# Reasoning and Logical Proofs of the Fundamental Laws: “No Hope” for the Challengers of the Second Law of Thermodynamics

## Abstract

**:**

“If your theory is found to be against the second law of thermodynamics, I give you no hope; there is nothing for it but to collapse in deepest humiliation”.—Arthur Eddington

Impasse: “Perhaps, after all, the wise man’s attitude towards thermodynamics should be to have nothing to do with it. To deal with thermodynamics is to look for trouble”.

Anecdotal Laws of Thermodynamics (LT) [bracketed terms added]:♦[0LT]: You must play the game [equilibrium]. ♦[1LT]: You can’t win [conservation]. ♦[2LT]: You can’t break even [dissipation]. ♦[3LT]: You can’t quit the game [0 K impossible].—Thermodynamics-WikiQuote

“The Second Law of thermodynamics can be challenged, but not violated—Entropy can be decreased, but not destroyed at any space or time scales. […] The self-forced tendency of displacing nonequilibrium useful-energy towards equilibrium, with its irreversible dissipation to heat, generates entropy, the latter is conserved in ideal, reversible processes, and there is no way to self-create useful-energy from within equilibrium alone, i.e., no way to destroy entropy”.—[2LT.mkostic.com (accessed on 30 June 2023)]

## 1. Introduction

**Selected Essential Abbreviations and Notes**(in logical order for usage convenience):

_{ThVP}: Number of Thermal Virtual Particles, ThVP. It may be considered as the “particle dimensionless entropy.”

_{th}: Number of thermal moles is the number of ThVP per the Avogadro’s number, i.e., n

_{th}= N

_{ThVP}/N

_{A}. It may be considered as the “molar dimensionless entropy.”

**= η**

_{max}**= W**

_{C}**/Q**

_{C}**= [1 − Q**

_{H}**/Q**

_{L}**]**

_{H}**= F**

_{rev}**(T**

_{C}**,T**

_{H}**) = 1 − T**

_{L}**/T**

_{L}**), see “Carnot Cycle” above.**

_{H}**/Q**

_{L}**= T**

_{H}**/T**

_{L}**, or Q**

_{H}**/T**

_{H}**=Q**

_{H}**/T**

_{L}**= Q**

_{L}**/T**

_{ref}**=**

_{ref}**Q/T**

**=**constant).

## 2. Ubiquity and Conjugation of “Energy Forcing and Displacement”

**Table 1.**Typical Energy Intensive and Extensive Conjugate Properties (Energy Force and Energy Displacement).

Generic Name | Customary Name | Definition | Unit |
---|---|---|---|

Energy Force (or intensity)(intensive property, conjugate with energy displacement) | Generalized force (intensity) | Energy intensity or energy density is energy per unit of energy displacement, by definition, it is the conjugate property with energy displacement, see next. | [F] =J/[ δ] |

Energy displacement(or energy space or extensity)(extensive property conjugate with energy force) | Generalized displacement (extensity) | Energy extensity or energy space is energy per unit of energy intensity, by definition, it is the conjugate property with energy force, see above. | [δ]see specifics below |

Mechanical force (Newtonian) | Force (Newtonian) | Newtonian bulk force or total pressure force, or energy per unit of bulk displacement. | N = J/m |

Mechanical displacement (Newtonian) | Displacement (linear) | Linear displacement of bulk body or Energy per Newtonian bulk force. | m |

Mechanical compression force | Pressure | Mechanical compression energy per space volume. | J/m^{3}=N/m ^{2} |

Mechanical compression displacement | Volume | Compressible volume. | m^{3} |

Thermal force | Temperature | Thermal energy per unit of entropy (or average thermal energy per dynamic thermal particle). | K (or J/[k _{B}] = J/[1] ^{(+)}) |

Thermal displacement ^{(}*^{)} | Entropy or number of thermal virtual particles | Thermal energy per absolute temperature (or number of dynamic, thermal virtual-particles; irreversibly generated, include thermal-particle chaotic-dynamics in space, non-conserved). | J/K (or [1]) |

Chemical force | Chemical potential | Chemical energy per unit of number or moles of species (or per number of chemical species). | J/Mole =J/[1] |

Chemical displacement | Number of moles or species | Number of species or number of moles of chemical species (conserved). | [1] |

Electrical force | Voltage | Electrical energy per unit of electrical charge (or per number of charged particles). | V = J/C (or J/[1]) |

Electrical displacement | Capacity or number of charged particles | Electrical energy per unit of electrical force (or number of electrically charged particles; conserved). | C = J/V (or [1]) |

Etc., for other energy types (the above are not inclusive) | Etc. | Magnetization, nuclear, radiation, etc. | Etc. |

^{(+)}NOTE that [1] is dimensionless unit and not a Reference.

^{(}*

^{)}NOTE that all but thermal energy displacements are conserved, while thermal displacement (entropy or number of thermal virtual particles, N

_{ThVP}) is irreversibly generated due to dissipation of all other energy types to heat.

**Key Point 1.**Mass energy, or energy in general, is the underlying, building block of all energy fields and material existence in space (“activity” of all fundamental particles, including field-equivalent particles, and “inertia” of their bulk interactions) with a spontaneous tendency to displace in time towards mutual, stable equilibrium, thus defining space and time existence. During its displacement, energy is conserved (1LT).

**Key Point 2.**Force or forcing is the spontaneous (by-itself or of-itself) energy tendency to displace, directionally from higher to the “adjoining” locality of lower intensity (from higher to lower energy density). Since displacement is the mutual interaction of competing particles and systems, the force duality is mutually exhibited and balanced between the interacting systems, the action and reaction forces as described by the Third Newton Law (3NL), including the acceleration force (the Second Newton Law, 2NL), and including its special case of uniform motion without acceleration (uniform velocity, including zero velocity or resting), with balanced external forces as described by the First Newton Law (1NL).

**Key Point 3.**Useful-energy or work-energy potential (or free energy, or work potential, or work for short) is the non-equilibrium energy within interacting systems, capable of displacing spontaneously (by itself) out of a system while it is coming at the mutual equilibrium with the most efficient processes (without dissipative conversion to heat). In an ideal reverse-process, such original work, as the formation work, would create the original non-equilibrium. If the surrounding reference system is well defined (P

**, T**

_{0}**, µ**

_{0}**, V**

_{0,i}**, …), then such work potential (WP) of a given system state (P, T, µ**

_{0}**, V, …) is a unique [quasi-] property of the system state and is defined as exergy. Therefore, useful energy, work potential, or exergy are essentially the same concepts and conserved during ideal, reversible interactions. In real, irreversible processes, the work (i.e., exergy) will be dissipated (converted) to heat and diminished. The WP as energy cannot be generated but only displaced (transferred) and is not conserved since it is irreversibly dissipated to heat with entropy generation (2LT).**

_{i}**Key Point 4.**The driving cause and source of any and all process forcing, manifested by energy displacement, is due to non-equilibrium WP, or the exergy difference between any two process states.

**Key Point 5.**The energy process (i.e., energy interaction displacement, or process for short) is caused or driven by directional forcing due to non-equilibrium WP. Ideally, in the most possible efficient, reversible processes, the WP (or exergy) is conserved; however, in real processes, the exergy is dissipated to heat with entropy generation due to diverse causes of directional work dissipation, i.e., chaotic energy redistribution in all possible directions, known as dissipation of WP into randomized thermal-energy, or dissipation of work to heat during a process. If all WP is dissipated, then the mutual equilibrium is achieved with no mutual work potential, with maximum entropy, and with no possibility of any further energy displacement, unless external exergy (i.e., WP) is applied.

**Key Point 6.**There is no perfect equilibrium, nor perfect absolute zero temperature, nor reversible process, nor any other ideal, perfect system nor process. However, such perfect systems and ideal processes are very useful and often necessary to describe and define fundamental concepts of natural phenomena, and to quantify properties and relevant equivalences.

