Momentum work and the energetic foundations of physics. IV. The essence of heat, entropy, enthalpy, and Gibbs free energy

Momentum work enables a complete shift from kinematics to dynamics. This involves changes in the very fundamentals of physics, not only in mechanics, statistical mechanics, and special relativity, as shown in Papers I–III [G. Kalies and D. D. Do, AIP Adv. 13 (6), 065121 (2023); G. Kalies, D. D. Do, and S. Arnrich, AIP Adv. 13 (5), 055317 (2023); and G. Kalies and D. D. Do, AIP Adv. (in press) (2023)] of this series, but also in thermodynamics. In this paper, we challenge the narrative that classical phenomenological thermodynamics is completed and show that it represents an efficient interim solution that hides essential information. The essence of heat transfer and entropy is revealed, and an answer is given to the question of why entropy had to remain abstract and elusive in the past. Furthermore, we uncover the specific forms of energy behind thermodynamic state variables, such as enthalpy, Helmholtz free energy, and Gibbs free energy, which play a central role in describing chemical reactions and phase transitions. We thereby lay the foundation for thermodynamics to evolve from a framework theory valid for macroscopic systems to vivid quantum-process thermodynamics.


I. INTRODUCTION
Classical phenomenological thermodynamics (TD), which is also called Gibbs thermodynamics, 1 is an efficient theory that has many applications in describing processes in industry and engineering.However, the theory is difficult to understand, as, for example, the famous theoretical physicist Sommerfeld 2 noted.The following statement by Sommerfeld is popularly prefixed to textbooks today: Thermodynamics is a funny subject.The first time you go through it, you don't understand it at all.The second time you go through it, you think you understand it, except for one or two points.The third time you go through it, you know you don't understand it, but by that time you are so used to the subject, it doesn't bother you anymore. 3 Since the mathematics of classical thermodynamics is simple, there is no problem here.The low level of understanding results rather from the abstractness of variables, such as heat, entropy, enthalpy, and Gibbs free energy.Although they have their firm place and play an important role in basic equations, it has remained unclear; for example, what form of energy the Gibbs free energy actually represents.Another question frequently asked by scientists and engineers alike is "What is heat?," and the current answer "disordered energy" 4 remains vague, as scientists rightly state. 5In particular, entropy, introduced by Clausius in 1865, 6 has remained mysterious, which is responsible for its continuing diverse interpretations and history of fascination. 7The problem of this state variable is pointed out repeatedly, 5,[8][9][10][11] and the statistical-mechanical interpretation of entropy by Boltzmann 12 cannot justify irreversibility at the micro level. 13,146][17][18][19] Nevertheless, ARTICLE pubs.aip.org/aip/adv as the habituation effect is significant, many scientists have adopted Sommerfeld's attitude that "it doesn't bother you anymore."This is the fourth part of a series of papers that challenge the interaction concept of modern physics on a fundamental level.1][22] Classical thermodynamics is also becoming more understandable, evolving from a theory that applies only to macroscopic systems to one that applies to individual objects at the quantum level.
We first present the basic contents of Gibbs thermodynamics (Sec.II) and review its strengths and weaknesses.In Sec.III, we briefly present the results from Papers I-III needed for the reformulation of classical thermodynamics.Finally, in Sec.IV, we reinterpret thermodynamic state and process variables.Heat exchange is described as a composite process, which consists of various microscopic subprocesses with their own cause-effect principles, and entropy is recognized as a state variable with multiple meanings.We assign specific forms of energy to the various thermodynamic potentials, followed by a discussion of the enthalpy H and the Gibbs free energy G. Since the connection between thermodynamics and quantum physics becomes immediately clear, we lay the foundations for a new quantum-process thermodynamics (QPT)-with an unconditional validity of both the principle of energy conservation and irreversibility.

II. CLASSICAL GIBBS THERMODYNAMICS
Classical Gibbs thermodynamics (TD) is a mathematical formalism with energetic variables, so-called "thermodynamic potentials," mainly the internal energy U, the enthalpy H, the Helmholtz free energy A, and the Gibbs free energy G.We briefly describe the current formalism and its origin, and then point out its strengths and weaknesses.

A. The status quo: The Gibbs formalism
The formalism is based on two empirically confirmed axioms, the so-called laws of thermodynamics.The first law [23][24][25][26] was formulated in the mid-19th century for a closed system, where ∆eU is the change in the internal energy U of a system due to exchange processes (index e), Q is heat exchange, W is work, while mole number exchange is not allowed in a closed system.∆ i U = 0 is the principle of conservation of energy, according to which energy can neither be created nor be destroyed by internal processes (index i) in an isolated system.
The second law was formulated by Clausius, who linked the heat exchange Q with the change in the extensive state variable S, which he called entropy in 1865. 6The current notation of the second law according to Nobel laureate Prigogine, 27 p.35, is where deS is the entropy exchange, while d i S is the entropy production in the system, which is zero in equilibrium or for reversible processes.Thus, the reversible heat exchange Q is Instead, in any natural spontaneous (irreversible) process, S is produced, 6,27 and the differential heat exchange δQ is described with Eq. ( 2) by Starting from the first and second laws, Gibbs developed his formalism 1 in the period 1876-1878.Using only volume work W V in Eq. ( 1) and setting d i S = 0 in Eq. ( 2), he formulated the Gibbs fundamental equation for U and then, via Legendre transformation, the Gibbs fundamental equations for H, A, and G, dU = δQ + δW V = T dS − p dV, dH = T dS + V dp, dA = −S dT − p dV, dG = −S dT + V dp. ( If irreversible processes are described, the term −Td i S according to Eq. ( 4) is added to each of the equations in Eq. ( 5), 27 p.37.For an open system, the mol number exchange δWn of a pure substance (or of a mixture of several components) is added to each of the equations in Eq. ( 5), Accordingly, the total differentials of U, H, A, and G are By coefficient comparison of Eqs. ( 6) and (7), definitions for state variables, such as T, p and V, are obtained, e.g., four expressions for the chemical potential μ of a pure substance, AIP Advances the (integral) Euler equations of the thermodynamic potentials follow, where we include the grand potential or Landau free energy Ω to the list, Other commonly used equations derived from the above formalism are Gibbs free reaction energy.( 14)

B. Critical review of the Gibbs formalism
The formalism was deduced from the laws of thermodynamics (TD).Starting from the internal energy U, Gibbs created new energetic functions that depend on variables that can be more easily controlled and regulated than S and V. Specifically, the Gibbs free energy G has proven to be important [cf.Eqs. ( 11)-( 14)].In the following, we identify and briefly discuss the strengths and weaknesses of TD.

