Absolute Internal Energy of the Real Gas

The internal energy U of the real, neutral-gas particles of total mass M in the volume V can have positive and negative values, whose regions are identified in the state chart of the gas. Depending on the relations among gas temperature T , pressure p and mass-specific volume v V M = , the mass exists as a uniform gas of freely-moving particles having positive values U or as more or less structured matter with negative values U . In the regions 0 U > above the critical point [ ] , , c c c T p v it holds that ( ) , c p T v p > and c v v > , and below the critical point it holds that ( ) , c p T v p < and v v v > , where v v is the mass-specific volume of saturated vapor. In the adjacent regions with negative internal energy values 0 U < the mean distances between particles are short enough to yield negative energy contributions to U due to interparticle attraction that exceeds the thermal, positive energy contributions due to particle motion. The critical isochor c v is the line of equal positive and negative energy contributions and thus represents a line of vanishing internal energy 0 U = . At this level along the critical isochor the ever present microscopic fluctuations in energy and density become macroscopic fluctuations as the pressure decreases on approaching the critical point; these are to be observed in experiments on the critical opalescence. Crossing the isochor c v from 0 U > to 0 U < , the change in energy 0 U ∆ > happens without any discontinuity. The saturation line v v also separates the regions between 0 U > and 0 U < , but does not represent a line 0 U = . The internal-energy values of saturated vapor v U and condensate l U can be determined absolutely as functions of vapor pressure p and densities ( )v M V and ( )l M V , repectively, yielding the results 0 l v U U < < , 0 l v U U U = + < for c T T < and 0 l v U U U = = = at the critical point. Crossing the line v v from 0 v U U = > to 0 v l U U U = + < requires the energy 0 l U U ∆ = − > to be removed from the particle system. The thermodynamic and quantum-mechanical formulations of the internal energy of a particle system only agree if both the macroscopic and microscopic energy scales have the same absolute energy reference value 0. How to cite this paper: Elsner, A. (2017) Absolute Internal Energy of the Real Gas. Engineering, 9, 361-375. https://doi.org/10.4236/eng.2017.94020 Received: March 22, 2017 Accepted: April 27, 2017 Published: April 30, 2017 Copyright © 2017 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access

requires the energy 0 l U U ∆ = − > to be removed from the particle system.
The ther-modynamic and quantum-mechanical formulations of the internal energy of a particle system only agree if both the macroscopic and microscopic energy scales have the same absolute energy reference value 0.
Arguments for the energy reference value 0 U = in the state of transition from bound to freely-moving particles in macroscopic classical and microscopic quantum particle systems are discussed.

Introduction
Investigated here are thermodynamic properties of a real single-component gas of mass M in the volume V .In accordance with Gibbs [1], the internal energy U is assigned to the gas, composed of a very large number of particles, and the entropy S to its distribution in V .It is not the total energy of M in V that is considered, i.e. not the contributions from relativistic or kinetic and potential energies of a mass system moving in external fields, but only the contributions resulting from all time-independent kinetic and potential energies in V averaged over all particles.The sum of energies exchanged between the particles can take a positive, negative or else vanishing value U , depending on the conditions to which the gas is subjected.

Dependence of the Sign of the Internal Energy of the Real Gas on the Aggregate State of Its Mass
Phenomenologically, a distinction is to be made between three thermodynamic conditions under which the gas is in equilibrium.Thermodynamic equilibrium states that the fluid particles are homogeneously distributed in the particular phase volumes and satisfy there the Boltzmann equation, i.e. the equation of kinetic theory.By means of variation of the external parameters, temperature and pressure, the mean distance d between fluid particles, which are continually in motion, can be varied; this distance determines the statistical mean value of both the thermal energy and the attractive energy of the particles in each phase volume and is thus related to the aggregate state of the particular fluid mass.In the following, the ratio of the mean distance between fluid particles at given temperature, pressure and volume to that assumed at the critical state, is denoted as 1) Above the critical point for pressures  ( ) ( ) .The internal energies are likewise equally large, i.e.
( ) ( ) ( ) . Under the condition 1 r = the energy contributions of motion and binding cancel out in sum, making it impossible to distinguish between positive and negative values, so that . At the critical point, however, not only do the values of the internal energies of vapor and condensate vanish but, since , the energy value of the total fluid mass does as well.The result is ( ) ( ) 3) The sign of the internal energy U of the real gas depends on the aggregate state of its mass M .The aggregate state, on the other hand, is determined by the mean distances d between fluid particles, which are qualitatively described by the value of the relative number , , , ,

