# Action and Entropy in Heat Engines: An Action Revision of the Carnot Cycle

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}ω). However, this property is scalar rather than vectorial, including a dimensionless phase angle (@ = mr

^{2}ωδφ). We have recently confirmed with atmospheric gases that their entropy is a logarithmic function of the relative vibrational, rotational, and translational action ratios with Planck’s quantum of action ħ. The Carnot principle shows that the maximum rate of work (puissance motrice) possible from the reversible cycle is controlled by the difference in temperature of the hot source and the cold sink: the colder the better. This temperature difference between the source and the sink also controls the isothermal variations of the Gibbs potential of the working fluid, which Carnot identified as reversible temperature-dependent but unequal caloric exchanges. Importantly, the engine’s inertia ensures that heat from work performed adiabatically in the expansion phase is all restored to the working fluid during the adiabatic recompression, less the net work performed. This allows both the energy and the thermodynamic potential to return to the same values at the beginning of each cycle, which is a point strongly emphasized by Carnot. Our action revision equates Carnot’s calorique, or the non-sensible heat later described by Clausius as ‘work-heat’, exclusively to negative Gibbs energy (−G) or quantum field energy. This action field complements the sensible energy or vis-viva heat as molecular kinetic motion, and its recognition should have significance for designing more efficient heat engines or better understanding of the heat engine powering the Earth’s climates.

## 1. Introduction

## 2. Revisiting the Carnot Cycle

_{source}) of a hot source (such as a coal fire in a steam engine) during which heat is transferred from the source to the working fluid and external work is done. This stage provides heat isothermally (we identify as Carnot’s caloric a) with equivalent work performed as a logarithmic function of the expansion in volume.

_{sink}) during which further external work is done at the expense of the heat content of the gases (consistent with Carnot’s caloric b′), but no heat enters or leaves the piston chamber.

_{2}/V

_{1})(T

_{source}− T

_{sink})]/[R ln(V

_{2}/V

_{1})(T

_{source})] = (T

_{source}−T

_{sink})/(T

_{source}).

_{source}− Q

_{sink})/Q

_{source}= (T

_{source}− T

_{sink})/(T

_{source}).

## 3. Action and Entropy

_{t}), rotational (@

_{r}), and vibrational forms (@

_{v}) forms of relative motion as action. This approach allows easier calculation of the entropy of ideal gases and the Gibbs energy, as shown in Section 5 and Section 6. To obtain the total entropic heat energy required to reversibly bring a gas from absolute zero (where the entropy is zero) to the current temperature, it is sufficient to multiply each of these partitioned entropies by the temperature T and then to take their sum (ST). This includes the heat required for all isothermal phase changes such as melting and vaporizing as well as for other forms of disaggregation. For simplicity of expression, each partition for entropy contains within the logarithm an exponential term that accounts for enthalpy (e.g., e

^{3/2}, e

^{5/2}) depending on the complexity and degrees of kinetic freedom of the molecule. A relative action ratio (e.g., @

_{t}/ħ = n

_{t}) is also included, indicating the mean quantum state or molecular configuration. The logarithm of this term accounts for latent heat in ST or negative Gibbs energy that varies with volume and temperature.

^{−Ej/kT}), indicating the total occupation number of different energy states as a fraction of the total number of possible molecular systems N. Then, klnZ is equal to (S–E/T), and it can be shown that Z equals (n

_{t}

^{3}e)

^{N}, where n

_{t}is the relative molecular translational action (@/ħ = mrv = n

_{t}) for each of N molecules of a monatomic gas. Note that quantum numbers n

_{t}given are mean values that would be integral for particular molecules. Furthermore, the microcanonical or molecular partition function (z) is justified, using statistical mechanics, as equal to Nn

_{t}

^{3}for N molecular systems, in an addendum attached to this paper. This Supplementary Material also considers the status of vibrational action comparing N

_{2}to CO

_{2}and how its excitation can be considered as the translational action of activated molecules.

_{2}, including the sensible heat or enthalpy (H) for constant pressure systems, as follows:

_{2}, even at the most elevated temperatures. However, its rotational entropy is highly significant.

^{7/2}{(3kTI

_{t}/ħ

^{2})

^{3/2}/z

_{t}}(2kTI

_{r}/ħ

^{2}σ)]

_{t}= mrv = mr

^{2}ω = Iω) and rotational action (@

_{r}= Iω). The symbol @

_{t}represents the relative translational action, which is a functional property of molecular momentum and radial separation equal to [(3kTI

_{t})

^{1/2}/z

_{t}

^{1/3}] [7]; I

_{t}is a translational moment of inertia calculated for a cubic distribution of molecules, and z

_{t}(z

_{t}= 10.2297) is a factor avoiding double counting of molecules and correcting for the ratio of their mean speed and their root-mean-square velocity. The moment of inertia for translational motion (I

_{t}) is equal to mr

^{2}—the molecular mass multiplied by the square of the mean radial separation of similar molecules. The one-dimensional translational action (@

_{t}) varies at each of the four stages of the Carnot cycle as pressure and temperature vary. The rotational action of two-dimensional molecules such as N

_{2}is obtained as (2kTI

_{r})

^{1/2}, where I

_{r}is the moment of inertia of unvarying radius as in chemically bonded structures. Then, entropy action partitions are given in (6)–(9).

