T ) are prominent in the current literature. The energy equations can be written in the form

u ( v c , T ) = [ d ( μ / T ) d ( p / T ) v c ] d ( p / T ) d ( 1 / T ) 0 , u v , l ( T ) = [ d ( μ / T ) d ( p / T ) v v , l ] d ( p / T ) d ( 1 / T ) . (3.1)

According to Equations (1.20)-(1.22) the diffential quotient d ( μ / T ) / d ( p / T ) is a positive volume quantity monotonically decreasing with increasing T from high values v v ( T ) near absolute zero to the lowest value vc at the critical point. The expression of measurable volumes v v ( T ) and v l ( T ) instead of d ( μ / T ) / d ( p / T ) as published in (2015) [3] reads:

d ( μ / T ) d ( p / T ) = v v + v l v v v l l n ( v v / v l ) . (3.2)

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

u ( T , v c ) = [ v v v l ln ( v v / v l ) v v v v + v c ] [ d ( p / T ) d ( 1 / T ) ] 0 , u v , l ( T ) = [ v v v l ln ( v v / v l ) v l , v ] [ d ( p / T ) d ( 1 / T ) ] . (3.3)

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 u v ( T c ) = u l ( T c ) = 0 since ( v v v l ) / ln ( v v / v l ) = v c and v v = v l = v c at T = T c .

If the functions uv and ul are expressed as dependent on the volume ratio z = v v / v l , one obtains for the energy ratios u v / ( u v u l ) and u l / ( u v u l ) according to Equations (1.28) the following relations:

0 u v u v u l = 1 ln z 1 z 1 1 2 for 0 T T c and z = v v ( T ) v l ( T ) 1 , 1 2 u l u v u l = 1 ln z + z z 1 1. (3.4)

The sum of them is, of course, equal to 1. Equations (3.4) allow one to calculate the relation between the energy ratio η = u v / u l and volume ratio z of fluid particles in the vapor and liquid spaces, viz.

0 η = u v u l = z 1 ln z z 1 z ln z 1 for 0 T T c and z = v v ( T ) v l ( T ) 1. (3.5)

According to Equations (1.17) 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 (where z = 1 ). The relation η ( z ) represents a universal law of the two-phase equilibrium of real gases.

Calculation of the energy ratio ( u v + u l ) / ( u v u l ) as a function of z starts from Equation (1.20) or from ( η + 1 ) / ( η 1 ) and ends in any case with the result

1 u v + u l u v u l = 2 ln z z + 1 z 1 0 for 0 T T c and z = v v v l 1. (3.6)

Rigorous thermodynamic calculations combine Equations (1.8) and (1.44) and yield the following relations for the internal energy and entropy:

1 u v / u l < 0 , 1 s v / s l v v / v l for T T c . (3.7)

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

v v + v l u v v v u l v l u v u l = u v v l u l v v u v u l = v v + v l v v v l ln ( v v / v l ) , u v v v u l v l u v u l = v v v l ln ( v v / v l ) . (3.8)

Algebraic rearrangement of Equation (3.8) leads to a data criterion for consistent thermodynamic values v v ( T ) , v l ( T ) , u v ( T ) and u l ( T ) , which is valid for T T c and reads:

[ u v u v u l + v l v v v l ] ln ( v v v l ) = [ u l u v u l + v v v v v l ] ln ( v v v l ) = 1. (3.9)

In turn, from Equations (3.9) one obtains

u v = u v u l ln ( v v / v l ) v l u v u l v v v l 0 , u l = u v u l ln ( v v / v l ) v v u v u l v v v l 0 , ( u v u l ) = ( h v h l ) ( v v v l ) p = ( s v s l ) T ( v v v l ) p = ( v v v l ) d ( p / T ) d ( 1 / T ) 0. (3.10)

Equations (3.9) and (3.10) clearly state that, if table data T, p, vv, vl, ( h v h l ) , or ( s v s l ) and ( u v u l ) are thermodynamically consistent, then the internal energies, uv, ul, u = u v ( v c v l ) / ( v v v l ) + u l ( v v v c ) / ( v v v l ) , 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.

s v , l + μ / T [ v v + v l ( v v v l ) / ln ( v v / v l ) ] p / T s v s l = u v , l u v u l (3.11)

to obtain with u v , l / ( u v u l ) = 1 / ln ( v v / v l ) v l , v / ( v v v l ) the criterion desired:

[ s v , l s v s l + v l , v v v v l + μ ( s v s l ) T [ v v + v l v v v l 1 ln ( v v / v l ) ] p / T d p / d T ] ln ( v v v l ) = 1. (3.12)