Macroscopic Energy and Density Fluctuations in the Fluid on Approaching the Critical Point
In a fluid, macroscopic density fluctuations are observed in a narrow temperature range before the critical temperature is reached [2].Andrews discovered these for the first time in 1869 when he irradiated a gas (CO 2 ) with visible light (wavelengths [ ] In the following, the physical processes occuring when the gas approaches the critical point are investigated.In the course of cooling the kinetic energy of all gas particles decreases and at the same time their binding energy increases.This also raises the probability that particles with diameters of the order 10 −10 [m] aggregate to more voluminous mass conglomerations with diameters of light wavelengths.From their surfaces the light is reflected and scattered proportionally to 4   λ in different directions.The deflection intensity increases in the course of cooling with the number and size of the scattering centres [2].The illuminated fluid appears opaque and does not become transparent again till the critical point is reached because there a single aggregate with about 10 −2 [m] diameter and vapor particles with diameters of less than 10 −9 [m] exist from which visible light is no longer diffusely scattered. The reason why there are mass fluctuations in the scattering range between  ) ( ) The square root

Thermodynamic Verification of the Value U = 0 for the Internal Energy at the Critical Point
In accordance with the thermodynamic theory of Willard Gibbs [1] it is possible to calculate every thermodynamic property of a real gas.The quantities uniquely determining the gas are the absolute values of mass M , volume V , internal energy U and entropy S , which represent the external or extensive parameters of the gas system.The likewise absolute quantities, temperature T , pressure p and chemical potential µ , are referred to as intensive parameters.
They are defined by T p µ one knows the equilibrium state of the gas in V , which is also described by Gibbs's fundamental equation [3] 0.
Although Gibbs's theory [1] was already known in the scientific world at the and the chemical potential U ST Vp − + were indeterminable functions owing to the missing thermodynamic reference points, so that the fundamental equation could not be applied to every thermodynamic question.Meanwhile the situation has been changed by the proof of 0 U = found at the critical point [6].
The theory provides the Gibbs-Duhem equations as the sum of the products of the extensive parameters and the differentials of the corresponding intensive parameters [3].By enlisting the fundamental equation one then obtains the explicit expressions for the internal energy and entropy [3]: and . 1 1 This relation states that in the case of the saturated fluid there is no difference between a heat capacity at constant volume and one at constant pressure.
In the following, it is to be proved once again that the zero value of U is thermodynamically fixed by Gibbs's theory as 0 U = at the critical point.
We consider a saturated fluid which according to Gibbs's phase rule has one thermodynamic degree of freedom fewer than the single phase gas and is thus The relative mass distributions with the value 1 2 at the critical point are In accordance with the masses the values of the internal energy and entropy in the partial volumens v V and l V are divided as follows: where we abbreviate by putting, for example, U M V T .The mass-specific quantities are

