# A Simple Method to Estimate Entropy and Free Energy of Atmospheric Gases from Their Action

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{i}) of atmospheric gases based on physical action is proposed. This realistic approach is fully consistent with statistical mechanics, but reinterprets its partition functions as measures of translational, rotational, and vibrational action or quantum states, to estimate the entropy. With all kinds of molecular action expressed as logarithmic functions, the total heat required for warming a chemical system from 0 K (ΣS

_{i}T) to a given temperature and pressure can be computed, yielding results identical with published experimental third law values of entropy. All thermodynamic properties of gases including entropy, enthalpy, Gibbs energy, and Helmholtz energy are directly estimated using simple algorithms based on simple molecular and physical properties, without resource to tables of standard values; both free energies are measures of quantum field states and of minimal statistical degeneracy, decreasing with temperature and declining density. We propose that this more realistic approach has heuristic value for thermodynamic computation of atmospheric profiles, based on steady state heat flows equilibrating with gravity. Potentially, this application of an action principle can provide better understanding of emergent properties of many natural or evolving complex systems, including modelling of predictions for global warming.

## 1. Introduction

_{Total}= S

_{t}+ S

_{r}+ S

_{v}+ S

_{e}+ S

_{n}+ ….,

## 2. Materials and Methods

^{−23}J·K

^{−1}), Planck’s quantum of action (h = 6.626 × 10

^{−34}J·sec) and the system volume (V) and moments of rotational inertia (I). For ease of use and consistency in dimensions, all modelling and calculations have been performed in centimetre–gram–second (cgs) units, before conversion to Systéme Internationale (SI) units where required.

_{t}= (2πmkT/h

^{2})

^{3/2}V

_{r}= 8πIkT/h

^{2}

_{r}= 8π

^{2}(8π

^{3}I

_{A}I

_{B}I

_{C})

^{1/2}(kT/h

^{2})

^{3/2}

_{vi}= ∏

_{i}[1 − exp

^{−hν/kT}]

^{−1},

_{i}indicates a product of i functions, for each mode of vibration.

#### 2.1. Translational Entropy and the Sackur–Tetrode Equation

_{t}= R[ln(2πmkT)

^{3/2}V/h

^{3}N + 2.5]

_{t}and internal S

_{int}parts as follows [5].

_{V}+ RlnQ − klnN! = S

_{t}+ S

_{int}

_{tr}/∂T) + (dlnQ

_{int}/dT)] + R[lnQ

_{tr}+ lnQ

_{in}] – klnN!

_{int}in the above equation can be ignored.

_{t}= RT[(∂lnQ

_{tr}/∂T)] + R[lnQ

_{tr}] − klnN!

_{t}is (2πmkT/h

^{2})

^{3/2}V as given in Equation (1), and that the Stirling approximation for lnN! is effectively NlnN−N, we have from [5]:

_{t}= 3/2R + Rln[(2πmkT/h

^{2})

^{3/2}V] − Rln(N − 1) = R[ln(2πmkT)

^{3/2}V/(h

^{3}N) + 2.5].

_{t}ω

_{t}, as follows. The three-dimensional kinetic energy (½mv

^{2}) for motion with polar coordinates are shown in Equation (10), with translational angular velocity (ω

_{t}or dӨ/dt) given in radians per second.

^{2}= 3/2kT = ½mr

_{t}

^{2}ω

_{t}

^{2}= ½I

_{t}ω

_{t}

^{2}

_{t}ω

_{t}= 3kTI

_{t}/ω

_{t}= (3kTI

_{t})

^{1/2}

_{t}is thus defined as equal to (3kTI

_{t})

^{1/2}, although a correction factor—as indicated by the prime and discussed below—is required because 3kT is a statistical result from the three-dimensional Maxwell distribution, equal to twice the most probable kinetic energy ½mv

^{2}for the root mean square velocity v in three dimensions. Moreover, 50% of molecules have speeds greater than the root mean square velocity, which is 1.085 times the mean speed of the ideal gas molecules ([9], Table 8). In the Maxwell distribution, the most probable velocity is slightly less than either of these speeds.

^{3}, a cell for each gas molecule. Then, for r

_{t}arbitrarily taken as the mean value of the half-distance between the centres of any two nearest neighbour gas molecules, a

^{3}is equal to (2r

_{t})

^{3}or 8r

_{t}

^{3}. Considering a mole of gas at 298.15 K and 1 atmosphere pressure (N = 6.022169 × 10

^{23}molecules in 24465.1 mL), then r

_{t}or (V/N)

^{1/3}/2 is equal to 1.7188 × 10

^{−7}cm.

_{t}

^{3}term inside the brackets, we have

_{t}= Rln[8e

^{5/2}(2πmr

_{t}

^{2}kT)

^{3/2}/h

^{3}].

_{t}

^{2}kT is taken as equal to kTI

_{t}and @’

_{t}, an uncorrected version of the translational action equal to (3kTI)

^{1/2}.

_{t}= Rln[8(2π/3)

^{3/2}e

^{5/2}(@’

_{t}/h)

^{3}]

_{t}= Rln[e

^{5/2}(2/3π)

^{3/2}(@’

_{t}/ħ)

^{3}]

^{3/2}were to be incorporated into the action term, the inertial radius would be decreased to 7.92287 × 10

^{−8}cm rather than 1.7188 × 10

^{−7}cm. However, for computation, we initially assume a translational resonance symmetry factor 1/z

_{t}of 1/10.22967, replacing (2/3π)

^{3/2}. In a preprint lodged in Cornell’s arXiv [10], a z

_{t}factor was retained; more recently, we have identified this z

_{t}constant as being exactly equal to the inverse of (2 × 1.0854)

^{3}, providing two corrections to prevent double counting in neighbouring molecular couples, and for the ratio of the root mean square velocity to the mean velocity respectively [11], as noted above. This allows for calculation of a mean translational action value, n

_{t}, corresponding to a quantum field state for the molecule, equilibrated with gravity.