## 3. Reasoning Logical-Proofs of the Fundamental Laws

**, is balanced by (equal magnitude to) reaction force, F**

_{A}**, (the 3NL, including inertial force—the 2NL) along the shared, interaction displacement, dL**

_{R}**= dL**

_{A}**, then, the amount of “action energy out” would equal to the amount of “reaction energy into”, i.e., the energy is conserved during any and all interactions (First Law of Thermodynamics, 1LT):**

_{R}**Key Point 7.**All interactions in nature are mechanistic, and during forced interactions, energy is directionally transferred (2LT) and conserved (1LT). In cases without interaction, if a particle (or a body, a bulk system) is bounded by an enclosure boundary (thus restricting displacement), or not encountering resisting particle (or resisting body; no reaction force), the particle or body will stay at rest or continue with its “free motion,” or an expanding gas without any resisting interaction will undergo “free expansion” without transferring any energy, and therefore, the energy will be conserved internally within (Figure 1).

**Key Point 8.**The forced-displacement interaction is a process of energy transfer from the acting particle (or body) with higher energy density onto a reacting particle (or body) of lower energy density, displacing (transferring) its energy during the interaction, i.e., diminishing its energy (figuratively “decelerating” its structure) while increasing energy of the reacting body (figuratively “accelerating” its structure) until the energy densities (or intensities) are equalized when mutual self-sustained equilibrium is achieved.

## 4. Ubiquity of Thermal Motion and Heat“, Thermal-Roughness,” and Indestructibility of Entropy

#### 4.1. Ubiquity of Thermal Motion, Thermal Energy and Heat, Temperature, and Entropy

**Key Point 9.**As an adjective, “thermal,” implies a chaotic, randomized motion, kind of “thermal turbulence.” Average thermal energy per particle is temperature (or intensity of ThM energy), and extensive randomness of the bulk ThM is entropy (or extensity of ThM energy; or the total ThM energy per temperature, since intensity and extensity are the conjugate thermal-energy properties, see Table 1).

#### 4.2. Thermal Particles, “Thermal Virtual-Particles,” and “Thermal-Moles” or Dimensionless Entropy

**Key Point 10.**Thermal virtual particles (ThVP) are non-conserved dynamic particles (as opposed to the conserved, physical ThP) and they increase with entropy increase, i.e., with an increase in thermal randomness of the physical ThP. The Avogadro’s number (N

_{A}) of the ThVP represents a “thermal mole,” i.e., both are “dimensionless entropy,” per ThVP or the mole, respectively.

_{ThVP}, which may be considered as the “particle dimensionless entropy,” i.e.,:

_{ThVP}= S/k

_{B}= ln(Ω) = U

_{th}/(k

_{B}·T).

_{th}, which may be considered as the “molar dimensionless entropy,” i.e.,:

_{th}= N

_{ThVP}/N

_{A}= S/(k

_{B}N

_{A}) = S/R

_{u}

_{th}is the internal thermal energy; Ω is the number of the “possible thermal, microscopic states”; N

_{A}is the Avogadro’s number; k

_{B}is the particle Boltzmann constant; and R

_{u}is the molar, universal gas constant.

_{ThVP}, is non-conserved, as opposed to conserved number of physical thermal particles, N

_{ThP}(atoms, molecules, electrons, and similar). The former increases with entropy, i.e., with increase in thermal randomness.

#### 4.3. Thermal Energy Is a Distinguished Part of Internal Energy (“Pond Analogy” Demystified)

_{M}add together to give dU, the energy U of a state cannot be considered as the sum of “work” and “heat” components … the sum is the energy difference ΔU, which alone is independent of the process.” Cullen continued [11]: “The concepts of heat, work, and energy may possibly be clarified in terms of a simple analogy. A certain farmer owns a pond, fed by one stream and drained by another. The pond also receives water from an occasional rainfall and loses it by evaporation, which we shall consider as negative rain.… In this analogy the pond is our system, the water within it is the internal energy, water transferred by the streams is work, and water transferred as rain is heat. … The strict analogy of each of these procedures with its thermodynamic counterpart is evident.”

**False Point 1:**Callen’s “Pond analogy” in which the change in internal energy is independent, whether heat or work is added into a system [11] (p. 20), is misleading and generally erroneous, since the pond water is at the same surrounding T and P, which is not the case if we reversibly store internal energy by heating or working. Adding the same reversible amount of work or heat will result in different forms and quality of energy with different final states (with different entropy, volume, etc.), i.e., different WP to be extracted. Therefore, the quality of internal energy is not “the same form and not independent of the process,” as claimed [11]. Namely, the water streams representing work in “pond analogy” undergo full dissipation (called here “complete irreversibility,” such as during the famous “Joule’s 1843 experiments (work-heat equivalency; 1LT only)” or isochoric heating only. Only for the “completely irreversible” processes, the outcome is the same internal energy, regardless of whether either the work or heat source of different temperatures are used, since all WP would be completely dissipated within such a system—however, the claim is erroneous in general.

_{i}, V

_{i}, S

_{i}, …), the final state will be (U′ = U

_{i}+ Q′, V′ = V

_{i}, S′≠ S

_{i}, …); however, if the same amount of work W″ = Q′ is reversibly stored instead, the final state would be different (U″ = U

_{i}+ W″ = U′, V″≠ V′, S″ = S

_{i}, …). If the processes are reversed back to the original initial state, it would be ideally possible to retrieve the original work W″ from U″ but that work could not be obtained from U′ (even though U′ = U″), which proves that the internal energies U′ and U″ are not the same quality (not the same states, different WPs and exergies; the 2LT), regardless of being the same quantity (same U-amounts; the 1LT).

**Key Point 11.**Claiming that storing Q′ or W″ (if Q′ = W″) would indistinguishably increase the internal energy U, is only convenient for easy bookkeeping (it sidesteps difficulties of distinguishing energy quality), but it is deceptive since U′ = U + Q′ and U″ = U + W″ are not truly (reversibly) equivalent (not the same free energies nor WPs, see Section 5). Namely, there is the specific and distinguishable quantitative measures of stored work, i.e., the work potential (WP or available-energy or exergy, or “stored-work”) within internal energy (U), and of stored heat (thermal energy, U

_{th}, or “stored-heat”) associated with temperature and entropy (T, S). Both the exergy and U

_{th}(being uniquely defined for a specified reference state) may be considered as (quasi-) properties, to be further elaborated in a separate writing.

_{B}T, manifests also as mechanical compression energy PV, as expressed by its equation of state: PV ≡ Nk

_{B}T, see Section 6.2.

#### 4.4. Thermal Roughness Ubiquity, and Inevitability of Thermal Friction

#### 4.5. Inevitability and Conjugation of Work dissipation and Entropy generation

**Key Point 12:**In summary, any irreversible “entropy generation” is caused by and related to “heat generation” due to irreversible work (or WP) dissipation, and vice versa; any irreversible work dissipation to heat is always accompanied with irreversible “entropy generation” at any space and/or time scale, without exception. If entropy is generated during any process, then, to reverse the final to the initial state, the irreversible generated entropy has to be removed from the system, which would require removal of the commensurate heat (thus reduction of internal energy); the latter has to be compensated with external work (ideally equal to the prior work dissipated or even more due to unavoidable process irreversibilities) to make up for the prior work dissipation loss reflected in reduced internal energy after the generated entropy (and commensurate heat) is removed from the system. The heat and entropy generation should not be confused with reversible entropy transfer, like during phase change and chemical reaction in equilibrium processes where entropy is conserved.

**False Point 2.**Plank’s statements regarding “the same form of energy […] diffusion mixing with appreciable increase in the entropy accompanied by no perceptible transference of heat, nor by external work, nor by any noticeable transformation of energy, [14] (pp. 103–104)” are misleading and erroneous, as well as Uffink’s endorsement of the Plank’s claim: “This view on irreversibility, which focuses on the ‘dissipation’ or ‘degradation’ of energy instead of an increase in entropy was still in use … Planck’s work extinguished these views … [15]). However, “to reverse irreversible diffusion,” external work would be required (regardless of the amount), without exception, to compensate for the WP dissipation loss during the prior diffusion.