T U M V T M u T U T M T u T U T M T s v T S M V T M s T S T M T s T S T M T
Whereas u is a function of the variables v and T , the quantities v u and l u are pure temperature functions since, for example, it holds that The quantity ( ) This representation is possible if, by analogy with Equation (4.2) for U , also The system of Equation (4.10) provides three relations for determining the reference value for the internal energy.Discussion of this is conducted through the mass-specific quantities u , v u and l u , whose temperature dependences are described as follows: The difference ( ) At the critical point one has v l u u = , i.e. the value of the evaporation energy ( )( ) u v T is smaller than that of v u and larger than l u and at c T takes the value v l u u = , which is denoted here as c u .This gives the relations ( ) ( ) ( ) It is now to be shown that 0 c u = is valid.Because the stated extensive and intensive parameters are absolute quantities, c u has an absolute value.It is certainly reasonable to assume this to be zero for the time being since otherwise choosing a positive or negative value should be physically justified so as not to appear arbitrary.With the choice 0 c u = it follows from the three Equation  The mass-specific energies can thus be put in the simpler form: It can be shown that the volume function v can be explicitly represented by the following thermodynamic expression [7]: The symmetric occurrence of the variables v v and l v in v marks the phase independence of the volume function ( ) ( ) . The internal energies can now be expressed in terms of the measurable quantities p and Energies as extensive parameters can be added and subtracted.From follows in addition to relations (4.13) and (4.19) that ( ) ( ) ( ) Particularly interesting is the ratio of v u to l u , which as a function of the volume ratio v l z v v = assumes universally for every gas values between −1 at the critical point and 0 at absolute zero: From this it follows that the ratio of the phase-specific energies to the evaporation energy is likewise universal: It can readily be shown that the internal energy of the saturated fluid 1 0 for 0 and 1 0.
The above mentioned relations (4.14)-(4.23)are a consequence of fixing the critical value 0 c u = .If, on the other hand, the assumption 0 c u = ∆ ≠ were correct, relations (4.13) would read l v u u u + ∆ ≤ + ∆ ≤ ∆ ≤ + ∆ and a unique transformation between thermodynamic parameters would no longer be given; one need only think of the functions, free energy and chemical potential, which would then remain undetermined because of the constant 0 ∆ ≠ , whereas it is just these functions, which regulate all processes in nature, that should be determinable.0 ∆ ≠ is not compatible with the extensivity of U formulated by Gibbs, but it holds that ( ) This then also means that the absolute tempera- ture value ( ) , , U M V T can be measured.If a cooling process starts at c T and ends at c T T < and the energy decrease is measured in the form of the heat removed from the system by the amount Q ∆ , ( ) , , U M V T is negative and is determined from U M V T can also be calculated from the heat capacity ( ) , ,

U M V T and ( )
, , C M V T have a discontinuity at the triple point and increase monotonically as T : , , d 0 for 0 .
The value of the internal energy of the saturated fluid is negative for c T T < .

Thermodynamic Verification of the Value S = 0 for the Entropy at Absolute Zero
In the context of quantum theory the known Planck form of the Nernst postulate, viz.
is interpreted as a macroscopic entropy state of a particle system whose value is proportional to the logarithm of the number of its realized microstates.At the temperature absolute zero the particle system is in the ground state of lowest energy, at which it assumes just a single microstate.In the context of the thermodynamic theory of Willard Gibbs the result (5.1) can likewise be derived.

Relation between the Internal Potential and Thermal Energies of the Particle System
By definition, at a given configuration of a particle system the internal energy U of the system is the sum of the binding or potential energy pot U and the kinetic or thermal energy th U of all interacting particles, i.e.
For a certain particle configuration the sum pot th U U + thus expresses the law of internal energy conservation.In turn, with the internal energy value given the particle arrangement in the system is fixed and stable.From relations (6.1) it follows that and .
According to relations (2.1)-(2.3) the value of U depends on the mean inter-particle separation r : Relations (6.2) and (6.3) are represented, for example, in Figure 2 of ref. [8] for the two-phase water fluid.In this special case it holds that 0, where 0, 0 .
Relations (6.3) and (6.5) are plotted in the volume-vs.-pressurechart of water in Figure 3 of ref. [8].The statement reported there that the isotherm c T is a line 0 U = has to be corrected; instead the isochor c v is the line with 0 U = ; this is shown in the following section.