_{t}= Rln[e

^{5/2}(@

_{t}/ħ)

^{3}] = Rln[e

^{5/2}(n

_{t})

^{3}] (translation)

_{r}= Rln[e(@

_{r}/ħ)

^{2}] = Rln[e(j

_{r})

^{2}] (rotation-diatomic or linear molecule)

_{r}= Rln[π

^{1/2}e

^{3/2}(@

_{A}@

_{B}@

_{C}/ħ

^{3})] (rotation-polyatomic molecule)

_{vi}= Σ[Rx/(e

^{x}− 1) − Rln(1 − e

^{−x})], where x = hcν

_{i}/kT (vibration of each bond)

_{Total}T = [Σ(S

_{t}+ S

_{r}+ S

_{vi})]T

_{t}in Equation (4) corrects the magnitude of the translational action to match precisely the field energy required to sustain it; for the translation of ideal gases at 1 atmosphere pressure, this energy-sparing factor z

_{t}is of constant magnitude of 10.2297 for all species of molecules [7,11]. The correction has now been logically interpreted [12] as involving inverted sub-factors of (1/2)

^{3}to prevent double counting of molecular partners and (1/1.0854)

^{3}to correct the root-mean-square velocity (from kT = mv

^{2}/3) to the mean molecular velocity, which is required to calculate the translational action (mvr = mr

^{2}ω). It is noteworthy that this means establishing molecular entropy in a reversible process as defined by Clausius [13], which requires that the entropy per molecule is dependent on gas density, increasing logarithmically as the mean volume (a

^{3}) occupied by each molecule increases. Such a relationship with density for heat content was also proposed by Carnot.

## 4. Gibbs Energy

_{n}T

_{n}= RT

_{n}ln[e

^{5/2}(@

_{t}/ħ)

^{3}] = 2.5RT

_{n}+ RT

_{n}ln[(@

_{t}/ħ)

^{3}] = H

_{n}+ RT

_{n}ln[(@

_{t}/ħ)

^{3}]

_{n}) is equal to the molar internal energy (E

_{n}), plus the pressure-volume function (RT

_{n}) for atmospheric work, which is obligatory for a system open to air, although not in the Carnot cycle where internal pressure varies with volume, which is reversibly equal to external pressure. Negative signs are given for the two so-called free energies, which are actually inversed potential energies—the Helmholtz energy (A

_{n}) used in constant volume systems and the Gibbs energy (G

_{n}or g

_{n}per molecule) used with systems open to the atmosphere requiring pressure–volume work also affecting heat content. These calculations shown in (12) and (13) give exact values, not differences.

_{n}= RT

_{n}ln[e(@

_{t}/ħ)

^{3}] = RT

_{n}ln[e(n

_{t})

^{3}]

_{n}= RT

_{n}ln[(@

_{t}/ħ)

^{3}] = RT

_{n}ln[(n

_{t})

^{3}]

_{t}as shown in (12) and (13), which are related to the molecular Gibbs energy (g

_{t}) as an indicator of the field energy at temperature T

_{n}.

_{n}/N = kT

_{n}ln(n

_{t})

^{3}= −g

_{t}

_{n}= RT

_{n}ln[(ħ/@

_{t})

^{3}]

_{n}= RT

_{n}ln[(ħ/@

_{t})

^{3}/e] = RT

_{n}ln[(ħ/@

_{t})

^{3}] − RT

_{n}

_{n}= A

_{n}+ RT

_{n}

_{v}is 1.5R, separating the Gibbs energy from the enthalpy, we have (18) at constant pressure.

_{n}T

_{n}= RT

_{n}ln[(@

_{t}/ħ)

^{3}] + 2.5RT

_{n}= −G

_{n}+ H

_{n}

_{n}= H

_{n}− S

_{n}T

_{n}or, to indicate spontaneous change, we have:

_{n}= ΔH

_{n}− ΔS

_{n}T

_{n}.

_{n}T

_{n}= RT

_{n}ln[e

^{7/2}(@

_{t}/ħ)

^{3}(@

_{r}/ħ)

^{2}] = RT

_{n}ln[(n

_{t})

^{3}(j

_{r})

^{2}] + 3.5RT

_{n}= −G

_{n}+ H

_{n}.

_{r}for a diatomic molecule is taken as equal to (2kTI

_{r}/σ)

^{1/2}, where I

_{r}is the molecular moment of inertia and σ is a symmetry factor preventing excess counting of indistinguishable conformations; this factor has the value of 2 for diatomic molecules such as N

_{2}, where each end of the molecule presents the same but reaches 12 in the case of methane (CH

_{4}, σ = 4!/2!). The symmetry indicated by σ is considered as a statistical factor adjusting for the likelihood of an encounter by a quantum of energy with an indistinguishable species of molecule that is proportional to its symmetry. Since there is no way to distinguish one end of an N

_{2}molecule from the other, except isotopically, the concentration of such symmetric molecules is effectively doubled by comparison with NO, and the distance and elapsed time between encounters shortened. Any two systems having the same difference between enthalpy and entropic energy (H-ST) will have the same Gibbs free energy (G) and will be at equilibrium if opposed to each other. We should be aware that mechanistically, the negative-entropy energy term (−S

_{n}T

_{n}) contains both the other terms, so Equation (19) can be seen as a tautology. Some of the confusion regarding the nature of molecular entropy results from a lack of awareness of this fact. Equations (18) and (20) are more informative—they express the heat content of a polyatomic gas as the sum of the latent or potential heat, a logarithmic function of pressure or volume and temperature, plus the sensible heat or enthalpy, which is a function of the temperature alone.

_{n}) expresses the non-sensible heat content, whereas the enthalpy term expresses the sensible or kinetic heat. The influence of both these functions was clearly identified by Carnot [1,2]. The Gibbs and Helmholtz functions are greatest when the internal potential energy is least, conversely to the entropy. Thus, higher quantum states, achieved as more quanta are absorbed by the field, correspond to increased entropy and decreased free energy, as stated by Planck [14]. By contrast, molecules in their ground states at the lowest temperatures have minimum entropy. Paradoxically, the Gibbs energy or function is only a potential to acquire field energy and is greatest when the latter is least. Often referred to as free energy—perplexing generations of students—this is actually true in the sense of being a measure of a molecule’s inaction and relative freedom from sustaining field energy. Its alternative name as recommended by the IUPAC of the Gibbs function is a neutral description, but we prefer Gibbs potential as even more descriptive. The following Section 5 and Section 6 summarize the key results of this paper.