_{t}= Rln[e

^{5/2}(@’

_{t}/ħ)

^{3}/z

_{t}] = Rln[e

^{5/2}(@

_{t}/ħ)

^{3}] = Rln[e

^{5/2}(n

_{t})

^{3}]

^{−7}cm for r

_{t}at 298.15 K and 1 atm, or 1.6994 × 10

^{−7}cm at the Earth’s global average surface temperature of 288.15 K. However, it may be possible to make experimental determinations of the dynamic structure, allowing for more accurate estimates of the action. In any case, the sensitivity to variations is low, given its logarithmic nature and any errors would only cause a slight displacement in the entropy value. Initially, the absolute value of the translational symmetry factor z

_{t}was rarely of importance because, in nearly all cases, differences in the entropy of free energy were taken, or the system is isothermal

_{.}In such cases, the z

_{t}factor disappeared.

_{t}and entropy, given a suitable choice of action radius and by incorporating the translational correction factor (z

_{t}).

_{t}= Rln[e

^{5/2}(@

_{t}/ħ)

^{3}] = 2.5R + 3Rln(@

_{t}/ħ)

#### 2.2. Rotational Action and Entropy

_{r}= R + Rln(8π

^{2}kTI

_{r}/σ

_{r}h

^{2}), or Rln[e(8π

^{2}kTI

_{r}/σ

_{r}h

^{2})] = Rln[e(2kTI

_{r}/ħ)

^{2}/σ

_{r}].

_{2}or CO

_{2}. The rotational partition function is 8π

^{2}kTI

_{r}/h

^{2}; clearly, this can also be recast as an action ratio. Here, the moment of inertia I

_{r}is given by (m

_{1}m

_{2}/(m

_{1}+m

_{2}))r

_{r}

^{2}and r

_{r}is the average bond length, σ

_{r}is the rotational resonance symmetry number (e.g., σ

_{r}= 2 for O

_{2}and σ

_{r}= 1 for NO). In this equation, we can recognise that the rotational action of a gas molecule, @

_{r}, is equal to (2kTI)

^{1/2}—derived from the rotational energy equal to ½mr

_{r}

^{2}ω

_{r}

^{2}or I

_{r}ω

_{r}

^{2}/2. Hence, I

_{r}ω

_{r}

^{2}equals 2kT and I

_{r}ω

_{r}is given by (2kTI

_{r})

^{1/2}, and is equal to @

_{r}, by definition. Therefore, using similar notation as for translational action and entropy, we have

_{r}= Rln[e(@

_{r}/ħ)

^{2}/σ

_{r}] = R + Rln[(@

_{r}/ħ)

^{2}/σ

_{r}].

_{r}is a constant. It is shown elsewhere that action is a function of volume and temperature [3], but volume or pressure changes have little or no effect on rotational entropy as long as the temperature is not too high. All other terms in this equation are constant for a given gas molecule.

_{r}, is given from statistical mechanics as

_{r}= Rln[{8π

^{2}(8π

^{3}I

_{A}I

_{B}I

_{C})

^{1/2}(kT)

^{3/2}}/σ

_{r}h

^{3}] + 3/2R,

_{A}, I

_{B}, and I

_{C}in Equation (19) correspond to the three principal moments of rotational inertia with respect to three perpendicular axes (see Glasstone [5]). In terms of action ratios analogous to those used above, we rearrange this equation to read

_{r}= Rln[π

^{1/2}{(8π

^{2}kTI

_{A}/h

^{2})

^{1/2}(8π

^{2}kTI

_{B}/h

^{2})

^{1/2}(8π

^{2}kTI

_{C}/h

^{2})

^{1/2}}/σ

_{r}] + 3/2R.

_{r}of a diatomic molecule as (2kTI

_{r})

^{1/2}for each inertial axis of a linear molecule with more than one atom, we can express the rotational entropy contribution of a non-linear molecule as

_{r}= Rln[{π

^{1/2}(@

_{A}/ħ)(@

_{B}/ħ)(@

_{C}/ħ)}/σ

_{r}] + 3/2R = Rln[{π

^{1/2}e

^{3/2}(@

_{A}@

_{B}@

_{C}/ħ

^{3})}/σ

_{r}],

_{A}, @

_{B}, and @

_{C}are the three principal rotational actions for non-linear molecules. Hence, once again, it is possible to express changes in entropy as a simple function of action alone, as all other terms in the equation are constant for a given gas molecule. Given that the product of entropy and temperature ST indicates the thermal energy required, there is obviously an exact logarithmic relationship between the total energy required to sustain a system of molecules at a given temperature and the action of each mode of rotation.

_{r}for polyatomic molecules depends on the point group of the molecule as defined by the Nobel laureate Herzberg [12]: “A possible combination of symmetry operations that leaves at least one point unchanged is called a point group”. This is a term derived from crystallography, and the characteristic symmetry number σ of each point group can be shown to be equal to “the number of indistinguishable positions into which the molecule can be turned by simple rigid rotations”. Table 1, adapted from Herzberg [13], gives the symmetry number for the more important point groups. Note that methane in the T point group has a rotational symmetry of 12, indicating how its quantum field indicated by its rotational entropy is economical for energy in view of its indistinguishable structure regarding its orientation in space. This situation for methane can be contrasted with a similar tetrahedral carbon molecule having only one hydrogen atom in its structure together with three different halogens such as fluorine, chlorine, and bromine. In this case, the symmetry is unity (1.0), such that the energy field has a 12-fold lower frequency of encountering an identical structure in action space.