The Conditions
This relation can immediately be derived from Equation (4.2).For  ( ) . This, in turn, yields the Nernst relations (5.4) valid at 2) The arrows of µ in Figure 9.4 and Figure 9.5 of ref. [3] have to be put in the opposite direction to get the correct relation d d 0 p µ < , since for c T T < the chemical potential µ is a negative and concavely curved function of pressure p [6].
The thermodynamically established result 0 U = states that the statistical mean value of the internal energy of a particle system vanishes under the condition of equal energy contributions of binding and repulsion.The same result must be afforded by quantum mechanical calculation of stationary energy states of a particle system because thermodynamics and quantum mechanics are based on the same suppositions when treating a particle system, viz. the conservation of energy and momentum and application of statistical laws [3] [9].Cohesion of all system particles is accomplished by interatomic interaction of the particles, which are continually in motion.Quantum mechanical calculation of the stationary and stable energy states provides as result the revealing picture of discrete energy steps arranged above one another.The lowest energy step constitutes the ground state of the particle system and the energy levels above specify the particle system completely.A particle system is then in the ground state of lowest energy when its cohesion by binding energy is maximal and the kinetic energy components are minimal.A particle system may consist of a single atom or a multitude of particles.For example, a Bose-Einsteincondensate. From a statistical point of view, bosons are to be treated as undistinguishable particles; consequently, the sum of the lowest particle energy of N condensed bosons, viz.Nu boson < 0, yields the zero-momentum, ground state energy of the particle system BEC, which is negative and denoted by BEC U .
If thermal energy is supplied to a structured particle condensate which can absorb it in discrete energy quanta and step by step can then change the arrangement of its particles, with sufficient excitation energy the energy level is reached at which the binding and excitation energies have equal values and particle cohesion is no longer given.This energy level is assigned the internal energy zero.Above this level system particles move freely and have positive internal energy.
The energy zero of a particle system, 0 U = , is determined by the transition from the bound state particles to freely moving particles.In relation to the number N of a particle system, the negative energy value of the condensate is in a volume V .The thermal properties of the system are characterised by introducing an internal energy U and entropy S (Equation (4.2)), and the temperature T , pressure p and chemical potential µ are defined by partial derivatives of U and S .The fundamental equation ( ) solved by means of Equation (4.1) in the explicit form 0 From this follow the negative sign of the free energy, 0 Because the ratio v l u u can be given as an analytic function of the argument v l z v v = , its evaluation according to Equation (4.21) at the same z is equal for every gas and yields the values −1 at the critical point and 0 at absolute zero.It thus follows that the energy ratios ( )

>
exists as a uniform gas of freely-moving particles having positive values U or as more or less structured matter with negative values U .In the regions 0 U > above the critical point [ ] , and below the critical point it holds that ( ) is the mass-specific volume of saturated vapor.In the adjacent regions with negative internal energy values 0 U < the mean distances between particles are short enough to yield negative energy contributions to U due to inter-particle attraction that exceeds the thermal, positive energy contributions due to particle motion.The critical isochor c v is the line of equal positive and negative energy contributions and thus represents a line of vanishing internal energy 0 U = .At this level along the critical isochor the ever present microscopic fluctuations in energy and density become macroscopic fluctuations as the pressure decreases on approaching the critical point; these are to be observed in experiments on the critical opalescence.Crossing the isochor c v from 0 U > to 0 U < , the change in energy 0 U ∆ > happens without any discontinuity.The saturation line v v also separates the regions between 0 U > and 0 U < , but does not represent a line 0 U = .The internal-energy values of saturated vapor v U and condensate lU can be determined absolutely as functions of vapor pressure p and densities ( )

2 )
> , where v v is the mass-specific volume of vapor, the mass M exists as a gas homogeneously distributed in V .For values 1 r > , the averaged kinetic energies of all freely moving particles exceed their mutual bindings, which as potential attractive energy contributions are to be regarded as negative, so that Starting from the condition at the critical point [ ] mass M disintegrates into a vapor mass v M with particle volume ( ) The internal energy of all freely moving vapor particles is positive, i.e. ( ) 0 v U M > , and that of the more closely bound, not freely moving, condensed particles is negative, i.e. ( ) 0 l U M < .The internal energy of the gas mass M is likewise negative owing to the excess of more closely bound condensed particles.From phenomenon has since been known as critical opalescence.Macroscopic fluctuations of mass indicate instability of the gas as the consequence of the vanishing internal energy 0 U = on approaching the critical point.

