Absolute Reference Values of the Real Gas

With his publication in 1873 [1] J. W. Gibbs formulated the thermodynamic theory. It describes almost all macroscopically observed properties of matter and could also describe all phenomena if only the free energy U ST − were explicitly known numerically. The thermodynamic uniqueness of the free energy obviously depends on that of the internal energy U and the entropy S, which in both cases Gibbs had been unable to specify. This uncertainty, lasting more than 100 years, was not eliminated either by Nernst’s hypothesis 0 S = at 0 T = . This was not achieved till the advent of additional proof of the thermodynamic relation 0 U = at c T T = . It is noteworthy that from purely thermodynamic consideration of intensive and extensive quantities it is possible to derive both Gibbs’s formulations of entropy and internal energy and their now established absolute reference values. Further proofs of the vanishing value of the internal energy at the critical point emanate from the fact that in the case of the saturated fluid both the internal energy and its phase-specific components can be represented as functions of the evaporation energy. Combining the differential expressions in Gibbs’s equation for the internal energy, ( ) ( ) d d 1 T T μ and ( ) ( ) d d 1 p T T , to a new variable ( ) ( ) d d T p T μ leads to a volume equation with the lower limit vc as boundary condition. By means of a variable transformation one obtains a functional equation for the sum of two dimensionless variables, each of them being related to an identical form of local interaction forces between fluid particles, but the different particle densities in the vapor and liquid spaces produce different interaction effects. The same functional equation also appears in another context relating to the internal energy. The solution of this equation can be given in analytic form and has been published [2] [3]. Using the solutions emerging in different sets of problems, one can calculate absolutely the internal energy as a function of temperature-dependent, phase-specific volumes and vapor pressure.

energy obviously depends on that of the internal energy U and the entropy S, which in both cases Gibbs had been unable to specify.This uncertainty, lasting more than 100 years, was not eliminated either by Nernst's hypothesis 0 S = at 0 T = .This was not achieved till the advent of additional proof of the thermodynamic relation 0 U = at

Introduction
The purpose of this paper is to show that Gibbs's theory [1] contains intrinsically "natural" reference values of entropy and internal energy that are reciprocal and thus represent thermodynamic reference values [4].Nevertheless, it was and is still accepted till the recent past that the value of the internal energy of the real gas cannot be given absolutely, e.g.[5] [6].Instead, the calculation of entropy and internal energy could be based on a so-called fiducial reference value [7].To put an end to the diversity of individually, arbitrarily chosen fiducial values, it was decided at conferences in the 1950s to assign the values of entropy and internal energy (or enthalpy) of a liquid at the triple point the reference values zero, which in any case the Nernst hypothesis contradicts.On this basis mathematically complex equations were put forward, but an analysis (see Appendix) shows that they contain thermodynamic inconsistencies.These then yielded incorrect thermodynamic data published for a large number of gases, e.g.[8]- [15].None of these skeleton tables presents data on the chemical potential, which could have been derived from the differences , , v l v l h s T − .A paper that specifically investigates calculation of the chemical potential of the generally accepted conference agreements comes to the surprising conclusion that the chemical potential increases as the temperature [16].This result cannot, however, account for daily observation that flow of freely-moving matter occurs from cold to warm regions [17].One has to take the consequence from the numerous thermodynamic discrepancies ensuing from calculating entropy and internal energy from the assumptions mentioned, ( ) Here, too, as so often in the history of physics, one has to abandon a trusted hypothesis.Here it is the assumption of the naive addition of fiducial constants to thermodynamic fundamental quantities.
The paper shows four possible ways of finding directly the thermodynamic reference values of entropy and internal energy and thus comply with the correct treatment of real properties of matter called for by Gibbs.First there is the possibility of studying the characteristic features of intensive and extensive quantities in order to describe thermodynamically the real properties of matter.A second investigation deals with the question what follows from representation of the internal energy of the fluid as a function of the evaporation energy.The third possibility is concerned with solution of a functional equation for the sum of two dimensionless variables, the one referring to the local interaction potential of fluid particles in the vapor space and the other to that in the liquid.
The functional equation emerges from the original equation for the internal energy and has been solved [3].The functional equation is, on the other hand, also encountered when one represents the particular phase-specific internal energy as a funtion of the evaporation or condensation energy.It has of course the same (physically unique) solution and allows the internal energy to be explicitly calculated as an absolutely determined temperature function of the measurable quantities: phase-specific volumes and vapor pressure [2].