Strengths of the Gibbs formalism
1.A few energetic state variables, such as U, H, A, and G, and other macroscopically measurable state variables, such as the volume V or the mole number n, are sufficient to describe the behavior of macroscopic matter with sufficient accuracy.2. The approach has proven useful for the design and control of systems in chemistry, engineering, and physics under specific conditions (constraints).It describes not only exchange processes but also internal processes, such as chemical reactions within a system by means of ∆ R G ≠ 0 [see Eq. ( 14)]. 3. The cause-effect principle inherent in process equations δYi = ξidXi is always respected, where dX i is the effect; see the Gibbs fundamental equation for U in Eq. ( 6). 4. Conservation of energy is respected at any time, irrespective of how long a process is.
5. The directionality of spontaneous natural processes toward equilibrium, also called irreversibility or the "arrow of time," can be described by −Td i S, e.g., the experimentally proven fact that heat does not pass by itself from a colder body to a warmer body.6.The change in U is usually described by simultaneous processes.For instance, if the number of particles in a system is increased via Wn, volume work W V usually occurs at the same time because V cannot be kept artificially constant in condensed matter.If d i S ≠ 0, internal processes can be quantified by TD of irreversible processes.
The principles in points 3-6 are by no means self-evident in other physical theories.For instance, mechanics and quantum mechanics allow for the violation of the energy conservation at the reversal point of motion in a collision [20][21][22] or when quantum fluctuations arise from and pass into nothing.Other theories, such as special and general relativity (GR), do not use process equations at all, 28 and all theories except thermodynamics describe processes as reversible, which gives thermodynamics a unique selling point.

Weaknesses of the Gibbs formalism
1.The formalism (designed for technical purposes) provides only limited information: it can describe processes but cannot explain them.][10][11] Boltzmann and Gibbs contributed to its statistical-mechanical interpretation 12,29 but built on Newtonian mechanics that describes interaction via forces and not via processes. 20-223. A huge shortcoming is that no specific forms of energy were assigned to each term on the right-hand side of the first equation of Eq. (10).While TD is superior to mechanics in process description, it is inferior in the attribution of energy amounts. 20There is no doubt that TS, −pV, and μn are energetic contributions to U but what are the specific forms of energy behind them?How are these terms related to the kinetic or potential energies of particles in a system?The question also arises regarding the physical meaning of negative energy −pV contributing to U.

Generations of students (and teachers) have been introduced
to abstract "thermodynamic potentials."While the internal energy U is reasonably descriptive, although the distinction between internal and external energies has remained a problem, 20,30 the newly created energetic variables H, A, G, and Ω, mostly named after scientists, allow even less concrete notions.Their formulas are known, nothing more.
For example, what is the physical meaning of the enthalpy H = U + pV?Does this mean that H is a larger energy amount than the whole energy U of a system? 5.In particular, the meanings of the Gibbs free energy G and the chemical potential μ = Gm have remained unclear.Is there a specific form of energy or molar energy behind G or Gm?Why exactly this form of energy is so important in the description of chemical reactions and phase transitions?The current Gibbs formalism only shows that it is so but does not explain it.
6. Equation (10) implies the assertion of energetic completeness. 22ince the number of terms on the right-hand side of U = TS − pV + μn depends on the number of processes considered in dU = ΣδYi in Eq. ( 6), energetic completeness cannot be guaranteed.Clausius included U 0 in the expression for U, where "U 0 represents the value of the energy for an arbitrarily chosen initial state of the body," 26 p. 386.This is correct because changes in U cannot be used to deduce the integral value of U.For pragmatic reasons, Gibbs omitted U 0 in his formalism, which is often not noticed by users.
It becomes evident that the Gibbs formalism has weaknesses in terms of explaining the variables and terms used.Users in industry and engineering might take the position that the end justifies the means, especially since the deductive method can be elegant.However, a purely mathematical approach carries risks: incorrect physical conclusions may be drawn if the underlying axioms are not entirely correct, and hence logical inconsistencies cannot be identified.The applicability of an equation does not mean a priori that its physical content is fully understood, as has been shown, for example, for F = ma or the velocity-dependent mass m(v) = γm 0 . 20here are justifiable grounds for assuming that Gibbs formalism is so far still "mathematical phenomenology" 31 and an efficient interim solution.We show in Secs.IV and V that the formalism can be freed from its weaknesses while preserving its strengths when moving from kinematics to dynamics at the fundamental level.We also show why the Gibbs formalism works.