**Key Point 13:**The “Principle of the increase of entropy” is complementary with the “Principle of (unavoidable) energy degradation” due to the dissipation of WP to heat accompanied with entropy generation (irreversible “entropy increase, not to be confused with reversible entropy transfer”): δI

_{rr}= δ(WP)

_{diss}= δQ

_{gen}= TδS

_{gen}(for a variable process temperature, the differential quantity, TδS

_{gen}, has to be properly integrated along the process path).

#### 4.6. Irreversibility of Entropy Generation and Indestructibility of Entropy—Essence of the 2LT

_{LOSS}= W

_{diss}= Q

_{gen}. The work dissipation is directly related to the entropy generation, S

_{gen}, at relevant reference, absolute temperature, T

_{ref}, (the Gouy–Stodola correlation, Equation (5). The work dissipation and related entropy generation are two sides of the same coin (“half empty vs. half full”), i.e.,

_{LOSS}≡ W

_{diss}] = Q

_{gen}} = T

_{ref}·S

_{gen}≥ 0

**Key Point 14.**The generated entropy is the irreversible “final transformation”: the “lost or dissipated” work is actually compensated with or converted into the generated heat (the 1LT). Furthermore, along the generated heat, the accompanying generated entropy, conjugate to relevant temperature, is the “final and indestructible quantity” since there is no way (no process possible) to convert entropy into nor to compensate entropy with anything at all, nor to annihilate it—the entropy is truly indestructible, the “final transformation” (the 2LT).

**Key Point 15.**Since all real, irreversible processes generate heat and entropy due to the unavoidable dissipation of work to heat (ultimately instigated by the “thermal roughness” as elaborated and named here, Section 4.4), and all ideal, reversible processes conserve entropy, then, there are no other processes left to miraculously generate WP without a due WP-source forcing and transfer, nor any “imaginary process” could destroy (or annihilate) entropy, since it would be a “self-reversal of dissipation” and contradiction impossibility against the natural forcing—it would imply self-generation of non-equilibrium (and its WP); therefore, rendering a logical proof of indestructibility of entropy (the 2LT). Therefore, there is no process possible (no heat nor work transfer process) to destroy entropy—the thermal entropy cannot be converted to anything else nor destroyed, but it will be always irreversibly generated, without exception, at any relevant space or time scale, where the macro-properties could be defined.

**Key Point 16.**A non-equilibrium (i.e., its WP) may be increased only by forcing on the expense of another WP, as a necessary WP-source. During such forced interactions the WP in ideal reversible processes would be reorganized, i.e., transferred and conserved (1LT and 2LT), or, in part, it would irreversibly dissipate to heat, i.e., the WP would be irreversibly diminished (2LT)—however, the totality of energy (WP and the generated-heat) would be conserved (1LT, again). Therefore, there is no way to self-create non-equilibrium work potential against the natural forcing towards equilibrium. The former would be a contradiction of the latter.

## 5. Carnot Maximum Efficiency, Reversible Equivalency, and Work Potential

#### 5.1. Carnot Cycle Maximum Efficiency: Proof by Contradiction Impossibility

**Key Point 17.**If the critical and ingenious discoveries by Clausius and Kelvin make them “fathers of thermodynamic,” then, Sadi Carnot was the “grand-father of thermodynamics-to-become”.

_{C}is the Carnot cycle efficiency, maximum possible and equal for any and all reversible cycles. The cycle is converting heat, Q

_{H}, from high-temperature thermal reservoir at T

_{H}, extracting cycle work, W

_{C}, and passing heat to a low-temperature thermal reservoir at T

_{L}.

**False Point 3.**Some references cite that Sadi Carnot derived the maximum cycle efficiency, η

_{C}= 1 − T

_{L}/T

_{H}, named in his honor, is false since Carnot wrongly assumed conservation of caloric (Q

_{L}= Q

_{H}), and the absolute temperature were not defined in his time. Regardless, Carnot ingeniously, considering the knowledge at his time, deduced completely and correctly, although implicitly, that the efficiency depends on the two thermal reservoirs’ temperatures, t

_{H}and t

_{L}, only; see Equation (6). The explicit, maximum cycle efficiency was derived later by Kelvin [4] (1850, using IG) and generalized by Clausius [5] (1854), based on Carnot’s work in 1824 [3], and named it in his honor; see [5].

**False Point 4.**Some references also cite that Sadi Carnot stated that “the maximum cycle efficiency depends on the temperature difference of the two thermal reservoirs, implying it is a function of the temperature difference only [η

_{C}= f(t

_{H}− T

_{L})]. However, it is misplaced, since Carnot’s statement was “in principle,” and he was fully aware that the maximum efficiency depends implicitly on the two temperatures only, but not their difference directly, as Carnot stated accurately [3]; see related Equation (6).

**Key Point 18.**Proof by “contradiction-impossibility” of an established fact is, by definition, the logical proof of the stated fact. If a contradiction of a fact is possible then that fact would be void and impossible. It is illogical, absurd, and impossible to have both, “the one-way and the opposite-way.” For example, if heat self-transfers from higher to lower temperature, it would be “contradiction-impossibility” to self-transfer in the opposite direction, from low to high temperature.

#### 5.2. Carnot Cycle, Carnot Efficiency, and Carnot Equality (CtEq)

**Figure 3.**Carnot Equality (as named here), Q/Q

_{0}= T/T

_{0}, or Q/T = constant, for reversible cycles (different from Carnot Theorem), is much more important than what it appears at first. It is probably the most important correlation in Thermodynamics and among the most important equations in natural sciences. Carnot’s ingenious reasoning unlocked the way (for Kelvin, Clausius, and others) for generalization of “thermodynamic reversibility,” definition of absolute thermodynamic temperature and a new thermodynamic property “entropy” (Clausius Equality is generalization of Carnot Equality), as well as the Gibbs free energy, one of the most important thermodynamic functions for characterization of electro-chemical systems and their equilibriums, resulting in formulation of the universal and far-reaching Second Law of Thermodynamics (2LT) (as originally stated by this author in 2008 [16] and 2011 [17]).

**Key Point 19.**The Carnot Equality (CtEq), Q/T = constant, the well-known correlation, the precursor for the famous Clausius Equality (CsEq), CI(dQ/T) = 0 (the cyclic integral for variable temperature reversible cycles), is specifically named here “as such” by this author in Carnot’s honor. The CtEq was based on Carnot’s 1824 discovery [3] that was finalized later by Kelvin (1850 using ideal gas) and generalized by Clausius (1854; see [5], pp. 69–109). The CtEq was also precursor for discovery of thermodynamic temperature and entropy. It is among the most important correlations in natural sciences, on par with Einstein’s famous, E = mc

**correlation, see Figure 3 (as originally stated by this author in 2008 [16] and 2011 [17]).**

^{2}**Key Point 20.**The Carnot Efficiency, CtEf, η

_{C}= (1 − T

_{L}/T

_{H}), a.k.a. Carnot Theorem (not to be confused with the Carnot Equality, CtEq) was originally established implicitly by Carnot, Equation (6), “as independent of cycle design and mode of operation,” therefore, in fact, not dependent on cycle per se, but dependent on the thermal-reservoirs’ temperatures (T

_{H}and T

_{L}) only. Therefore, in fact, the CtEf represents the WP of the heat Q

_{H}, transferred from T

_{H}-reservoir while interacting with T

_{L}-reservoir only, i.e., it represents the work potential of heat, WP

_{Q}= (1 − T

_{L}/T

_{H})Q

_{H}, realized by ideal, reversible Carnot cycle or any other, reversible steady-state device (so that any transient accumulation of heat or WP within the devices are excluded); see also Key Point 19.

_{H}= W

_{H}) while increasing volume and entropy (process 1–2), or in reverse, where work is entirely (100%) converted to heat (W

_{L}= Q

_{L}) and there is decreasing volume and entropy (process 3–4). Note that the isothermal ideal gas heating is accompanied with the expansion work-out equal to heat-in, W

_{H}= Q

_{H}), while the quantity of its internal energy is unchanged. However, its quality is degraded (part of its work potential replaced with heat), as manifested by the increase in entropy (i.e., U = constant, but decrease in the WP and free energy G = U − TS)!

