A Reference Equation of State for Heavy Water

Heavy water or deuterium oxide (D₂O, CAS no. 7789-20-0) is a liquid at ambient conditions. It is not radioactive and, if not taken in unreasonably large amounts, nontoxic. It differs from ordinary, light water in its hydrogen isotopes. The heavy water molecule contains two deuterium atoms instead of two ordinary hydrogen atoms. The nucleus of ordinary hydrogen, also called “protium” (¹H), consists of only one proton. The isotope deuterium (²H, D) has one additional neutron. As a result, the molecular mass of heavy water is higher than that of ordinary water by a factor of roughly 20/18. The resulting higher density is vividly presented in Fig. 1, which shows a photograph of heavy-water ice cubes that sink in ordinary water due to their higher density, whereas the ordinary-water ice cubes float due to the well-known expansion of water upon freezing.

Heavy water should not be confused with other heavier forms of water such as “super-heavy water” (tritium oxide, T₂O) containing the hydrogen isotope tritium (³H, T), “semi-heavy water” (deuterium hydrogen oxide, HDO), or “heavy-oxygen water” (H₂¹⁷O or H₂¹⁸O) enriched in heavier oxygen isotopes. The equation of state (EOS) presented here was developed to describe the properties of deuterium oxide with the oxygen isotopes ¹⁶O, ¹⁷O, and ¹⁸O in the standard proportions as defined by “Vienna Standard Mean Ocean Water” (V-SMOW), discussed by Kell,¹ and adopted by the International Association for the Properties of Water and Steam (IAPWS).² These standard (molar) proportions are x_16O = 0.997 620 6, x_17O = 0.000 379, and x_18O = 0.002 000 4. However, experimentally investigated samples of heavy water never contain 100% D₂O but are contaminated by a varying amount of H₂O and HDO. Since the EOS was fitted to experimental data, the D₂O content of the samples investigated within the corresponding references was considered in order to estimate the experimental uncertainty of the data.

In 1932, Urey et al.³ were the first to prove the existence of deuterium and thus of heavy water. Soon after this discovery, heavy water became a target for many nuclear physicists and played an important role in the first research on nuclear fission. It was therefore regarded as a compound of high commercial and military potential. Consequently, interest in the production of D₂O increased significantly during World War II. A summary of the interesting history of this almost mystical substance is given within the essay of Waltham.⁴ 

Over the past decades, D₂O has mostly been used as a neutron moderator in nuclear reactors. It slows free neutrons down to thermal energies, which is necessary for a self-sustaining chain reaction. The process of moderation can also be done with light water, but due to the additional neutrons in its hydrogen atoms, heavy water absorbs significantly fewer free neutrons.⁵ Nowadays, the interest in heavy water is as much scientific as commercial. In biological and medical research, heavy water is used in the “doubly labeled water method” to measure the average daily metabolic rate of an organism.^6,7 Another application is “boron neutron capture therapy” for the treatment of brain tumors, in which the ability of heavy water to moderate neutrons is useful (see, for instance, the work of Fairchild et al.⁸).

The equilibrium geometry of the D₂O molecule is almost identical to that of H₂O.⁹ While normally isotopic substitution has little effect on the thermal properties of fluids, this is not the case when hydrogen bonding is important. Quantum delocalization has a net weakening effect on water’s hydrogen bonds,¹⁰ so the heavier deuterium atoms make the hydrogen bonding stronger in D₂O than in H₂O. The effect is large enough that a separate EOS for D₂O must be developed, rather than a small perturbation to the H₂O EOS. The differences in thermodynamic behavior between H₂O and D₂O as quantified by their EOS therefore provide useful insights into quantum effects and hydrogen bonding.

The previous reference EOS for the thermodynamic properties of D₂O was published in 1982 by Hill et al.¹¹ and became a standard of the International Association for the Properties of Steam (IAPS, now IAPWS) in 1984. This formulation was later adjusted to the International Temperature Scale of 1990 (ITS-90)¹² as discussed in the corresponding Release of IAPWS.¹³ It is explicit in the specific Helmholtz energy with temperature T and density ρ as independent variables. The equation is valid from the triple-point temperature up to 800 K and for pressures up to 100 MPa. The equation is not recommended for calculations in the critical region bordered by |T − T_c| ≤ 10 K and |ρ/ρ_c − 1| ≤ 0.3, with the critical temperature T_c and critical density ρ_c. For calculations within this region, IAPWS¹³ refers to the much more complex crossover EOS by Kostrowicka Wyczalkowska et al.,¹⁴ which is only valid in a small region around the critical point and is therefore not discussed in this article. In comparison to modern EOS for other fluids, the formulation by Hill et al.¹¹ has a quite long functional form with a total of 50 terms. This relatively complex mathematical structure frequently leads to numerical problems. Due to both the great advances in the development of EOS and modern computer technology, it is now possible to develop equations with a reduced number of terms without loss of accuracy. Based on these factors, IAPWS initiated the development of a new EOS for heavy water in 2013, although at that point new experimental data since the publication of Hill et al. were relatively few. During the development of the EOS, additional accurate data were contributed by various groups associated with IAPWS. Thus, the new formulation is based on the most up-to-date thermodynamic database. A preliminary version of this formulation was adopted as an IAPWS Release in 2017,¹⁵ and a Revised Release based on the formulation presented in this paper is expected to be adopted by IAPWS in 2018.

An overview of the most important physical constants and characteristic properties of D₂O as relevant for the EOS presented here is given in Table 1. Aside from the information taken from literature references, Table 1 contains values determined from the new equation.

In the development of an EOS, the critical temperature and density are essential thermodynamic properties. With regard to the structure of modern equations, the reason for this is obvious. Most of these formulations are explicit in the reduced Helmholtz energy with the critical temperature and density as the reducing parameters of the independent variables. However, measurements of the critical point are difficult and thus rare and with wide variation between sources. The experimental determination of the critical density poses a special challenge because of the extreme sensitivity of density to changes in temperature and pressure near the critical point. For the critical point of heavy water, only four references are available and only two of them include information about the critical density. The available critical parameters are listed in Table 2. All temperature values were converted to ITS-90. The critical temperature and density used to develop the new EOS are recommended in the IAPWS Release on the critical values of ordinary and heavy water.¹⁷ The critical temperature in Ref. 17 corresponds to the measurement of Blank¹⁹ converted to ITS-90. The critical pressure and density were obtained numerically by Levelt Sengers et al.²⁰ by a scaled analysis of pρT data measured by Rivkin and Akhundov.²¹ The recommended critical parameters and their corresponding uncertainties (reflecting the correlation between errors in critical temperature and critical pressure) are T_c,IAPWS/K = 643.847 + δ with δ = 0.000 ± 0.200, p_c,IAPWS/MPa = 21.671 + 0.27δ ± 0.01, and ρ_c,IAPWS/(kg m⁻³) = 356 ± 5 or ρ_c,IAPWS/(mol dm⁻³) = 17.775 55 ± 0.25 on a molar basis. We note that the critical pressure recommended by IAPWS slightly differs (by 0.0092 MPa) from the value obtained from the new EOS as given in Table 1. However, this deviation is within the uncertainty of the IAPWS value, which is 0.01 MPa at the chosen critical temperature.