## 5. An Action-Based Calculation for the Carnot Cycle

#### 5.1. Isothermal Stage 1=>2

_{source}(1) = −W

_{rev}(1) = RT

_{source}ln(V

_{2}/V

_{1}); And ΔS (1) = Q

_{source}(1)/T

_{source}

_{t}= mr

^{2}ω) varies proportional to kln(r

^{3}) as the volume increases. Then, the external work per molecule is equal to the internal configurational or quantum work—the mean decrease in translational Gibbs energy per molecule (-δg

_{t}). The convention that internal work implies a change in Gibbs potential is applied. We have:

_{source}ln[(r

_{2})

^{3}/(r

_{1})

^{3}] = kT

_{source}ln[@

_{t2}/@

_{t1}]

^{3}= kT

_{source}ln[n

_{2}/n

_{1}]

^{3}= −δg

_{t}.

^{3}= kT = mv

^{2}/3 where V = Na

^{3}and v is the root-mean-square velocity), so that in an isothermal expansion, the product of pressure and specific volume (pv, pa

^{3}) remains constant with the increasing volume per molecule, while the pressure decreases from its maximum. Performing work reversibly requires that the external pressure or mechanical resistance should always be equal to the internal pressure.

_{v}δT), as the temperature remains constant and the kinetic energy is a function of temperature independent of volume for an ideal gas. The intensity of field quanta required to maintain a constant temperature while external work is being done is a logarithmic function of the increase in volume, as Carnot stated prominently in his memoir. The amount of heat Q

_{source}is acquired by the working fluid, and the resultant increase in entropy is Q

_{source}/T

_{source}(see also Table 1 and Table 2 below). Carnot designated Q

_{source}equal to an amount of caloric a [2]; the external work performed in his discussion is shown in Equations (21) and (22).

_{f}, which can be equated to the internal change in action (@) shown in 3kT

_{f}ln(@

_{2}/@

_{1}). This heat has been absorbed to sustain the working fluid in its new higher quantum state. For a reversible system, more field energy is needed to sustain molecules with greater spatial separation if external work is done.

#### 5.2. Adiabatic Stage 2=>3

_{v}(T

_{source}− T

_{sink}).

_{3}/V

_{2}) + C

_{v}ln(T

_{sink}/T

_{source}) = ΔS = 0.

_{t}/ħ = n

_{t}), despite the fall in temperature. Thus:

_{3}

^{2}ω

_{3}) − (mr

_{2}

^{2}ω

_{2}) = 0.

_{internal}= kT

_{sink}ln[(mr

_{3}

^{2}ω

_{3})/ħ)]

^{3}− kT

_{source}ln[(mr

_{2}

^{2}ω

_{2})/ħ)]

^{3}

_{sink}− T

_{source})ln[(n

_{t})]

^{3}= −δg

_{t}

_{2}, to include rotation as well as translation in the reversible adiabatic process, we will have the quantum number product shown in (28).

_{t})

^{3}(j

_{r})

^{2}]

_{sink}= [(n

_{t})

^{3}(j

_{r})

^{2}]

_{source}

_{2}, we will have the following equation:

_{total}= k(T

_{sink}− T

_{source})ln[(n

_{t})

^{3}(j

_{r})

^{2}] = −(δg

_{t}+ δg

_{r}).

#### 5.3. Isothermal Stage 3=>4

_{sink}, the lower the heat that is extracted during the compression and the greater the external work that is possible in each cycle. Hence:

_{rev}(3) = −W

_{rev}(3) = RT

_{sink}ln(V

_{4}/V

_{3}); And ΔS (3) = Q

_{rev}(3)/T

_{sink}.

_{r}), although without changes in internal energy. We have:

_{sink}ln[(mr

^{2}

_{4}ω

_{4})/(mr

^{2}

_{3}ω

_{3})]

^{3}= kT

_{sink}ln[@

_{t4}/@

_{t3}]

^{3}= −δg

_{t}= δsT

_{sink}.

_{r}= 3kT

_{r}ln(@

_{4}/@

_{3}), which has a negative value as the action declines to less than half. Note that @

_{4}/@

_{3}) is equal to @

_{1}/@

_{2}, so the changes in entropy for stages 1 to 2 and for stages 3 to 4 are equal but opposite. Given that the temperature is much greater for the first of these changes, the same change in relative action states costs much less in energy in stage 3 to 4. Carnot designated Q

_{r}or 3kT

_{r}ln(@

_{4}/@

_{3}) in his discussion as a′, the caloric removed by the refrigerator body B, in stage 3=>4.

#### 5.4. Adiabatic Stage 4=>1

_{v}for monatomic gases like argon of 1.5k

_{external}= c

_{v}T

_{source}− c

_{v}T

_{sink}

_{internal}= kT

_{sink}ln[(mr

^{2}

_{1}ω

_{1})/ħ)]

^{3}− kT

_{source}ln[(mr

^{2}

_{4}ω

_{4})/ħ)]

^{3}

= [T

_{source}− T

_{sink}]kln(n

_{t})

^{3}= −δg

_{t}

_{2}, both translational and rotational action will change as temperature and pressure increase. We also have the increase in internal energy with molecular c

_{v}of 2.5k, which accompanies the decrease in Gibbs energy and the increase in entropic energy:

_{internal}= c

_{v}T

_{source}− c

_{v}T

_{sink}.

_{t})

^{3}(j

_{r})

^{2}] remains constant, the Gibbs energy declines. Hence:

_{internal}= kln [1/(n

_{t})

^{3}(j

_{r})

^{2}](T

_{source}− T

_{sink}) = (δg

_{t}+ δg

_{r}).