This relation allows, in principle, to give the chemical potential in terms of p, v v , l and s v , l as follows:

μ = s v , l T v l , v d p d T T + v v v l ln ( v v / v l ) d p d T T + [ v v + v l v v v l ln ( v v / v l ) ] p . (3.13)

The entropy expressions read

0 s ( v c , T ) = d μ d T + v c d p d T = s l + ( v c v l ) d p d T = s v ( v v v c ) d p d T . (3.14)

It holds that s v , l T v l , v d p d T T = s T ( v v + v l v c ) d p d T T and

μ = s T [ v v + v l v v v l ln ( v v / v l ) ] d p d T T + v c d p d T T + [ v v + v l v v v l ln ( v v / v l ) ] p .

According to relations (1.39) one has s T = T 0 T d s d T d T = T 0 T c T d T . The

chemical potential can thus be determined, on the one hand, by measuring the two-phase heat capacity, phase-specific volumes and vapor pressure:

μ ( T ) = T 0 T c ( T , v c ) T d T + v c d p d T T [ v v + v l v v v l l n ( v v / v l ) ] [ d ( p / T ) d ( 1 / T ) ] , (3.15)

or, on the other, by measuring the phase-specific volumes and vapor pressure only:

μ ( T ) = T [ μ ( T c ) T c p / T p ( T c ) / T c [ v v + v l v v v l l n ( v v / v l ) ] d p T ] < 0 , μ c T c = s c + v c p c T c . (3.16)

The energy sum ( μ + s T ) is also measurable and calculable and it holds that

μ + s T = [ v v v l l n ( v v / v l ) v v v l ] [ d ( p / T ) d ( 1 / T ) ] + v c d p d T T = u + v c p . (3.17)

From d ( μ + s T ) / d T = c + v c d p / d T > 0 and d 2 ( μ + s T ) / d T 2 = d c / d T + v c d 2 p / d T 2 > 0 it is concluded that the energy sum μ + s T = u + v c p is a convex temperature function strongly increasing from μ ( 0 ) = ( u v u l ) ( 0 ) < 0 at absolute zero to μ c + s c T c = v c p c > 0 at the critical point.

5. Phase-Specific Energy as a Function of the Evaporation Energy

The Carnot-Clapeyron-Clausius equation

u v u l = ( v v v l ) [ d ( p / T ) / d ( 1 / T ) ] (4.1)

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 u v , l depending on the density coefficient and temperature function. As density coefficient, a function ρ of the density variable z = v v / v l is now chosen, and as temperature function the evaporation energy ( u v u l ) . The ansatz proposed for the energy in the vapor phase is then u v = ρ ( 1 / z ) ( u v u l ) . The energy in the condensed phase must be u l = ρ ( z ) ( u v u l ) 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 ( u v u l ) to ( u l u v ) = ( u v u l ) , while the form of ρ is preserved because the functional density dependence of the microscopic interaction forces is of course phase-invariant. The conjectures [2]

u v = ρ ( 1 / z ) ( u v u l ) , u l = ρ ( z ) ( u v u l ) (4.2)

then yield, on the one hand, the reference values

0 = u v ( T c ) = u l ( T c ) = 1 2 [ u v ( T c ) + u l ( T c ) ] = u ( v c , T c ) and, on the other, for the density coefficient the equation

ρ ( 1 / z ) + ρ ( z ) = 1 (4.3)

and because u v ( 0 ) = 0 and u v ( T c ) / u l ( T c ) = 1 the boundary conditions

0 ρ ( 1 / z ) / ρ ( z ) 1. (4.4)

The functional Equation (4.3) for ρ under condition (4.4) is satisfied by

0 ρ ( 1 / z ) = 1 ln z 1 z 1 1 2 ρ ( z ) = 1 ln z + z z 1 1. (4.5)

From Equation (4.2) it follows that

0 u v u v u l = 1 ln z 1 z 1 1 2 1 ln z + z z 1 = u l u v u l 1. (4.6)

The relations (4.6) and (3.4) are identical and valid for 0 T T c and z = v v ( T ) / v l ( T ) 1 .

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 T [ T 1 , T 2 ] 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, c v ( T ) , and one in the liquid space, c l ( T ) . The measurable heat capacity of the fluid, C ( v c , T ) = M c ( v c , T ) = M v c v + M l c l , can also be calculated. Further results are the concavity of the positive, measurable temperature function ( u v u l ) and the convexity of the negative temperature functions ( u v + u l ) . The chemical potential is a negative, measurable and calculable temperature function. The ratios of phase-specific energies to isothermal transient

energies obey the relations 0 u v / ( u v u l ) 1 2 u l ( u l u v ) 1 , which should

be heeded when a state chart of the fluid under consideration is constructed.