Intensive and Extensive Quantities of the Saturated Fluid
The homogeneity of the fluid allows its macroscopic properties to be described by intensive and extensive quantities.
The intensive quantities are the temperature T, vapor pressure p and chemical potential μ; vapor pressure and chemical potential are pure temperature functions below the critical point of the fluid.With the finite critical values p c and For 0 T ≥ the vapor pressure is a positive and convexly curved temperature function increasing with T from 0 to p c .Also the temperature derivatives increase with T from ( ) , and, on the other hand, the divergent terms d d Thermodynamics treats the quantity X as an extensive quantity, which means that X is proportional to the fluid mass M. temperatures below the critical point the fluid mass M in the volume V is additively composed of the vapor mass M v and the condensed mass M l in the sub-volumes V v and V l : It is worth mentioning that the temperature variation of the ratio of the differences ( ) of quantities such as the volume, entropy, internal energy, enthalpy, free energy, and specific heat is the same and equal to that of the ratio of vapor to condensed masses, viz.
where the equality signs are valid for 0 T = and c T T = , respectively.From Equation (1.4) one arrives at the interdependence of two extensive quantities x and y and their phase-specific values , v l x and , v l y in the following form: This relationship can also be deduced from the correlations The decomposition of mass M into v M and l M below the critical point occurs within limits and is given by While the mass-specific quantity x constitutes an average of the quantity X in V in relation to the total mass M and is thus a function of T and V/M, the massand phase-specific quantity x v describes the quantity X v in the volume V v and is related to the vapor mass M v , and the quantity x l describes X l in V l and is related to M l .The quantities , v l x then give thermodynamic information on the masses V , which are subject to equal values of temperature T, vapor pressure ( ) , and different density values Since the densities of the vapor and condensate in V v and V l are functions of the temperature alone, the quantities , T v .The information from x v and x l implicitly contains all particle interactions that can be expressed in terms of various imaginable types of descriptions of thermodynamic properties. A The critical value x c of the quantity { } , , , x v s u f = is finite.In fact, when x v approaches the finite value x c from below, then x l approaches x c from above and vice versa; in any case, one has x T x T at the critical point is then different for the two possible cases of equal or opposite sign of ( ) x T and ( ) The relations of the first line of (1.7) are valid for the quantities The consequence from relations (1.7) for the quantity x u = shall be investigated.It can be stated that the vapor energy v U at low temperatures is positive since vapor particles are so far apart that their (negative) interaction potentials are vanishingly small in comparison with their (positive) thermal energies.At low temperatures one thus has 0 From ( ) This gives an estimate of the mass-specific energy u l in the form of ( ) − ≤ , i.e. the vapor energy u v is not negative and the condensate energy u l is not positive.The second line of relations (1.7) then states 0 c u = .This yields the important results, From Equations (1.3) one obtains the thermodynamic relation of the mean fluid quantity x (which is a function of T and v) to the phase-specific quantities x (which are pure temperature functions) as follows: Differentiation of ( ) Thus the fluid quantity ( ) , .
If the saturated fluid does not have the critical volume v c , but the volume v, the following conversion has to be made: T T = the fluid takes the critical volume v c .From Equation (1.9) it is immediately obvious that the critical values ( ) x T , ( ) x T , and the zero-point values ( ) and from Equation (1.10) it follows that In the theory it is not only the difference of the phase-specific quantities x v and x l , i.e.

(
) ( ) ( ) that is of importance, but also their sum The sign of the function ( ) x x + will subsequently be of interest.It is the same as that of the function ( ) has a positive sign, while the signs of ( ) The critical mean fluid value ( ) x T v is thus equal to the mean of the phase-specific critical values ( ) ( )  and, if the fluid value ( ) 0, x v at absolute zero is given by the condensation energy value ( )( ) is the Planck constant, k the Boltzmann constant, m the particle mass and n the particle density in the condensate.The atomic densities achieved in experiments range from 10 -14 to 10 -15 cm -3 and transition temperatures from 100 nK to a few μK [18].The internal energy of the dilute gas is positive for tr T T > and vanishes at tr T T = , whereas that of the condensate is negative [19].