## 6. Results: Carnot Cycle Calculations for Argon and Nitrogen

_{f}) and 288 K (T

_{r})—the Earth’s average surface temperature as the coldest refrigerator or sink usually available.

_{2}), which is a molecule with five degrees of freedom by including rotation and operating between 640 and 288 K, which is the average temperature. The tables also contain calculated data related to thermal energy content, pressure, volume, action, Gibbs energy, entropy, and internal energy. Using action ratios, it is also possible to express the negative Gibbs energies or their equivalents in field energy as quantum numbers n

_{t}and j

_{r}.

^{3}is accompanied by a mean translational quantum number of 80.042. At 640 K after isothermal expansion, these quantum states have increased in density to a number of 111.708 per molecule. According to Clausius’ definition of entropy for reversible heat exchange, δsT is equal to Q

_{source}/T

_{source}, but this is also equal to 3kTln(@

_{2}/@

_{1}), which is effectively a change in quantum state n

_{t}of about 80 to 112, as shown in Table 1.

_{v}) of argon and nitrogen, as shown in line 4a and 4, respectively. The heat absorbed to reach these thermodynamic states as internal work is now field energy sustaining these molecular temperatures and pressures. In action mechanics [4], heat is more than molecular motion but includes the field energy sustaining the molecular motion. One is not possible without the other. Given that the mean kinetic energy (I

_{t}ω

^{2}/2) of each species of molecule is the same at a particular temperature, the mean pressure exerted by each molecular species is inversely proportional to its specific volume (a

^{3}). Thus, pa

^{3}= kT is an average statistical property of each ensemble of molecules. For a volume V, pV = NkT, with V equal to Na

^{3}with N, which is the average number of molecules per unit volume. Where N is a mole of molecules, Nk is equal to the gas constant R. For more realistic consistency with physical models, it is convenient to make thermodynamic calculations as mean values per molecule. Then, values per mole are easily calculated multiplying by Avogadro’s number (6.022 × 10

^{23}). Note that the terms (3kTI

_{t})

^{1/2}and (2kTI

_{r})

^{1/2}used to calculation translational and rotational action include temperature (T) and radius (r

_{t}); the latter can act as a surrogate for the mean specific volume of each molecule. The ideal gas equation p = kT/a

^{3}or NkT where N indicates number density can be used to substitute for variations in temperature, number density, or volume and pressure.

## 7. Discussion and Key Points

#### 7.1. Sadi Carnot’s Legacy

_{v}T) remains constant. The heat absorbed isothermally from the hot source is regarded as consumed in the field energy sustaining the molecular orbits and maintaining the kinetic energy of the molecules as constant, allowing external work to be done via their pressure. This configurational entropic energy was also defined by Clausius in 1875 [10] as work heat or the ergal. There was no need for the editor Mendoza (see his foreword in Dover edition of Carnot’s book [2]) to have rejected the significance of Carnot’s conclusion on page 29 of his memoir that the chaleur specifique (specific heat) varied with the logarithm of the volume. Indeed, the editor of the Dover edition, Mendoza, claimed in 1960 that Carnot was mistaken, having been misled by faulty data produced by Delaroche and Bérard to calculate the effect of pressure on the specific heat of a gas. In fact, Mendoza’s criticism of their data was in error, as discussed next.

#### 7.2. Caloric as Negative Gibbs Potential

_{v}or C

_{p}). Furthermore, in Thurston’s 1890 (Macmillan) translation of Carnot’s book Reflections on the Motive Power of Fire, the translator (pointed out by Mendoza [2] Dover edition 1960) often used the same term heat for both of Carnot’s terms calorique and chaleur, providing lingering confusion regarding Carnot’s account. However, Carnot specifically states that while he is indifferent to the use of terms chaleur or calorique as a quantity of heat, he does reserve chaleur as a measure for the sensible heat of fire and is consistent in using calorique for changes in the state of the working fluid; we would consider the latter as variations in Gibbs energy or configurational entropic energy, not sensible heat.

_{p}− C

_{v}= R), it is reasonable to conclude that U represented the internal energy (E) or the enthalpy (H = E + RT), and thus, Equation (38) is closely analogous to Equations (16) and (17) that express the 2nd law of thermodynamics and statistical mechanics. Our modern version of this equation has the benefit of both the clarifying work of Clausius [15] on correctly establishing the fundamental principles of the mechanical theory of heat, the statistical mechanics of Willard Gibbs [8], and the quantum theory of Planck [14] and even Einstein soon after 1900.

_{n}T

_{n}= RT

_{n}ln[e

^{5/2}(@

_{t}/ħ)

^{3}]

= 2.5RT

_{n}+ RT

_{n}ln[(@

_{t}/ħ)

^{3}] = 2.5RT

_{n}+ 3RT

_{n}ln[(n

_{t})]

^{1/2}in adiabatic stages 2 and 4 exactly offsets the change in density and inertia (I

_{t}= mr

^{2}), a function of volume, so the relative action mrv remains the same. In the adiabatic processes, the change in Gibbs or Helmholtz energies is a linear function of the change in temperature only, as is the internal energy. These changes in energy (E or H) in the adiabatic stages cannot result in net work, as stage 2=>3 is the reverse of stage 4=>1, ensuring that the working fluid returns to the same stage at the completion of each cycle. So, in Equation (38), taken with his conviction that as chaleur specifique or specific heat would also change with temperature, unlike the constant heat capacity now designated as C

_{v}, we recognize that Carnot proposed the first version of the second law of thermodynamics.

#### 7.3. Heat Capacity versus Specific Heat

_{1}) and then, expanded with heat, added b units at constant temperature to a larger volume (V

_{2}) versus expanding first at 1 degree at constant temperature to the same volume V

_{2}with b′ units and then heated to 100 degrees with a′ units must be equivalent. “As the final result of these two processes is the same, the quantities of heat employed for both should be equal:

_{v}for argon remaining the same throughout as 1.5R. The energy change for an enclosed cylinder is equal to C

_{v}δT in both cases.