_{H}, to a “Low-intensity Energy Reference System (LERS or L-reservoir)” at a lower temperature T

_{L}, for an open or closed, steady-state or quasi-steady-cyclic process, respectively, including the irreversible loss of work potential to heat with entropy generation.

_{LOSS}= W

_{diss}= Q

_{gen}= T

_{L}S

_{gen}(the Gouy–Stodola correlation), is due to the dissipation of work (W

_{diss}) into the generated heat (Q

_{gen}). Note that work as useful energy cannot be lost per se (1LT) but is dissipated, i.e., irreversibly converted to heat as a degraded form of energy. Additionally, for closed-mass and cyclic processes, m

_{L}= m

_{H}= 0$,$ and for adiabatic turbine (Q

_{H|L}= 0), W

**= E**

_{OUT}**− E**

_{mH}**, see Figure 5.**

_{mL}**Key Point 21.**During any steady-state process or quasi-steady-cyclic process, see Figure 5, the entropy input S

**, with heat Q**

_{H}**(and with mass m**

_{H}**if any) at T**

_{H}**>T**

_{H}**, and any irreversible generated entropy S**

_{L}**within, must be discharged with heat Q**

_{gen}**(and with mass m**

_{L}**if any), as entropy S**

_{L}**at T**

_{L}**.**

_{L}_{H}= Q

_{H}/T

_{H}}

_{IN}= {Q

_{L}/T

_{L}= S

_{L}}

_{OUT}, which demonstrates a logical proof of the Carnot Equality), and also to be “compensated mechanically” (by bringing a cyclic process to the initial volume), before repeating the cycle. Therefore, the heat rejected during a reversible cycle process, Q

_{L,R}= T

_{L}ΔS

_{R}, is the necessity and therefore “useful quantity”, not a loss as sometimes mispresented, see False Point 5.

**False Point 5.**Citation in some references, that, “the heat rejected to the lower-temperature reservoir during a reversible cycle process, is a lost energy” is false, since it is necessary to remove the entropy input, in order to complete the cycle. Therefore, the rejected heat in a reversible cycle is the necessity and ‘useful quantity’, not a loss as mistakenly stated in some literature.

_{C}= Q

_{H}− Q

_{L,R}< Q

_{H}, i.e., the Carnot cycle efficiency, η

_{C}, is always smaller than 100% but bigger than a real cycle efficiency, η, i.e.,

_{C}− W

_{LOSS})/Q

_{H}<{η

_{C}= W

_{C}/Q

_{H}= 1 − (T

_{L}/T

_{H})} < 100%.

_{2LT}= η/η

_{C}} ≤ {η

_{C,2LT}= 1 =100% = η

_{R,2LT}}. The curled term on the right of the inequality being the perfect 100% 2LT reversible efficiency for the Carnot cycle or any reversible process.

#### 5.3. Carnot “Reverse-Cycle” and Thermodynamic “Reversible-Equivalency”

**to T**

_{L}**), while external work would be consumed. Therefore, all processes and energy flows would be in reverse direction, resulting in the “Carnot reverse-cycle” with regard to the original (power-producing) “Carnot cycle”, with infinitesimally different or in limit all equivalent, respective properties and energy flows, but in reverse directions; see Figure 4 and Equation (8). Such reverse cycles will provide cooling (refrigeration) of the low-temperature reservoir (any ambient; A/C or refrigeration cycle) and/or heating of the high-temperature reservoir (any ambient; heat-pump cycle), by effectively transferring heat from low to high temperature with utilization of external work.**

_{H}**Key Point 22.**Sadi Carnot proved the equivalency and maximum efficiency of reversible processes by logically demonstrating that otherwise they will violate the contradiction impossibility of “logical criteria,” in his case, the mistaken conservation-of-caloric criteria; still it resulted in the correct conclusion due to the ingenious logic by Carnot. With rectified criteria and energy conservation, Carnot’s logic implied the impossible self-transfer of non-conserved caloric from low to high temperature, the contradiction of valid criteria used by Clausius.

**:**Since ideal, reversible processes may effortlessly be self-reversed “back-and-forth in perpetuity” (Equation (8) and Figure 6), that imply they do not degrade their “energy quality,” and therefore, they have maximum possible efficiency and are equivalent—they are lossless or dissipationless. However, dissipative degradation of WP (energy quality) will diminish the WP and efficiency, and prevent “perpetual reversibility” or self-reversal.

#### 5.4. Work Potential, Formation Work, and Exergy

**Key Point 23.**The work potential (WP) of a system state with regard to a reference state, is the unique, “energy quality” that could be reversibly retrieved as “useful-energy” if a system state is reversibly brought to a lower, reference equilibrium state, while interacting with respective reference surroundings. Such retrieved WP could be used in reverse as formation work to re-form the system state from that equilibrium to the original state, with ideal, reversible processes, thus defining the “reversible equivalency,” see Figure 6. Furthermore, if the “lower energy” state is chosen as a well-defined, standard reference state, then the WP becomes the “unique quantity” of such state, and hence, it may be considered as a system (quasi-) property, already defined as exergy. Some do not consider exergy as a property since it depends on the reference state, but that is also the case with some other properties. All WP-related quantities (work potential, useful energy, formation work, exergy), as asserted here and elsewhere, are directly interrelated and essentially the same concepts, they all irreversibly dissipate to heat in real processes, and they become zero at equilibrium.

**Key Point 24.**Non-equilibrium, useful energy or WP is directionally transferred (from higher to lower energy density) and conserved in ideal reversible processes (1LT), while in real processes the WP is irreversibly dissipated (converted) into heat with entropy generation (2LT); however, conserved “as work-and-heat” in totality (1LT), as detailed elsewhere.

## 6. Thermal Transformers: Carnot–Clausius Heat–Work Reversible Equivalency (CCHWRE)

#### 6.1. Thermal Transformers and CCHWRE

**Key Point 25.**Heat transfer (“thermal poking”) requires a higher temperature source and is always accompanied by entropy transfer (δQ = TdS). However, if an energy is transferred in an orderly manner, without entropy transfer, then it is not heat but adiabatic work transfer—it increases energy and temperature but is ideally isentropic, without entropy transfer. That is how the temperature could be increased without heating. In reverse, adiabatically extracting work would lower temperature without cooling, otherwise, the latter would require a lower temperature heat sink — namely, only work could increase temperature above or decrease below the available source or sink temperatures, respectively.

**Key Point 26.**Thermal transformers: With all relevant processes, working in sequence as the cycle, the reversible heat transfer from any to any temperature level could be achieved, functioning as a “reversible thermal-transformer.” Namely, the reversible heat transfer from higher T

_{H}to lower T

_{L}temperature with W, Carnot cycle work output; or in reverse, the reversible heat transfer from lower T

_{L}to higher T

_{H}temperature with W, Carnot cycle work input. Likewise, the real thermal transformers, as combined power heating and refrigeration cycles (including heat-pump cycles) also transfer heat from any to any temperature level, except with reduced efficiency due to the unavoidable dissipation of WP into generate heat and entropy (Equations (5) and (7)).

_{C}) and rejected heat at lower temperature (Q

_{L}) are obtained from heat at higher temperature (Q

_{H}) alone; and in-reverse, where the utilized work in refrigeration reverse cycles (W

_{C}) is added (in) to a heat at low temperature (Q

_{L}), resulting in the heat at higher temperature (Q

_{H}) alone, see Figure 7 and Equations (9) and (10).

_{H}at high temperature T

_{H}is equivalent with the sum of heat Q

_{L}at lower temperature T

_{L}and Carnot’s work W

_{C}, Equation (9) Left; or any other relevant rearrangement, Equation (9) Center or Right, along with the reversible Carnot Equality, as formalized and named here, Figure 3 and Equation (10).

#### 6.2. Proof of Ideal Gas Equation of State and CCHWE Confirmation

_{IG}≡ E

_{ThM}] =

**E**= N(k

_{th}_{B}T) = nR

_{u}T, where, N is number of particles, k

_{B}[J/K] is the Boltzmann constant (or energy-temperature conversion factor), and T is particle-average absolute temperature, i.e., k

_{B}T is the particle average energy, n is number of moles, and R

_{u}is the universal, molar gas constant.