#### 2.3. Vibrational Action and Entropy

_{vi}is given by Glasstone [5] as

_{vi}= Rx/(e

^{x}− 1) − Rln(1 − e

^{−x}), where x = hν

_{i}/kT.

_{i}is the wave number, which equals the number of vibrations per second divided by the velocity of light in cm per second. It therefore has the physical dimensions of cm

^{−1}. Ultimately, the total contribution to the vibrational entropy is the sum of all vibrations, taking into account any degeneracy where more than one mode of vibration has the same frequency. Then, Equation (22) is derived as follows. According to Moore [6], the vibrational energy E is given as

^{2}∂lnQ

_{vib}/∂T = Lhν/2 + Lhνe

^{−hν/kT}/(1 − e

^{−hν/kT}).

_{o}remaining at absolute zero Kelvin, where L is Avogadro’s number for the number of molecules in a mole. Thus, taking hν/kT as equal to x, as used above,

_{o})/T = Rxe

^{−x}/(1 − e

^{−x}) (A

_{vib}− E

_{o})/T = Rln(1 − e

^{−x}) since A

_{vib}= −kTlnQ

_{vib}= G

_{vib}

_{vib}= (E − G)/T = Rxe

^{−x}/(1 − e

^{−x}) − Rln(1 − e

^{−x})

_{vib}= Rx

^{2}e

^{−x}/(1 − e

^{−x})

^{2}= Rx

^{2}/(e

^{x}+ e

^{−x}− 2) = Rx

^{2}/2(coshx − 1), given (e

^{x}+ e

^{−x})/2 = coshx

_{vib}T contributes mainly as the enthalpy calculated with C

_{vib}, giving both kinetic and potential vibrational energy but with a relatively smaller negative Gibbs energy. It is only at elevated temperatures—when the ratio hν/kT is less than 1.0 and as C

_{vib}approaches 2.0—that −G, reflecting positive change in the ‘sum of states’, exceeds C

_{vib}T. Then, the statistical Gibbs energy of higher quantum states emerges as more dominant, so that heat is consumed doing quantum work without raising the temperature.

_{i}contributes its own entropy.

_{ν}= ΣSν

_{i}

^{−Er/kT}/Σ

_{i}e

^{−Er/kT}= N

_{r}/Σ

_{i}N = N

_{r}/N

_{r}divided by the total number density of all microstates. This implies that the ratio of the probable number of particles in any two microstates is (neglecting degeneracy)

_{r}/N

_{s}= e

^{−Er/kT}/e

^{−Es/kT}= e

^{−δE/kT}.

_{r}/N

_{s}) = −kTln(r

_{s}/r

_{r})

^{3}= −kTln(@

_{s}/@

_{r})

^{3}

## 3. Results

#### 3.1. Direct Computation of Molar Action and Entropies from Physical Properties of Atmospheric Gases

_{t}= Rln[e

^{5/2}(@

_{t}/ħ)

^{3}] (translation)

_{r}= Rln[e(@

_{r}/ħ)

^{2}/σ

_{r}] (rotation—diatomic or linear molecule)

_{r}= Rln[π

^{1/2}e

^{3/2}(@

_{A}@

_{B}@

_{C}/ ħ

^{3})/σ

_{r}] (rotation—polyatomic molecule)

_{vi}= Rx/(e − 1) − Rln(1 − e

^{−x}), where x = hν/kT (for each vibrational mode)

_{t}/ħ will vary as a function of temperature affecting velocity, but also with volume. Thus, action acts as a surrogate for the effects of both temperature and volume or density for translational entropy. Normally, these variables are considered separately. At extremely low temperatures near absolute zero, the action ratio will tend to a minimum and the entropy will tend to zero, as required by the third law of thermodynamics. Near zero, only vibrational energy remains significant, expressed as the zero-point vibrational energy of hν/2 per bond, proposed as essential by Planck and Einstein [5,6]. In Figure 3, the variation in vibrational energy above the ground state at absolute zero that contributes to molecular entropy is shown for the first three energised modes of carbon dioxide of wavelength 667 cm

^{−1}. The radii of each state in action phase space exponentially increase. Given that action (mrv) at a given temperature of constant mean momentum (mv) is proportional to radius, the ratios of the probable volumes from the number density (N

_{n}) can be expressed one-dimensionally in terms of the inertial radius r and then to the relative action compared to the ground state.

_{n}/@

_{o})) is computed for the 667 cm

^{−1}infrared resonance of CO

_{2}; similar calculations can be performed for the other infrared resonances at shorter wavelengths, or for those of water molecules. Each successive state of increasing energy occurs with decreasing frequency and, therefore, has a declining number density and increasing inertial radius and action ratio. Trajectories of increasing radius may therefore have greater impact in collisions corresponding to greater translational action. Then, the entropy for the activated states can be computed from the equation

_{vi}= Rx/(e

^{x}− 1) − Rln(1 − e

^{−x}).

#### 3.2. Considering Water’s Phase Changes

_{3}), have an exceptionally large vibrational contribution to entropy. Replacing the hydrogen atoms of methane with these two halogen atoms also significantly increases both the rotational and vibrational entropy; this should lessen absorptivity in the longer infrared region significantly, since this is relatively excited at 298 K as a result of longer bond lengths and greater ease of dissociation of atoms. According to Glasstone [5], the formula for calculating vibrational entropy strictly applies only to divalent molecules. However, this cautionary note may not be required for polyatomic molecules after all. By applying the formula to each bond separately and summating as shown in Table 3b, including any degeneracy, the agreement with experimentally determined entropies using the third law approach is just as good as for other molecules where only translational and rotational contributions are significant.