_{2}/V

_{1}) + R ln(T

_{2}/T

_{1})

^{3/2}= R ln[(V

_{2}/V

_{1})(T

_{2}/T

_{1})

^{3/2}

_{i}includes an extra amount for the change of temperature from T

_{1}to T

_{2}, indicating a change in state and entropy. By contrast, expanding with b units at the lower temperature of 1 degree Celsius involves less heat required, given that it equals RT ln(V

_{2}/V

_{1}).

_{v}and C

_{p}for heat capacity of gases are constants, although this is not true at either high or low temperatures because of quantum effects. If we consider 3.5R as the heat capacity of air C

_{p}given its predominant composition of nitrogen, the difference in entropic energy as negative Gibbs energy −G per mole between 298.15 and 297.15 K at the standard pressure of 1 atmosphere can be calculated as follows.

_{297.15}I

_{t}/3kT

_{298.15}I

_{t})]

^{3/2}(2kT

_{298.15}I

_{r}/2kT

_{297.15}I

_{r})

^{5/2}

^{3}) for an ideal gas at the same temperature will be 1/64 that at 1 atmosphere, so separation of the molecules will be one-quarter of that at one atmosphere, with a decrease in the translational moment of inertia to one-sixteenth. Will this change the Gibbs work factor and the heat capacity measured as the change in entropic energy between 298.15 and 297.15 K?

_{t})

^{3/2}/ħ

^{3}][(2kTI

_{r})/ħ

^{2}]

_{297.15}I

_{t}/3kT

_{298.15}I

_{t})]

^{3/2}(2kT

_{297.15}I

_{r}/2kT

_{297.15}I

_{r})

^{5/2}= 0.0084k

_{2}is constant, heat capacity will vary proportional to kln(T

_{2}/T

_{1})

^{5/2}or 3.49160k per molecule or 29.02917 J per mole per degree Kelvin, which is the accepted value for C

_{p}of N

_{2}.

#### 7.4. Quantum State Numbers

_{t}) supporting the molecular morphology of the field have been calculated in the tables, including Table 3, where these are summarized. These mean quantum numbers are calculated simply from the ratio of the action values (@

_{t}, @

_{r}) with Planck’s quantum of action (ħ). It is a feature of such translational quantum states that their magnitude decreases with the quantum number, as pointed out by Schrödinger [9], with only the levels very near the average energy occupied, explaining why the Maxwell–Boltzmann distribution has a sharp maximum. The average occupation number of the quantum cells in Table 1 and Table 2 approaches one in a million.

_{t}

^{3}and rotational quantum j

_{r}

^{2}products are also shown. For isentropic states, these are expected to be equal as adiabatic, although a small variation after four significant figures is shown in the table. Heat engines might function by irradiation with resonant quanta of specific long wavelength, varying according to the physical stage in the cycle, providing more efficient work than hitherto achieved. Such an experimental model should now be tested.

^{3}would increase proportional to the increase in volume (r

^{3}), but the temperature would remain constant, with no effect on mean molecular velocity (v). While the Gibbs energy would decrease by the increase in entropic energy on expansion, the action field would contain the same amount of field energy as before, since no heat is needed given the vacuum, but with a larger quantum number of smaller quanta n

_{t}cubed. Thus, the size of the associated quanta must be diminished by the same amount as the radial separation is increased. The frequency of impulses would be decreased by the increased radial separation, but the torques developed would remain the same. Viewing kinetic energy as the statistical consequence of mean value of torques exerted in exchanges of quanta, the temperature will remain the same if well insulated from the environment at large.

## 8. Implications for Climate Science and Future Research

_{v}δT) is only a small fraction (ca. 10–15%) of the increase in Gibbs energy in the adiabatic expansions in Table 1 and Table 2. To ignore the transfer of such a substantial source of energy in the Carnot heat engine to external work is a significant omission. In fact, in Table 1 and Table 2, the density of translational molecular kinetic energy has been found to be very similar to the density of translational quantum energy, which is remarkably consistent as the molecular and quantum pressures (kT/a

^{3}= RT/V

_{m}). Action resonance [4] proposes that there is just such an equation between the rate of change of momentum of molecules and the action force field of quantum exchanges. Demonstrated here as operative in the cylinders of the Carnot cycle, we can expect a similar relationship between molecular and quantum pressures in all physical environments. As explained earlier in the text, for monatomic gases and nitrogen (N

_{2}) discussed in this paper, vibrational entropy can be neglected, but in the general case, it will need to be considered for low frequencies. Furthermore, in the context of climate processes, we introduce another kind of entropy, the vortical entropy described below, to account for climate specific physical phenomena.

#### Vortical Entropy

_{vort}= mR

^{2}Ω

_{vort}= Nk ln(mR

^{2}Ω/ħ)

_{v}T).

## 9. Conclusions

- “The maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable by any means whatever… there should not occur any change of temperature which may not be due to a change of volume.”
- “The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures between which it is effected by the transfer of caloric.”
- “When a gas varies in volume without change in temperature, the quantity of heat absorbed or liberated is in arithmetical progression if the increments or decrements of volume are in geometrical progression.”
- “The temperature is higher during the movements of dilatation than during the movements of compression. During the former the elastic force of air is found to be greater and consequently the quantity of motive power during dilatation is more considerable than that consumed to produce movements of compression.”
- “The quantities of heat absorbed or set free in these different transformations are exactly compensated.”

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Carnot cycle flow sheet to estimate thermodynamic outputs. Run on Windows TRS32 emulator, Astrocal. See supplementary coding Carnot6/cal or contact the corresponding author.

