cycle of a Carnot heat enginefictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium.

The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot’s principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamicsand before the mathematical expression of the concept of entropy.

Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today.

The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance.

A Carnot engine operated in this carathedoory is the most efficient possible heat engine using those two temperatures. The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in by examining the relation between heat transfer and thdrmodynamics.

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Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.

The statement by Clausius uses the concept of ‘passage of heat’. As is usual in thermodynamic discussions, this means ‘net transfer of energy as heat’, and does not refer to contributory transfers one way and the other. Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example.

In a tyeorem, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.

Lord Kelvin expressed the second law as. Suppose there is an engine violating the Kelvin statement: Now pair it with a reversed Carnot engine as shown by the figure. Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent. Planck offered the following proposition as derived directly from experience.

This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law. It is almost customary in textbooks to speak of the ” Kelvin-Planck statement ” of the law, as for example in the text by ter Haar and Wergeland.

The Kelvin—Planck statement or the heat engine statement of the second law of thermodynamics states that. In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S. With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. InMax Planck wrote an important paper on the basics of thermodynamics.

This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law.

Second law of thermodynamics – Wikipedia

A closely thoerem statement is that “Frictional pressure never does positive work. Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin gheorem given just above. Nevertheless, this principle of Planck is not actually Planck’s preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.

Second law of thermodynamics

A statement that in a sense is complementary to Planck’s principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:. Differing from Planck’s just foregoing principle, this one is explicitly in terms of entropy change.

Removal of matter from a system can also decrease its entropy. The second law has been shown to be equivalent to the internal energy U being a weakly convex functionwhen written as a function of extensive properties mass, volume, entropy, Before the establishment of the second law, many people who were interested theoerm inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine.

Such a machine is called a “perpetual motion theore of the second kind”. The second law declared the impossibility of such machines.

Carnot’s theorem is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility.

Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle ‘s reversibility and was condemned to be less efficient. Though formulated in terms of caloric see the obsolete caloric theoryrather than entropythis was an early insight into the second law. The Clausius theorem states that in a cyclic process. The equality holds in the reversible case [54] and the strict inequality holds in the irreversible case.

The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.

Carnot’s theorem states that all reversible engines operating between the same heat rheorem are equally efficient. Thus, any reversible heat engine operating between temperatures T 1 and T 2 must have the same efficiency, that is to say, the efficiency is the function of temperatures only: In addition, a reversible heat engine operating between temperatures T 1 and Craatheodory 3 must have the same efficiency as one consisting of two cycles, one between T 1 and another intermediate temperature T 2and the second between T 2 and T 3.

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This can only be the case if. According to the Clausius equalityfor a reversible process. So we can define a state function S called entropy, which for a reversible process or for pure heat transfer [12] satisfies. With this we can only obtain the difference of entropy by integrating the above formula.

entropy – Principle of Caratheodory and The Second Law of Thermodynamics – Physics Stack Exchange

For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.

Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop. An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system called the total system or universemade up of two parts: Whatever changes to dS and dS R occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S tot of the isolated total system must not decrease:.

It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem. The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact. In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.

This expression together with the associated reference state permits a design engineer working at the macroscopic scale above the thermodynamic limit to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. Also, see process engineer. Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state.

An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found See second law efficiency.

This approach to the Second Law is widely utilized in engineering practice, environmental accountingsystems ecologyand other disciplines. Thus, a negative value of the change in free energy G or A is a necessary condition for a process to be spontaneous.

This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.

He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.

Recognizing the significance of James Prescott Joule ‘s work on the conservation of energy, Rudolf Clausius was the first to formulate the second law duringin this form: While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy Established during the 19th century, the Kelvin-Planck statement of the Second Law says, “It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work.

The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is therrmodynamics to the volume of this region, i. Equivalently, it says that time average and average over the statistical ensemble are the same. Theroem is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular ‘disorder’ within a macroscopic system. This doctrine is obsolescent.