_{2}(6.329 μm) (Table 2 and Table 4). For nitric oxide (NO), a large discrepancy in total entropy between the data calculated here from translation, rotation, and vibration would occur if the electronic (Q

_{e}= 4) term was neglected, as a result of its free radical nature containing unpaired electrons; these add Rln2

^{2}or 11.53 extra entropy units per mole, giving a total value of 211.1, in agreement with the Aylward and Findlay value [15].

_{e}). In the case of O

_{2}, the ground state electronic partition function Q

_{e}is 3 at STP because this molecule has two unpaired electrons that can have their two spins oriented three ways with respect to the nuclear spin —both up, both down, and oppositely. Since they can be distinguished, the three different oxygen species have three times the volume per particle, affecting their action because of the greater radial separation than if only a single species existed. This gives an additional electronic entropy contribution of S

_{e}= RlnQ

_{e}or Rln3.

_{e}factor is included in the Sackur–Tetrode equation as S

_{t}= Rln[Q

_{e}e

^{5/2}(V/N)(2πmkT)

^{3/2}/(h

^{3})], or in the action form of the equation as Rln[e

^{5/2}(@

_{t}/ħ)

^{3}Q

_{e}], which is equal to RlnQ

_{e}+ 2.5R + 3Rln(@

_{t}/ħ). Thus, asymmetry (Q

_{e}) increases entropy by increasing the spatial distance between molecular interactions, and symmetry decreases it by reducing spatial distances since less field energy is needed to sustain a symmetrical molecule than an asymmetrical one.

^{5}Pa

^{)}. To adjust these results to the actual gas pressures in the atmosphere at the same temperature, only the translational action and entropy will vary. In terms of centimetre–gram–second (cgs) units, the pressure is equal to kTa

^{−3}or kT/8r

^{3}at 1.013 × 10

^{5}pascals, being the product of the mass of air per square cm of the Earth’s surface (ca. 1 kg) and the acceleration of gravity (9.807 m∙s

^{−2}).

^{2}is equal to 3kT and Boltzmann’s constant k is also taken as dimensionless. It is also instructive to be aware that the product of entropy and absolute temperature (ST) is always a significant multiple of the kinetic energy since that is merely one of its components—the sustaining field energy corresponding to decreases in free energy from absolute zero must be added to its kinetic energy while heating the molecules, absorbing any heat that becomes latent during this process and in doing any work, such as breaking H-bonded aggregated structures or pressure–volume work against the atmosphere. In this connection, much of the magnitude of ST is generated together with increased enthalpy during phase changes when parent solid or liquid matter is melting or vaporising. Gibbs energy does not change when these reversible processes occur isothermally. The chemical potential of the liquid water is equilibrated with that of the vapour at the boiling temperature, with the increase in enthalpy on vaporisation being effectively an increase in internal entropies associated with increasing the internal vibration and rotation of the declustered water molecules. Such increases in internal action and entropy are actually increases in enthalpy.

#### 3.3. Phase Space as Action Space

_{pq}), claiming that “the quantity … which corresponds to entropy is log V, the quantity V (not volume) being defined as the extension in phase”. Hence, we can conclude that according to Gibbs, even before Planck identified his quantum of action, any equipotential contour in phase space of equal translational action (V

_{p}× V

_{q}= mv × r) would also correspond to states of equal translational entropy. In effect, changes in the momentum mv and a linear coordinate r would lead to no change in their product action and its logarithm, entropy. We can now recognise such contours as adiabatic, differing by a minimum of Planck’s quantum of action h, giving a scale for estimating maximum uncertainty in momentum or position. Hence, this paper can be considered as a 21st century quantum revision of Gibbs’ 19th century suggestion.

#### 3.4. Boltzmann’s Realistic Collision Model of Entropy

^{−1}T

^{3/2}) + const.], where ρ is the number density of gas molecules. We can observe that this result only lacks the quantum of action h as a suitable divisor in the logarithmic term to remove the physical dimensions of action per unit mass.

_{t2-t1}= Rln[(e

^{5/2}@

_{t2}/ħ)

^{3}] − Rln[e

^{5/2}(@

_{t1}/ħ)

^{3}] = 3Rln[(@

_{t2}/(@

_{t1})]

#### 3.5. Under Isothermal Conditions Gibbs Energy Varies with Translational Action and Entropy

_{t}/ħ for a gas at this minimum temperature must be slightly less than 1, if it could exist as such, since the translational entropy can then be considered as equal to 5/2R + Rln[(@

_{t}/ħ)

^{3}], equivalent to the Gibbs expression for entropy of ST = H−G, where G is the free energy or work potential of a monatomic gas at constant pressure. This suggests that the magnitude of the function RTln[(@

_{t}/ħ)

^{3}] has the same value as the free energy, although opposite in sign, so that G = −RTln[(@

_{t}/ħ)

^{3}] or RTln[(ħ/@

_{t})

^{3}] and ΔG = 3RTln[(@

_{tr}/(@

_{tp})] for changes in action state at constant temperature.

_{t}T) is equal to the change in Gibbs energy—as well as changes in the internal action and entropy representing changes in enthalpy as a result of revised bonding energies. It is important to understand that the enthalpy term designated H refers to the sensible heat that tends to change temperature. Thus, if a chemical reaction results in products where atoms or electrons are more firmly bound with shorter radii, the reduced potential energy will be compensated by increased internal kinetic energy and equal quantities of emitted quanta, resulting in a release of heat as a reduction in Gibbs energy and an increase in entropy of the surrounding system. In the absence of such chemical reactions, the enthalpy change can be measured by the changes in kinetic energy and pressure–volume work alone. This is also true with monatomic noble gases like argon.