10 − 6
[m] and 10 −4 [m] [2] is associated with the macroscopic fluctuations of the internal energy of the real gas at the critical point.The deviation of the fluctuating quantity is considered, U U −  ; its mean value vanishes owing to equally large positive and negative deviations from U , i.e. the amount of scatter of the value U is the mean value of the always positive quadratic deviation ( )the mean value of the square of the energy fluctuations occurring is[3]

( ) 2 U
 gives the standard deviation of U near the critical point.The fluctuations of energy are thermodynamically associated with fluctuations of mass.These spread out in the fluid on approaching the critical point, becoming stronger and slower because under the condition of decreasing pressure in the fluid there is an increasing volume for fluctuations disposable.In a Bose gas the phenomenon of critical opalescence is also observed.The shining of the gas cloud illuminated by laser light appears immediately before its condensation at the critical temperature.Cooling of a Bose gas from room temperature to values of the critical temperature of 10 −6 [K] within minutes occurs through interaction between laser light and gas atoms.By virtue of the fact that the ensuing aggregating high-energy particles are immediately removed from the available volume V there is no coexistent liquid phase.As soon as the low-energy gas particles remaining in V have on average the separation at which attractive and repulsive interatomic forces balance one another, the state of vanishing internal energy sets in.The enhanced macroscopic energy and particle density fluctuations show critical opalescence in the laser light.On the assumption that the fluctuations occur above the critical value BEC T condensation of the total gas mass at a temperature of just 10 −9 [K] before the transition temperature BEC T is reached.The values of M , V , U and S of both the gas and condensate are needed to describe the real Bose-Einstein gas.These values are thermodynamically defined by the experimental conditions and are given by Equation (7.3) in Section 7. In 1925 condensation was previously predicted by Einstein for Bose particles: in the lowest quantum mechanical state these assume vanishing entropy and take up equal, lowest possible energy.The first macroscopic quantum phenomenon, A. Elsner which was interpreted in 1938 by H. London as Bose-Einstein-condensate, is suprafluid helium.In 1995 Wieman, Cornell and Ketterle succeeded in the first experimental realisations of heavier alkali atoms.The laboratory conditions for a Bose-Einstein-condensate (BEC) call for temperatures in the range of 10 −6 [K] and particle densities of 10 20 [part/m 3 ].The particles are isolated from vessel walls in a magneto-optical trap (diameter 10 −2 [m]) and then cooled by means of laser light ( end of the 19th century and the question of absolute values of U and S had been posed, apparently few scientists were interested in tackling it.It was Nernst, Planck, Boltzmann and physicists in quantum mechanics who solved the problem of the thermodynamic reference point for S .The corresponding problem for U , on the other hand, remained unsolved[2] [5].Instead, it sufficed to present determinations of arbitrary zero points, e.g. with = at the triple point of a liquid without ever taking thermodynamic conditions into account.Consequently, inconsistencies cropped up; in particular, the free energy U ST −

2 )
For the saturated fluid, whose two phases vapor and condensate have the same values ( ) p p T = and ( ) T µ µ = everywhere in V , it is immediately possible to give the temperature dependence of the heat capacity:

.
theoretically simpler to treat.The saturated fluid with the two phases, vapor and condensate, then has the critical density ( ) As the fluid values [ ] , , T p µ in V are locally equal in each of the particular volumes, the two phases can only be distinguished by the densities or mass-specific volumes v one has the relations be enlisted as definition of the critical volume, one can give as upper and lower limits for v the temperature functions c T have the value c v and with decreasing temperature increase monotonically.This shows that the differential quotient of the chemical potential ( ) ( )