Equations of state as presented here cannot be used to calculate the properties of solid phases. Thus, separate knowledge of the melting and sublimation curves is needed to set the lower temperature limit of the range of validity of such formulations. For (heavy) water, the triple point is not the lowest temperature at which the substance remains liquid due to the anomalous shape of its melting curve, which exhibits a negative initial slope (dp/dT). Nevertheless, the triple-point parameters are important natural constants, which are known extremely accurately for ordinary water²² but less satisfactorily for heavy water. Although triple-point parameters of D₂O can be found in a number of publications, almost all of these studies obtained the triple-point pressure by extrapolating vapor-pressure data measured at higher temperatures down to a given triple-point temperature. Furthermore, the triple point is frequently equated with the melting point at atmospheric pressure. Only Jones²³ and Markó et al.¹⁸ carried out “real” triple-point experiments. The data from these studies are given in Table 2 with temperatures converted to ITS-90. In addition to these references, the data of Bartholomé and Clusius²⁴ are included in Table 2. This reference is given since no earlier work providing triple-point parameters of heavy water was found. However, the authors do not describe a direct experimental investigation of the triple point; thus, the source of the given values is unclear. For the development of the new EOS, the most recent value for the triple-point temperature by Markó et al.¹⁸ was adopted. The triple-point pressure given in Table 1 was calculated at this temperature from the final EOS, since the pressure reported by Markó et al.¹⁸ deviates by about 0.35% from the most accurate vapor-pressure data at slightly higher temperatures.

Within the scope of this work, auxiliary equations for the vapor pressure, the saturated-liquid and saturated-vapor densities, the melting pressure, and the sublimation pressure were developed. The first three equations can be used to quickly obtain good estimates for the thermal properties along the phase boundaries of the vapor–liquid equilibrium region. However, the results calculated from these correlations are not identical to the saturation properties obtained from our reference EOS. In order to determine such reference-quality properties, iterations based on the Maxwell criterion are required. Fulfilling this criterion ensures that at any fixed temperature bubble- and dew-point pressures as well as the Gibbs energies of both coexisting phases are equal. Since accurate saturation properties are obtained by this iterative procedure, auxiliary equations for the saturated liquid and vapor are not necessarily needed but provide excellent starting values for phase-equilibrium calculations in typical engineering applications such as process simulations. With the purpose of achieving high consistency between the reference equation and the corresponding auxiliary equations, the latter were fitted to values calculated at 1 K intervals from our EOS between 238 K (which corresponds to an extrapolation far below the triple point) and the critical point.

As discussed in Sec. 2, the lower temperature limit of the EOS is not defined by the triple-point temperature, but by the lowest temperature along the melting curve. Additional auxiliary equations for the melting pressure of the relevant ice structures of heavy water were developed. The description of fluid-solid boundaries is completed by an auxiliary equation for the sublimation pressure. The correlations for the solid–fluid equilibria were fitted to experimental results available in the literature. The complete phase diagram, as calculated with the new EOS and the auxiliary equations for the melting and sublimation curves, is presented in Fig. 2.

The auxiliary equation for the vapor pressure p_v is given by

with $θ=(1−T/Tc)$ and the coefficients n₁–n₆ as given in Table 3. The critical parameters are discussed in Sec. 2 and listed in Table 1. The first two temperature exponents were not included in the fitting process but set to 1 and 1.5, respectively. In addition, the coefficient n₁ was kept negative and n₂ positive. A detailed explanation of these constraints for the first two exponents and coefficients is given by Lemmon and Goodwin.²⁸ Deviations between the calculated values from Eq. (1) and values obtained from the reference EOS by means of the Maxwell criterion are below 0.01% over the entire temperature range. However, these deviations (and the ones given for the auxiliary equations for the saturated-liquid and -vapor density) must not be equated with uncertainties of saturation properties calculated from our EOS, which are discussed in Sec. 5.1. For checking computer implementations, at 293.15 K Eq. (1) gives a vapor pressure of 0.199 914 326 × 10⁻² MPa.

The saturated-liquid density equation is

with the saturated density $ρ′$ and the critical density ρ_c as given in Table 1. The coefficients n₁–n₆ are given in Table 3. In order to ensure the correct behavior of the correlation and its derivatives at the critical point, the first temperature exponent must be between zero and one, with a positive coefficient n₁.^29,30 The calculated values from Eq. (2) deviate from the results obtained from our EOS by less than 0.03% at temperatures up to 641 K. In the vicinity of the critical point, deviations increase to a maximum of 0.06%. For checking computer implementations, at 293.15 K Eq. (2) gives a saturated-liquid density of 0.551 959 089 × 10² mol dm⁻³.

For the saturated-vapor density $ρ″$⁠, the auxiliary equation reads

with the coefficients n₁–n₆ as listed in Table 3. In fitting Eq. (3), the first temperature exponent was kept between zero and one with a negative coefficient n₁, ensuring a correct description of the critical region.²⁹ Deviations between calculated values from Eq. (3) and our EOS are below 0.03% at temperatures up to 590 K. The deviations increase at higher temperatures with a maximum of 0.07% in the vicinity of the critical point. For checking computer implementations, at 293.15 K Eq. (3) gives a saturated-vapor density of 0.821 136 767 × 10⁻³ mol dm⁻³.

Solid water forms different crystalline structures depending on the temperature and pressure. For ordinary water, five ice structures are known that are bordered by the liquid phase (ice Ih, III, V, VI, VII). The corresponding melting-pressure curves are limited by triple points, in which two ice structures and the liquid phase are in thermodynamic equilibrium. An exception to this is the lower pressure limit of the melting curve of ice Ih, which is the “normal” solid–vapor–liquid triple point. The five ice structures are described by equations that were developed by Wagner et al.³¹ and also included in the IAPWS-95 publication by Wagner and Pruß.³² The correlation for ice Ih was later updated by Wagner et al.³³ For heavy water, the same ice structures are believed to exist, although to our knowledge the high-pressure structure VII has not been experimentally observed. We adopted the structures of the original correlations and refitted them to the available experimental melting-pressure data by Bridgman³⁴ (for ice structures Ih, III, V, and VI) and Henderson and Speedy³⁵ (only for ice structure Ih). The triple-point data required to define the range of validity for each ice-structure correlation were taken from the work of Bridgman³⁴ for the solid-solid-liquid triple points and from Table 1 for the “normal” triple point. All applied triple-point parameters of the different ice structures are listed in Table 4. Since no data are available along the ice-VII melting curve, it was ensured that the equation for ice VI yields reasonable results up to the upper pressure limit of the new EOS, p_max = 1200 MPa. All experimental pressure values of Bridgman³⁴ were multiplied by a correction factor of 1.0102. This factor corresponds to the ratio between the melting pressure of mercury at 0 °C obtained by Bridgman³⁶ (749.2 MPa), which was used to calibrate the apparatus for his light and heavy water melting-pressure measurements, and the reference value reported by Molinar et al.³⁷ (756.84 MPa).