_{t}T + S

_{r}T + S

_{vi}T)

_{t}/ħ)

^{3}] = 1.5RT + RT − RTln[e

^{5/2}(@

_{t}/ħ)

^{3}]

_{t}/ħ)

^{3}Q

_{e}][(@

_{r}/ħ)

^{2}/σ

_{r}]} = 2.5RT + RT − RTln{[e

^{7/2}(@

_{t}/ħ)

^{3}Q

_{e}] [(@

_{r}/ħ)

^{2}/σ

_{r}]}

_{t}/ħ)

^{3}Q

_{e}][(@

_{r}/ħ)

^{2}/σ

_{r}]} = 2.5RT − RTln{[e

^{5/2}(@

_{t}/ħ)

^{3}Q

_{e}] [(@

_{r}/ħ)

^{2}/σ

_{r}]}

_{i}/N = e

^{−ε1/kT}+ e

^{−ε2/kT}+ e

^{−ε3/kT}+ e

^{−ε4/kT}…+ e

^{−εn/kT}; nkTlnζ is the thermodynamic potential (or free energy) for n molecules, a function of an inverse action ratio ζ. He defined the factor 1/ζ as a function of the translational partition function (2πmkT/h

^{2})

^{3/2}V, divided by the number of particles (n)—that is, as a translational action ratio as defined in this paper. By contrast, its inverse ζ is an ‘inaction’ ratio, indicating the free energy, and Schrödinger’s insightful equation precedes, by at least 70 years, the action potential theory of free energy given here. For a perfect monatomic gas, PV is equal to RT, and so U + PV is equal to the enthalpy H, which does not change for individual molecules of a chemical species unless the temperature changes.

#### 3.6. Greenhouse Gases and Temperature Equilibration in the Gravitational Field

^{−1}= 1 μm, 1000 cm

^{−1}= 10 μm; 100 cm

^{−1}= 100 μm) whereas sunlight is confined to the 0.3–5 μm range. The longer wavelength of terrestrial radiation compared to sunlight is a result of absorption of sunlight by surface materials, and the re-equilibration of quanta with the much cooler surface of the Earth, compared to the boiling ocean of hydrogen atoms of the Sun. Obviously, polyatomic molecules absorb in the 5–30 μm wavelength band of the infrared, and the more complex the molecules are, the greater the number of absorptions.

^{2}) with respect to time and its kinetic (T) and potential energy (V); these can be considered as surrogates for heat and work in a gravitational or central force system.

^{2}I/dt

^{2}= 2T + V

_{2}, N

_{2}O, and CH

_{4}will absorb. However, a CO

_{2}molecule activated by IR absorption to vibrate more vigorously will transfer most of this energy to other air molecules in the next collisions, thus increasing their action and entropy while dissipating the activated internal state and increasing their Gibbs energy. Furthermore, the more dilute the gas (e.g., N

_{2}O and CH

_{4}), the greater its translational entropy, although its vibrational and rotational entropies will be purely a function of temperature. Thus, on absorbing a specific quantum of IR radiation (exciting molecular vibration), such a dilute gas will have a larger disequilibrium between its vibrational action and its translational action. In a subsequent collision, the greater inertia and amplitude of the vibrating atom should cause a more efficient transfer of momentum to surrounding air molecules, irrespective of whether they are greenhouse gases or not.

_{i}/cδt) tending to selectively elevate the greenhouse gases, compared to the non-absorbing gases, N

_{2}(78%), O

_{2}(21%), and Ar (1%), but the thermodynamic action potential to elevate greenhouse gases outlined here—as a result of the opposite function of internal and external translational action and entropy—is much greater.

_{2}and O

_{2}as a result of collisions, and the heated gases expand to higher altitude, exchanging their increased kinetic energy for increased gravitational energy and cooling as a result. Perhaps it is more apt to consider that the greenhouse gases, such as water, play an important role in holding up the sky, enabling reversible gravitational work, thereby cooling the atmosphere. These adjustments of temperature of the troposphere allow the outgoing longwave radiation to balance the incoming solar shortwave radiation, providing temperature equilibrium. The average temperature at the surface is automatically adjusted to ensure this balance, fluctuating according the rate of heat flow from the Earth.

#### 3.7. Adiabatic Processes

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Calculation of translational and rotational action (@). Mean translational action @

_{t}(

**a**) is estimated, as explained in the text, from the average separation of a = 2r by allocating each molecule a space of a

^{3}= V/N, where V is the total volume and N is the total number of diatomic molecules like dinitrogen (N

_{2}). Relative angular motion dӨ/dt = ω is estimated for molecules exhibiting the root mean square velocity, taking 3kT = mv

^{2}= mr

^{2}ω

^{2}. Then translational action @

_{t}is equal to [(3kTI

_{t})

^{1/2}/2.170806]. Rotational action @

_{r}(

**b**) for linear molecules such as N

_{2}, O

_{2}, and CO

_{2}is similarly estimated, and equated to (2kTI

_{r})

^{1/2}. I is the moment of inertia equal to mr

^{2}.

**Figure 2.**Flow diagram for computing absolute entropy and Gibbs energy. I

_{a}, I

_{b}, and I

_{c}refer to the inertial moments of inertia. A fully annotated description of the relevant algorithms and subroutines to compute entropy and free energy is available online at the Entropy site, or on request to the corresponding author.