III. PROCESSES AND ENERGIES
In Paper I, 20 we have introduced the mechanical momentum work W P , where v is the velocity, P the momentum, and E P the momentum energy of an object due to its motion in the Euclidian space, written in terms of its instant mass m and velocity v, E P is larger than the kinetic energy E kin of an object because m depends on v, Equation ( 15) describes a real dynamic process Yi in time (no observer effect) and follows the general form of a process equation, where the driving force ξi is an intensive variable, while the effect is the change of its conjugated extensive variable Xi, Each process changes a specific type of energy Ei of an object or system (cf.Table I), which, as we shall see, is important information.
Using momentum work and displacement work, the elastic collision can be described via process = counter-process, whereby the conservation of energy is ensured at any point in time. 20A complete departure from kinematics allows us to develop the process theory of gases (PTG) 21 and to formulate the unified energy equation, which does justice to the wave nature of matter. 22Here, there are no "pointlike particles" or "point masses" with "rest mass" and "rest energy" moving in "empty space" as suggested in special relativity (SR), but all particles and photons are real, spatially extended wave objects.
Real waves can only exist in a medium.Consequently, elementary particles are emergent entities with emergent properties, such as mass, charge, flavor, and spin, created by the agitation of a condensed medium, i.e., from the medium's energy by energy conversion in compliance with the conservation of energy. 22,30he momentum energy E P of a particle or body is 22 where m is the total mass, m d is the dynamic mass due to the motion of the object as a whole, and m 0 is the intrinsic mass of the particle or body due to its intrinsic motion in Euclidian space.Thus, any measurable mass of a particle or body is dynamic and generated by motion.Objects moving with v < c have both dynamic mass m d and intrinsic mass m 0 .For photons, their momentum is 22 The unified energy equation describes the possible components of the total energy E of a moving real wave object (photon, particle, body, and system), 22 where E 0 is the internal momentum energy or intrinsic excitation energy of the wave object due to its intrinsic motion (rotation of wave packets, oscillations, helical motion, etc.), and it is proportional to the intrinsic mass m 0 of the object.Epot is the potential energy of the object, which does not contribute to its mass, [20][21][22]28,30,32 and E M is the mass energy or (total) excitation energy of an object, which is proportional to its total mass m. The intrnsic mass m 0 , intrinsic motion, and intrinsic excitation energy E 0 of a photon are zero, see Eq. ( 20).
The intrinsic motion of a spatially extended wave object includes all intrinsic degrees of freedom, excluding the motion of the object as a whole in space.E 0,j of an electron j, for example, includes the internal momentum energies of the hitherto hidden motions in its strongly agitated and, thus, most massive wave core, the momentum energy E hf = Σhfj of all photons absorbed in the core, and the long-range wave extensions of the wave core.E 0 of an atom covers, for example, ΣE 0,j of all electrons in the atom ΣE P,j of all electrons, e.g., due to their motion around the atomic nucleus, E 0 and E P of all gluons and quarks in the hadrons, E P of the hadrons (protons and neutrons) in the nucleus and of the nucleus itself, and E hf = Σhfj of the photons outside electrons but present in the atom.E 0 of a moved body or system in turn covers E 0 and E P of all particles (atoms, molecules, etc.) and E hf of all photons in the body/system.It can be seen that the distinction between internal momentum energy E 0 and (external) momentum energy E P depends on how the object is described and what is the focus in the description or measurement.
In a binding process, e.g., between two atoms forming a molecule, all energies contributing to E 0 of an atom are affected by the binding process, a theory of which will be developed in detail in Part 5.In this work, we lay the foundations for Paper V by showing that the Gibbs formalism, already describes a reduction of the intrinsic excitation energy E 0 of atoms or molecules during a binding process, e.g., by the release of photons (see Sec. V C).

IV. THE ESSENCE OF HEAT, ENTROPY, AND SO ON
The findings of Papers I-III [20][21][22] allow us to reveal the essence of basic thermodynamic variables.We shortly repeat the meaning of the temperature T, derived in Papers II and III, and then turn to heat Q, entropy S, chemical potential μ, and pressure p.

A. The temperature T
The temperature is the driving force ξi = T of the heat exchange δQ = TdeS and is closely related to ξi = v of the momentum work δW P = vdP (cf.Table I).The faster the particles move in a system, the higher their velocities vj and masses mj and thus T and m of a system.While vj of a particle j is [cf.Eq. ( 16)] a major insight of Paper II of this series 21 is that the temperature Tj of a single particle j in an ideal gas can be defined.Just as a particle has a velocity vj, it has a temperature Tj at a given moment.vj and Tj are very similar.While vj describes the momentum energy E P,j of a particle related to the momentum Pj of the particle, i.e., related to Newton's "quantity of motion," 33 Tj describes the momentum energy E P,j of a particle related to one particle (N = 1), i.e., E P,j of the particle.Thus, vj and Tj are intensity measures of that type Ei of energy that is contained in the external degrees of freedom of the particle motion.
In contrast to today's interpretation via E kin , the temperature T of a system is a measure of the average momentum energy per particle.The definitions of Tj of a single particle j and T of a system with N particles are 21 where E P is the momentum energy of all particles in the system, and the Boltzmann factor k B is a historical factor for unit conversion. 34,35t is well-known that the supply of photons increases T of a system.A major insight of Paper III 22 is that electromagnetic energy E hf is momentum energy E P [cf.Eq. ( 20)].Accordingly, the electromagnetic energy hfj of a photon j is the momentum energy E P,j of this photon.Since the temperature is an intensity measure for momentum energy, we can define Tj of one photon j as well as T of a system with N photons completely analogous to Eq. ( 23), 22 k B Tj = h fj; where hfj is the electromagnetic energy of the photon j, and E hf is the total electromagnetic energy in the system, which corresponds to the momentum energy E P in the photon system.As the momentum energy is always related to a number (N or N hf ), it is quantized.Due to the various momentum energies E P,j = k B Tj of particles and hf j = k B Tj of photons with different frequencies fi, there is usually a Tj distribution in a system.Normally, every system has particles and photons, i.e., T becomes a mixed variable.

B. Heat exchange Q
In the 19th century, as the first and second laws of TD with the process Q were formulated, the existence of atoms and photons was still disputed.Hence, it was a great achievement of Maxwell, Boltzmann, and others to attribute macroscopic phenomena to the microscopic behavior of particles.However, since the momentum work W P was unknown, they applied force interactions, and the essence of Q remained hidden.In this section, we show that it is the knowledge of the momentum work W P and the momentum energy E P that gives us an avenue to provide an in-depth understanding of the heat exchange Q.
It is known that Q can be realized, e.g., via heat conduction and heat radiation.While heat conduction occurs, for example, when vibrating atoms collide, i.e., by momentum work, heat radiation means the transfer of photons, where E hf = E P is exchanged.Hence, Q is a macroscopic concept for the microscopic momentum work of colliding particles or the exchange of photons.In every real substance, there are photons, and each quantum leap of an electron from a higher to a lower shell of an atomic nucleus is associated with emitted photons.
With logical extrapolation, reversible heat exchange Q describes all quantum processes in which E P is exchanged.Thus, Q can be understood as the sum of α different microscopic subprocesses Yi, each with its own driving force ξi and effect ΔXi, The α quantum subprocesses Yi with the exchange of E P can occur simultaneously, e.g., via the collisions of N vibrating particles (heat conduction), the exchange of E hf = E P (heat radiation), the transport of E P with particles (heat convection), or even the exchange of E P by gravitational waves, which do not represent oscillations of an assumed flexible space-time but make a vanishingly small contribution to Q, where N hf k is the number of photons of frequency f k , and μ hf k is their thermal potential, quite analogous to the chemical potential of particles.If moles of photons of one frequency are reversibly exchanged, one can even write in perfect analogy to the mole number exchange δWn = μdn of a pure substance, The analogy to δWn = μdn was chosen intentionally for the following reasons: (i) Just like other particles, photons are discrete and countable.(ii) Matter and light can be understood analogously.Indeed, matter particles have both E 0 and E P , while photons only have E P . 22However, both intrinsic excitation energy E 0 and momentum energy E P are (internal and external, respectively) momentum energy and mass-proportional.Particles and photons can be transformed into each other, i.e., E 0 and E P can be converted into each other.(iii) The definition in Eq. ( 27) is closest to that of temperature in Eqs. ( 23) and ( 24).(iv) If only radiation would be considered, the old idea of a flowing "heat substance" (phlogiston, caloricum) would be even quite justified.Of course, this does not apply to other forms of heat exchange Q such as heat conduction.
Equation (28) shows that Q involves different microscopic work processes.In other words, heat exchange Q is also work, or more precisely, it is, in each case, microscopic momentum work W P .Therefore, the vague distinction made today between Q (exchange of "disordered energy" 4 ) and work W (exchange of "ordered energy") does not apply.We can only distinct microscopic work (described by Q) and macroscopic work.
The long-awaited answer, what specific type of energy is transferred during heat exchange Q, is now exposed: momentum energy E P !This will change our understanding of the entropy S (see Sec. IV C) and clarify thermodynamic nomenclature.Today, for example, the one-dimensional Fourier's law for heat conduction is written in the form of Eq. (29a), where Q is called "heat" or "heat energy" transported in time.This naming represents a stopgap solution because the variable Q is defined as "heat exchange," i.e., as a process variable δQ.It is illogical to transport a process, and conceptual imprecision and multiple meanings of variables hinder understanding.It is an amount of energy that is transported in the process Q, namely, the momentum energy E P in Eq. (29b).