**Figure 2.**Action revision Carnot cycle for argon showing the thermodynamic energy in each stage commencing at 20 atmospheres pressure and 640 K temperature, with kT equal to Q

_{f}initiating changes in action. All values for negative Gibbs energy (−g

_{t},) entropic energy (sT), and internal energy (e) are ×10

^{−13}ergs per molecule. Heat Q

_{f}is added from the heat source in the isothermal transition stage 1=>2 as shown in the increased values of sT and -g, and heat Q

_{r}is removed to the sink in the isothermal stage 3=>4, showing markedly decreased value of sT, and −g. The value of the entropy per molecule is ×10

^{−17}ergs/K, for temperature degrees Kelvin, pressure in Pascals, and volume per molecule ×10

^{−22}cm

^{3}. Pressure, volume, and temperature conform to pa

^{3}= kT. The small increase in entropy in stage 1=>2 of kln(V

_{2}/V

_{1}) is equal to the decline in entropy in stage 3=>4, kln(V

_{3}/V

_{4}).

**Figure 3.**Action revision Carnot cycle for nitrogen showing thermodynamic energy in each stage; action is initiated in stage 1 at 40 atmospheres pressure and 640 K source temperature, declining to 298 K in stage 2. All values for negative Gibbs energy (−g

_{t},) entropic energy (sT), and internal energy (e) are ×10

^{−13}ergs per molecule. The heat Q

_{f}of kT is added from the heat source in the isothermal transition stage 1=>2 shown in the increased values of sT and −g, and heat Q

_{r}is removed to the sink in the isothermal stage 3=>4, showing markedly decreased values of sT, and −g. The value of the entropy per molecule is ×10

^{−17}ergs/K, for temperature degrees Kelvin, pressure in Pascals, and volume per molecule ×10

^{−22}cm

^{3}. Pressure, volume, and temperature conform to pa

^{3}= kT. The small increase in entropy in stage 1=>2 of kln(V

_{2}/V

_{1}) is equal to the decline in entropy in stage 3=>4, kln(V

_{3}/V

_{4}).

**Figure 4.**Carnot cycle for argon showing changes in mean molecular energy (e), Gibbs potential (g), pressure (p

_{m}), and quantum field intensity (p

_{g}). While the energy is stationary in stages 1=>2 and 3=>4, the Gibbs potential varies (a=>b′=>a′=>b in Carnot’s model), and thus, the maximum work possible is equal to (a − a′) = (b − b′).

**Figure 5.**By analogy with translational, rotational, and vibrational relative action and entropy of N

_{2}(see Equations (6), (7) and (9) respectively) shown here, we introduce vortical relative action in rotating air masses of anticyclones and cyclones, further increasing the action and entropy of masses of air and its heat capacity ST. Vortical action is modeled here simply as one-dimensional, but it would also contain a smaller fraction of second dimensional action but no third locally.

Thermodynamic Property (Cgs Units per Molecule) | Stage 1=>2 Isothermal | Stage 2=>3 Adiabatic or Isentropic | Stage 3=>4 Isothermal | Stage 4=>1 Adiabatic or Isentropic | |
---|---|---|---|---|---|

1 | Degrees Kelvin | 640 K | 640 K=> 288 K | 288 K | 288 K => 640 K |

3 | Pressure (kT/a^{3}) at t_{o} | 4.191891 × 10^{7} | 1.542111 × 107 | 2.094820 × 10^{6} | 5.694312 × 10^{6} |

4 | Volume (a^{3}) cm^{3} at t_{o} | 2.107880 × 10^{−21} | 5.729813 × 10^{−21} | 1.898111 × 10^{−20} | 6.9827762 × 10^{−21} |

4a | Relative volume | ×2.718 | ×3.313 | ×1/2.718 | ×1/3.313 |

5 | pa^{3} initial ergs | 8.836006 × 10^{−14} | 8.836006 × 10^{−14} | 3.976203 × 10^{−14} | 3.976203 × 10^{−14} |

6 | δpv ergs | 0 | −4.8615 × 10^{−14} | 0 | +4.8615 × 10^{−14} |

7 | Translational inertia mr^{2} = I_{t} (g.cm^{2}) | 2.749749 × 10^{−37} | 5.355780 × 10^{−37} | 1.190173 × 10^{−36} | 6.110553 × 10^{−37} |

8 | Action @_{t} (Iω erg.sec) | 8.441202 × 10^{−26} | 1.178065 × 10^{−25} | 1.178065 × 10^{−25} | 8.441202 × 10^{−25} |

9 | Quantum number | n = 80.042 | n = 111.708 | n = 111.708 | n =80.042 |

9 | r_{t} = a/2 (cm) | 6.410895 × 10^{−8} | 8.947125 × 10^{−8} | 13.373586 × 10^{−8} | 9.556798 × 10^{−8} |

10 | −g/T (erg/K) | 1.815201 × 10^{−15} | 1.953263 × 10^{−15} | 1.953264 × 10^{−15} | 1.815201 × 10^{−15} |

11 | −δg/T ergs/K | +1.38062 × 10^{−16} | 0 | −1.38062 × 10^{−16} | 0 |

12 | −Gibbs energy (-g) | 11.617287 × 10^{−13} | 12.500888 × 10^{−13} | 5.625399 × 10^{−13} | 5.227779 × 10^{−13} |

13 | δGibbs energy (δg) | −0.8837 × 10^{−13} a | +6.875489 × 10^{−13} b′ | +0.39762 × 10^{−13} a′ | −6.389508 × 10^{−13} b |

14 | Energy density (ergs/cm^{3}) | 5.5113594 × 10^{8} | 2.181727 × 10^{8} | 2.963682 × 10^{7} | 7.486693 × 10^{7} |

15 | Total entropy | 2.160358 × 10^{−15} | 2.298420 × 10^{−15} | 2.298420 × 10^{−15} | 2.160358 × 10^{−15} |