The four melting-pressure equations developed within this work are as follows:

The melting-pressure equation for ice Ih (temperature range from 276.969 K to 254.415 K) is

with reduced temperature θ = T/T_n and the reducing parameters T_n = 276.969 K and p_n = 0.000 661 59 MPa. For checking computer implementations, at 270 K Eq. (4) gives a melting pressure of 0.837 888 413 × 10² MPa.

The melting-pressure equation for ice III (temperature range from 254.415 K to 258.661 K) is

with θ = T/T_n, T_n = 254.415 K, and p_n = 222.41 MPa. For checking computer implementations, at 255 K Eq. (5) gives a melting pressure of 0.236 470 168 × 10³ MPa.

The melting-pressure equation for ice V (temperature range from 258.661 K to 275.748 K) is

with θ = T/T_n, T_n = 258.661 K, and p_n = 352.19 MPa. For checking computer implementations, at 275 K Eq. (6) gives a melting pressure of 0.619 526 971 × 10³ MPa.

The melting-pressure equation for ice VI (temperature range from 275.748 K to 315 K) is

with θ = T/T_n, T_n = 275.748 K, and p_n = 634.53 MPa. For checking computer implementations, at 300 K Eq. (7) gives a melting pressure of 0.959 203 594 × 10³ MPa.

Comparisons between the melting-pressure curves calculated from our correlations and the available experimental data are shown in Fig. 3. Especially for the ice structures III, V, and VI, the correlations are in very close agreement with the experimental data; the deviations are clearly below 1%. However, these low deviations are less meaningful because the correlations were exclusively fitted to the only available data by Bridgman.³⁴ Since these data are relatively self-consistent, quite low deviations can be achieved when fitting the correlations to these values. The melting-pressure equations for ordinary water of Wagner et al.³¹ were also mainly fitted to Bridgman’s data. The estimated uncertainty of melting pressures calculated from these equations is stated to be 3% for the ice structures III, V, and VI. Due to the comparable database and the same basic mathematical structure, this uncertainty estimate can reasonably be adopted for our melting-pressure equations for heavy water. Thus, the estimated uncertainty of melting pressures calculated from Eqs. (5)–(7) is 3%.

Deviations between the experimental data and calculated melting pressures for ice Ih are considerably higher than those for the other ice structures. Two datasets are available for ice Ih, namely, the already discussed data by Bridgman³⁴ and also the data of Henderson and Speedy.³⁵ The authors of Ref. 35 state an uncertainty of 0.1 K in temperature and 0.5 MPa in pressure. Considering these uncertainties, the total combined expanded (k = 2) uncertainties in melting pressure range from approximately 1% at 258.6 K to 11% at 275.1 K, where uncertainties are higher due to the steep slope of the melting-pressure curve. The correlation describes the data within their uncertainty for temperatures above 265 K. For lower temperatures, the calculated melting curve represents a compromise between the best possible description of the datasets by Henderson and Speedy³⁵ and Bridgman.³⁴ Although the latter data are less accurate, they were used to fit the correlation in order to achieve a low deviation from the ice Ih-ice III-L triple point. An accurate description of this state point is necessary to ensure a continuously consistent description along the ice-Ih and ice-III melting curve. Attempts to fit the complete dataset of Henderson and Speedy³⁵ together with the upper triple point led to unreasonable changes in curvature along the saturation curve. Thus, the correlation was fitted to a carefully weighted selection of data points from both references. Additionally, the curvature and third derivative (d³p_m/dT³) were carefully constrained to avoid any unreasonable shape of the melting-pressure curve. The calculated initial slope of the melting curve at the “normal” solid–vapor–liquid triple point is −13.96 MPa K⁻¹. As a reliability check, this slope is compared to the result of the Clapeyron equation

with the enthalpy of fusion Δh_m and the liquid-solid volume change (v_L − v_S). Considering the experimental results of Long and Kemp³⁸ for Δh_m = 1501 cal mol⁻¹ = 6280.2 J mol⁻¹ and of Timmermans et al.³⁹ for (v_L − v_S) = −1.62 cm³ mol⁻¹, Eq. (8) yields a slope of −13.96 MPa K⁻¹ at T = T_tp =276.969 K. Thus, the initial slope calculated from the new melting-pressure equation is in perfect agreement with the result of the Clapeyron equation. With regard to the representation of the data of Henderson and Speedy,³⁵ the uncertainty of melting pressures of ice Ih calculated from Eq. (4) is conservatively estimated to be 4%.

To complete the description of the phase equilibria of heavy water, an auxiliary equation for the sublimation pressure of ice Ih is presented. The equation uses the structure of the correlation for ordinary water ice Ih as given in the work of Wagner et al.³¹ and Wagner and Pruß.³² The parameters were fitted to sublimation pressures obtained by Pupezin et al.⁴⁰ and Bottomley.⁴¹ The data of Pupezin et al.⁴⁰ range from the triple-point temperature down to about 210 K, whereas Bottomley’s measurements do not cover temperatures below 261 K. Thus, the lower temperature limit of the equation is defined only by the dataset of Pupezin et al.⁴⁰ The pressure and temperature are reduced by the “normal” triple-point parameters discussed in Sec. 2.

The sublimation-pressure equation for ice Ih (temperature range from 210 K to 276.969 K) is

with θ = T/T_n, T_n = 276.969 K, and p_n = 0.000 661 59 MPa. For checking computer implementations, at 245 K Eq. (9) gives a sublimation pressure of 0.327 390 934 × 10⁻⁴ MPa.

Comparisons between sublimation pressures calculated from Eq. (9) and the experimental data are shown in Fig. 4. For the sake of completeness, the data of Niwa and Shimazaki⁴² are shown in addition to the datasets mentioned above, although they comprise only three data points of lower accuracy. Within the deviation plot in the bottom panel (as in all other deviation plots in this article), data points shown at the upper or lower vertical limits of the diagram indicate that the points are off scale.

At temperatures above 255 K, the data of Pupezin et al.⁴⁰ and Bottomley⁴¹ are in agreement and are represented within 0.5% in sublimation pressure. At lower temperatures, where the sublimation pressures become quite low (p_sub ≤ 10⁻⁴ MPa), the data exhibit significantly more scatter. Most of the data between 210 K and 255 K are described within 5%. The experimental uncertainty of the data of Bottomley⁴¹ is not stated clearly and is difficult to estimate since the publication presents differences between the sublimation pressure and the vapor pressure of the metastable subcooled liquid. The data were calculated with vapor pressures obtained from our EOS extrapolated below the triple-point temperature. The experimental setup applied to measure the data of Pupezin et al.⁴⁰ was presented by Jancsó et al.⁴³ together with experimental results for the sublimation pressure of ordinary water. These results deviate from the most accurate sublimation-pressure equation of Wagner et al.³³ (IAPWS standard correlation for the sublimation pressure of ordinary water⁴⁴) by up to 0.5% at temperatures above 250 K. At lower temperatures, the deviations increase significantly to more than 10% at about 200 K. In this temperature range, the uncertainty of the sublimation-pressure equation for ordinary water is below 0.5%.^33,44 Thus, the experimental uncertainty of the data for D₂O of Pupezin et al.⁴⁰ can be reasonably estimated based on the deviations between the data for H₂O of Jancsó et al.⁴³ and the reference correlation by Wagner et al.³³ Considering these deviations, the estimated expanded (k = 2) uncertainties of sublimation pressures of D₂O calculated from Eq. (9) are 0.5% at temperatures above 255 K, 5% at 225 ≤ T/K ≤ 255, and 10% at 210 ≤ T/K < 225. Qualitative comparisons with the equation for H₂O show that Eq. (9) can be reasonably extrapolated down to temperatures of 150 K or lower.