C. The entropy S
The entropy S is an extensive state variable Xi of a system changed via reversible heat exchange δQ = TdeS [cf.Eq. ( 3)].According to Eq. ( 26), S represents a makeshift collective term that stands for k different extensive state variables Xi of a system such as the total momentum P of the particles and the total number N hf of photons in a system, S = P; N hf ; . . .
The larger the entropy S of a system at V, n = const., the larger E P and thus the internal energy U and the mass m of the system.Before discussing and reinterpreting S, we briefly review the historical interpretation.

Historic interpretation of S
In 1824, Carnot called the extensive state variable Xi that is changed via heat transfer Q "calorique" or "chaleur," 36 i.e., heat.The naming is analogous to that of other Xi changed via exchange processes Yi and corresponds to the intuitive understanding: The heat of a system is changed via heat exchange, just as the volume of a system is changed via volume work.A furnace possesses heat.Today, there are still scientists who point out that Xi = S changed via Q should be called "heat" (e.g., Refs.5, 8-11, and 37).
In 1865, Clausius searched for an expression for the observation that a spontaneous process is directed, e.g., that energy flows from a hotter reservoir to a colder one.In order to account for this asymmetry, he proposed that S has more meaning than being only exchanged. 6He coined the word "entropy" (Greek: entropía) in analogy to "energy" and suggested that S is additionally increased by internal processes, namely by the disgregation, which is to be regarded as the transformation value of the ongoing arrangement of the components, 6 p. 390 He thus combined the entropy exchange ΔeS via Q with the rearrangement of the components of a system, known in the literature as the entropy production Δ i S [cf.Eq. ( 2)], While ΔeS can be positive or negative, Δ i S is always positive when a natural (irreversible) process occurs.Δ i S > 0 is thus a criterion for the spontaneous evolution of an isolated system toward equilibrium (S = max).The extrapolation to the universe by postulating socalled "heat death of the universe" made Clausius' entropy S famous overnight.
Δ i S > 0 in an isolated system (ΔeS = 0) is explained today for "the heat flow" from subsystem A to B with T A > T B (using a discontinuous T-profile) by the following balancing: which Prigogine called "the entropy production per unit time," 38 p. 20.The direction of the heat flow is determined by the positive sign of entropy production (cf.Fig. 1).Boltzmann and Planck suggested the following statistical-mechanical interpretations of S, where S is the number of the microscopic realization possibilities of a macroscopically defined state, P is the number of possible permutations, and W is the thermodynamic probability.While S in Eq. ( 34a) is a number (without a unit), the use of the conversion factor k B in Eq. (34b) provides S the unit (J/K).Due to the openness to interpretation of Clausius' and Boltzmann's entropy S, there are various other statistical-mechanical and information-theoretical definitions of entropy, e.g., suggested by Gibbs and Shannon.

Interpretation evaluation
Clausius' idea of "disgregation," 6 i.e., the dispersion of matter toward uniform distribution with time, corresponds to measurements.Nevertheless, it is questionable whether one is allowed to link Δ i S > 0 to heat exchange Q, thereby creating a new state variable Xi = S with more meaning than other Xi changed in exchange processes Yi.The meanings of S in ΔeS and Δ i S (statistical interpretation) are compared in Table II.
There is no question that the number of microstates changes if particles are redistributed.Nevertheless, a microstate number [cf.Eq. (34a)] cannot be combined with, e.g., the total momentum P of the particles in a system in one and the same state variable S. While ΔeS changes the amount of U and the mass m of a system, Δ i S does not.Two completely different meanings are mixed, i.e., S in deS and S in d i S definitely are not identical variables.
The reason for choosing and accepting the contents of Eqs. ( 31)-( 34) was the unsettled nature of Q and T. Knowing that momentum energy E P is transported, not the process Q itself, we rewrite Eq. ( 33) of Prigogine to make the concept clearer, Now, we recall that δQ = dE P = TdeS stands, for instance, for heat radiation δW n,hf = dE hf , cf.Eq. ( 28).If a monochromatic photon gas is considered, S A stands for n hf,A and S B for n hf,B , while T B stands for μ hf,B and T A for μ hf,A .Equation (35a) consequently becomes We realize that Eq. (35a) and thus Eqs. ( 31)-( 33) demand something impossible.The mole number n hf of photons in an isolated system without absorption remains constant.n hf is neither increased nor only increased by simple transport.Consequently, it is the lack of knowledge about momentum work W P and momentum energy E P that mystified Q, T, and S. Clausius' idea to combine energetic and geometric issues in one and the same state variable S involves an impermissible mixing of different things, which contributes to conceptual confusion.Again, multiple meanings in a single variable hinder understanding and lead to scientific stagnation.In reality, every dynamic process is associated with spatial redistribution, and d i S in Boltzmann's interpretation has to be decoupled from T and Q.

Reinterpretation of the entropy S
Equations ( 2)-( 4) are useful but not physically justified and, therefore, not set in stone.Based on momentum work and its consequences, [20][21][22] we propose another interpretation.