Acknowledgements

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) [13] [14] 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 v v , l , internal energies u v , l , and entropies s v , l 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 [13] [14] ,

u v , l = α v v , l d ( p / T ) d ( 1 / T ) , s v , l = ϕ + v v , l d p d T , (A1)

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,

u v , l s v , l T = μ v v , l p , (A2)

where μ is the chemical potential of the saturated fluid. Equations (A1) and (A2) are related as follows:

α ϕ T = μ , α = d ( μ / T ) d ( 1 / T ) , ϕ = d μ d T . (A3)

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

u v u l v v v l = d ( p / T ) d ( 1 / T ) , s v s l v v v l = d p d T . (A4)

These are satisfied by Equations (A1). The Gibbs-Duhem equations read

d u v , l d T + d v v , l d T p = d s v , l d T T . (A5)

They are satisfied by Equations (A1) if the following relation between α and ϕ is valid:

d α d T = d ϕ d T T . (A6)

Note that Equation (A6) is implicitly contained in relations (A3) since d μ / d T = d α / d T d ϕ / d T T ϕ = ϕ and d ( μ / T ) / d ( 1 / T ) = μ d μ / d T T = ( α ϕ T ) + ϕ T = α . 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 [13] [14] for α ( T ) and ϕ ( T ) , whose temperature dependences are expressed as follows:

α [ J / g ] = d α + d 1 θ 19 + d 2 θ + d 3 θ 4.5 + d 4 θ 5 + d 5 θ 54.5 , where θ T T c , (A7)

ϕ [ J / g K ] = [ d ϕ + 19 20 d 1 θ 20 + d 2 ln θ + 9 7 d 3 θ 3.5 + 5 4 d 4 θ 4 + 109 107 d 5 θ 53.5 ] 1 T c , (A8)

where [ d α , d ϕ ] = [ 1.135905627715 E 3 , 2.3195246 E 3 ] , T c = 647.096 K ,

[ d 1 , d 2 , d 3 , d 4 , d 5 ] = [ 5.65134998 E 8 , 2.69066631 E 3 , 1.27287297 E 2 , 1.35003439 E 2 , 9.81825814 E 1 ] .

The vapor pressure p and chemical potential μ obey for temperatures T T c the following relations:

p > 0 , d p d T > 0 , d ( p / T ) d ( 1 / T ) < 0 , d 2 p d T 2 > 0 , (A9)

μ < 0 , d μ d T < 0 , d ( μ / T ) d ( 1 / T ) < 0 , d 2 μ d T 2 < 0. (A10)

Relations (A10) then lead to further conditions for α and ϕ:

α < 0 , d α d T > 0 , ϕ > 0 , d ϕ d T > 0. (A11)

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 d α / d T > 0 , but not that α < 0 for all T in the range [ T t , T c ] ; similarly, it does hold that d ϕ / d T > 0 , but not that ϕ > 0 for all T [ T t , T c ] . 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 u v , l shows that the sum ( u v + u l ) yields a concavely, but not a convexly curved function, as should be.

The fit function for measured vapor pressure data reads [13] [14]

ln ( p p c ) = T c T [ a 1 τ + a 2 τ 1.5 + a 3 τ 3 + a 4 τ 3.5 + a 5 τ 4 + a 6 τ 7.5 ] , where τ 1 θ (A12)

and p c = 22.064 [ MPa ] . The terms τ 1.5 , τ 3.5 , τ 7.5 produce divergent temperature derivatives d n p / d T n at T c for n 2 , whereas all vapor pressure derivatives have finite values. In contrast, the derivatives of the chemical potential d n μ / d T n diverge at T c for n 2 and determine the divergence of the heat capacity C ( M , V , T ) = M c ( V / M , T ) . 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, c ( V v / M v , T ) , and one in the liquid space, c ( V l / M l , T ) , and it holds that C ( M , V , T ) = M v c ( V v / M v , T ) + M l c ( V l / M l , T ) = C ( M v , V v , T ) + C ( M l , V l , T ) .

Numerous thermodynamic deficiencies have been mentioned, viz. violation of the conditions α ( T ) < 0 and ϕ ( T ) > 0 for every temperature T [ T t , T c ] , the incorrect temperature dependence of ( u v + u l ) , the proposed vapor pressure fit formula with a divergent term d 2 p / d T 2 at T c 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 [13] [14] are in need of basic correction.

Conflicts of Interest

The authors declare no conflicts of interest.

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