_{IG}≡ E

_{ThM}] =

**E**= PV, where P is mechanical pressure (defined as relevant energy per unit of volume), and V volume of IG. Therefore, we may express the IG equation of state (i.e., the constitutive correlation of its mechanical and thermal properties), as the equivalence (“

_{me}**≡**”) of the two forms of the same energy (Equation (11)).

**Key Point 27.**The reasoning here presents a logical proof of the IG equation of state. The duality of manifestation of IG’s ThM energy, either as mechanical (via pressure and volume) or as thermal (via temperature of particles), demonstrate why the IG structure (random ThM of its particles) enables interchangeability of heat or work storage and transfer, depending if energy is stored or transferred via thermal-motion (ThM), by “jiggling” across a boundary surface (at constant volume) and thus changing the temperature and entropy, or by mechanical displacement of the boundary and changing the pressure and volume.

_{H}= W

_{H}) while increasing volume and entropy (process 1–2), or in reverse, where work is entirely (100%) converted to heat (W

_{L}= Q

_{L}) while decreasing volume and entropy (process 3–4). Note that the isothermal ideal gas heating is accompanied with the expansion work-out equal to heat-in, W

_{H}= Q

_{H}), while the quantity of its internal energy is unchanged. However, its quality is degraded (part of its work potential replaced with heat), as manifested by the increase in entropy (i.e., U = constant, but decrease in the WP and free energy G = U − TS). Furthermore, the Carnot cycle comprises reversible adiabatic processes where internal energy (stored heat in IG) is entirely (100%) converted to work, −Cv(T

_{L}− T

_{H}) = W

_{23}, by lowering temperature and increasing volume (process 2–3), or in reverse, where work is entirely (100%) reversibly converted to internal energy (stored heat in IG), W

_{41}= Cv(T

_{H}− T

_{L}), by increasing temperature and decreasing volume (process 4–1 on Figure 8).

_{23}| = |W

_{41}|) and cancel out for the whole cycle (they have the main purpose to change temperature levels for the reversible heat transfer to be possible). Consequently, the net cycle work is the result of the isothermal works’ difference due to the respective temperature difference, W = |W

_{H}| − |W

_{L}| = (T

_{H}− T

_{L})|ΔS|, while exchanging the same entropy in and out, |ΔS

_{12}| = |ΔS

_{34}| = |ΔS|, so that entropy cancels out, enabling the completion of the cycle (note that the isentropic works do not contribute to the entropy balance). For more details and all the specific equations, see Table 1 in [17] (p. 344).

#### 6.3. Reversible Cycles Are 100% Efficient and Carnot Efficiency Is Essentially the “Measure” of Heat-WP

**Key Point 28.**All reversible processes (including cyclic processes) under the same conditions must have equal and maximum efficiency, as demonstrated by relevant “contradiction impossibility.” As a matter of fact, the reversible processes and cycles were a priori “specified” as ideal, with maximum possible efficiency, with a priory 100% 2LT reversible efficiency, not dependent on their design or mode of operation (independent of their quasi-stationary cyclic path or any other, reversible stationary process path). Actually, as the ideal ‘work-extraction measuring-devices’, all reversible processes and cycles, in fact, determine the WP (as % or ratio efficiency with reference to relevant total energy) of an energy-source system with another reference system (such as with the two thermal reservoirs with the Carnot cycle, so their WP ratio is dependent on their temperatures only).

**Key Point 29.**The maximum-possible work potential (WP) of a system (thermal-reservoir or any other), between any two states (its initial and final (reference) states), is independent of the process path or the process device properties or design (cyclic or otherwise), that reversibly brings the initial energy state to another reference state (by reversibly interacting with reference surroundings towards an equilibrium state), but it only depends on the two states’ relevant properties, e.g., it only depends on the temperatures of the two thermal energy reservoirs (as ingeniously deduced by Sadi Carnot in 1824 [3]).

**Key Point 30.**The ideal heat-engine cycle is just a (cyclic) process path between high- and low-temperature thermal reservoirs, and the maximum possible reversible efficiency is not dependent on the cycle device and process path, but only dependent on the two temperatures (as originally reasoned by Sadi Carnot in 1824 [3]), and elsewhere, including by this author [16,17]. The reversible cycles are “used” to evaluate (“measure”) mutual, maximum efficiency of the thermal reservoirs, not the cycle per se, since it is independent of the cycle design and mode of operation, therefore not dependent on the cycle per se, as implicitly postulated by Carnot, “maximum work is obtained by any reversible cycle, independent form the medium used or mode of operation, is dependent only on the temperatures of the two heat reservoirs [hence, not dependent on the cycle but the reservoirs’ properties/temperatures only].”

**Key Point 31.**The cycles are only intermediary devices, such as different “paths of operations” and all deductions and correlations derived refer to “the heat from the high temperature reservoir being transformed [i.e., converted] to “extracted work and remaining heat transferred to the lower temperature reservoir”; and in reverse, with all relevant quantities having equal magnitude in opposite directions. The Carnot work refers not only to thermal cycles but also to the thermoelectric and other steady-state devices, i.e., it refers to thermal work potential of an energy source in general. This rationalization will require further elucidation in separate writings.

## 7. “No Hope” for the Challengers of the Second Law of Thermodynamics

“It is hard to believe that a serious scientist nowadays, who truly comprehends the Second Law and its essence, would challenge it based on incomplete and elusive facts […] However, sometimes, even highly accomplished scientists in their fields do not fully realize the essence of the Second Law of thermodynamics [16,17,18,19,20,21,22,23].”

**Key Point 32.**The current frenzy about violation of the 2LT, of getting “useful energy” from within equilibrium alone (with the “Perpetual-motion machine of the second kind”, PMM2), is in many ways similar, but more elusive and opportunistic, than the prior frenzy about violation of the First Law of Thermodynamics (1LT), of obtaining “useful-energy” from nowhere (with the “Perpetual-motion machine of the first kind”, PMM1).

**Key Point 33.**The driving force of any process (or change) is the non-equilibrium or useful energy (or WP, or related free energy, or exergy) that exhibit a forced directional tendency for its displacement towards mutual equilibrium and not in the opposite direction (i.e., “energy ability to do work [and transfer heat]” or to produce change). It is illogical and pointless “to insist on the impossible-possibility” for the self-producing non-equilibrium from within the equilibrium alone without required forcing. A new WP cannot be created since it only can be displaced (or transferred), and it is always diminished due to its irreversible dissipation (or its conversion to heat with entropy generation) until mutual equilibrium with uniform properties and maximum entropy, is asymptotically achieved—a “dead-state” without WP required for any further change. Even the heat transfer at finite temperature difference is caused by its Carnot thermal WP (“heat exergy”).

#### 7.1. “Perpetual-Motion Watch” Deceptive Example

#### 7.2. Three “Primary Deception Structures” of Hypothetical Violation of the Second Law of Thermodynamics

**Key Point 34.**In fact, any process that perpetually self-sustain its own macro-structure, regardless of whether it is with uniform or non-uniform macro-properties, is in equilibrium or quasi-equilibrium, respectively; and it is “in its own right”, reversible (perpetually self-sustained). Furthermore, as a matter of logic, the reverse-process perpetuity implies maximum efficiency and reversible equivalency—it is the required condition and definition of reversibility. That is why the reversible processes are called quasi-static or quasi-equilibrium processes; see next Key Point.

**Key Point 35.**The ideal gas (IG) micro-structure consists of chaotic ThM of perfectly-elastic particles, and in equilibrium, their collisions are reversible and without dissipation (as if the ThM “micro-dissipates to itself”). However, during an adiabatic free expansion (no heat nor work transfer), its energy will not change but entropy will be irreversibly generated due to its volume increase, regardless that the ThM collisions are elastic. Furthermore, if during a phase change of a system, its expansion is isothermal–isobaric while in equilibrium with its surroundings, such process will be reversible: work-out (to displace surrounding) would be equal to heat-in from the surroundings, and without entropy generation (its entropy increase will equal to entropy decrease in the surroundings, or vice-versa).