ARTICLE
pubs.aip.org/aip/advS in ΔeS S in Δ i S

Meaning
Collective term standing for P of particles The number of microstates compatible with a and N hf of photons in a system, cf.Eq. ( 30) macrostate of a system ("microstate number") Measure of the momentum P and number N hf in the system the arrangement of particles or photons in the system

Influence on
Quantity of U Quality of U If the macroscopic behavior of a system is to be described, one can simply accept the composite process Q without the need to specify the individual quantum subprocesses that contribute to Q.Then, our work supports the proposal of Carnot and others 5,[8][9][10][11]36,37 to call the extensive state variable Xi changed via Q simply "heat," which is descriptive. We prpose to distinguish "heat" from the entropy production and suggest using the symbol Σ (the Greek character for S) to show that heat Σ is a composite variable.The heat Σ of a system is changed via δQ = TdΣ.The new understanding and naming are listed in Table III.
In the following, we deal with the sounding neologism "entropy S" coined by Clausius and with Δ i S that only affects the quality of U (cf. Table II).Δ i S > 0 is called today "dissipation," "degradation," or "devaluation" of energy because with increasing uniform distribution, less work can be done.It is known that Planck initially disagreed with Eq. (34a) because arrangement possibilities do not explain irreversibility at the quantum level. 39Only when Planck, after many years, did not succeed in finding an irreversible quantum process, he reluctantly agreed to Boltzmann's interpretation.Indeed, Eq. (34b) does not explain irreversibility.Boltzmann, Gibbs, and others were influenced by the kinetic theory of gases (KTG), where the point-like particles in an ideal gas system have only E kin and can only change their position.Thus, the observed macroscopic irreversibility is seen as an emergent property of statistical-mechanical origin, not as a quantum effect.
The process approach suggests a different picture: All photons and particles, even those of an ideal gas, are real wave objects created by excitation of a condensed medium, 22,30,40 whereby the energy of the medium is conserved (also if quantum fluctuations arise and pass away).As real waves, particles are not limited to changing their positions but can change themselves.Even during simple collisions, E P of a particle is converted into Epot, whereby the mass m of the particle is changed. 20,21Even during the increasing uniform distribution of particles in an isolated system or during the motion of particles or photons through the medium, dynamic processes with energy conversion occur, which means a complete departure of kinematics.
Below, we would like to outline some plausible dynamic, genuinely irreversible quantum processes that will be substantiated in Paper V, devoted to interaction.
The energy in a system is "devalued" when the excitation energy or mass energy E M of particles (real wave objects) is (minimally) reduced via energy conversion, We give two examples.
1. Spontaneous spreading of particles occurs if the interaction via collisions, which drive them apart, is stronger than that via their long-range wave extensions holding them together.
In reality, the wave extensions always interact, albeit very weakly, also in an ideal gas, especially since each quantum object "occupies the entire space available to it," 41 p. 30.
With increasing uniform distribution, E M = mc 2 of gas particles imperceptible decreases, while their Epot increases because the connecting network is stretched, so to speak, and their motion is more hindered.Spreading cannot be reversed along the same path.It is spontaneous, while the reversal process must be induced.2. Real wave objects interact with the condensed medium when propagating, which is, for example, a plausible explanation for the measurable cosmological redshift.The momentum energy E hf = E P = m d c 2 of photons is converted progressively into the non-ponderable energy of the medium.With increasing wavelength, the photons lose dynamic mass m d and E P .Due to this "aging process," 30,40 photons are less able to perform work and gradually pass into the highly entropic medium.The aging cannot be reversed along the same path, but only along another, if photons, e.g., interact with other particles.
The "devaluation" of the energy via ∆E M < 0 is no one-way street.Along another path, the mass energy E M is confined in a smaller spatial area and enhanced, We give again two examples.
1. Spontaneous self-organization of particles takes place if the interaction via entanglement is stronger than that via collisions.In this case, E M = mc 2 is focused in a smaller spatial area, mostly under the release of photons with E P .
The cluster enables a stronger local differentiation from the highly entropic condensed medium, associated with stabilization-the principle of binding (see Paper V of this series).Binding cannot be reversed along the same path.It is spontaneous while unbinding must be induced.2. It is known that via quantum fluctuations, unstable particles appear and disappear.They do not arise out of nothing or out of the "empty space" introduced in special relativity. 42Already Nernst proposed the emergence of particles from a condensed medium, 43 which, before 1905, was called ether or light ether. 44et Nernst's ideas have been reinterpreted as the precursor of today's understanding of particles arising out of nothing.
Promising candidates for the formation of stable particles with E M are phase transitions in specific medium regions, e.g., in black holes 45,46 being neither "spacetime singularities" nor "holes," but densely packed matter with E M , i.e., "enhanced energy."Once created, a stable elementary particle does not disappear along the same path in the medium but starts its evolution.
Accordingly, irreversible quantum processes, i.e., processes that cannot be reversed along the same path, are characterized by (i) energy conversion due to unbinding or binding processes of real wave objects, (ii) energy conversion due to dynamic interaction with the medium, and (iii) the unconditional conservation of energy at any point in time.
Here, S can not only be increased ("devaluation of energy") but also decreased ("enhancement of energy"), whereby the energetic assessment is human-made.Hence, several quantum processes are hidden behind Δ i S, 40 where Δ i S D > 0 is Clausius' disgregation, Δ i S A > 0 is the aging, Δ i S S < 0 is the self-organization, and Δ i S C < 0 is the creation of elementary particles.Equation (38) suggests that (i) irreversibility is a law of nature rooted in quantum processes, and (ii) there is no "heat death of the universe."

