Clausius Statement Research Articles

A Symmetric Form of the Clausius Statement of the Second Law of Thermodynamics.

Bridgman once reflected on thermodynamics that the laws of thermodynamics were formulated in their present form by the great founders of thermodynamics, Kelvin and Clausius, before all the essential physical facts were in, and there has been no adequate reexamination of the fundamentals since. Thermodynamics still has unknown possibilities waiting to be explored. This paper begins with a brief review of Clausius's work on the second law of thermodynamics and a reassessment of the content of Clausius's statement. The review tells that what Clausius originally referred to as the second law of thermodynamics was, in fact, the theorem of equivalence of transformations (TET) in a reversible cycle. On this basis, a new symmetric form of Clausius's TET is proposed. This theorem says that the two transformations, i.e., the transformation of heat to work and the transformation of work from high pressure to low pressure, should be equivalent in a reversible work-to-heat cycle. New thermodynamic cyclic laws are developed on the basis of the cycle with two work reservoirs (two pressures), which enriches the fundamental of the second law of thermodynamics.

Interdependence of the First and Second Laws of Thermodynamics

Historically, classical thermodynamics was developed during attempts to improve the efficiency of steam engines. Those investigations led to the well-known zeroth, first, and second laws of thermodynamics. Although these laws are presented independently, linkages between the zeroth and second laws have been pointed out in the physics teaching literature. In our 2020 paper, “What if Energy Flowed from Cold to Hot? Counterfactual Thought Experiments,” we imagined a universe in which energy spontaneously flowed from colder to hotter objects. That paper inspired us to explore the connections between the first law and the Clausius statement of the second law in the present paper.

Clausius Statement of Second Law of Thermodynamics: A Review and Engineering Perspective of the Law

In this paper, the different forms of the Clausius statement of second law of thermodynamics presented by various authors have been critically reviewed and compared with the original statement of Clausius to identify the variations. The notable deviations from the original Clausius statement have been found regarding the conditions and constraints for applicability of this law, which can be of great concern for engineers. It has been found that the original statement has no ambiguity and it addresses all the issues relating to heat transfer between cold and hot bodies and hence it has been concluded that the original Clausius statement of the law must be always stated verbatim. Since the engineers regard two-way radiation heat transmission for net heat transfer calculation, i.e., from hot to the cold as well as from cold to the hot body, the law has been reviewed from a heat transfer engineer's point of view for the radiation heat transfer and modification in the original Clausius statement or inclusion of an explanation has been suggested for the sake of clarity or to avoid any ambiguity.

A Review of the System-Intrinsic Nonequilibrium Thermodynamics in Extended Space (MNEQT) with Applications.

The review deals with a novel approach (MNEQT) to nonequilibrium thermodynamics (NEQT) that is based on the concept of internal equilibrium (IEQ) in an enlarged state space involving internal variables as additional state variables. The IEQ macrostates are unique in and have no memory just as EQ macrostates are in the EQ state space . The approach provides a clear strategy to identify the internal variables for any model through several examples. The MNEQT deals directly with system-intrinsic quantities, which are very useful as they fully describe irreversibility. Because of this, MNEQT solves a long-standing problem in NEQT of identifying a unique global temperature T of a system, thus fulfilling Planck’s dream of a global temperature for any system, even if it is not uniform such as when it is driven between two heat baths; T has the conventional interpretation of satisfying the Clausius statement that the exchange macroheatflows from hot to cold, and other sensible criteria expected of a temperature. The concept of the generalized macroheat converts the Clausius inequality for a system in a medium at temperature into the Clausius equality, which also covers macrostates with memory, and follows from the extensivity property. The equality also holds for a NEQ isolated system. The novel approach is extremely useful as it also works when no internal state variables are used to study nonunique macrostates in the EQ state space at the expense of explicit time dependence in the entropy that gives rise to memory effects. To show the usefulness of the novel approach, we give several examples such as irreversible Carnot cycle, friction and Brownian motion, the free expansion, etc.

Towards understanding the mathematics of the \(2^{nd}\) law of thermodynamics

In this paper, the mathematical formulation of \(2^{nd}\) law of thermodynamics has been explained, and the mathematical formulation of \(1^{st}\) law has been revisited from this noble perspective. It is not claimed that the \(2^{nd}\) law of thermodynamics is a redundant of the \(1^{st}\) law, rather I shown here how we can extract the mathematical formulation of the \(2^{nd}\) law from the mathematical formulation of the \(1^{st}\) law of thermodynamics. The Clausius statement of the \(2^{nd}\) law of thermodynamics is, it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. An alternative statement of the law is, "All spontaneous processes are irreversible" or, "the entropy of an isolated system always increases". Having strong experimental evidences, this empirical law is obvious, which tells us the arrow of time and the direction of spontaneous changes.