Interdependence of Extensive and Intensive Quantities
Since the two-phase equilibrium can be described by extensive as well by intensive quantities, an interdependence between these quantities exists.
Thermodynamics yields for the quotients ( ) ( ) Correspondingly, for the quotients ( ) ( ) Equations (1.18) and (1.19) allow one to define volume functions − , which can be represented in different ways: In evaluating the critical value of a volume function (1.20) one should give heed, in respect of Equations (1.7) in the case x T x T = , to whether a finite limiting value exists; in the case x T x T = − the limiting value is v c .Hence the result is: From Equations (1.22) it immediately follows that the critical value is finite for { } , , x f s u = and divergent for x c = .

Entropy and Internal Energy Relations
According to relation (1.5), the interdependence of volume and entropy is ( ) ( )

Thus one gets
Gibbs's entropy relations, including the thermodynamic reference value 0: ( ) The value ( ) and is positive for 0 T > and vanishes for 0 T = .The same is valid for the phase-specific entropy ( ) l s T ; and since ( ) ( ) ( ) The interdependence of volume and internal energy is gives Gibbs's internal energy relations and with respect to relation (1.21) the thermodynamic reference value 0: , where it holds that ( ) ( ) ( ) ( ) ≤ and therefore ( ) , u v T is not positive.Furthermore, the phase-specific internal energies obey the relations Hence one gets ( ) ( ) The relations ( ) ( ) In other words: For 0 c T T < < the vapor internal energy u v is positive and always lower than half the evaporation energy, ( ) − , and the liquid internal energy u l is negative and lower than half the condensation energy, ( ) The critical values of the entropy are obtained from the relations , the functions ( ) , u v T and ( ) , s v T increase monotonically with increasing T, and so the critical values present the maximum internal energy and entropy of the saturated fluid.
The interdependence of entropy and internal energy is calculated from the equation ( ) ( ) and leads to the following identities and estimates: The estimates follow from The two obviously equivalent Equations (1.30) present the opportunity for proving the correctness of the reference data mentioned in Equations (1.8), (1.13), (1.17), (1.21) and (1.27).For example, it follows from ( ) The data mentioned are thus reciprocal to one another.These data are thermodynamic reference values.

Heat Capacity Relations
The measurable heat capacity is defined by As the values of ( ) diverges at the critical point, the fluid heat capacity and the specific heat capacities also diverge there: Further expressions for the heat capacity and temperature derivatives of internal energies can be given as follows: ( ) ( ) In order to show that ( ) one has to prove that ( ) or that the straight line ( )( ) + .The condition of convexity for ( ) , which means that ( )( ) ( )( ) and thus confirms that the condition for concavity of ( ) It is of interest to take 0 T = in entropy and heat capacity relations.This is immediately possible in the case of the entropy because Nernst's theorem states that the entropy vanishes at absolute zero, where only the condensed phase exists, and increases with the temperature: Similarly, in the case of the heat capacity one gets for 0 The temperature derivatives of the chemical potential function ( ) and the phase-specific entropies , v l s can be determined from measurements of ( ) With the result ( ) These state that the entropy values are always positive and greater than the product of the volumes and vapor pressure coefficient.From relations (1.43), in turn, one can derive the following relations: Temperature properties of the phase-specific heats are similarly derived.With the result ( ) from which in turn the relations ⋅ also proves confirmation of ( )

Chemical Potential Relations
The identities , , , are now used to put the chemical potential functions in explicit form as energy functions: , , The relations state that μ is a negative, concavely curved function, decreasing with increasing T. For μ as a function of measurable quantities see Equations (3.15) and (3.16) below.

Internal Fluid Energy as a Function of the Condensation Energy
It can be shown that the fluid energy ( ) , u T v can be expressed in terms of the condensation energy ( ) Since the temperature coefficients ( ) ( ) It is symmetric in the variables and linear in both v v and v l , and at the critical point it yields v c .Then inserting the solution (3.2) in the internal energy Equations (3.1) yields the results ( ) These equations state that the internal energies can be expressed in terms of the measurable quantities, phase-specific volumes and vapor pressure, and are given by absolute figures; in particular, it holds that ( ) ( ) If the functions u v and u l are expressed as dependent on the volume ratio ( ) ( ) for 0 and 1, ln 1 2 ( ) ( ) According to Equations (1.17 Rigorous thermodynamic calculations combine Equations (1.8) and (1.44) and yield the following relations for the internal energy and entropy: 1 0, 1 for .
The constraints (3.7) state that the ratios of absolute energy and entropy for vapor and liquid are restricted within certain limits for temperatures in the two-phase region.And combining Equations (1.20), ( . ln In turn, from Equations (3.9) one obtains absolutely.An entropy data criterion of the same kind as for the internal energy can also be formulated.
This relation allows, in principle, to give the chemical potential in terms of p, , v l v and , v l s as follows: The entropy expressions read According to relations (1.39) one has . The chemical potential can thus be determined, on the one hand, by measuring the two-phase heat capacity, phase-specific volumes and vapor pressure: or, on the other, by measuring the phase-specific volumes and vapor pressure only: ( ) ( ) ( ) ( ) The energy sum ( ) is also measurable and calculable and it holds that ( ) ( ) ( )