The equation developed in this work is a fundamental EOS explicit in the Helmholtz energy as a function of density and temperature. A fundamental EOS enables calculations of all thermodynamic properties by combining derivatives of its functional form. Formulating the equation explicitly in the Helmholtz energy with the independent variables temperature and density yields the additional advantage of a clear description over the whole fluid surface, including the vapor–liquid equilibrium region. Thus, most modern reference EOS were developed with the same basic mathematical structure as presented in this section. Excellent examples are the equations for R-125 of Lemmon and Jacobsen,⁴⁵ for propane of Lemmon et al.,²⁹ for R-1234ze(E) of Thol and Lemmon,⁴⁶ and for sulfur dioxide of Gao et al.³⁰ 

In general, EOS in terms of the Helmholtz energy a can be formulated as

The function a^o describes the behavior of the hypothetical ideal gas, whereas a^r represents the residual Helmholtz energy that results from molecular interactions in the real fluid. Since it is more convenient to work with dimensionless equations, density and temperature are reduced by their values at the critical point and the form of the resulting dimensionless function is

where α is the reduced Helmholtz energy and R is the molar gas constant as given in Table 1. The reciprocal reduced temperature is τ = T_c/T, and the reduced density is δ = ρ/ρ_c. For D₂O, the critical-point parameters are given in Table 1.

The ideal-gas part of the reduced Helmholtz energy can be written as

where $cpo$ is the isobaric heat capacity of the ideal gas and τ₀ and δ₀ are the reduced reciprocal temperature and reduced density at any arbitrary reference state defined by T₀ and ρ₀. For heavy water, this reference state was chosen following the conventions of IAPWS, where the internal energy u′ and entropy s′ of the real fluid at the saturated liquid state are set to zero at the triple point T_tp = 276.969 K. A correlation for the ideal-gas heat capacity $cpo$ is necessary in Eq. (12). The functional form used here is

where c₀ = 4.0, v₁ = 0.010 633, v₂ = 0.997 87, v₃ = 2.1483, v₄ = 0.354 90, u₁ = 308 K, u₂ = 1695 K, u₃ = 3949 K, and u₄ = 10 317 K. The value of c₀ is physically meaningful, since it corresponds to the internal-energy contributions of translational and rotational motions of the molecule at low temperatures. A nonlinear triatomic molecule, such as D₂O, has three translational and three rotational degrees of freedom, leading to the isochoric ideal-gas heat capacity $cvo=6/2R$⁠. From Eq. (13), this leads to the constant $c0=(cvo+R)/R=8/2=4$⁠. As temperature increases, thermal energy additionally contributes to vibrational excitations that yield an increase in $cpo$⁠. This temperature dependency is represented by four so-called “Planck-Einstein terms.” Because this approach is empirical and only loosely based on physical considerations, the individual terms do not represent the contributions of specific vibrational frequencies. The adjustable coefficients v_k and exponents u_k were mainly fitted to the recently published ideal-gas heat capacities obtained from a highly accurate partition function by Simkó et al. [the temperature in these $cpo(T)$ values is the thermodynamic temperature; for our purposes, this is negligibly different from temperatures on the ITS-90 scale].⁴⁷ However, since the ideal-gas part contributes to all caloric properties calculated from the EOS, it was simultaneously fitted with the residual part to additional experimental data for the speed of sound and heat capacity (c_p and c_v) of the real fluid. The ideal-gas part of the EOS, derived from Eqs. (12) and (13), is

where a₁ and a₂ are defined to yield the above-discussed values of u′(T_tp) = s′(T_tp) = 0. The values of these constants are a₁ = −8.670 994 022 646 00 and a₂ = 6.960 335 784 587 78. All other parameters are the same as given for Eq. (13).

The data of Simkó et al.⁴⁷ are part of a comprehensive IAPWS-associated project in which thermochemical functions for the ideal-gas properties of water and its different isotopologues are being developed. The results for H₂¹⁶O were published by Furtenbacher et al.⁴⁸ Because the single-molecule partition functions developed in this project do not account for the dissociation that occurs in real water at high temperatures, they produced artificial maxima in the heat capacity. For heavy water, this maximum occurs near 4100 K. Since this is far beyond the upper temperature limit of the new EOS, the correlation given in Eq. (13) was only fitted, and is consequently only valid, up to this temperature. Comparisons between the new EOS and the available calculations^47,49,50 for the ideal-gas heat capacity are presented in Fig. 5. On the $cpo$⁠, T-diagram, the maximum in $cpo$ is quite distinct. The new equation yields reasonable information about the ideal-gas state over a much broader temperature range than the EOS of Hill et al.,¹¹ which was exclusively fitted to the old data of Friedman and Haar⁴⁹ up to about 1500 K.

Based on comparisons to the data of Simkó et al.,⁴⁷ the uncertainty of calculated ideal-gas heat capacities from the new EOS is estimated to be smaller than 0.01% at temperatures below 300 K and within 0.02% over the whole range of validity of the new EOS (with T_max = 825 K). The uncertainty of calculated ideal-gas heat capacities from this correlation is still within a maximum uncertainty of 0.25% up to 4100 K. Our uncertainties do not match the standard uncertainties of the data as given by Simkó et al.,⁴⁷ which are stated to be below 0.01% up to 1800 K. However, since the present ideal-gas correlation led to the best representation of the real-fluid properties, minor concessions were made in the description of the ideal-gas properties at elevated temperatures.

The residual part of our EOS for heavy water consists of six polynomial terms, six exponential terms, and 12 “Gaussian bell-shaped” terms. The complete equation with a total of 24 terms reads

All parameters (coefficients n_i, temperature exponents t_i, density exponents d_i and l_i, and the parameters of the Gaussian bell-shaped terms η_i, ε_i, β_i, and γ_i) are listed in Table 5. The formulation is valid for all stable fluid states from T_min = 254.415 K, which corresponds to the minimum temperature along the melting-pressure curve, to T_max = 825 K at pressures up to 1200 MPa.

As previously mentioned, all thermodynamic properties can be calculated through the derivatives of Eqs. (14) and (15) with respect to the reduced density and reciprocal reduced temperature. Detailed explanations of the calculation of thermodynamic properties, including the required derivatives, are given by Span⁵¹ and Lemmon et al.²⁹ To assist users in computer-program verification, three tables with test values are provided. Table 6 contains values of the ideal and residual part of the reduced Helmholtz energy together with values of their relevant derivatives. Table 7 gives various calculated single-phase properties at selected temperatures and densities, whereas Table 8 provides values of saturation properties at selected temperatures.