D. The chemical potential μ
The chemical potential μ of a pure substance is the driving force ξi = μ of the mole number exchange δWn = μdn and similar to T as the driving force ξi = T of the heat exchange δQ = TdeS described in Sec.IV A. The common features are as follows (cf.Table I): (i) Both μ and T are intensity measures of quantizable and mass-proportional energies.(ii) Both μ and T can refer to the number N of particles in a system.
μ of a pure substance is an intensity measure of the intrinsic excitation energy E 0 (the internal momentum energy) of the particles, whereas T is an intensity measure of the momentum energy E P of the particles in a system.To illustrate the analogy, we list μ and T of particles together.In case of photons, which do not have E 0 , the terms μ hf and k B T can be used equivalently [cf.Eqs. ( 24) and ( 27)], where k B is merely a historical factor for unit conversion, and Accordingly, μ describes the average intrinsic excitation energy E 0 of the particles of a pure substance.If the number of particles is given in mol, as is usual for describing macroscopic systems, μ of a pure substance becomes the molar variable, Since μ and T refer to discrete and countable objects, E 0 and E P are quantized, and μj and Tj are given by μj = E 0,j (particle); k B Tj = E P,j (particle); k B Tj = h fj (photon), (42)   where μj (T, p) has a clear microscopic meaning, as already proposed by Falk and Ruppel. 47In a macroscopic system with a uniform particle distribution, it is reasonable to assume a uniform E 0,jdistribution, which simplifies calculations.If there is no particle j (N = 0) in a system, then there is no E 0 and no μ in the system.We add a note on conceptual accuracy.Although n is defined as a particle number, n is often called "amount of substance" and δWn = μdn is called "substance exchange," "matter exchange," or "mass transfer."These terms are misleading because they suggest that real substances with volume V, momentum P, etc. would be exchanged by Wn, which is not correct.V and P are changed by W V and Q.If "mass transfer" is used, an error in thinking immediately creeps in because the mass of a system is also changed by heat transfer Q.

E. The pressure p
The pressure p is the driving force ξi = p of the volume work δW V = dEpot = −pdV (cf.Table I).We briefly present below the meaning of p, derived in Paper II. 21 In contrast to the kinetic theory of gases (KTG), where p is connected with the kinetic energy E kin of the particles in an ideal gas, the process theory of gases (PTG) interprets p as an intensity measure of the positional energy Epot, Accordingly, p represents an Epot density and describes how much Epot is on average at a point in the volume V. Since Epot refers to a spatial state variable, namely, the volume V of a system, p of a system is not particle-related but depends on the available space for the particles, which are confined by walls.The pressure p has an antithesis meaning to the temperature T, i.e., they represent driving forces of oppositely directed processes. 21e add a remark for the next section.To compress a system, i.e., V 2 < V 1 , potential energy Epot has to be supplied.If one integrates dEpot = −pdV from the state of minimal Epot at V 1 → ∞ to the state of the maximal Epot at V 2 → 0, keeping p constant, the result corresponds to Eq. ( 43).With inverse limits, one obtains If only −pdV were integrated from V 1 → 0 to V 2 → ∞, while the minus sign on the left side before Epot was not considered, one would be tempted to write an integral expression −pV.However, negative absolute values of an energy cannot be concluded from a change in energy.

V. VIVID THERMODYNAMICS TOWARD QUANTUM-PROCESS THERMODYNAMICS
We apply the results of Sec.IV to the Gibbs formalism presented in Sec.II and then discuss the enthalpy H and the Gibbs free energy G.

A. The Gibbs formalism
Instead of Eq. ( 6), the Gibbs fundamental equation for U of a system reads as where Q stands for all microscopic subprocesses, in which the momentum energy E P of the system is changed, and Σ is the heat of the system (cf.Table III).The operator d stands for exchange and d i for internal processes.With d i S = 0, irreversible energy conversion, which accompanies each process, is neglected (cf.Sec.IV C 3).
Instead of the total differential of U in Eq. ( 7), we write or by using the composite process δQ = TdΣ in Table III, It becomes evident that the process approach in Eqs. ( 47)-( 49) provides much more physical information on the variables than the total differential in Eq. ( 7) because the type of energy Ei changed in a system via the respective process Yi is specified, and simultaneous processes are allowed to occur, which is discussed in more detail in the Appendix.
The integral Euler equations in Eq. ( 10) can now be associated with specific energy terms, Consequently, one can write With Eqs.(50)-( 52), the energies TΣ, pV, and μN and, thus, the energetic meaning of H, A, G, and Ω are no longer elusive.The results are summarized in Table IV.We highlight key new findings: (i) If δQ = TdΣ is separated from the entropy idea, then Q, T, and Σ can be vividly described.The form of energy transferred, namely momentum energy E P , is identified.(ii) The "entropy S" is decoupled from Q, T, and Σ. Superficially, ∆ i S > 0 describes the measurable tendency toward uniform distribution.Irreversible quantum processes with ∆ i S > 0 mean energy conversion, where the mass energy E M in the system, i.e., the excitation energy of the medium in this region of space, is reduced.(iii) Specific forms of energy can be assigned to the terms TΣ, −pV, and μN.Here, it is important to realize that the minus sign in −pV is a mathematical artifact that only refers to the direction of energy transfer.In the Gibbs formalism, it is needed, but negative absolute values of Epot = Ω cannot be concluded from a process (see Sec. IV E).

State variable Name Expression
Energetic meaning The enthalpy H = E M = E P + E 0 is a smaller amount of energy than U, i.e., H does not go beyond the scope of thermodynamics.Equation (50) shows that each of the thermodynamic energies H, A, G, and Ω is smaller than the total internal energy U. (v) The thermodynamic energies H, A, G, and Ω become descriptive.For example, G is the intrinsic excitation energy E 0 of all particles in the system, which explains why G is so important for describing chemical reactions, phase transitions, etc. and confirms the particle concept presented in Paper III of our series. 22

