The purpose of this paper is to show that Gibbs’s theory  contains intrinsically “natural” reference values of entropy and internal energy that are reciprocal and thus represent thermodynamic reference values  . 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.   . Instead, the calculation of entropy and internal energy could be based on a so-called fiducial reference value  . 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.  -  . None of these skeleton tables presents data on the chemical potential, which could have been derived from the differences . 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  . This result cannot, however, account for daily observation that flow of freely-moving matter occurs from cold to warm regions  . One has to take the consequence from the numerous thermodynamic discrepancies ensuing from calculating entropy and internal energy from the assumptions mentioned, and (or ). 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  . 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 pc and ( ) the ò-expansion of p to second order yields
For the vapor pressure is a positive and convexly curved temperature function increasing with T from 0 to pc. Also the temperature derivatives increase with T from to finite values (where ). In contrast, the chemical potential is a negative and concavely curved temperature function decreasing with T. With the finite critical values and , and, on the other hand, the divergent terms ( ) an ò-expansion of μ is not possible:
Thermodynamics treats the quantity X as an extensive quantity, which means that X is proportional to the fluid mass M. The validity of the relation leads to x having the property of additivity and ensures its uniqueness. At temperatures below the critical point the fluid mass M in the volume V is additively composed of the vapor mass Mv and the condensed mass Ml in the sub-volumes Vv and Vl: in . 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 and the corresponding mass-specific quantities by and , one obtains the following definitions:
It is worth mentioning that the temperature variation of the ratio of the differences and 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 and , respectively. From Equation (1.4) one arrives at the interdependence of two extensive quantities x and y and their phase-specific values and in the following form:
This relationship can also be deduced from the correlations and with .
The decomposition of mass M into and 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 mass- and phase-specific quantity xv describes the quantity Xv in the volume Vv and is related to the vapor mass Mv, and the quantity xl describes Xl in Vl and is related to Ml. The quantities then give thermodynamic information on the masses in , which are subject to equal values of temperature T, vapor pressure and chemical potential in V, and different density values in Vv and in Vl. Since the densities of the vapor and condensate in Vv and Vl are functions of the temperature alone, the quantities in are likewise pure temperature functions. As vv is different to vl for , the value xv is different to xl for , and as vv and vl are equal to the critical value vc at the critical point , the values xv and xl are equal to the critical value xc at . The information from xv and xl implicitly contains all particle interactions that can be expressed in terms of various imaginable types of descriptions of thermodynamic properties.
If a thermodynamic quantity is represented in its domain of definition by a thermodynamic function , a thermodynamic quantity is always an absolute quantity. For example, the physics of the real gas operates in the temperature range in the limits and .
The critical value xc of the quantity is finite. In fact, when xv approaches the finite value xc from below, then xl approaches xc from above and vice versa; in any case, one has . The approach is determined by . The ratio value of at the critical point is then different for the two possible cases of equal or opposite sign of and . One has
The relations of the first line of (1.7) are valid for the quantities .
The consequence from relations (1.7) for the quantity shall be investigated. It can be stated that the vapor energy 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 . Under these conditions the evaporation energy is very much larger than . From one obtains or, with , . This gives an estimate of the mass-specific energy ul in the form of , i.e. the vapor energy uv is not negative and the condensate energy ul is not positive. The second line of relations (1.7) then states . 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 (which are pure temperature functions) as follows:
Differentiation of with respect to v at fixed T yields the coefficient of isothermal phase transition,
Thus the fluid quantity can be expressed in terms of , , , and :
If the saturated fluid does not have the critical volume vc, but the volume v, the following conversion has to be made:
At the fluid takes the critical volume vc. From Equation (1.9) it is immediately obvious that the critical values , , , and the zero-point values and are respectively equal,
and from Equation (1.10) it follows that
In the theory it is not only the difference of the phase-specific quantities xv and xl, i.e.
that is of importance, but also their sum . Since , one has
The sign of the function will subsequently be of interest. It is the same as that of the function if the product function has a positive sign, while the signs of and are opposite if is negative. The latter can be the case if either the product function or the difference is negative (see Equations (1.25) and (1.26) below). At the critical point one has and at absolute zero and ; for the vapor-phase quantity one then obtains
The critical mean fluid value is thus equal to the mean of the
phase-specific critical values and, if the fluid value
at absolute zero is given by the condensation energy value , the vapor value vanishes, i.e. it holds, for example, that .
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 , where 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  . The internal energy of the dilute gas is positive for and vanishes at , whereas that of the condensate is negative  .
2.1. 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 of the particular quantities the well-known vapor pressure relations,
Correspondingly, for the quotients the chemical potential relations are
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 , to whether a finite limiting value exists; in the case the limiting value is vc. Hence the result is:
With one obtains
From Equations (1.22) it immediately follows that the critical value is finite for and divergent for .