The essence of the fitting procedure can be described quite simply: The adjustable parameters of the ideal-gas correlation and of the residual part of the reduced Helmholtz energy [Eqs. (13) and (15)] are varied by a fitting algorithm to reach the best agreement between the input data and the properties calculated from the EOS. The mathematical structure of the ideal-gas part is relatively simple compared to the residual part. Because the residual part is fitted to a significantly more comprehensive database, most of the time required to fit an EOS is spent on the development of the residual part. The fitting algorithm used in this work was originally developed by Lemmon and Jacobsen⁴⁵ and is continuously improved at the National Institute of Standards and Technology (NIST). It is based on so-called “nonlinear fitting methods” that enable a simultaneous optimization of all parameters to different types of data. It is also possible to apply thermodynamic constraints to ensure that all properties behave correctly and extrapolate well even in regions beyond the data available. The EOS is fitted to a carefully chosen selection of points from the most accurate and consistent datasets. Each selected data point and constraint is weighted individually depending on the type, pressure and temperature region, target accuracy, and experimental uncertainty. Deviations of the calculated properties from the weighted data points and constraints contribute to an overall sum of squares that is minimized by varying the adjustable parameters of the EOS.

The number of each type of term in Eq. (15) is not varied by the fitting algorithm. The correlator has to find the optimum number and combination of terms to obtain a good fit. The number of terms should be as small as possible but as large as necessary. Equations with a large number of terms are very flexible and allow the correlator to find special terms to describe particular areas with high accuracy, but the extrapolation behavior of these equations can be unmanageable due to the many mathematical degrees of freedom. By contrast, short equations can be shaped more easily to extrapolate well and to exhibit smooth derivative behavior. However, it can be difficult to find a solution that covers the whole fluid region and yields a good compromise among all different data types. As discussed in Sec. 1, a short and thus numerically stable functional form was one of the main requirements for the new EOS for heavy water. In comparison to the equation of Hill et al.,¹¹ the new formulation has fewer than half the number of terms, 24 instead of 50 (excluding the ideal-gas correlations).

The 24 terms of the residual part of the present equation contain a total of 126 adjustable parameters as presented in Table 5. With regard to this number, it is apparent that during the fitting process the number of possible parameter sets is almost unmanageable. For this reason, some guidelines given by Lemmon and Jacobsen⁴⁵ were followed. The derivatives of the residual part of the equation must go to zero for small densities, because in this limit the fluid behaves like an ideal gas. Therefore, the density exponents d_i and l_i in Eq. (15) must be positive integers and are not adjusted in the fitting algorithm but are defined by the correlator while choosing the optimal set of terms. Special restrictions are defined for the first polynomial term, which should have a density exponent of d₁ = 4 and a corresponding temperature exponent of t₁ = 1. This ensures that the isotherms converge for high densities and do not cross or diverge. All other temperature exponents do not have to be integers but should be positive. Negative exponents might result in unreasonable extrapolation behavior at high temperatures. For low temperatures and in the vapor–liquid equilibrium region, the overall behavior of the equation is essentially defined by terms with high temperature exponents. In order to ensure proper behavior in these regions, the exponents t_i should be as small as possible. As discussed in Sec. 5.3, at low temperatures the slope of the second virial coefficient B with temperature is much steeper for (heavy) water than for most other fluids. The correct representation of this behavior and of the experimental data in the metastable subcooled-liquid region required some higher temperature exponents than are generally needed. Figure 6 shows that at low temperatures the derivative of the residual Helmholtz energy with respect to density $αδr$⁠, which in the zero-density limit yields the second virial coefficient B, is basically defined by the second exponential term of Eq. (15) (i = 8). The temperature exponent of this term is t₈ = 4.6. Thus, no unreasonably large exponent was needed to describe the special behavior of B. The highest temperature exponent is t₁₁ = 5.4644, which is still relatively low compared to older reference equations such as those for ordinary water,³² carbon dioxide,⁵² or nitrogen.⁵³ In those equations, the highest temperature exponents range from 16 up to 50.

The description of the critical region is particularly challenging in the development of an EOS. The reference equations for ordinary water³² and carbon dioxide⁵² contain so-called “non-analytical” terms that were developed to model the special physical characteristics at the critical point such as the extreme increase in c_v or the global minimum of the speed of sound w. Since these terms frequently lead to numerical problems, especially when the particular equation is evaluated within mixture modeling, they are no longer used for the development of new EOS. The approximate description of the critical point is normally enabled by Gaussian bell-shaped terms. For heavy water, two such terms (i = 23 and i = 24) were set up, with similar parameters but one positive and one negative coefficient. Consequently, these two terms cancel each other out over most of the stable fluid region, except in the vicinity of the critical point, where they have their maximum contribution. The impact of these terms on properties calculated near the critical point is visualized in Fig. 7, where the residual part of the isochoric heat capacity is calculated first with all terms of Eq. (15) and then with all terms except for the two critical-region terms. Both resulting surfaces are plotted over temperature and density. The steep increase in the isochoric heat capacity is modeled by the two Gaussian bell-shaped terms. However, as an analytic EOS, Eq. (15) is not capable of reproducing the nonclassical critical exponents that govern fluid behavior in the asymptotic limit of the critical point.

In general, Gaussian bell-shaped terms can be used to describe not only the critical region but also any region of the fluid surface. However, these terms need to be applied carefully, since they might lead to unreasonable results for derivatives of the reduced Helmholtz energy and thus to wrong qualitative behavior of certain properties. For heavy water, Gaussian bell-shaped terms were important to describe the anomalous behavior of the liquid at low temperatures, especially its density maximum. The final EOS was achieved by fitting its adjustable parameters to a representative selection of the reliable data from the literature and various mathematical and thermodynamic constraints. The fitting process was completed by carefully rounding all exponents and coefficients. This was done successively while refitting the equation. This guaranteed that the rounding off of each group of parameters was compensated by the coefficients and exponents that were still included in the fit.

As briefly discussed in Sec. 4, the EOS was fitted to a carefully weighted selection of the most reliable experimental data. Nevertheless, the fit was continuously evaluated by comparisons to all available data points. In this section, such comparisons to the final EOS are discussed in order to validate the EOS and to estimate the uncertainties of calculated values. The relative deviation of every data point from the value obtained from the EOS is calculated. This deviation for any property X can be written as

Comparisons of the equation to complete datasets are based on the average absolute relative deviation (AAD). This property is defined as the arithmetic average of all percentage absolute deviations of a dataset (excluding clear outliers). It reads

where n is the number of data points used in the calculation. In many cases, calculating an overall AAD for one dataset would lead to false conclusions. For example, the AAD of a dataset including many excellent measurements in the liquid phase can be significantly affected by a small number of inaccurate data points in the vapor phase. Therefore, it is meaningful to separate the fluid range into parts and calculate the AAD for each region. This separation needs to be different for thermal saturation data than for other types of data and is discussed in Secs. 5.1 and 5.2. The relative deviation defined by Eq. (16) is not useful for properties such as virial coefficients whose values cross zero; see Sec. 5.3 for the discussion of deviations for virial coefficients.

All experimental values considered in the present work were converted to molar-based SI units, with temperatures on the ITS-90 scale.