Non-Clausius heat transfer: The method of the nonstationary Langevin equation.

Compared to other formulations of the second law of thermodynamics, the Clausius statement that heat does not spontaneously flow from cold to hot concerns a system in nonequilibrium states, and in that respect is more ambitious but also more ambiguous. We discuss two scenarios when the Clausius statement in its plain form does not hold. First, for ergodic systems, the energy transfer may be consistent with the statement on a coarse-grained timescale, but be anomalously directed during time intervals shorter than the thermalization time. In particular, when an initially colder system is brought in contact to a hotter bath, the internal energy of the former increases with time in a long run but not monotonically. Second, the heat transfer may not respect the Clausius statement on any timescale in nonergodic systems due to the formation of localized vibrational modes. We illustrate the two scenarios with a familiar model of an isotope atom attached to a semi-infinite harmonic atomic chain. Technically, the discussion is based on a Langevin equation for the isotope, using the initial condition when the isotope and chain are initially prepared in uncorrelated canonical states under the constraint that the boundary atom between the isotope and chain is initially fixed and later released. In such setting, the noise in the Langevin equation is nonstationary, and the fluctuation-dissipation relation has a nonstandard form.

What Is the Real Clausius Statement of the Second Law of Thermodynamics?

In this paper, we first analyze the difference between the second law of thermodynamics and the laws in other disciplines. There are some phenomena in other disciplines similar to the Clausius Statement of the second law, but none of them has been accepted as the statement of a certain law. Clausius’ mechanical theory of heat, published in the nineteenth century, is then introduced and discussed in detail, from which it is found that Clausius himself regarded “Theorem of the equivalence of the transformation of heat to work, and the transformation of heat at a higher temperature to a lower temperature”, rather than “Heat can never pass from a colder to a warmer body without some other change”, as the statement of the second law of thermodynamics. The latter is only laid down as the fundamental principle for deriving the theorem of the equivalence of transformations. Finally, based on the theorem of the equivalence of transformations and the average temperature method, a general quantitative relation among the heat, the work, and the temperatures is obtained for arbitrary cycles, which is thus recommended as an alternative mathematic expression of the second law. Hence, the theorem of the equivalence of transformations is the real Clausius Statement of the second law of thermodynamics.

The misunderstanding of Clausius statement of the second law of thermodynamics and its aftermath

As with the first law of thermodynamics, the second law of thermodynamics has been proven by countless examples. The Clausius statement of the second law of thermodynamics given in most of the literatures is “heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time”. After carefully studying Clausius’ mechanical theory of heat published from 1850 to 1865, we found that this is not true. Actually, Clausius already clearly stated that the second law of thermodynamics is “Theorem of the equivalence of the transformation of work to heat and the transformation of heat at a higher to a lower temperature”. The former statement is only a basic principle that can lead to the theorem of the equivalence of transformations. In addition, all processes in derivation of the theorem of the equivalence of transformations and proof of the equivalence between Clausius statement and Kelvin statement are reversible processes. The statement “Heat can never pass from a colder to a warmer body without some other change” only illuminates the spontaneous natural phenomenon in opposite direction that heat flows from high temperature to low temperature, which tells that heat at different temperatures are nonequivalent. Admittedly, Clausius discussed the process irreversibility and thereby obtained Clausius Inequality. But he only used the state change to measure the process irreversibility, which did not involve the concept of space, time and speed. Clausius just expressed that the process irreversibility will reduce heat efficiency and did not focus exclusively on the laws of process irreversibility. The misunderstanding of viewing Clausius statement as “Heat can never pass from a colder to a warmer body without some other change”, easily produces such a viewpoint that the second law of thermodynamics is the rule about process direction and irreversibility, consequently leading to the faulty assumption that the minimum entropy generation principle can be used to derive the transport law of heat conduction and optimize the heat transfer process. However, theory and practice analyses have proven that it is problematic. The action of heat conduction process is just half of entransy dissipation, whose variation points to the linear Fourier’s heat conduction law, which thereby has a better application for the optimization of heat transfer process. Transfer process has different laws from conversion process, so they should be treated distinguishingly and analyzed with different methods and indexes. It is not advisable to simply put heat transfer issues into category of non-equilibrium thermodynamics.

Role of quantum coherence in the thermodynamics of energy transfer.