2.2. Entropy and Internal Energy Relations
According to relation (1.5), the interdependence of volume and entropy is . Taking relations (1.18) and (1.19) into account, viz. and , this can be transformed to . Thus one gets Gibbs’s entropy relations, including the thermodynamic reference value 0:
The value is the sum of the terms and and is positive for and vanishes for . The same is valid for the phase-specific entropy ; and since one obtains the following sequences
The interdependence of volume and internal energy is , which with and gives Gibbs’s internal energy relations and with respect to relation (1.21) the thermodynamic reference value 0:
For the value is the sum of the negative term and the positive term , where it holds that and therefore is not positive. Furthermore, the phase-specific internal energies obey the relations and ; the last relation
since . Hence one gets
The relations and lead to the limits of the energies in relation to the transient energies :
In other words: For the vapor internal energy uv is positive and
always lower than half the evaporation energy, , and the liquid internal energy ul is negative and lower than half the condensation energy, . The critical values of the entropy are obtained from
the relations , yielding
Because , the functions and 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 (where the equality sign is valid for ) and (where the equality sign is valid for ).
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 that with and from that . The data mentioned are thus reciprocal to one another. These data are thermodynamic reference values.
2.3. Heat Capacity Relations
The measurable heat capacity is defined by
Calculation of the specific heat capacities requires the temperature derivatives of the quantities given in Equations (1.23) and (1.25). Taking into consideration Equations (3.1)-(3.3) below, the heat capacity relations are
As the values of , , and vanish at absolute zero, those of and also vanish there according to Equations (1.18) and (1.19). And as the value 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:
It holds that and for that
In order to show that is a convexly curved temperature function, one has to prove that or that the straight line is above . The condition of convexity for then reads , which can be transformed to
Indeed, for the terms in Equation (1.37) are positive since and , which meets the condition mentioned. Likewise, the function is concavely curved when its values are above the straight line , which means that . Division by gives the correct relations and thus confirms that the condition for concavity of is met.
From and it follows that
It is of interest to take 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:
For one thus gets
Similarly, in the case of the heat capacity one gets for
The temperature derivatives of the chemical potential function and the phase-specific entropies can be determined from measurements of , and since
With the result one obtains from Equations (1.23) the relations
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 one obtains from Equations (1.32) the relations
from which in turn the relations
can be derived. Experimental verification of also proves confirmation of .
2.4. Chemical Potential Relations
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.
3. Internal Fluid Energy as a Function of the Condensation Energy
It can be shown that the fluid energy can be expressed in terms of the condensation energy :
Since the temperature coefficients and are represented by and , respectively, the mass distributions and at give the three relations , , , and the distributions at give the three relations , , . One thus obtains the following characteristic thermodynamic reference values, which are valid for every gas:
4. Internal Fluid Energy as an Expression of Measurable Quantities
Endeavors to publish data of the functions and are prominent in the current literature. The energy equations can be written in the form
According to Equations (1.20)-(1.22) the diffential quotient is a positive volume quantity monotonically decreasing with increasing T from high values near absolute zero to the lowest value vc at the critical point. The expression of measurable volumes and instead of as published in (2015)  reads:
It is symmetric in the variables and linear in both vv and vl, and at the critical point it yields vc. 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 since and at .
If the functions uv and ul are expressed as dependent on the volume ratio , one obtains for the energy ratios and according to Equations (1.28) the following relations:
The sum of them is, of course, equal to 1. Equations (3.4) allow one to calculate the relation between the energy ratio and volume ratio z of fluid particles in the vapor and liquid spaces, viz.
According to Equations (1.17) and (1.8), the energy ratio η assumes the value 0 at absolute zero (where ) and, respectively, the value -1 at the critical point (where ). The relation represents a universal law of the two-phase equilibrium of real gases.
Calculation of the energy ratio as a function of z starts from Equation (1.20) or from and ends in any case with the result
Rigorous thermodynamic calculations combine Equations (1.8) and (1.44) and yield the following relations for the internal energy and entropy:
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), (1.22) and (3.2) gives
Algebraic rearrangement of Equation (3.8) leads to a data criterion for consistent thermodynamic values , , and , which is valid for and reads:
In turn, from Equations (3.9) one obtains
Equations (3.9) and (3.10) clearly state that, if table data T, p, vv, vl, , or and are thermodynamically consistent, then the internal energies, uv, ul, , can be given absolutely. An entropy data criterion of the same kind as for the internal energy can also be formulated. To this end one has to confirm the validity of the relation.
to obtain with the criterion desired:
This relation allows, in principle, to give the chemical potential in terms of p, and as follows:
The entropy expressions read
It holds that and
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
From and it is concluded that the energy sum is a convex temperature function strongly increasing from at absolute zero to at the critical point.