An overview of the available data for thermal saturation properties, namely, for the vapor pressure and the saturated-liquid and saturated-vapor densities, is given in Table 9 along with the AAD for each dataset. Aside from the overall AAD that was calculated for the complete dataset, separate values for the AAD for the low (T/T_c ≤ 0.6), medium (0.6 < T/T_c ≤ 0.98), and high (T/T_c > 0.98) temperature ranges are provided.

Most of the available saturation data for heavy water are measurements of the vapor pressure. In some cases, the vapor pressure was reported as a difference or ratio relative to that of ordinary water; in these cases, we used the IAPWS-95 formulation^32,73 to calculate the vapor pressure of ordinary water. Comparisons of the available data for D₂O with values calculated from the EOS are shown in Fig. 8. Figure 8 contains two deviation plots. The first one shows deviations of all available measurements and thus provides an overview of the entire database on this property. The second plot (with a smaller scale) illustrates comparisons with datasets included in the fitting process.

Most of the available data at temperatures higher than the triple-point temperature are represented within deviations of 0.2%. The database includes vapor pressures of the metastable subcooled liquid at temperatures below the triple-point temperature, published by Kraus and Greer⁶¹ and Bottomley.⁴¹ This scientifically interesting region is discussed in detail in Sec. 5.5. For the sake of completeness, deviations of these data are included in the top panel of Fig. 8. Most of the data of Bottomley⁴¹ deviate from the EOS by less than 0.15%, although they were not used in the fitting process. Thus, these data demonstrate good extrapolation behavior of the vapor-pressure curve calculated from our EOS. The triple-point measurement of Markó et al.¹⁸ is specifically highlighted in the top panel of Fig. 8. As noted in Sec. 2, the corresponding pressure value is less accurate than the available vapor-pressure data at slightly higher temperatures; the pressure deviates by −0.35% from the EOS. Therefore, the triple-point pressure provided in Table 1 was calculated from the EOS. Between the triple-point temperature (T_tp = 276.969 K) and about 300 K, the EOS was fitted to the experimental data of Besley and Bottomley,⁵⁴ which are the most accurate data available in this temperature range. The authors state uncertainties of 3 mK in temperature and about 2.7 Pa in pressure. Considering this information, we calculated relative combined expanded (k = 2) uncertainties in vapor pressure for every state point that range from 0.02% to 0.04%. Except for very few data points, the EOS represents all the data within their estimated experimental uncertainty. The AAD of the dataset from the equation is 0.024%. We conservatively estimate the relative uncertainty of calculated vapor pressures to be 0.05% at temperatures up to 300 K. As a further reliability check, the vapor pressures for H₂O provided by Besley and Bottomley⁵⁴ were compared to the reference equation of Wagner and Pruß³² (IAPWS-95). Although the data were not used in the fitting process of IAPWS-95,³² they are represented with an AAD of 0.056%.

In the temperature range between 300 K and 350 K, the data situation is less satisfactory than for the rest of the vapor-pressure curve, although the amount of data available is relatively large. However, all of the datasets that exhibit an acceptable level of self-consistency come from the same group of experimentalists, namely, the Central Research Institute for Physics of the Hungarian Academy of Sciences in cooperation with the Chemistry Department of the University of Tennessee.^40,56–58 Although these data were obtained with the same basic experimental setup, they exhibit considerable differences, indicating that the data are not as accurate as the best measurements at lower and higher temperatures. The earliest study from this group was published in 1972 by Pupezin et al.⁴⁰ As shown in Fig. 8, these data exhibit considerable scatter. However, the EOS of Hill et al.¹¹ was obviously fitted to this dataset, which led to a less reliable description of vapor pressures in this temperature range than with the equation presented here. The later published datasets of Jákli and Illy⁵⁶ and Jákli and Van Hook⁵⁸ are more consistent, but still exhibit scatter of up to 0.09% and an offset of about 0.08% from the very accurate data of Zieborak⁶⁶ at higher temperatures. The newest dataset was published in 1995 by Jákli and Markó⁵⁷ within a study of excess properties of ordinary and heavy water solutions of tetrabutylammonium bromide. The vapor pressures of pure heavy water obtained in this work were not provided in the corresponding publication, but were later provided in a personal communication to Harvey and Lemmon who included them in their article.⁷² The reproducibility of these data is roughly 0.05%. Considering that the data were published relative to H₂O and that the uncertainty of vapor pressures calculated from IAPWS-95 is between 0.01% and 0.02% within this temperature range,^32,73 the uncertainty of the data is probably not much less than 0.1%. Summing up the data situation between 300 K and 350 K, the estimated relative uncertainty of calculated vapor pressures in this temperature range is 0.1%. However, this estimate is quite conservative. Considering the experimental uncertainty of the data at lower and higher temperatures, the uncertainty of calculated values is probably lower than that estimated here.

At temperatures between 350 K and 495 K, the vapor-pressure curve calculated from the EOS was mostly defined by fitting the equation to the experimental results of Zieborak.⁶⁶ The author states an uncertainty in pressure of about 2.7 Pa but does not provide comparably clear information about the uncertainty in temperature. Nevertheless, azeotropic temperatures of the H₂O + D₂O mixture are reported with an accuracy of 3 mK. Based on this information, we estimated combined expanded (k = 2) uncertainties in vapor pressure ranging from 0.01% to 0.03%. The maximum deviation of these data from the EOS is 0.02% and their overall AAD is 0.008%. Considering these deviations and the experimental uncertainty of the data, the estimated relative expanded uncertainty of calculated vapor pressures in this range is 0.03%.

Between 495 K and the critical temperature (T_c = 643.847 K), the most reliable data were published by Oliver and Grisard.²⁷ Above 548 K, the data overlap with the vapor pressures of Rivkin and Akhundov,²¹ who provided no clear information about experimental uncertainties. However, the data confirm the vapor pressures of Oliver and Grisard²⁷ within about 0.03%. Oliver and Grisard²⁷ measured the saturation temperature relative to ordinary water within 0.01 K. The reported pressures were obtained with the H₂O vapor-pressure correlation of Osborne and Meyers,⁷⁴ which is negligibly different from IAPWS-95 in this region. Since the uncertainty of IAPWS-95 in this region is 0.02%,^32,73 we consider this as a good estimate for the uncertainty in saturation pressure of the data by Oliver and Grisard.²⁷ The estimated relative combined expanded (k = 2) uncertainties of these data are thus within 0.06%. However, at temperatures up to 642 K, deviations between these data and the EOS are below 0.05%, and the data are confirmed by the values of Rivkin and Akhundov.²¹ Very close to the critical temperature, the deviations of the data increase up to 0.1%, but the critical pressure recommended by IAPWS¹⁷ is represented within 0.05%. As discussed in Sec. 2, for a fixed critical temperature, the uncertainty of the IAPWS value is 0.01 MPa, which corresponds to a relative uncertainty of 0.05%. The EOS consequently represents the critical pressure within its uncertainty. We therefore estimate an expanded relative uncertainty in calculated vapor pressures at temperatures between 495 K and T_c of 0.05%.

A complete overview of the estimated relative uncertainties of calculated vapor pressures as discussed in this section is provided in Fig. 9.