Recent research on the thermodynamic arrow of time, at the microscopic scale, has questioned the universality of its direction. Theoretical studies showed that quantum correlations can be used to revert the natural heat flow (from the hot body to the cold one), posing an apparent challenge to the second law of thermodynamics. Such an "anomalous" heat current was observed in a recent experiment (K. Micadei et al., arXiv:1711.03323), by employing two spin systems initially quantum correlated. Nevertheless, the precise relationship between this intriguing phenomenon and the initial conditions that allow it is not fully evident. Here, we address energy transfer in a wider perspective, identifying a nonclassical contribution that applies to the reversion of the heat flow as well as to more general forms of energy exchange. We derive three theorems that describe the energy transfer between two microscopic systems, for arbitrary initial bipartite states. Using these theorems, we obtain an analytical bound showing that certain type of quantum coherence can optimize such a process, outperforming incoherent states. This genuine quantum advantage is corroborated through a characterization of the energy transfer between two qubits. For this system, it is shown that a large enough amount of coherence is necessary and sufficient to revert the thermodynamic arrow of time. As a second crucial consequence of the presented theorems, we introduce a class of nonequilibrium states that only allow unidirectional energy flow. In this way, we broaden the set where the standard Clausius statement of the second law applies.

Second Law Redux

Classical thermodynamics, as usually presented to engineering students, offers a midcourse impediment known as the Second Law. The Second Law suffers from a lack of intuitive clarity. The objective of the present work is to provide clarity, for simple systems to aid students and teachers. In this effort, it is demonstrated that Planck’s statement of the Second Law can be viewed as a corollary of a more fundamental observation, namely that work must always be done against friction. Also, Clausius's statement is demonstrated, entropy is defined, and Caratheodory’s statement is validated.

The local approach to quantum transport may violate the second law of thermodynamics

Clausius statement of the second law of thermodynamics reads: Heat will flow spontaneously from a hot to cold reservoir. This statement should hold for transport of energy through a quantum network composed of small subsystems each coupled to a heat reservoir. When the coupling between nodes is small, it seems reasonable to construct a local master equation for each node in contact with the local reservoir. The energy transport through the network is evaluated by calculating the energy flux after the individual nodes are coupled. We show by analyzing the most simple network composed of two quantum nodes coupled to a hot and cold reservoir, that the local description can result in heat flowing from cold to hot reservoirs, even in the limit of vanishing coupling between the nodes. A global derivation of the master equation which prediagonalizes the total network Hamiltonian and within this framework derives the master equation, is always consistent with the second law of thermodynamics.

Efficiency of heat engines coupled to nonequilibrium reservoirs

We consider quantum heat engines that operate between nonequilibrium stationary reservoirs. We evaluate their maximum efficiency from the positivity of the entropy production and show that it can be expressed in terms of an effective temperature that depends on the nature of the reservoirs. We further compute the efficiency at maximum power for different kinds of engineered reservoirs and derive a nonequilibrium generalization of the Clausius statement of the second law.

Motion machines of second kind

The second law of thermodynamics, i.e. the law stating that the entropy in isolated macroscopic system can never decrease, is tightly connected to the work of the device called perpetual motion machine of second kind. Often this law is also defined as the inability to construct such a device. In the current paper we give complete, independent and consistent definitions of static, stationary and changing physical field. Based on that for the first time we give summarising, correct and complete definitions of natural resource machine and perpetual motion machine of second kind as well as motion machine of second kind in the set of tardyons and luxons. We present a principal structure of a motion machine of second kind using which we show that the Clausius statement and its equivalent statements in the thermodynamics can be violated for a practically big interval-time even under equilibrium fluctuations.

A Missing Deduction of the Clausius Equality and Inequality

The two main statements of the second law of thermodynamics were given by Clausius, and Kelvin and Planck. Other statements followed well into the 20th century. It is known that the two main statements are not exactly equivalent, although their differences may never show outside academic environments or very specific areas of science. At negative absolute temperatures, a phenomenon studied mainly in connection to the constitution of matter and to its behavior under specific electromagnetic conditions, the non-equivalence of the two major statements, appears. Whereas the Kelvin−Planck statement must be modified to remain true, the Clausius statement remains true unaltered. This article proposes a deduction of the Clausius equality and inequality from the Clausius statement only, thus, inheriting its validity in the domain of negative absolute temperatures. The deduction developed here uses a simple arrangement of Carnot engines and a conceptualization of a system with very loose constraints. As a result, the level of generality achieved is high and includes both positive and negative absolute temperatures.

COMMENT ON "FALSIFICATION OF THE ATMOSPHERICCO2GREENHOUSE EFFECTS WITHIN THE FRAME OF PHYSICS"

In this journal, Gerhard Gerlich and Ralf D. Tscheuschner claim to have falsified the existence of an atmospheric greenhouse effect.1Here, we show that their methods, logic, and conclusions are in error. Their most significant errors include trying to apply the Clausius statement of the Second Law of Thermodynamics to only one side of a heat transfer process rather than the entire process, and systematically ignoring most non-radiative heat flows applicable to the Earth's surface and atmosphere. They claim that radiative heat transfer from a colder atmosphere to a warmer surface is forbidden, ignoring the larger transfer in the other direction which makes the complete process allowed. Further, by ignoring heat capacity and non-radiative heat flows, they claim that radiative balance requires that the surface cool by 100 K or more at night, an obvious absurdity induced by an unphysical assumption. This comment concentrates on these two major points, while also taking note of some of Gerlich and Tscheuschner's other errors and misunderstandings.

