The density of water is particularly important in metrology. It is used as a reference in most cases to determine the density and volumes of solids. The value of the density of water is calculated with a formula in which the pressure and the temperature of the fluid are input parameters.

1. Available formulas

Several standards and documents used in legal metrology provide charts or simplified polynomial formulas to calculate the density of water (e.g. ). Generally these charts and formulas were developed from measured data from the first half of the twentieth century and are not provided with uncertainties, i.e. an uncontrolled accuracy and traceability. From the 1990s, several experiments to measure the density of water were made by several national metrology institutes around the world [2, 3, 4, 5]. This work brought the basis for the development of two formulas to calculate the density of water with a known uncertainty. One was issued under the aegis of the International Committee for Weights and Measures (Comité international des poids et mesures, CIPM) and the other of the International Association for the Properties of Water and Steam (IAPWS).

Both formulas apply to water called VSMOW (Vienna Standard Mean Ocean Water), that is a reference material defined by the IAEA in 1968. This is a pure water containing no air, with the following isotopic composition:
- 0,000 155 76 mole of $$^{2}\text{H}$$ per mole of $$^{1}\text{H}$$ ;
- 0,000 379 9 mole of $$^{17}\text{O}$$ per mole of $$^{16}\text{O}$$ ;
- 0,002 005 2 mole of $$^{18}\text{O}$$ per mole of $$^{16}\text{O}$$.

2. CIPM formula

2.1. VSMOW water

In 2001, the CIPM has released a formula to calculate the density of water from 0 °C and 40 °C . The density of water is modeled using Thiesen formula (1) which was proposed in the early twentieth century:

 $$\rho(t) = a_{5} \cdot \left[ 1 - \dfrac{\left(t + a_1\right)^{2} \cdot \left(t + a_{2}\right) }{a_{3} \cdot \left(t + a_{4}\right)} \right] \text{,}$$ (1)

with t the water temperature in degrees Celsius and a1, …, a5 five coefficients. A redetermination of these coefficients has been performed more recently in  by considering the results of four experiments:

• measures of relative density of Takenaka and Masui  (measurements of the volume change of water between 0 °C and 85 °C with a quartz dilatometer);
• measures of relative density of Watanabe  (a measure of the buoyancy of a quartz mass in water from 0 °C and 44 °C);
• the density measurements of Patterson and Morris  (a measurement of the buoyancy force on a sphere with a known mass value submerged in water between 1 °C and 40 °C);
• the measures of density by Masui, Fujii and Takenaka  (a measurement of buoyancy on a quartz sphere of known mass and volume immersed in water at 16 °C).

The following values were determined by the chi-square method:

 $$\left\{\begin{array}{l} a_{1} = -3,983~035~\text{°C} \\ a_{2} = 301,797~\text{°C} \\ a_{3} = 522~528,9~\text{°C}^{2} \\ a_{4} = 69,348~81~\text{°C} \\ a_{5} = 999,974~950~\text{kg} / \text{m}^{3} \end{array} \right.$$ (2)

The uncertainty on the density calculated from (1) and (2) was modeled in  by the following 4th degree polynomial (k = 2):

 $$U_{p} = b_{1} + b_{2} \cdot t + b_{3} \cdot t^{2} + b_{4} \cdot t^{3} + b_{5} \cdot t^{4} ~\text{,}$$ (3)

with

 $$\left\{\begin{array}{l} b_{1} = 8,394 \times 10^{-4}~\text{kg} \cdot \text{m}^{-3} \\ b_{2} = -1,28 \times 10^{-6}~\text{kg} \cdot \text{m}^{-3} \cdot \text{°C}^{-1} \\ b_{3} = 1,10 \times 10^{-7}~\text{kg} \cdot \text{m}^{-3} \cdot \text{°C}^{-2} \\ b_{4} = -6,09 \times 10^{-9}~\text{kg} \cdot \text{m}^{-3} \cdot \text{°C}^{-3} \\ b_{5} = 1,16 \times 10^{-10}~\text{kg} \cdot \text{m}^{-3} \cdot \text{°C}^{-4} \end{array} \right.$$ (4)

2.2. Additional corrections for "usual" water

These corrections are included with the CIPM formula .