Figure 8 shows that the new EOS enables a better description of the vapor-pressure curve than the EOS of Hill et al.¹¹ The most significant improvements were obtained at temperatures below 400 K, where the equation of Hill et al.¹¹ was obviously fitted to the less reliable data of Pupezin et al.⁴⁰ Furthermore, Hill et al. did not consider the very accurate data of Besley and Bottomley.⁵⁴ Figure 8 also highlights that the previous standard EOS should not be used in the critical region. In fact, numerical issues with the EOS of Hill et al.¹¹ prevented us from calculating vapor pressures at temperatures higher than approximately 638 K.

In comparison to the previously discussed situation for vapor pressures, the data for saturated-liquid and saturated-vapor densities are quantitatively and qualitatively quite limited. Thus, none of the available datasets was used to fit the EOS. Deviations of the data for both properties are shown in Fig. 10.

The saturated-liquid density data cover almost the entire phase boundary from the triple point up to the critical temperature. The new EOS represents most of these experimental data within 0.5%, except for some points close to the critical temperature. The newest reference was published by Mursalov et al.,⁷¹ covering the complete temperature range of vapor–liquid equilibrium. The authors state an uncertainty of 0.05% in temperature and 0.04% in volume. Based on this information, the estimated combined uncertainties of these data range from 0.1% to 0.4% for temperatures up to 640 K and increase closer to the critical temperature. Except for one point, our EOS represents this dataset within its experimental uncertainty. The most valuable experimental results were measured by Grossmann-Doerth, who published two remarkably accurate datasets in 1955 and 1956.^68,69 The measurements were carried out relative to H₂O. The author states an uncertainty in the ratio between the mass density of D₂O and H₂O of 3 × 10⁻⁵. The saturation temperature is reported with an uncertainty of 0.1 K. Hence, the combined expanded (k = 2) uncertainty of the saturated density is estimated to be below 0.03%. Although the EOS was not fitted to these data, all points exhibit deviations less than 0.02%, which underlines the high accuracy of the data and their consistency with the vapor pressures and the homogeneous-density data used to fit the EOS. Hebert et al.⁷⁰ published their saturated-liquid data with an uncertainty estimate of 1%, which seems too pessimistic. Although none of the data points was used for fitting the EOS, all deviations are within 0.5% (except close to the critical temperature). The publication of Costello and Bowden⁶⁷ does not provide any statement on the experimental uncertainty. The maximum deviation of the data is 0.5%.

Aside from the data of Grossmann-Doerth,^68,69 none of the available datasets is accurate enough to yield appropriate uncertainty estimates for saturated-liquid densities calculated from the EOS. It is, therefore, important to consider that the density at saturation is not a completely independent property, but a subset of the homogeneous density close to the phase boundary ρ(T, p). This implies that fitting the EOS to accurate homogeneous densities leads to an accurate description of saturated densities if high-quality saturation pressures p_v(T) are available. Hence, the combined expanded uncertainty in saturated density can be estimated reasonably by adding in quadrature the uncertainty in homogeneous density u_c(ρ) near the phase boundary and the uncertainty contribution of the vapor pressure. The latter can be determined from the relative uncertainty in vapor pressure estimated in Sec. 5.1.1 [U_r(p_v) = Δp_v/p_v in Fig. 9] multiplied by p_v and the isothermal sensitivity coefficient (∂ρ/∂p)_Tsat calculated from the EOS for the homogeneous phase in the limit of the saturation boundary. The uncertainty of the homogeneous density close to the phase boundary can be reasonably estimated by multiplying the relative uncertainty of the homogeneous density by the saturated density. The final combined uncertainty in saturated density u_c(ρ_sat) is then

Over most of the temperature range, the liquid density is not very sensitive to the pressure, and the uncertainty in saturation pressure can be neglected; thus, the uncertainty in the saturated-liquid density should, to a good approximation, be no larger than the uncertainty in the homogeneous-liquid density. An uncertainty analysis for this property is discussed in detail in Sec. 5.2. Between the triple-point temperature and 315 K, the uncertainty in liquid density is 0.04%, which we adopt as the uncertainty of the saturated-liquid density in this temperature range. The data of Grossmann-Doerth^68,69 cover a temperature range from 333 K up to 434 K. Due to the high quality of these data, the relative uncertainty of calculated saturated-liquid densities in this region is estimated to be 0.04%. We therefore define the uncertainty from the triple-point temperature up to 435 K to be 0.04%. We can reasonably extend the 0.04% uncertainty region up to 600 K, corresponding to the uncertainty in liquid density (see Sec. 5.2). At higher temperatures, the sensitivity coefficient (∂ρ/∂p)_Tsat becomes larger, which leads to slightly higher combined uncertainties. Our estimated uncertainty is therefore increased to 0.05% between 600 K and 630 K. The phase boundary at temperatures above 630 K is within the critical region (discussed in Sec. 5.2), where the uncertainty in density increases considerably. The uncertainty of the critical density recommended by IAPWS¹⁷ and given in Sec. 2 is about 1.5%. Because the EOS represents this value with a negligible deviation, the uncertainties in calculated saturated-liquid densities are estimated to increase linearly from 0.05% at 630 K to 1.5% at the critical temperature. The results of the uncertainty analyses for calculated saturated-liquid densities over the entire temperature range of the phase boundary are summarized in Fig. 9.

The saturated-vapor density was measured by Hebert et al.⁷⁰ at temperatures above 448 K. At lower temperatures, no experimental data are available. Hebert et al.⁷⁰ state that between 200 °C and 370 °C their data for light water deviate by about 10% from previous literature values. At higher and lower temperatures, these deviations are even larger. Consequently, comparisons of their data for heavy water provide no valuable information on the uncertainty of the EOS. The above-discussed publication by Mursalov et al.⁷¹ contains densities of the saturated vapor at temperatures above 572 K. Considering the given information on their accuracy in temperature and volume, the relative expanded (k = 2) uncertainties of their data are within 2% below 630 K. At higher temperatures, the experimental uncertainties increase significantly. The EOS represents almost all data points within their uncertainty. Analogous to the experimental saturated-liquid data, the overall data situation for the saturated-vapor density does not allow for reasonable uncertainty estimates for values calculated from the EOS. Therefore, our uncertainty estimates are based on Eq. (18). Unlike for the saturated-liquid density, the uncertainty of the saturation pressure is essential for the accuracy of calculated saturated-vapor densities. We considered the estimated uncertainties in homogeneous vapor density (see Sec. 5.2), the uncertainty in vapor pressure (summarized in Fig. 9), and the sensitivity coefficient (∂ρ/∂p)_Tsat calculated from the EOS. The uncertainties in saturated-vapor densities calculated in this way are 0.06% between the triple-point temperature and 300 K [where U_r(p_v) = Δp/p = 0.05% and U_r(ρ) = Δρ/ρ = 0.03%], 0.15% between 300 K and 350 K [U_r(p_v) = U_r(ρ) = 0.1%], 0.5% between 350 K and 450 K [U_r(p_v) = 0.03%, U_r(ρ) = 0.5%], and 0.15% between 450 K and 600 K [U_r(p_v) = 0.05%, U_r(ρ) = 0.1%]. At temperatures above 600 K, the sensitivity of the saturated-vapor density to the vapor pressure, (∂ρ/∂p)_Tsat, and the total uncertainty in density, U_r(ρ) × ρ, increase significantly. Therefore, we estimate a linear increase in the uncertainties of calculated saturated-vapor densities from 0.15% at 600 K to 1.5% at the critical temperature. All results of the uncertainty analysis for calculated saturated-vapor densities are illustrated in Fig. 9.