4 )
because the differential quotients d d p T and d dT µ vanish at absolute zero, The entropy relation analogous to Equation (4.25) is

U = 0 and S = 0
As the mass M in V can assume both positive and negative values of the internal energy, the question of the course of the line 0 U = in the state chart arises.In the case 0 U = the mass M has the entropy value yield not only the already known relation (4.14) valid at c T T = , but they are also valid along the critical isochor ( ) c V M , where we have the following relation between specific volume V M and the other intensive parameters, temperature, pressure and chemical potential: 0 U > .In the case of the saturated fluid it holds that 0 relations (4.16) and (4.18) are valid.If, on the other hand, the mass M in V has the entropy value 0 S = , then it has the internal-energy value ( ) ( )

0 U) 2 U
and the relation U ST < , which can be satisfied by the relations 0 U ST ≤ < in the state chart [ ( ) , p T v -vs.-v ] of a fluid, the ranges of existence of 0 U < and 0 U ≥ are separated from one another, on the one hand, by the critical isochor c v and, on the other hand, by the vapor volume line v v at the boundary of the two-phase region; both lines meet at the critical point.In the regions of ≥ with, respectively, [ ] as a gas, while in the adjacent regions of 0 U < the mass shows more or less distinct structures.Along the line c v the fluid has the mean energy value 0 U U = = and shows near the critical point large fluctuations (  , which become detectable as macroscopic density fluctuations.Crossing the line c v from 0 U > to 0 U < occurs at a continuous change in energy 0 U ∆ > to be removed and crossing the line v = − > be extracted from the fluid.The values of the internal energy of vapor v the two-phase region can be given in acccordance with Equations (4.6) and (4.19) as functions of the measured values of the vapor pressure ( ) p T and densities of the condensate value of the internal energy of the two-phase fluid is always negative,

1 r
be universally represented as functions of z for every gas.The entropy values of the two-phase fluid can likewise be calculated absolutely in acccordance with Equations (4.3), (4.23) and (5.5) from the heat capacity data ( ) , , C M V T .The following four observations are an indication of the thermodynamic zero of a real gas becoming evident: 1) above [ ] macroscopic density fluctuations appear within a very narrow temperature range (critical opalescence), 2) at [ ] , , c c c T p v the mass decomposes into a vapor component with density lower than before and a condensate component with higher density, 3) at [ ] any heat of evaporation vanishes, and 4) below [ ] , , c c c T p v an interface layer between vapor and condensate appears where on approaching c T T → the surface tension of the condensate decreases asymptotically to 0v = is thus experimentally proved.The fundamental Equation (4.1) is valid not only for a gas conforming to the Maxwell-Boltzmann statistics, but also for a gas subject to the Bose statistics.The statement that the internal energy of a condensate is negative therefore also applies to the internal energy of the Bose-Einstein condensate.The quantum mechanical, microscopic and thermodynamical, macroscopic formulations of the internal energy are then only valid if the two energy scales have the common absolute energy reference value 0. The energy values above 0 are positive and those below 0 negative.A system whose particles are homogeneously distributed in the volume available has in the ground the smallest interparticle distance min d , or equivalently, the number min < , the entropy and the lowest energy.By virtue of this most dense particle arrangement the contribution of the binding energy to the internal energy is maximal, and since binding energy is negative energy it constitutes the lowest energy of the system.In an excited state the corresponding discrete energy level is higher.As the energy level increases stepwise, the boundary to the continuum of positive energies is reached; this boundary is assigned the energy value 0, at which the energy contributions by attractive, binding inter-particle forces and repulsive forces are equal.The system is then in the critical state with the critical value = .The critical value 0 U = is the common boundary value 0 of the microscopic and macroscopic energy scales.In the case of a Bose gas, the energy values at the critical transition temperature BEC T are zero for the gas and negative for the Bose-Einstein condensate and above BEC T they are positive.A macroscopic system has the thermodynamically determined energy zero at the critical point [ ] the lowest temperature the lowest system energy is equal to the negative value of the heat of vaporization.