Phase-Specific Energy as a Function of the Evaporation Energy
The Carnot-Clapeyron-Clausius equation this it is concluded that the published data [13] [14] are in need of basic correction.
is noteworthy that from purely thermodynamic consideration of intensive and extensive quantities it is possible to derive both Gibbs's formulations of entropy and internal energy and their now established absolute reference values.Further proofs of the vanishing value of the internal energy at the critical point emanate from the fact that in the case of the saturated fluid both the internal energy and its phase-specific components can be represented as functions of the evaporation energy.Combining the differential expressions in Gibbs's equation for the internal energy, ( ) ( ) leads to a volume equation with the lower limit v c as boundary condition.By means of a variable transformation one obtains a functional equation for the sum of two dimensionless variables, each of them being related to an identical form of local interaction forces between fluid particles, but the different particle densities in the vapor and liquid spaces produce different interaction effects.The same functional equation also appears in another context relating to the internal energy.The solution of this equation can be given in analytic form and has been published[2] [3].Using the solutions emerging in different sets of problems, one can calculate absolutely the internal energy as a function of temperature-dependent, phase-specific volumes and vapor pressure.
n =  ).In contrast, the chemical potential is a negative and concavely curved temperature function decreasing with T. With the finite critical values c µ and d d c T µ as v v and v l are equal to the critical value v c at the critical point () , the values x v and x l are equal to the critical value x c at ( ), c c

1 )
According to Equations (1.20)-(1.22) the diffential quotient ( ) ( ) d d T p T µ is a positive volume quantity monotonically decreasing with increasing T from high values ( ) v v T near absolute zero to the lowest value v c at the critical point.The expression of measurable volumes (1.28) the following relations:

4 )
them is, of course, equal to 1. Equations (3.4) allow one to calculate the relation between the energy ratio the vapor and liquid spaces, viz.

where 1 z
) and(1.8), the energy ratio η assumes the value 0 at absolute zero (where z → ∞ ) and, respectively, the value -1 at the critical point (= ).The relation ( )z ηrepresents a universal law of the two-phase equilibrium of real gases.Calculation of the energy ratio (

8 )
Algebraic rearrangement of Equation (3.8) leads to a data criterion for consistent thermodynamic values

17
) then lead to further conditions for α and ϕ: must be a negative function increasing as T, and ϕ must be a positive function likewise increasing as T. It is found that d Equations (A1), despite their correct formal structure, are therefore not thermodynamically appropriate for justifying the chemical potential according to Equations (A3).Further consideration of the temperature dependence of , , but not a convexly curved function, as should be.The fit function for measured vapor pressure data reads[13] [14] whereas all vapor pressure derivatives have finite values.In contrast, the derivatives of the chemical potential , it should be noted here that, if the saturation state is maintained, it is not possible to distinguish between a constant pressure and a constant volume condition.There is thus only one phase-specific heat capacity in the vapor space, ( ), v vc V M T , and one in the liquid space, ( ), l l c V M T , and it holds that T M c V M T M c V M T C M V T C M V T cT and, finally, tables which list two different phase-specific heat capacities for vapor as well for liquid.From all The validity of the relation X xM = The same applies to other extensive quantities such as the entropy S, internal energy U, enthalpy H, free energy F, and heat capacity C. Denoting such quantities by X and For example, the physics of the real gas operates in the temperature range [ ] At this place the quantum state of the Bose-Einstein condensation should be noticed.In contrast to the thermodynamic temperature absolute zero the lowest temperature available is the transition temperature the vapor value ( ) 0 v x vanishes, i.e. it holds, for example, that ( ) DOI: 10.4236/eng.2018.105019276 Engineering