REPLY TO "COMMENT ON 'FALSIFICATION OF THE ATMOSPHERIC CO2GREENHOUSE EFFECTS WITHIN THE FRAME OF PHYSICS' BY JOSHUA B. HALPERN, CHRISTOPHER M. COLOSE, CHRIS HO-STUART, JOEL D. SHORE, ARTHUR P. SMITH, JÖRG ZIMMERMANN"

It is shown that the notorious claim by Halpern et al. recently repeated in their comment that the method, logic, and conclusions of our "Falsification Of The CO2Greenhouse Effects Within The Frame Of Physics" would be in error has no foundation. Since Halpern et al. communicate our arguments incorrectly, their comment is scientifically vacuous. In particular, it is not true that we are "trying to apply the Clausius statement of the Second Law of Thermodynamics to only one side of a heat transfer process rather than the entire process" and that we are "systematically ignoring most non-radiative heat flows applicable to Earth's surface and atmosphere". Rather, our falsification paper discusses the violation of fundamental physical and mathematical principles in 14 examples of common pseudo-derivations of fictitious greenhouse effects that are all based on simplistic pictures of radiative transfer and their obscure relation to thermodynamics, including but not limited to those descriptions (a) that define a "Perpetuum Mobile Of The 2nd Kind", (b) that rely on incorrectly calculated averages of global temperatures, (c) that refer to incorrectly normalized spectra of electromagnetic radiation. Halpern et al. completely missed an exceptional chance to formulate a scientifically well-founded antithesis. They do not even define a greenhouse effect that they wish to defend. We take the opportunity to clarify some misunderstandings, which are communicated in the current discussion on the non-measurable, i.e., physically non-existing influence of the trace gas CO2on the climates of the Earth.

Analysis of open system Carnot cycle and state functions

A first principle analysis of an open system thermodynamical Carnot cycle is provided and the results are compared to those proposed by Gibbs for open systems. The Kelvin–Clausius statement concerning heat transfer for reversible cycles is taken as an axiom, from which several rigorous theorems are proven. An equation is derived that resembles a Gibbs–Duhem relation equating convected entropies, from which two distinguishable forms of entropy are proven to exist for such systems; this questions prevailing developments which presume a singular or characteristic entropic form that couple all work and heat flows, such as in Onsager first-order thermodynamics. In particular, a closed path undergone by the system does not return the environment to the initial state for one of these entropic forms. The Biot assertion that the entropy contribution due to diathermal heat transfer does not form a state function is therefore contradicted and a local entropy is shown to exist. Several other new composite state functions for work and heat flow are shown to exist for open systems. From Gibbs’ results, it is suggested that the ensuing chemical potentials used routinely may possibly ignore heat effects. The functions developed here are suitable for application since the functions are proven to exist, rather than presumed to exist.

Connections between the various forms of the second law of thermodynamics

The forms of the second law due to Caratheodory, Kelvin and Clausius are examined. The possible forms of thermodynamics which may be deduced from each of these statements are found before the connections between the separate forms of the law itself are examined. It is found that the usual proof of the equivalence of the Kelvin and Clausius statements is invalidated at negative absolute temperatures. However, it is shown that the Clausius statement is equivalent to the modified form of the Kelvin principle which is applicable for both positive and negative absolute temperatures.

Kelvin and Caratheodory—A Reconciliation

In conventional thermodynamics, the two most important consequences of the second law of thermodynamics, namely, the existence of an absolute temperature scale and the existence of an entropy function are deduced from the Kelvin-Planck statement or its equivalent, the Clausius statement, with the aid of the Carnot engine and Carnot refrigerator. In the Caratheodory method, a new statement of the second law is made; the Carnot cycle is dispensed with and purely analytical methods are used to derive the absolute temperature and the entropy. It is shown in this paper how the analytic methods of Caratheodory may be used without replacing the traditional statements of the second law by a new axiom. Both the Caratheodory statement of the second law and the Caratheodory theorem on Pfaffian differential forms are unnecessary.

The Second Law of Thermodynamics

Derivations of the mathematical statement of the second law of thermodynamics from the physical statements of Kelvin and Clausius are presented. Caratheodory's principle is obtained as a direct consequence of either Kelvin's or Clausius's statement.