**Table 1.**Symmetry numbers for various point groups. (Modified from Herzberg [13] p. 508).

Point Group | Symmetry No. | Point Group | Symmetry No. | Point Group | Symmetry No. |
---|---|---|---|---|---|

σ_{r} | σ_{r} | σ_{r} | |||

C_{1}, C_{i}, C_{s} | 1 | D_{2}, D_{2d}, D_{2h} | 4 | C_{∞} | 1 |

C_{2}, C_{2v}, C_{2h} | 2 | D_{3}, D_{3d}, D_{3h} | 6 | D_{∞h} | 2 |

C_{3}, C_{3v}, C_{3h} | 3 | D_{4}, D_{4d}, D_{4h} | 8 | T, T_{d} | 12 |

C_{4}, C_{4v}, C_{4h} | 4 | D_{6}, D_{6d}, D_{6h} | 12 | O_{h} | 24 |

C_{6}, C_{6v}, C_{6h} | 6 | S_{6} | 3 |

Gas | MW | I_{t} × 10^{40}(g·cm ^{2}) | @_{t}/ħ = n_{t} | S_{t} = Rln[e^{5/2}(n_{t})^{3}] (J·K^{−1}) | Radius × 10^{10} (cm) | I_{r} × 10^{40} (g·cm^{2}) | @_{r}/ħ = n_{r} | S_{r} = Rln[e(n_{r})^{2}/σ] (J·K^{−1}) | 1/λ cm^{−1} | x = h ν/kT | S_{v} | Q_{e} | S_{e} | ΣS |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

H_{2} | 2 | 98.83 | 104.757 | 117.48 | 74 | 0.458 | 1.8413 | 12.7 | - | - | - | 1 | 0 | 130.18 |

N_{2} | 28.013 | 139.02 | 392.901 | 150.45 | 110 | 14.235 | 10.2654 | 41.27 | - | - | - | 1 | 0 | 191.73 |

O_{2} | 31.999 | 158.12 | 419.02 | 152.06 | 121 | 19.59 | 12.0426 | 43.93 | 1580 | 7.63 | 0.04 | 3 | 9.13 | 205.16 |

CO | 28.011 | 138.33 | 391.923 | 150.39 | 113 | 14.643 | 10.4115 | 47.27 | 2170 | 10.47 | 0 | 1 | 0 | 197.67 |

NO | 30.006 | 148.54 | 406.134 | 151.28 | 115 | 16.555 | 11.0706 | 48.29 | 1904 | 9.188 | 0.01 | 4 | 11.5 | 211.12 |

CO_{2} | 44.01 | 217.42 | 491.354 | 156.03 | 244 | 79.665 | 24.2846 | 54.72 | See | below | 2.99 | 1 | 0 | 214.61 |

N_{2}O | 44.013 | 215.9 | 489.65 | 155.94 | 66.9 | 22.255 | 59.9 | See | Below | 3.05 | 0 | 218.89 |

Gas | MW | I_{t} × 10^{−40} (g·cm^{2}) | @_{t}/ħ | S_{t} (J·K^{−1}) | I_{rA} × 10^{−40} | I_{rB} (g·cm^{2}) | I_{rC} × 10^{40} | @_{rA}/ħ | @_{rB}/ħ | @_{rC}/ħ | σ_{r} | S_{r} (J·K^{−1}) | Point Group |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

H_{2}O | 18.015 | 88.372 | 313.268 | 144.8 | 1.024 | 1.92 | 2.947 | 2.7533 | 3.7699 | 4.6709 | 2 | 43.74 | C_{2v} |

H_{2}S | 34.08 | 167.18 | 430.867 | 152.75 | 2.667 | 3.076 | 5.845 | 4.4435 | 4.7721 | 6.5779 | 2 | 52.52 | C_{2v} |

O_{3} | 47.998 | 235.45 | 511.336 | 157.02 | 7.877 | 62.865 | 70.9 | 7.6366 | 21.5796 | 22.9104 | 2 | 79.94 | C_{2v} |

SO_{2} | 64.063 | 314.25 | 590.74 | 160.62 | 13.807 | 81.328 | 95.356 | 10.1103 | 24.5377 | 26.5697 | 2 | 84.58 | C_{2v} |

H_{2}O | Wave Number | x = hcν_{i}/kT | S_{vi} | CO_{2} | Wave Number | x = hcν_{i}/kT | S_{vi} | Degen | ∑S_{vi} | |
---|---|---|---|---|---|---|---|---|---|---|

A_{1} | 3652 | 17.6235 | <0.0001 | σ_{g}^{+} | 1388 | 6.6981 | 0.079 | 1 | 0.079 | |

A_{1} | 1595 | 7.697 | 0.0329 | Π | 667 | 3.2188 | 1.4547 | 2 | 2.9093 | |

B_{2} | 3756 | 18.1254 | <0.0001 | σ_{u}^{+} | 2349 | 11.3356 | 0.0012 | 1 | 0.0012 | |