**Key Point 36.**Hypothesizing a violation of the 2LT, or worse, claiming that the existence of some unexplained structures or phenomena, “disapprove” universal validity of the 2LT, is misplaced and impossible since it would be the contradiction impossibility of the proven reversible equivalency: if the 2LT is not valid in a particular case, then it would not be valid in general due to reversible equivalency [17,18,19,20,21,22,23] (and elsewhere).

**Key Point 37.**Real thermal motion (ThM) with accompanying collisions (not perfectly elastic such as IG collisions), evidently “micro-dissipates-to-itself”, and the ThM is self-sustaining the perpetual, thermal macro-equilibrium; therefore, it is a reversible phenomenon. Similarly, the Brownian motion or chemical equilibrium reactions or electro-magnetic currents or any self-sustained quasi-equilibrium local phenomena, appears to be micro-dissipating into and are driven in reverse by the surrounding ThM (including thermal EM radiation) or are negligibly irreversible; and if in self-sustaining quasi-equilibrium, they have to be virtually macro-reversible.

**Key Point 38:**During the WP transfer and conversion/storage, a part of WP will always and everywhere, without exception, dissipate to heat and generate entropy, until all WP is ultimately dissipated, being zero at ultimate equilibrium. However, in the process towards ultimate equilibrium with no work potential (“thermodynamic death”), the new structural, temporary, and localized quasi-equilibriums may be established and self-sustained within certain bounded structures, with residual work potential related to their surroundings. Every such quasi-equilibrium is represented with self-sustained micro-fluctuations, or micro-perpetual motions, including ultimate thermodynamic equilibrium (e.g., residual cosmic radiation).

**Key Point 39.**Many creative hypotheses of wishful-inventions, to create useful energy from within the surrounding equilibrium, against the natural forcing, have never materialized, since it would be the “contradiction-impossibility” of existence of self-sustained stable equilibrium and natural forcing of non-equilibrium energy displacement towards mutual equilibrium of all interacting systems.

**Key Point 40.**Therefore, the spontaneous displacement of energy from lower to higher energy density, in opposite direction from natural forcing and self-creation of non-equilibrium, would be similar to forcing in one direction with acceleration in opposite direction. Such wishful thinking would be the natural contradiction impossibility. It would negate stable equilibrium existence and will imply self-creation of WP with entropy destruction. Consequently, a non-equilibrium, the source of WP and forcing, cannot be generated (contradiction impossibility of unavoidable dissipation), but could only be transferred and ideally conserved, while in realty, a WP will tend to dissipate to heat (within a complex micro-structure and fluctuating micro-processes), towards a mutual macro equilibrium.

#### 7.3. “Experimental Test of a Thermodynamic Paradox” Demystified

- 1.
- Most of the fundamental formulations of the phenomenological, classical thermodynamics (called “standard thermodynamics” in the last sentence in Section 2 in the paper [25]), are reasoned and derived for the “simple compressible thermodynamic system,” the latter structure allows for heat and mechanical work interactions and storage only, but not other interactions, as well as for the ideal, black-body cavities, with uniform thermodynamic properties in equilibrium (uniform temperatures and pressures in such simple material structures and systems). The experimental system described in the paper is not nearly closed to the ideal black-body cavity, and the described, dissociation/recombination interactions between heterogeneous devices within a controlled isothermal tube (of the same order of magnitude size as the devices inside) are much more complex than a simple thermo-mechanical interaction of a simple compressible system in an ideal black-body cavity.
- 2.
- The stationary quasi-equilibriums (with non-uniform properties) are abundant in nature, and do not violate the 2LT at all. For example, hydrostatic pressure distribution in a container, or adiabatic atmospheric temperature distribution, or non-uniform distribution of other properties in a stationary equilibrium, in gravity, electromagnetic or chemical fields, such as the presented results. I called the above a “structural equilibrium” (sustainable equilibrium but with non-uniform properties), as opposed to ideal thermodynamic equilibrium (with uniform properties) between the simple compressible systems with boundary heat and work interactions only, immune from any other structure or field interactions. This is one of several other problems of the paper’s judgments and conclusions.
- 3.
- The statements in Section 6 show a discussion in paper [25], “Within the traditional understanding of the second law, stationary temperature differentials such as those reported should not be possible.” This statement is arbitrary and not justified; see also comments above. Likewise, “Second, the temperature differences in DP experiments generated Seebeck voltages that can drive currents—and did, through their thermocouple gauges—thus, were capable of performing work like a heat engine.” This is pure speculation, since we do not know what kind of stationary process will re-establish if a heat engine (HE) or electrical load is interfaced to utilize temperature or Seebeck voltage differences within the described system and devices.

- A simple question arises: why have the authors not experimentally verified their hypothesis, if a stationary work extraction would be possible from within an environment in equilibrium? Such straightforward experiments could and should have been performed to experimentally check such a critical hypothesis. Based on classical thermodynamics, which allows transient processes, after an initial non-equilibrium is externally imposed (as in the paper experiments), the appropriate stationary structural equilibrium with property gradients will establish, as in the paper, a stationary process with perpetual work extraction outside of an equilibrium which is not possible, without external perpetual work source.

- 4.
- The last two concluding sentences of the paper were “In summary, Duncan’s temperature difference has been experimentally measured via differential hydrogen dissociation on tungsten and rhenium surfaces under high temperature blackbody cavity conditions. We know of no credible way to reconcile these results with standard interpretations of the second law.” The assumptions and conclusions are misleading and unjustified, as specifically described above.

## 8. Conclusions

_{ThVP}, as defined and named here) is irreversibly generated due to the dissipation of all other energy types to heat.

_{H}≡ W

_{C}+ Q

_{L}and Q

_{H}/T

_{H}= Q

_{L}/T

_{L}= W

_{C}/(T

_{H}− T

_{L}), Equations (9) and (10), are much more important than they appear at first, since they represent the “heat-work reversible equivalency and interchangeability” in general, for all reversible steady-state processes not only for cycles (see Figure 7). Namely, heat Q

_{H}at high temperature T

_{H}is equivalent with the sum of heat Q

_{L}at lower temperature T

_{L}and Carnot’s work W

_{C}.

_{ThM}=

**E**= N(k

_{th}_{B}T) = nR

_{u}T, along with temperature also exhibited the pressure on any hypothetical or real boundary surface and, therefore, its energy may also be represented as mechanical (pressure) energy: E

_{ThM}=

**E**= PV. Therefore, we may express the IG equation of state (i.e., the constitutive correlation of its mechanical and thermal properties) as the equivalence (“

_{me}**≡**”) of the two forms of the same energy, PV ≡ nR

_{u}T, thus rendering its logical proof.

_{H}to lower T

_{L}temperature with W, Carnot cycle work output; or in reverse, the reversible heat transfer from lower T

_{L}to higher T

_{H}temperature with W, Carnot cycle work input. Likewise, the real thermal transformers, combined power-and-heat cycles, and refrigeration cycles (including heat-pump cycles) also transfer heat from any to any temperature level, except for reduced efficiency due to unavoidable dissipation of WP to generated heat and entropy (Equations (5) and (7)).

**Challenge Point.**“Entropy of an isolated, closed system (or universe) is always increasing”, is a necessary but not sufficient condition of the second law of thermodynamics. Entropy cannot be destroyed (annihilated), locally or at a time, and “compensated” by generation elsewhere or later. It would be equivalent to allowing rivers to spontaneously flow uphill and compensate it by a more downhill flow elsewhere or later. Thermodynamic (macroscopic) entropy is generated everywhere and always, at any scale (where it could be defined) without exception, and it cannot be destroyed by any means at any scale. Impossibility of entropy reduction by destruction should not be confused with a local entropy decrease due to entropy outflow with heat [13,17,18,19,20,21,22].