16 | δ total entropy | +1.38062 × 10^{−16} | 0 | −1.38062 × 10^{−16} | 0 |

17 | Entropic energy ergs s_{n}T_{n} | 13.826288 × 10^{−13} | 14.709889 × 10^{−13} | 6.619450 × 10^{−13} | 6.221830 × 10^{−13} |

18 | δs_{n}T_{n} | +0.8836 × 10^{−13} | −8.090439 × 10^{−13} | −0.3976 × 10^{−13} | +7.604458 × 10^{−13} |

19 | Net heat input | +8.839 × 10^{−14} | +8.839 × 10^{−14} | +4.8615 × 10^{−14} | +4.8615 × 10^{−14} |

20 | Internal energy e | 1.325376 × 10^{−13} | 1.325376 × 10^{−13} | 0.59642 × 10^{−13} | 0.59642 × 10^{−13} |

21 | Δe | 0 | −7.28956 × 10^{−14} | 0 | +7.28956 × 10^{−14} |

22 | Heat transfer Q_{f}, Q_{r} | +0.8839 × 10^{−13} | 0 | −0.3976 × 10^{−13} | 0 |

23 | Entropy change= | +1.3812 × 10^{−16}= Q _{f}/T_{f}=3kln(@_{2}/@_{1}) | 0 | − 1.3812 × 10^{−16}= Q _{r}/T_{r}=3kln(@ _{4}/@_{3}) | 0 |

Thermodynamic Property (Cgs Units) | Stage 1=>2 Isothermal | Stage 2=>3 Adiabatic | Stage 3=>4 Isothermal | Stage 4=>1 Adiabatic | |
---|---|---|---|---|---|

1 | Degrees K | 640 K | 640 K => 288K | 288 K | 288 K => 640 K |

2 | Pressure (kT/a^{3}) | 4.191891 × 10^{7} | 1.542111 × 10^{7} | 9.42669 × 10^{5} | 2.56244 × 10^{6} |

3 | Volume (a^{3}) (cm^{3}) | 2.107881 × 10^{−21} | 5.729813 × 10^{−21} | 4.21803 × 10^{−20} | 1.551725 × 10^{−20} |

4 | Relative volume | ×2.718 | ×7.362 | ×1/2.718 | ×1/7.362 |

5 | pa^{3} (ergs) | 8.836006 × 10^{−14} | 8.836006 × 10^{−13} | 3.976203 × 10^{−14} | 3.976203 × 10^{−14} |

6 | δpv (ergs) | 0 | −0.48598 × 10^{−13} | 0 | +0.48598 × 10^{−13} |

8 | r_{t} (cm) | 6.410895 × 10^{−8} | 8.947125 × 10^{−8} | 1.740496 × 10^{−7} | 1.247120 × 10^{−7} |

9 | Initial translational inertia mr^{2} = I (g·cm^{2}) | 1.924824 × 10^{−37} | 3.749046 × 10^{−37} | 1.4187304 × 10^{−36} | 7.2840050 × 10^{−37} |

10 | Translational action @_{t} (Iω erg.sec) | 7.062416 × 10^{−26} | 9.856396 × 10^{−26} | 1.118839 × 10^{−25} | 9.216142 × 10^{−26} |

11 | Translational quanta | n_{t} = 66.968 | n_{t} = 93.462 | n_{t} = 121.963 | n_{t} = 87.391 |

12 | Translational –g_{t}/T | 1.741336 × 10^{−15} | 1.879398 × 10^{−15} | 1.989642 × 10^{−15} | 1.851580 × 10^{−15} |

13 | −g_{t} (ergs) | 1.114455 × 10^{−12} | 1.202815 × 10^{−12} | 5.730170 × 10^{−13} | 5.332550 × 10^{−13} |

14 | Rotational inertia I_{rn} | 1.416704 × 10^{−39} | 1.416704 × 10^{−39} | 1.416704 × 10^{−39} | 1.416704 × 10^{−39} |

15 | Rotational action @_{r} | 1.118839 × 10^{−26} | 1.118839 × 10^{−26} | 7.505400 × 10^{−27} | 7.50540 × 10^{−27} |

16 | Rotational quanta | j_{r} = 10.609 | j_{r} = 10.609 | j_{r} = 7.117 | j_{r} = 7.117 |

17 | Rotational entropy | 0.652131 × 10^{−15} | 0.652131 × 10^{−15} | 0.546600 × 10^{−15} | 0.546600 × 10^{−15} |

18 | Rotational Gibbs energy | 4.173639 × 10^{−13} | 4.173639 × 10^{−13} | 1.560635 × 10^{−13} | 1.560635 × 10^{−13} |

19 | −(g_{t +} g_{r}_{)}/T (erg/K) | 2.393467 × 10^{−15} | 2.531530 × 10^{−15} | 2.531530 × 10^{−15} | 2.393467 × 10^{−15} |

20 | −δg/T | 1.38063 × 10^{−16} | 0 | −1.38063 × 10^{−16} | 0 |

21 | −Gibbs energy ergs/molecule (−g _{t} −g_{r}) | 1.531819 × 10^{−12} | 1.620179 × 10^{−12} | 7.290805 × 10^{−13} | 6.893185 × 10^{−13} |

22 | δGibbs energy | −0.88360 × 10^{−13} a | +8.910985 × 10^{−13} b′ | +0.39762 × 10^{−13} a′ | −8.425005 × 10^{−13} b |

24 | Energy density | 5.287088 × 10^{8} | 2.099222 × 10^{8} | 1.358496 × 10^{7} | 3.436531 × 10^{7} |

26 | Energy density | 1.979977 × 10^{8} | 7.283929 × 10^{7} | 4.205456 × 10^{6} | 1.143162 × 10^{7} |

27 | Total entropy ergs/K | 2.876686 × 10^{−15} | 3.014749 × 10^{−15} | 3.014749 × 10^{−15} | 2.876686 × 10^{−15} |