Total | Total | 0.033 | Total | 2.9895 | ||||||

H_{2}S | x = hcν_{i}/kT | S_{v} | N_{2}O | x = hcν_{i}/kT | S_{v} | |||||

A_{1} | 2615 | 12.6193 | 0.004 | ∑ | 2224 | 10.7324 | 0.0002 | |||

A_{1} | 1183 | 5.7088 | 0.1856 | ∑ | 1285 | 6.2011 | 0.1216 | |||

B_{2} | 2626 | 12.6723 | <0.0001 | Π | 589 | 2.8423 | 1.4627 | |||

Total | 0.186 | Π | 589 | 2.8423 | 1.4627 | |||||

Total | 3.0473 | |||||||||

O_{3} | x = hcν_{i}/kT | SO_{2} | x = hcν_{i}/kT | |||||||

A_{1} | 1110 | 5.3565 | 0.2504 | A_{1} | 1151 | 5.5544 | 0.2117 | |||

A_{1} | 705 | 3.4021 | 1.2561 | A_{1} | 518 | 2.4997 | 2.5715 | |||

B_{2} | 1042 | 5.0284 | 0.3301 | B_{2} | 1352 | 6.5244 | 0.0919 | |||

Total | 1.8367 | Total | 2.8751 |

Gas | MW | I_{t} × 10^{40} g·cm^{2} | @_{t}/ħ | S_{t} (J·K^{−1}) | I_{rA} × 10^{40} | I_{rB}(g·cm^{2}) | I_{rC} × 10^{40} | @_{rA}/ħ | @_{rB}/ħ | @_{rC}/ħ | σ_{r} | S_{r}(J·K^{−1}) | Point Group |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

NH_{3} | 17.031 | 83.543 | 27.988 | 144.1 | 2.9638 | 2.9638 | 4.5176 | 4.6841 | 4.6841 | 5.783 | 3 | 48.36 | C_{3v} |

Vibrational | NH_{3} | cm^{−1} | Species | Wave number | x = hcν_{i}/kT | S_{v} | |||||||

A_{1} | 3337 | 16.1034 | 0.0001 | ||||||||||

A_{1} | 950 | 4.5844 | 0.4785 | ||||||||||

E | 3447 | 16.6343 | 0.0001 | ||||||||||

E | 1627 | 7.8514 | 0.0287 | ||||||||||

Total | 0.5074 |

Gas | MW Daltons | I_{t} × 10^{40} g·cm^{2} | @_{t}/ħ | S_{t}(J·K^{−1}) | I_{rA} × 10^{40} | I_{rB}(g·cm^{2}) | I_{rC} × 10^{40} | @_{rA}/ħ | @_{rB}/ħ | @_{rC}/ħ | σ | S_{r}(J·K^{−1}) | Point Group |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

CH_{4} | 16.401 | 78.678 | 295.623 | 143.35 | 5.27 | 5.27 | 5.27 | 6.2461 | 6.2461 | 6.2461 | 12 | 42.263 | T_{d} |

CFCl_{3} | 137.37 | 673.84 | 865.041 | 170.13 | 340.35 | 340.35 | 799.71 | 50.197 | 50.197 | 62.433 | 3 | 107.59 | C_{3v} |

CF_{2}Cl_{2} | 120.91 | 593.12 | 811.578 | 168.54 | 203.73 | 318 | 375.66 | 38.8364 | 48.5206 | 52.737 | 2 | 107.13 | C_{2v} |

CF_{3}Cl | 104.46 | 512.41 | 754.336 | 166.71 | 146.32 | 251.58 | 251.58 | 32.9126 | 43.1566 | 43.156 | 3 | 99.745 | C∞v |

CH_{4} | Wave number | x = hcν_{i}/kT | S_{v} | Degn. | S_{v} | CFCl_{3} | Wave Number | x = hcν_{i}/kT | S_{v} | Degn. | S_{v} | ||

A | 2914 | 14.063 | 0.0001 | 1 | 0.0001 | A | 1085 | 5.2359 | 0.2773 | 1 | 0.2733 | ||

E | 1526 | 7.364 | 0.0441 | 2 | 0.0882 | A | 535 | 2.5818 | 2.4105 | 1 | 2.4105 | ||

T | 3020 | 14.575 | 0.0001 | 3 | 0.0002 | A | 350 | 1.689 | 4.8792 | 1 | 4.8792 | ||

T | 1306 | 6.3034 | 0.1113 | 3 | 0.334 | E | 847 | 4.0874 | 0.7208 | 2 | 1.4416 | ||

Total | 0.4225 | E | 394 | 1.9013 | 4.121 | 2 | 8.242 | ||||||

E | 241 | 1.163 | 7.5121 | 2 | 15.024 | ||||||||

Total | 32.275 |

CF_{2}Cl_{2} Band | cm^{−1} | x = hcν_{i}/kT | S_{v} | CF_{3}Cl | Band | cm^{−1} | x = hcν_{i}/kT | S_{v} | Degeneracy | ∑S_{v} |
---|---|---|---|---|---|---|---|---|---|---|

A | 1101 | 5.3131 | 0.2598 | A | 1105 | 5.3324 | 0.2556 | 1 | 0.2556 | |

A | 667 | 3.2188 | 1.4547 | A | 781 | 3.7689 | 0.9344 | 1 | 0.9344 | |

A | 458 | 2.2108 | 3.2297 | A | 476 | 2.297 | 3.0163 | 1 | 3.0163 | |

A | 262 | 1.2643 | 6.8968 | E | 1212 | 5.8488 | 0.1646 | 2 | 0.3293 | |

A | 322 | 1.5539 | 5.4387 | E | 563 | 2.7169 | 2.1667 | 2 | 4.3334 | |

B | 902 | 4.3528 | 0.5796 | E | 350 | 1.689 | 4.8792 | 2 | 9.7584 | |

B | 437 | 2.1088 | 3.4981 | Total S | 18.627 | |||||

B | 1159 | 5.593 | 0.2048 | |||||||

B | 446 | 2.1523 | 3.3804 | |||||||

Total S | 24.945 |

Gas | St J·K^{−1} | Sr J·K^{−1} | Sv J·K^{−1} | S J·K^{−1} | Ref. [15] | IR Spectrum Vibration Wavelength Mm (Degeneracy Bracketed) |
---|---|---|---|---|---|---|