B. The enthalpy H
According to Table IV, the enthalpy H is smaller than U of a system, While U describes the sum of all forms of energy in a system, H describes a distinct part, namely, the mass energy E M of a system with N particles and N hf photons [cf.Eq. ( 21)], where m is the mass of the resting system.Thus, H describes both the intrinsic excitation energy E 0 of N particles with intrinsic mass m 0 and E P of all particles and photons with dynamic mass m d .
In 1907, Planck noted that the change in mass of a resting body corresponds to the change in the enthalpy H of the body, and in fact the increase of mass is always equal to the amount of heat, which is absorbed from the outside during an isobaric change of the body, divided by the square of the speed of light in vacuum 1 , 15 p. 566 Here, M is the rest mass m 0 of the body, E 0 is U, and p and V 0 are pressure and volume.The "amount of heat, which is absorbed from the outside during an isobaric change of the body," 15 corresponds to ΔH, which is another energy amount than ΔU.In addition, further physicists pointed out that only H is mass-proportional, such as Laue 48 or Falk, 49 "that the theory of relativity in the case of a nonzero pressure in rigor requires an enthalpy-mass equivalence -the energy-mass equivalence thus applies only approximately," 49 p. 83 In Planck's paper of 1907, the footnote to the above quotation reads 1 Essentially the same conclusion has already been drawn by Einstein (Ann.d.Phys.18, p. 639, 1905) from the application of the principle of relativity to a special radiation process, however, under the only in first approximation admissible condition that the total energy of a moving body is composed additively from its kinetic energy and from its energy for a reference system resting in it, 15 p. 566 The footnote shows that Planck knew both U ≠ H and Einstein's idealization Epot = 0 in special relativity. 42And yet, he agreed that the total internal energy U of a body is mass-proportional.Planck's advocacy of special relativity certainly has several reasons: 1. H = U + pV obscured the essence of H.It looked like H was even larger than U. 2. At this time, Planck had turned away from thermodynamics as he found no genuinely irreversible quantum process.Reluctantly, he agreed with Boltzmann's statistical interpretation of S. 3. Planck trusted Einstein, who had shown intuition with his light quantum hypothesis.
The above quotes show that there was an early awareness that E = mc 2 is incomplete.However, just the historical state of thermodynamics prevented the rejection of special relativity.The later acceptance of E = mc 2 , in turn, hindered the further development of thermodynamics and led to physics with many misinterpretations.For instance, the mass defect in an exothermic binding reaction or atom bomb explosion is caused by not by the release of (non-ponderable) potential energy Epot.A change in the positional energy Epot of the particles in the system also takes place but as a simultaneous side process such as displacement or volume work, which does not contribute to the mass defect.Table IV shows that Gibbs, by clever mathematical manipulations, succeeded in isolating the intrinsic excitation energy G (T, p, N) = E 0 (T, p, N) of a system with N particles, With Eq. ( 57), we gain access to the molar intrinsic excitation energies E 0,j of particles, i.e., to internal particle energies, which Einstein earlier called "rest energies."Writing Eqs. ( 12) and ( 13 we see that the molar intrinsic excitation energy μ = E 0,m (molar internal momentum energy) of a pure substance, and thus, the intrinsic excitation energies E 0,j of the particles in a system are variable, even for an ideal gas.Particles are by no means rigid, unchangeable, or point-like but highly complex wave entities 22 whose E 0,j change as a function of T and p, even in the case of an ideal gas.
The gas experiments hence confirm the process theory of gases 21 and contradict the idea U = E kin of the kinetic gas theory.If U = E kin were true, an ideal gas should have no chemical potential μ = E 0,m , which, however, is absolutely necessary to describe its behavior.In fact, only the access to the variable intrinsic excitation energies E 0,j of particles has so far enabled us to describe processes correctly.
The Tand p-dependencies of E 0,m will be the content of later papers.At this point, we will only briefly review chemical reactions at T, p = const.and rewrite Eq. ( 14), Gibbs free reaction energy, (60) where the partial molar variable Δ R E 0 is the intrinsic reaction energy, ζ is the extent of the reaction, and τi is the stoichiometric coefficient (cf.Fig. 2).
We see that a spontaneous reaction such as the binding of two atoms to form a molecule reduces the internal momentum energy E 0 of the binding partners, while a forced reaction increases E 0 of the particles.At T, p = const., the exchange processes Q and W V occur simultaneously.

ARTICLE pubs.aip.org/aip/adv
This important finding leads to a new binding theory and a reinterpretation of so-called potential curves, which will be presented in Paper V.

D. Future tasks for thermodynamics (TD)
The above findings are the basis for (i) a descriptive mixturephase, interfacial, and process TD, (ii) an improved TD of irreversible processes, and (iii) a new form of TD, which we call quantum-process thermodynamics.We outline the starting points in each case.

Descriptive phenomenological thermodynamics
Comparing ∆G = ∆H − T∆S in Eq. ( 11) with our findings, we see that today's interpretations, e.g., in mixture-phase thermodynamics, that ΔH is the "enthalpic contribution" to ΔG, while TΔS is the "entropic contribution" to ΔG fall short.TΔS = TΔΣ is the heat exchange Q = ΔE P , which is just not considered in ΔG.ΔG is simply the change in the intrinsic excitation energy ΔE 0 of all particles in a system.It becomes clear that there is an enormous need for clarification of classical TD, whose formulas work.This also applies to socalled "potential curves," which we will discuss in Paper V because released mass-proportional binding energy does not represent Epot [cf.Eq. ( 56)].We will discuss the many consequences of descriptive thermodynamics in more detail in our future correspondence.

Improved thermodynamics of irreversible processes
To specify the direction of processes up to equilibrium, the socalled basic equation of thermodynamics of irreversible processes can be used, 38 p. 40, where the thermodynamic flux Jα is the temporal transport of an extensive variable Xi, such as P or n, and the thermodynamic force Xα is the local gradient of an intensive variable ξi, such as v or μ.Here, d i S > 0 only describes the tendency toward uniform distribution of E M , not irreversibility.Irreversible quantum processes with ∆ i S > 0 mean that E M is gradually reduced.∆E M < 0 is predominant in moderate regions of the universe and compensated by ∆E M > 0 in other regions (cf.Sec.IV C 3).Thus, if one really wants to describe irreversible processes, a shift from kinematics to dynamics at the quantum level is necessary.The knowledge of the form of energy Ei transferred in the respective process Yi gives new input for transport processes (see Fig. 3).For instance, the mol number transport Wn from subsystem A to B, where E 0 is transported (e.g., via diffusion), is inevitably connected with transport of E P via Q.Every particle moving in space has E 0 and E P .Thus, Wn cannot be considered in isolation (cf.Appendix).This also applies at T A = T B. If the average velocity of the particles during transport remains constant, T remains constant, too.
A general principle is that transport processes occur whenever one form of energy Ei is unequally distributed, which is indicated by the spatial gradient of its intensity measure ξi, i.e., by the thermodynamic force Xα.Here, not only the amount of (internal and external) momentum energy E M = E 0 + E P but also the spatial nature of internal motion and the degree of entanglement play a role.A major task is to combine the transport laws of, e.g., Newton, Fourier, Fick, Ohm, and Maxwell-Stefan, with vivid thermodynamics.