5. Phase-Specific Energy as a Function of the Evaporation Energy
The Carnot-Clapeyron-Clausius equation
suggests a unique relation between the energy density and particle density in the coexisting bulk phases, vapor and liquid. As the local interaction potentials in the partial volumens Vv and Vl are unique functions of the local particle density and determine the values uv and ul, respectively, it should be possible by means of a density coefficient and a temperature function to find suitable ansatzes for the functions depending on the density coefficient and temperature function. As density coefficient, a function ρ of the density variable is now chosen, and as temperature function the evaporation energy . The ansatz proposed for the energy in the vapor phase is then . The energy in the condensed phase must be since the transfer of a particle from Vv to Vl causes the value uv to change to ul, the density variable from 1/z to z and the phase transition energy from to , while the form of ρ is preserved because the functional density dependence of the microscopic interaction forces is of course phase-invariant. The conjectures 
then yield, on the one hand, the reference values
and, on the other, for the density coefficient the equation
and because and the boundary conditions
The functional Equation (4.3) for ρ under condition (4.4) is satisfied by
From Equation (4.2) it follows that
The relations (4.6) and (3.4) are identical and valid for and .
6. Results and Discussion
The paper treats thermodynamic properties of the saturated fluid. It is shown that the fluid state is completely determined by the internal energy and entropy in the vapor and liquid spaces. The ratios of the absolute phase-specific internal energies and entropies are restricted within certain limits. If for temperatures measured saturation data of the vapor pressure, reciprocal phase-specific densities and isothermal transient energy obey the data criterion Equation (3.9), then the internal energy as a function of T calculated according to Equations (3.10) is an absolute thermodynamic quantity. This fundamental procedure in gaining thermodynamic data excludes any application of so-called fiducial reference data since they cannot yield correct values. As the state of saturation is maintained, it is not possible to distinguish between a constant pressure and a constant volume condition. This is the characteristic difference between the heat capacities of a two-phase and a one-phase fluid. There is only one phase-specific heat capacity in the vapor space, , and one in the liquid space, . The measurable heat capacity of the fluid, , can also be calculated. Further results are the concavity of the positive, measurable temperature function and the convexity of the negative temperature functions . The chemical potential is a negative, measurable and calculable temperature function. The ratios of phase-specific energies to isothermal transient
energies obey the relations , which should
be heeded when a state chart of the fluid under consideration is constructed.
The author is grateful to A. M. Nicol for the English translation.
Appendix: Comments on the Internationally Accepted Equations for the Saturation Properties of Water
The International Association for the Properties of Water and Steam (IAPWS)   provides internationally accepted formulations for the properties of water. There are special correlation equations for the vapor-liquid saturation properties of water. Formulas are given for the vapor pressure p, phase-specific volumes , internal energies , and entropies as functions of the saturation temperature T. This affords a unique description of the temperature dependence of every property of the saturated water.
This study treats the IAPWS equations in the framework of thermodynamics. The IAPWS equations read, on the one hand   ,
where the so-called auxiliary quantities α and ϕ are given as functions of the temperature T; and, on the other, in terms of the thermodynamic fundamental equation,
where μ is the chemical potential of the saturated fluid. Equations (A1) and (A2) are related as follows:
Let us now investigate the thermodynamic conditions that have to be satisfied by the temperature functions α and ϕ if they are to define the two-phase chemical potential μ according to Equations (A3). First a few thermodynamic relations are taken and then transformed into IAPWS parlance. The Carnot-Clapeyron-Clausius equations read
These are satisfied by Equations (A1). The Gibbs-Duhem equations read
They are satisfied by Equations (A1) if the following relation between α and ϕ is valid:
Note that Equation (A6) is implicitly contained in relations (A3) since and . Relations (A4) and (A5) therefore afford nothing new, but merely confirm relations (A3). The mathematical structures of Equations (A1) thus conform to the thermodynamic internal energy and entropy expressions in respect of Equations (A3).
Condition (A6) is numerically satisfied by the formulas in   for and , whose temperature dependences are expressed as follows:
The vapor pressure p and chemical potential μ obey for temperatures the following relations:
Relations (A10) then lead to further conditions for α and ϕ:
Accordingly, α must be a negative function increasing as T, and ϕ must be a positive function likewise increasing as T. It is found that , but not that for all T in the range ; similarly, it does hold that , but not that for all . The auxiliary 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 shows that the sum yields a concavely, but not a convexly curved function, as should be.
The fit function for measured vapor pressure data reads  
and . The terms , , produce divergent temperature derivatives at for , whereas all vapor pressure derivatives have finite values. In contrast, the derivatives of the chemical potential diverge at for and determine the divergence of the heat capacity . In addition, 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, , and one in the liquid space, , and it holds that .
Numerous thermodynamic deficiencies have been mentioned, viz. violation of the conditions and for every temperature , the incorrect temperature dependence of , the proposed vapor pressure fit formula with a divergent term at and, finally, tables which list two different phase-specific heat capacities for vapor as well for liquid. From all this it is concluded that the published data   are in need of basic correction.
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