For the EOS of Hill et al.,¹¹ no uncertainty analysis for saturated densities was published. Due to the poor data situation, a clear statement about the accuracy of that equation for these properties is difficult; however, the calculated saturated densities from both EOS agree within our estimated uncertainties. But the new EOS allows for a reliable description of saturated densities in the critical region where it is not recommended to calculate state properties from the previous standard EOS.

In Table 10, the available experimental data for homogeneous densities are summarized. In addition to the overall AAD, Table 10 provides separate AAD for the vapor, liquid, critical, and supercritical state regions. The supercritical region is subdivided into three areas: the region of low densities (ρ/ρ_c < 0.6), of medium densities (0.6 ≤ ρ/ρ_c ≤ 1.5), and of high densities (ρ/ρ_c > 1.5).

Aside from speed-of-sound measurements, homogeneous density data, also called “pvT data,” are often the most accurate experimental data available. The majority of the experimental values included in fitting the EOS are pvT measurements, and comparisons of the available pvT data and calculated values are important to evaluate the EOS. Due to the large amount of experimental data, only the most important datasets can be discussed here. An overview of the database and deviations from the new EOS and from the EOS by Hill et al.¹¹ is given in Fig. 11. The most obvious improvements in accuracy are shown in color, namely, the description of the two accurate and comprehensive vapor and liquid-phase datasets of Kell et al.^88,89 and the high-pressure data of Bridgman.³⁴ 

In the homogeneous vapor phase, the EOS was exclusively fitted to the experimental data of Kell et al.⁸⁹ This dataset comprises the largest number of data points (more than 600) and is considered to be one of the most reliable experimental studies on heavy water. The data range from 423 K to 774 K at pressures up to 37 MPa. Deviations of the data from the new EOS versus temperature and pressure are shown in Fig. 12.

Despite the overall high quality of these data, the corresponding publication has few details about experimental uncertainties. The authors state their uncertainties in density to be between 0.1 mol m⁻³ and 0.3 mol m⁻³,⁸⁹ which we interpret as standard uncertainties in the density measurement, not including any effects of temperature or pressure uncertainty. Earlier papers of Kell and co-workers,^105–107 in which their experimental apparatus is discussed in detail, state uncertainties of 2 mK in temperature and 100 Pa in pressure. Considering this information, we calculated combined expanded (k = 2) uncertainties for every state point that range from 0.0006 mol dm⁻³ to 0.0010 mol dm⁻³. Except for a few points, the EOS represents the data within their experimental uncertainties. Since the data cover a wide range of densities (0.03 mol dm⁻³ to 8.8 mol dm⁻³), their relative uncertainties as well as the deviations from the EOS vary considerably. However, excluding the two lowest isotherms, the EOS represents more than 95% of the data within 0.1%, which we adopt as an uncertainty estimate for the calculated vapor densities between 450 K and 775 K including the supercritical gas-like fluid at pressures up to 30 MPa. The measurements of the two lowest isotherms (423 K and 448 K) are known to be less accurate, as discussed in detail in the IAPWS-95 publication for ordinary water.³² Based on the deviations of these data, the estimated uncertainty of calculated vapor densities in this temperature range is 0.5%. At temperatures lower than the temperature range investigated by Kell and co-workers, there are no experimental vapor density data available. However, we can still obtain a reasonable estimate of the uncertainty of the EOS in this region. Because the pressure in this region does not exceed 0.5 MPa [p_v(423 K) = 0.46 MPa], the vapor density can be described by a virial equation truncated after the second virial coefficient. The EOS accurately reproduces the values of the second virial coefficient B(T) obtained from first principles (see Sec. 5.3). Any uncertainty in B(T) translates directly into a relative uncertainty in Z − 1, where Z is the compressibility factor p/ρRT. A conservative estimate of a 10% expanded uncertainty in B(T), taken at the maximum pressure p_v where the nonideality is greatest, produces the uncertainty estimates shown in Fig. 13. Between 350 K and 425 K, this estimate conveniently matches the uncertainty estimated based on the two lowest isotherms measured by Kell et al.⁸⁹ Thus, the estimate of a 0.5% uncertainty of calculated vapor densities is extended to the temperature range from 350 K to 450 K. At lower temperatures down to 300 K, the uncertainty is within 0.1%, and at temperatures below 300 K, it does not exceed 0.03%. In fact, the real uncertainties in density are even smaller at the lowest pressures shown in Fig. 13, because in the low-pressure limit the density approaches that of the ideal gas, which is known almost exactly. In Fig. 14, selected experimental data including the vapor-phase data of Kell et al.⁸⁹ are shown along various (quasi-)isotherms. The solid lines represent the results of the previous standard EOS of Hill et al.¹¹ Comparing the representation of the vapor-phase data of Kell et al.⁸⁹ with the new EOS and the EOS of Hill et al.¹¹ shows that the previous EOS enables a comparably accurate description of the data for the lowest three isotherms (423 K to 473 K). At higher temperatures, values calculated from the EOS of Hill et al.¹¹ exhibit larger deviations from this dataset. The new EOS yields considerably more accurate results (see also Fig. 11), especially at elevated pressures and densities. We consequently note that the description of densities in the homogeneous vapor phase is improved by the EOS presented here.

Highly accurate densities of liquid heavy water from its melting curve up to 315 K and at pressures up to 100 MPa were recently measured by Duška et al.⁷⁹ at the Institute of Thermomechanics in Prague. The dataset also includes measurements in the metastable subcooled-liquid region that are discussed in Sec. 5.5. In personal communications, the experimentalists estimated the so-far unpublished data to be accurate within an expanded (k = 2) uncertainty of 0.04%. Figure 14 shows that this uncertainty is confirmed by the accurate datasets of Emmet and Millero⁸⁰ and Aleksandrov et al.⁷⁶ Since the EOS describes all the data of Duška et al.⁷⁹ within their experimental uncertainty, the estimated uncertainty of liquid densities calculated from the present EOS is 0.04% at temperatures up to 315 K and pressures up to 100 MPa. At temperatures between 315 K and 423 K, the most reliable liquid densities at pressures exceeding 1 atm (0.101 325 MPa) were published by Tsederberg et al.¹⁰⁰ The authors state an accuracy in specific volume of 0.0015 cm³ g⁻¹, which we interpret as the standard uncertainty of their results. The corresponding relative uncertainties are between 0.05% and 0.06%, which would be equivalent to an expanded uncertainty of about 0.1%. However, although the EOS was not fitted to these data, all data points between 315 K and 423 K are represented within 0.06%. Thus, we estimate the expanded uncertainty of calculated liquid densities between 315 K and 425 K at pressures up to 100 MPa to be 0.07%.

Figure 14 shows that liquid densities calculated from the present EOS and the previous standard EOS are in quite good agreement within the region from the melting curve up to 425 K and pressures up to 100 MPa. In fact, deviations between the two EOS are within the uncertainties of the data.