28 | δs_{n} | +1.38063 × 10^{−16} | 0 | −1.38063 × 10^{−16} | 0 |

29 | Total entropic energy ergs/molecule s_{n}T_{n} | 1.841079 × 10^{−12} | 1.929439 × 10^{−12} | 8.682476 × 10^{−13} | 8.284856 × 10^{−13} |

30 | Energy density | 7.267105 × 10^{8} | 2.827630 × 10^{8} | 1.728488 × 10^{7} | 4.442273 × 10^{7} |

31 | δs_{n}T_{n} | +0.8836 × 10^{−13} | −1.031044 × 10^{−12} | −0.39762 × 10^{−13} | +0.983827 × 10^{−12} |

32 | Net heat input | +8.836006 × 10^{−14} | +8.836006 × 10^{−14} | +4.721736 × 10^{−14} | +4.721736 × 10^{−14} |

33 | Internal energy e | 2.208976 × 10^{−13} | 2.208976 × 10^{−13} | 1.028554 × 10^{−13} | 1.028554 × 10^{−13} |

34 | Δe | 0 | −1.180422 × 10^{−13} | 0 | +1.180422 × 10^{−13} |

35 | Heat transfer Q_{f}, Q_{r} | +8.836006 × 10^{−14} | 0 | −3.9762 × 10^{−14} | 0 |

36 | Entropy change= | +1.38063 × 10^{−16}= Q _{f}/T_{f} = 3kln(@_{2}/@)_{1}) | 0 | −1.38063 × 10^{−16}= Q _{r}/T_{r}= 3kln(@ _{4}/@)_{3}) | 0 |

_{f}= 1kT or 0.8836006 × 10

^{−13}ergs per molecule during stage 1 when the piston is released. Then, −0.411427 × 10

^{−13}ergs per molecule is transferred to the exterior refrigerator at 298 K during stage 3. Pressure is given from the perfect gas law, using a

^{3}to indicate the cubic volume available to each molecule. The inertial radius r used in I

_{t}is equal to a/2.

Stage 1 | Stage 2 | Stage 3 | Stage 4 | |
---|---|---|---|---|

Argon, translational −g_{t}, ergs | 11.6173 × 10^{−13} | 12.5009 × 10^{−13} | 5.6255 × 10^{−13} | 5.2278 × 10^{−13} |

Mean quantum number | n = 80.042 | n = 111.708 | n = 111.708 | n = 80.042 |

n_{t}^{3} | 5.12807 × 10^{5} | 1.393968 × 10^{6} | 1.393968 × 10^{6} | 5.12807 × 10^{5} |

Mean quantum, ergs | 2.2654 × 10^{−18} | 8.9679 × 10^{−19} | 4.0356 × 10^{−19} | 1.0194 × 10^{−18} |

N_{2}, translational −g_{t} ergs | 11.1446 × 10^{−13} | 12.0282 × 10^{−13} | 5.7302 × 10^{−13} | 5.3326 × 10^{−13} |

Mean quantum number | n_{t} = 66.968 | n_{t} = 93.462 | n_{t} = 120.584 | n_{t} = 86.402 |

n_{t}^{3} | 3.00346 × 10^{5} | 8.16404 × 10^{5} | 1.753352 × 10^{6} | 6.45017 × 10^{5} |

Mean quantum, ergs | 3.7106 × 10^{−18} | 1.4733 × 10^{−18} | 3.2681 × 10^{−19} | 8.2674 × 10^{−19} |

N_{2}, rotational −g_{r} ergs | 4.1736 × 10^{−13} | 4.1736 × 10^{−13} | 1.5606 × 10^{−13} | 1.5606 × 10^{−13} |

Mean quantum number | j_{r} = 10.609 | j_{r} = 10.609 | j_{r} = 7.239 | j_{r} = 7.239 |

j_{r}^{2} | 112.551 | 112.551 | 52.403 | 52.403 |

Mean quantum, ergs | 3.7082 × 10^{−15} | 3.7082 × 10^{−15} | 2.9781 × 10^{−15} | 2.9781 × 10^{−15} |

n_{t}^{3} × j_{r}^{2} | 3.3802660 × 10^{7} | 9.1887008 × 10^{7} | 9.1881105 × 10^{7} | 3.3800921 × 10^{7} |

288.2 K | Vibrational Action | Rotational Action | Translational Action | Vortical (ω = 5 × 10 ^{−5};r = 10 ^{8} cm | Vortical (ω = 5 × 10 ^{−5};r = 10 ^{5} cm | Vortical (ω = 5 × 10 ^{−5};r = 10 ^{2} cm |
---|---|---|---|---|---|---|

Action ratio (@/ħ) | <0.1 | 8.1 | 152.2 | 2.28259 × 10^{15} | 2.28259 × 10^{9} | 2.28259 × 10^{3} |

Entropy ln(@/ħ) | <0.01 | 4.18k | 15.07k | 35.364k | 21.549k | 7.333k |

Energy | kJ per mol | 10.017 | 36.115 | 84.749 | 51.642 | 18.532 |

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**MDPI and ACS Style**

Kennedy, I.R.; Hodzic, M. Action and Entropy in Heat Engines: An Action Revision of the Carnot Cycle. *Entropy* **2021**, *23*, 860.
https://doi.org/10.3390/e23070860

**AMA Style**

Kennedy IR, Hodzic M. Action and Entropy in Heat Engines: An Action Revision of the Carnot Cycle. *Entropy*. 2021; 23(7):860.
https://doi.org/10.3390/e23070860

**Chicago/Turabian Style**

Kennedy, Ivan R., and Migdat Hodzic. 2021. "Action and Entropy in Heat Engines: An Action Revision of the Carnot Cycle" *Entropy* 23, no. 7: 860.
https://doi.org/10.3390/e23070860