**Key Point 41.**The Second Law of Thermodynamics can be challenged, but not violated—Entropy can be decreased, but not destroyed at any space or time scale. […]. The self-forced tendency of displacing non-equilibrium useful energy towards equilibrium, with its irreversible dissipation to heat, generates entropy, the latter is conserved in ideal, reversible processes, and there is no way to self-create useful energy from within equilibrium alone, i.e., no way to destroy entropy.”—[http://2LT.mkostic.com (accessed 5 July 2023)].

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Outline: Reasoning Fundamental Laws and “No Hope” for the 2LT Challengers

Outline: Reasoning Fundamental Laws and “No Hope” for the 2LT Challengers |

1. Introduction 2. Ubiquity and Conjugation of “Energy Forcing and Displacement” 3. Reasoning Logical-proofs of the Fundamental laws 4. Ubiquity of Thermal-motion and Heat, “Thermal roughness” and Indestructibility of Entropy4.1. Ubiquity of Thermal-motion, Thermal-energy and Heat, Temperature and Entropy 4.2. Thermal particles, “Thermal Virtual-particles,” and “Thermal-moles” or Dimensionless entropy 4.3. Thermal-energy is Distinguished part of Internal-energy - “Pond analogy” demystified 4.4. Thermal-Roughness Ubiquity, and Inevitability of Thermal-Friction 4.5. Inevitability and Conjugation of Work-dissipation and Entropy-generation 4.6. Irreversibility of Entropy-generation and Indestructibility of Entropy - Essence of the 2LT 5. Carnot Maximum Efficiency, Reversible Equivalency, and Work Potential 5.1. Carnot Cycle Maximum Efficiency: Proof by Contradiction-Impossibility 5.2. Carnot Cycle, Carnot Efficiency, and Carnot Equality (CtEq) 5.3. Carnot “Reverse-cycle” and Thermodynamic “Reversible Equivalency” 5.4. Work-potential, Formation-work, and Exergy 6. Thermal Transformers: Carnot-Clausius Heat-Work Reversible-Equivalency (CCHWRE) 6.1. Thermal Transformers and CCHWRE 6.2. Proof of Ideal-gas Equation-of-state and CCHWE Confirmation 6.3. Reversible-cycles are 100%-efficient: Carnot-efficiency is essentially the “Measure of Heat-WP” 7. “No Hope” for the Challengers of the Second Law of Thermodynamics (2LT) 7.1. “Perpetual-Motion Watch” Deceptive Example 7.2. Three “Primary Deception structures” of hypothetical violation of the 2LT 7.3. “Experimental test of a Thermodynamic-paradox” Demystified 8. Conclusions In summary, the Second Law of Thermodynamics (2LT) References and Notes |

## Appendix B. Thermal-Transformer and Temperature-Oscillator: Dynamic and Structural Quasi-Equilibriums (including “Persistent-Currents Quasi-Equilibrium”)

_{gen}= dS = R

_{gas}dV/V; and similarly, it could be due to subtle and illusive micro- and sub-micro forced-field ‘displacements’, including particle ‘correlation’ and/or quantum entanglement in respective force fields, etc.). Also, note that fluctuating temperature does not always mean fluctuation of entropy as demonstrated by the isentropic compression-expansion on Figure A1, etc. Therefore, the fluctuation phenomena do not violate the 2LT since the temperature fluctuations could be adiabatic (in limit, isentropic) or due to heat transfer fluctuations.

**Figure A1.**Thermal-Transformer and Temperature-=Oscillator, in adiabatic piston-cylinder system isentropically produces perpetual temperature oscillations or temperature difference. If the depicted piston, from isothermal center-position, is compressing an ideal gas in partition B, thus isentropically increasing the gas temperature, then gas will expand in partition A and isentropically decrease its temperature, without any heat transfer. If displaced piston within the cylinder with ideal, inertial mass and elastic spring system is left free, it will perpetually oscillate, but without any perpetual work generation, similarly to an ideal pendulum oscillations, thus demonstrating a thermal ‘dynamic quasi-equilibrium’ with perpetual temperature oscillations (but not entropy oscillations); or, at any locked, stationary piston position, a self-sustained ‘structural equilibrium’ will establish with perpetual temperature difference, without violating the Second law of thermodynamics (2LT).

## References and Notes

- This comprehensive treatise is written for the special occasion of the author’s 70th birthday. It presents his lifelong endeavors and reflections with original reasoning and re-interpretations of the most critical and misleading issues in thermodynamics. Only selected and this author’s relevant publications are referenced, since the goal was not to review the vast thermodynamic literature.
- Entropy Special Issue, “Exploring Fundamentals and Challenges of Heat, Entropy, and the Second Law of Thermodynamics: Honoring Professor Milivoje M. Kostic on the Occasion of His 70th Birthday”. Available online: https://www.mdpi.com/journal/entropy/special_issues/70th (accessed on 5 July 2023).
- Carnot, S. Reflections on the Motive Power of Heat; Thurston, R.H., Translator; Chapman & Hall, Ltd.: London, UK, 1897; Available online: https://sites.google.com/site/professorkostic/energy-environment/sadi-carnot (accessed on 5 July 2023).
- Thomson, W. (Lord Kelvin). On the Dynamical Theory of Heat. Transactions of the Royal Society of Edinburgh, March 1851, and Philosophical Magazine IV. 1852. Available online: https://zapatopi.net/kelvin/papers/ (accessed on 5 July 2023).
- Clausius, R. The Mechanical Theory of Heat; Macmillan: London, UK, 1879; pp. 21–38, 69–109, 212–215. [Google Scholar]
- Leggett, A.J. Reflections on the past, present and future of condensed matter physics. Sci. Bull.
**2018**, 63, 1019–1022. [Google Scholar] [CrossRef] [PubMed] - Gyftopoulos, E.P.; Beretta, G.P. Thermodynamics Foundations and Applications; Dover Publications: Mineola, NY, USA, 2005. [Google Scholar]
- Irreversibility is “irreversible transformation”, or something permanently changed (accompanied by irreversible entropy generation), without possibility to fully (or “truly”) reverse all interacting systems back by any means (impossibility of entropy destruction or annihilation). It should not be confused with local change back to the original condition by “compensation” from elsewhere (increase or decrease due to transfer of relevant quantity from or into a local system if interacting with its surroundings). For example, the work-potential (WP) is irreversibly reduced by converting it to heat accompanied with related irreversible entropy generation (or entropy production). There is no way to fully reverse these irreversibilities, without exception, at any space or time scale. However, such prior irreversibilities may be compensated locally by transferring work into or heat with entropy out of a local system, on the expense of the surrounding systems’ relevant quantities, while inevitably producing further irreversibilities (further permanent dissipation of WP accompanied with heat and entropy generation).
- Reversibility or Reversible Equivalency is an ideal concept, represented by ideal processes without any energy degradation (with maximum possible efficiency or without irreversible dissipation) so that its final and initial states are truly-equivalent and may self-reverse-back completing a cycle, or may perpetually repeat back-and-forth in any manner, therefore, effectively representing “dynamic (quasi-) equilibrium”. In fact, any process that perpetually self-sustain its own macro-structure, regardless if stationary or dynamic, or it is with perpetual uniform or non-uniform macro-properties), is in equilibrium or quasi-equilibrium, respectively; and, it is in its own right reversible, since, as a matter of concept, “reverse-process perpetuity and reversible equivalency” is the definition of reversibility.
- Cropper, W.H. Carnot’s function: Origins of the thermodynamic concept of temperature. Am. J. Phys.
**1987**, 55, 120–129. [Google Scholar] [CrossRef] - Callen, H.B. Thermodynamics and Introduction to Thermostatics, 2nd ed.; John Wiley & Sons: New York, NY, USA, 1985. [Google Scholar]
- Kostic, M. Irreversibility and Reversible Heat Transfer: The Quest and Nature of Energy and Entropy. In Proceedings of the IMECE2004, Anaheim, CA, USA, 13–20 November 2004; ASME: New York, NY, USA, 2004. [Google Scholar]
- Kostic, M. Nature of Heat and Thermal Energy: From Caloric to Carnot’s Reflections, to Entropy, Exergy, Entransy and Beyond. Entropy
**2018**, 20, 584. [Google Scholar] [PubMed] [Green Version] - Planck, M. Treatise on Thermodynamics, 3rd ed.; Dover: New York, NY, USA, 1969; pp. 103–104. [Google Scholar]
- Uffink, J. Bluff your way in the Second Law of Thermodynamics. Stud. Hist. Phil. Sci. Part B 2001 Stud. Hist. Philos. Mod. Phys.
**2001**, 32, 42–43. [Google Scholar] [CrossRef] [Green Version] - Kostic, M. Sadi Carnot’s Ingenious Reasoning of Ideal Heat Engine Reversible Cycles. In Proceedings of the 4th IASME/WSEAS International Conference on Energy, Environment, Ecosystems and Sustainable Development (EEESD’08), Algarve, Portugal, 11–13 June 2008; Available online: https://www.researchgate.net/publication/228561954 (accessed on 5 July 2023).
- Kostic, M. Revisiting the Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot’s Ingenious Reasoning to Holistic Generalization. AIP Conf. Proc.
**2011**, 1411, 327. Available online: http://www.kostic.niu.edu/kostic/_pdfs/Second-Law-Holistic-Generalization-API.pdf (accessed on 5 July 2023). [CrossRef] [Green Version] - Dissecting the 2nd Law Challenges-Kostic Web. Available online: https://sites.google.com/site/professorkostic/second-law-of-thermodynamics/dissecting-the-2nd-law-challenges (accessed on 5 July 2023).
- Thermodynamics—A section of Entropy. Available online: https://www.mdpi.com/journal/entropy/sections/thermodynamics (accessed on 5 July 2023).
- Kostic, M. Challenges to the Second Law Challengers: Reflections on the Universal Validity of the Second Law of Thermodynamics. In Proceedings of the Presentation at the Second Law Symposium, San Diego, CA, USA, 11–14 August 2016. [Google Scholar]
- Kostic, M. The Second Law and Entropy Misconceptions Demystified. Entropy
**2020**, 22, 648. [Google Scholar] [CrossRef] [PubMed] - Kostic, M. Maxwell’s Demon and its Fallacies Demystified. arxiv
**2001**, arXiv:2001.10083. [Google Scholar] - Kostic, M. Heat Flowing from Cold to Hot without External Intervention” Demystified: Thermal-Transformer and Temperature. arXiv
**2001**, arXiv:2001.05991. [Google Scholar] - Nikonov, A. The Law of Entropy Increase and the Meissner Effect. Entropy
**2022**, 24, 83. [Google Scholar] [CrossRef] - Sheehan, D.P.; Mallin, D.J.; Garamella, J.T.; Sheehan, W.F. Experimental Test of a Thermodynamic Paradox. Found. Phys.
**2014**, 44, 235–247. [Google Scholar] [CrossRef]