2.2.1. Correction due to the pressure

The formula giving the density of water was calculated for an atmospheric pressure equal to 101 325 Pa. Since water is compressible, it is necessary to correct the value of density calculated for different pressures. The multiplication factor given by the formula (5) can be used:

 $$C_{p} = 1 + (c_{1} + c_{2} \cdot t + c_{3} \cdot t_{2}) \cdot (p - 101~325)~\text{,}$$ (5)

with t water temperature, p atmospheric pressure and the following coefficients (6):

 $$\left\{\begin{array}{l} c_{1} = 5,074 \times 10^{-10}~\text{Pa}^{-1} \\ c_{2} = -3,26 \times 10^{-12}~\text{Pa}^{-1} \cdot \text{°C}^{-1} \\ c_{3} = 4,16 \times 10^{-14}~\text{Pa}^{-1} \cdot \text{°C}^{-2} \end{array} \right.$$ (6)

The validity domain of pressure and uncertainty associated with this correction is not provided.

2.2.2. Correction due to the presence of dissolved air

The formula giving the density of water was calculated assuming that there was no air dissolves in water. The correction for determining the density of air saturated water is obtained for a temperature t between 0 °C and 25 °C using the formula (7):

 $$C_{w} = d_{1} + d_{2} \cdot t~\text{,}$$ (7)

with

 $$\left\{\begin{array}{l} d_{1} = -4,612 \times 10^{-3}~\text{kg} \cdot \text{m}^{-3} \\ d_{2} = 0,106 \times 10^{-3}~\text{kg} \cdot \text{m}^{-3} \cdot \text{°C}^{-1} \end{array} \right.$$ (8)

Remark : in practice the water is not saturated, but knowledge of the maximum error to insert a contribution to the uncertainty calculated by a uniform law.

2.2.3. Correction due to the isotopic composition

The formula giving the density of water concerns VSMOW water. This water differs in isotopic composition of tap water. For tap water, it is common to substitute the coefficient in (2):

 $$a_{5} = 999,974~950~\text{kg}~/~\text{m}^{3}~\text{,}$$
by
 $$a_{5} = 999,972~\text{kg}~/~\text{m}^{3}~\text{.}$$

2.3. Limitation of the additional corrections

The main difficulty with these additional correction is that their domain and thus their uncertainty are not defined. This leads to loose the interest of the accuracy of the CIPM formula. In particular to work on larger ranges of temperature and pressure, it is advised to use the formula of IAPWS.

3. IAPWS formula

In 1995, IAPWS has developed a formula named IAPWS-95 to determine the Helmholtz free energy function. This formula is too complex to be detailed in these lines. It is provided in reference . By applying some algebraic operations, water parameters can be obtained: pressure, internal energy, entropy, enthalpy, heat capacity, sound velocity, density... The formula determined by the IAPWS is defined between the melting curve of water to a temperature of 1 273 K and a pressure equal to 1 GPa (Fig. 1). Fig. 1. - Phase diagram of water. The IAPWS formula covers different states
of water: gas, liquid and supercritical.

IAPWS-95 formula is provided with uncertainties in the form of a graph (Fig. 2). These uncertainties are not calculated with the GUM method . In fact, the values indicated by the IAPWS arise from differences found by comparisons between the formula and experimental data that are traceable to national standards. In other words, the intervals provided by the IAPWS ensure traceability, which is the primary function of uncertainty, but are not calculated by statistical methods. Since intervals have been chosen to take into account the maximum deviation, they can be considered to correspond to uncertainties with a coverage factor equal to 2. IAPWS-95 formula should not be confused with the industrial formulation of IAPWS, named IAPWS-IF97 . This second formulation contains simpler formulas than in IAPWS-95. These formulas are valid on limited areas of the phase diagram and are less accurate than the IAPWS-95 formulation. In other words, the IAPWS-IF97 is not suitable for use in metrology at the highest level of uncertainty. Fig. 2. - Phase diagram of water with the uncertainties on
the density obtained with the IAPWS-95 formula.