H_{2}O | 144.8 | 0.033 | 188.6 | |||

CO_{2} | 155.94 | 54.72 | 2.99 | 213.6 | 214 | 4.257, 7.2046, 14.993 |

H_{2}S | 152.81 | 52.54 | 0.19 | 205.5 | 206 | 3.808, 3.824, 8.453 |

N_{2}O | 155.94 | 59.8 | 3.05 | 218.9 | 220 | 4.446, 7.782, 16.978 |

O_{3} | 157.02 | 79.94 | 1.84 | 238.8 | 239 | 9.009, 9.597, 14.184 |

SO_{2} | 160.62 | 84.58 | 2.875 | 248.1 | 248 | 7.396, 8.688, 19.305 |

NH_{3} | 144.1 | 48.36 | 0.507 | 193 | 192 | 2.901, 2.997, 6.146, 10.526 |

CH_{4} | 143.36 | 42.26 | 0.423 | 186 | 186 | 3.311(2), 3.432(3), 6.553, 7.657(3) |

CFCl_{3} | 170.14 | 107.59 | 32.275 | 310 | 310 | 9.217, 11.806(2), 18.692, 25.381(2), 28.571, 41.494(2) |

CF_{2}Cl_{2} | 168.545 | 107.14 | 24.945 | 300.6 | 301 | 8.628, 9.083, 11.086, 14.999, 21.834, 22.421, 22.883, 31.056, 38.168 |

CF_{3}Cl | 166.721 | 99.75 | 18.627 | 285.1 | 286 | 8.251(2), 9.050, 12.804, 17.762(2), 21.008, 28.571(2) |

O_{2} | 162.07 | 43.93 | 0.035 | 206 | 205 | 6.329 |

CO | 150.31 | 47.19 | 0.0025 | 197.5 | 198 | 4.608 |

Gas | Pressure (Atm) | S_{t} | S_{r} | S_{v} | S Total | ∑S/Mole STP | J/Mole of Air/K | J Per m^{3} |
---|---|---|---|---|---|---|---|---|

H_{2}O | 0.00775 | 185.29 | 43.74 | 0.033 | 229.1 | 188.6 | 529.37278 | 21,637.48 |

CO_{2} | 0.000397 | 215.51 | 54.72 | 2.99 | 273.2 | 213.6 | 32.337468 | 1,321.76 |

H_{2}S | 2 × 10^{−10} | 338.49 | 52.52 | 0.19 | 391.2 | 205.5 | 0.0000023 | 0.000003 |

N_{2}O | 0.000000325 | 280.24 | 59.9 | 3.05 | 342.2 | 218.9 | 0.0331588 | 1.355326 |

O_{3} | 2.66 × 10^{−8} | 302.13 | 79.94 | 1.84 | 383.9 | 238.8 | 0.0032865 | 0.134332 |

SO_{2} | 3 × 10^{−10} | 343.01 | 84.58 | 2.875 | 430.5 | 248.1 | 0.0000039 | 0.000159 |

NH_{3} | 5 × 10^{−10} | 322.23 | 48.36 | 0.507 | 371.1 | 193 | 0.0000553 | 0.00226 |

CH_{4} | 0.0000017 | 272.21 | 42.26 | 0.423 | 314.9 | 186 | 0.1596086 | 6.523810 |

CFCl_{3} | 2.6 × 10^{−10} | 350 | 107.59 | 32.28 | 489.9 | 310 | 0.000038 | 0.000006 |

CF_{2}Cl_{2} | 5.5 × 10^{−10} | 345.9 | 107.14 | 24.945 | 478 | 300.6 | 0.0000784 | 0.000001 |

CF_{3}Cl | 1 × 10^{−10} | 353.08 | 99.75 | 18.627 | 471.5 | 285.1 | 0.0000141 | 0.000002 |

O_{2} | 0.2095 | 165.05 | 43.93 | 0.035 | 209 | 205.1 | 13,054.65 | 533,593.02 |

CO | 0.00000015 | 279.58 | 47.19 | 0.0025 | 326.8 | 197.5 | 0.0146153 | 0.597382 |

NO | 3 × 10^{−10} | 333.56 | 48.39 | 0.0087 | 382 | 199.6 | 0.0000034 | 0.000139 |

H_{2} | 0.0000005 | 238.11 | 12.7 | - | 250.8 | 130.2 | 0.037388 | 1.52819 |

N_{2} | 0.78084 | 152.45 | 41.27 | - | 193.7 | 191.7 | 45,094.80 | 1,843,195.93 |

Ar | 0.00934 | 193.7 | - | - | 193.7 | 154.8 | 539.400466 | 0.0442339 |

Total | 2,399,758.37 |

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## Share and Cite

**MDPI and ACS Style**

Kennedy, I.; Geering, H.; Rose, M.; Crossan, A.
A Simple Method to Estimate Entropy and Free Energy of Atmospheric Gases from Their Action. *Entropy* **2019**, *21*, 454.
https://doi.org/10.3390/e21050454

**AMA Style**

Kennedy I, Geering H, Rose M, Crossan A.
A Simple Method to Estimate Entropy and Free Energy of Atmospheric Gases from Their Action. *Entropy*. 2019; 21(5):454.
https://doi.org/10.3390/e21050454

**Chicago/Turabian Style**

Kennedy, Ivan, Harold Geering, Michael Rose, and Angus Crossan.
2019. "A Simple Method to Estimate Entropy and Free Energy of Atmospheric Gases from Their Action" *Entropy* 21, no. 5: 454.
https://doi.org/10.3390/e21050454