**Figure 1.**Reasoning concepts of forcing and energy displacement: Energy of a particle (or equivalent field particle) (

**Left**) or bulk body (

**Right**) will not change without forced interactions, i.e., interactive forcing (action–reaction) and energy displacement (energy transfer and conservation). A particle or bulk body in motion will uniformly move (

**Left**) or freely expand (

**Right**) unless interacting and exchanging energy with another particle or body. Elementary particle or ideal body interactions are reversible, but real, collective bulk-structure interactions of bounded collective-particles are irreversible due to dissipation of collective bulk, macro-energy within interacting micro-structures made up of interacting particles or equivalent field-particles.

**Figure 2.**Thermal roughness and thermal friction are the underlying cause and source of unavoidable irreversibility (2LT) since absolute-0 K temperature is unfeasible (3LT), i.e., perpetual “smooth surface” is impossible. Real surface is always “Dynamic-and-Rough” (chaotic dotted-line) since it is impossible to have a “Still-and-Smooth” surface (plane solid-line) due to perpetual and unavoidable, dynamic “Thermal-Motion (ThM)” of “Thermal Particles (ThP)” always above unachievable, absolute-0 K temperature (3LT)).

**Figure 4.**Carnot (steam power) Cycle (solid lines): heat Q

**at T**

_{H}**is reversibly converted to work W**

_{H}**=W**

_{Max}**− W**

_{T}**and to Q**

_{C}**at T**

_{L}**; and Carnot reverse cycle (dashed lines with reversed directions): work W**

_{L}**= W**

_{CR}_{|C}− W

_{|T}and heat Q

**at T**

_{L}**, are converted to Q**

_{L}**at T**

_{H}**. Thermal reservoirs’ high T**

_{H}**and low T**

_{H}**temperatures (dotted lines). T = turbine, C = compressor, |X = reverse of any X-quantity. All quantities are positive magnitudes [16,17].**

_{L}**Figure 5.**Converting heat and internal energy to work: In any steady-state process or quasi-steady-cyclic process, entropy input S

_{H}, with heat Q

_{H}(and with mass m

_{H}if any) at T

_{H}> T

_{L}, and if any irreversible generated entropy S

_{gen}within, must be discharged with heat Q

_{L}(and with mass m

_{L}if any), as entropy S

_{L}at T

_{L}. For ideal reversible process, ${Q}_{L,R}$ is “not a loss but necessity”, reducing maximum efficiency below 100%, such as in Carnot cycles (Carnot Equality). For real processes, irreversible work loss, ${W}_{LOSS}$ = ${Q}_{gen}$ = ${T}_{L}{S}_{gen}$, is due to dissipation of work to heat. For closed-mass and cyclic processes, m

_{L}= m

_{H}= 0, and for adiabatic turbine (Q

_{H|L}= 0), W

_{OUT}= E

_{mH}− E

_{mL}.

**Figure 6.**Reversible equivalency: Formation of “non-equilibrium state” requires “formation work-energy,” ideally all stored as “state work-potential (WP)” to be retrieved back in an ideal reversible process (Figure (

**Center**), “Reversible Equivalency”: ideal formation work equal to work potential). Due to irreversible dissipation of work to heat (work loss), real formation work-in is bigger than stored WP, and retrieved useful work-out is smaller (

**Left**). Formation of non-equilibrium state with less than its WP or obtaining more useful work than WP would require a “miracle Work-GAIN” without due WP source (violation of 2LT), being against natural forcing and existence of equilibrium, thus impossible (

**Right**). Therefore, all reversible processes must be maximally and equally efficient [17,18].

**Figure 7.**Carnot–Clausius Heat–Work Reversible Equivalency (CCHWRE) (as named here), established interchangeability and related equivalency between “Heat-and-Work”, based on the early work of Carnot (1824), that all reversible processes and cycles have equal and maximum efficiency, and among others, Kelvin and Clausius’ meticulous work, around 1850s, that finalized the thermodynamic temperature, reversible cycle efficiency, Carnot Equality, Clausius (In)Equality, Entropy, and generalized the Second Law of Thermodynamics (2LT).

**Figure 8.**Carnot cycle with ideal gas: Isothermal expansion and compression’s works result in cycle net-work out, while adiabatic expansion and compression’s works cancel out, but they change temperatures required for reversible heat transfer.

**Figure 9.**“Perpetual-motion-like” watch with 5–10 years battery life, as if its battery lasts forever. We could mistakenly hypothesize (as if we have proved experimentally), that it works without using energy (PMM1, 1LT violation), or it consumes energy from the surrounding thermal reservoir alone (PMM2, 2LT violation).

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kostic, M.
Reasoning and Logical Proofs of the Fundamental Laws: “*No Hope*” for the Challengers of the Second Law of Thermodynamics. *Entropy* **2023**, *25*, 1106.
https://doi.org/10.3390/e25071106

**AMA Style**

Kostic M.
Reasoning and Logical Proofs of the Fundamental Laws: “*No Hope*” for the Challengers of the Second Law of Thermodynamics. *Entropy*. 2023; 25(7):1106.
https://doi.org/10.3390/e25071106

**Chicago/Turabian Style**

Kostic, Milivoje.
2023. "Reasoning and Logical Proofs of the Fundamental Laws: “*No Hope*” for the Challengers of the Second Law of Thermodynamics" *Entropy* 25, no. 7: 1106.
https://doi.org/10.3390/e25071106