Quantum-process thermodynamics (QPT)
QPT differs from current versions of "quantum thermodynamics" insofar as principles, such as interaction via simultaneous processes, [20][21][22] causality, energy conservation at any time, and irreversibility, are applied at the micro level.QPT excludes relativistic spacetime and observer ideas from the outset and does not use current statistical mechanics or quantum mechanics based on force interaction, which describes the pressure p via kinetic energy.It is important to reclassify the forms of energies, which also concerns the Hamilton formalism that has the following weaknesses: (i) W P and E P are unknown.(ii) Epot is incorrectly interpreted as interaction or binding energy.(iii) E 0 is considered to be invariant, in strong contradiction to Eqs. (58)-(61).
Considering, e.g., an atom A consisting of N elementary particles, its total energy is where each contribution to E A can be changed by a process, and quantum processes usually occur simultaneously and cannot be considered in isolation.

E. Note on the history of theoretical physics
Current physics describes interaction via forces.At several historical bifurcation points, forces and kinematics were preferred to processes and dynamics, mostly for pragmatic reasons.Although often not noticed today, many theories do not use process equations, violate the conservation of energy, and disregard irreversibility.Examples are • the kinetic theory of gases by Maxwell and Boltzmann, • the theory of special relativity (SR) by Einstein, • the theory of general relativity (GR) by Einstein, • the Copenhagen interpretation (CI) by Bohr and Heisenberg, and • the standard models of particle physics and cosmology (with quantum field theories and the big bang hypothesis), which were developed based on SR, GR, and CI.
On the one hand, physics claims to describe processes and to respect energy conservation.On the other hand, ideas, such as quantum fluctuations created from nothing, the big bang as the beginning of the universe, or flexible and expanding spacetimes up to geometrodynamics are incompatible with dynamic processes and the principles of thermodynamics.If Newton's axioms had been questioned in the mid-19th century, when the conservation of energy was found, science would have developed differently.The non-existent momentum work W P and the acceptance of the total energy E = mc 2 have prevented the development of quantum-process thermodynamics, which, we believe, is the path to realistic physics.

VI. CONCLUSIONS
The use of momentum work leads to both descriptive thermodynamics and a deeper understanding of nature.By reinterpreting variables, such as heat, entropy, enthalpy, and Gibbs free energy, we were able to eliminate the weaknesses of Gibbs formalism while preserving its strengths, such as conservation of energy, irreversibility, and causality.The principles of thermodynamics can now be applied at the micro level.
The main results of our work are 1.Heat exchange was identified as a composite process consisting of different microscopic subprocesses, in which various forms of momentum energy are exchanged, e.g., the momentum energies of particles and/or those of photons.2. Entropy was identified as a composite state variable changed by genuinely irreversible quantum processes, connected with energy conversion and changing the quality of the internal energy in a system.Thus, entropy cannot be used as a variable in heat exchange.3. Enthalpy was identified as the mass-proportional part of internal energy, which we called mass energy or (total) excitation energy.It is the sum of the intrinsic excitation energies and the momentum energies of all particles and photons in a system.The enthalpy of one particle is the sum of its intrinsic excitation energy and momentum energy.4. Gibbs free energy G was identified as the intrinsic excitation energy of the particles in a system.This information provides access to the internal particle energies previously barred to us.The chemical potential of one particle is its variable molar intrinsic excitation energy.
The results not only affect thermodynamics.If thermodynamics were reintegrated into theoretical physics, a major gap that was created in the 19th and early 20th centuries could be filled.This gap cannot be filled by any other theory.

APPENDIX: TOTAL DIFFERENTIAL VS SIMULTANEOUS PROCESSES
In the following, we discuss the differences between the total differential of U, H, A, and G in Eq. ( 7) and the process approach in (47).
Gibbs identified his fundamental equations with the total differentials and obtained four expressions for the chemical potential μ of a pure substance in Eq. ( 8), which he described as equivalent.Accordingly, Prigogine and others equate them, 38  The four differential quotients should have the same physical content because they correspond to μ in each case.We check them one by one by using the energetic meanings listed in Table IV and interpreting S as heat Σ.
In Fig. 4(a), one isolated process Wn is described by where heat exchange Q and volume work W V are forbidden.The setup cannot be realized because a supply of E 0 via Wn is always associated with a supply of momentum energy E P via Q (heat convection).There are no particles having only rest energy E 0 .In Fig. 4(b), two simultaneous processes Wn and W V are described by where Q is forbidden.The setup cannot be realized because every particle also transports E P into the system, which is mass-relevant.Simultaneously, Epot is released by W V , but the mass energy H = E M is not affected by Epot.
In Fig. which can easily be realized.T = const.is realized by Q, and p = const.by W V .It becomes evident that the four differential quotients in Eq. (A1) are not equivalent or identical at all, even if they mathematically appear to be.The first two demand something that is physically impossible to realize; the third can be realized in special cases, and only the fourth is always measurable, where μ = Gm.
Thus, Eq. (A5) corresponds to nature.Accordingly, μ = Gm is used in the literature, and the Gibbs fundamental equation for U is often written in the form, dU = δQ + δW V + δWn = T dS − p dV + Gm(T, p) dn, (A6) which confirms that a special form of energy Ei, namely G, is transferred via Wn, which is a part of U.This corresponds to our process approach [cf.Eq. ( 48)], dU = T dΣ − p dV + E 0,m (T, p) dn.(A7) By contrast, the total differential of U does not provide this information.It is only a human-made idealization that makes arithmetic easier and allows us to choose the easiest integration path.However, every differential quotient that can be formulated mathematically must be examined to see to what extent it corresponds to reality.
Otherwise, an accurate understanding of state variables is hindered, and the simultaneity of processes, which is a hallmark of nature, is ignored.
Even though process equations are the stepchildren of mathematics, they provide more physical information than the total differential.Physics is more than pure mathematics.

FIG. 2 .
FIG. 2. Schematic representation of the minimum of G = E 0 during reactions in a closed system at T, p = const.Spontaneous reaction: Δ R G < 0, forced reaction: Δ R G > 0, and chemical equilibrium: Δ R G = 0.

TABLE I .
Coherent formulation of (reversible) processes.Reproduced with permission from G. Kalies and D. D. Do, AIP Adv.objects moving with c have only dynamic mass m d .The new insight that electromagnetic energy E hf is momentum energy E P leads to an analogous understanding of particles and photons and is crucial for the reinterpretation of the heat exchange Q.

TABLE II .
The two different meanings of the current entropy S.

TABLE III .
Suggested naming, symbols, and content of variables in δQ = dE P = TdΣ.

TABLE IV .
The energetic meaning of the thermodynamic energies U, H, F, G, and Ω.