The uncertainty values shown in figure 2 correspond to the formula uncertainty. To obtain the uncertainty on the density it is necessary to take into account the uncertainties on the temperature and pressure. The model is expressed as follows:

 $$\rho = \rho_{~\text{IAPWS}}(p,~t) + e_{\text{modélisation}}~\text{.}$$ (9)

Applying the law of propagation of variances to the formula (9), the uncertainty is given by the relationship:

 $$u(\rho) = \sqrt{ \left( \dfrac{ \partial \rho_{\text{IAPWS}}}{ \partial t} \right)^{2} \cdot u^{2}(t) + \left( \dfrac{ \partial \rho_{\text{IAPWS}}}{ \partial p} \right)^{2} \cdot u^{2}(p) + u^{2}(e_{\text{modélisation}}) }~\text{.}$$ (10)

Remark : in practice the values of the derivatives of the density can be calculated approximately with (10).

4. Agreement beteween the formulas

The domains of definition of the two formulas are given in Figure 3. Fig. 3. - Definition domains of CIPM formula
and IAPWS-95 formula.

Figure 4 shows the difference between the densities calculated by both formulas on their common domain (between 0 °C and 40 °C, 101 325 Pa) and shows that they are in close agreement with the uncertainties. Fig. 4. - Difference between the densities calculated by the CIPM formula and the formula
of IAPWS. Vertical bars represent the sum of the expanded uncertainties of the two formulas.
The compatibility between the two forms is not contradicted because the vertical bars
intersect the axis: y = 0.

5. Selection of a formula

A joint recommendation between the CIPM and IAPWS has been issued for the selection of formulas [10, 11]. Its main conclusions are summarized hereafter.

1. Between 0 °C and 40 °C and pressures close to atmospheric pressure (101 325 Pa), the CIPM formula should be used. This formula offers the best uncertainty. It must not be extrapolated outside its definition domain.
2. The densities calculated by the CIPM formula and the IAPWS-95 formula are in close agreement with the uncertainties on the common domain of definition that matches the domain of the CIPM formula.
3. In case it is necessary to calculate densities in an area where the CIPM formula is not fully valid, it is preferable to use the IAPWS-95 formula to avoid discontinuities.

6. DensiCal software

DensiCal software can calculate the density of VSMOW water with:

• the CIPM formula between 0 °C and 40 °C assuming that the pressure is approximately equal to 101 325 Pa;
• IAPWS-95 formula between the melting curve (approximately 273.15 K, i.e. 0 °C) and 1 273.15 K, i.e. 1 000 °C) for pressures between 20 MPa and 611.211 Pa (i.e. from 0.006 11 bar and 200 bar).

6.1. Calculation methods

Regarding the CIPM formula, calculations are made with (1) and (2). Regarding IAPWS-95, density calculations are performed using the formula for calculating the pressure from the density and temperature (, Table 3). By keeping the notations of , this formula reads:

 $$p(\delta, \tau) = \rho \cdot R \cdot T \left( 1 + \delta \phi^{r}_{\delta} \right)$$ (11)

This formula does not express in literal form of density as a function of pressure and temperature. The method consist in seeking the roots canceling the function for a given temperature T and a given pressure p.

 $$\psi(\rho) = p - \rho \cdot R \cdot T \cdot \left( 1 + \delta \cdot \phi^{r}_{\delta} \right)~\text{.}$$ (12)

The discontinuity of the density on the change of states corresponds to a change in root: Figure 5 shows this function and the roots corresponding to the density of the gas phase and liquid phase at conditions of pressure and temperature close to the vaporization curve. Fig. 5. - Drawing of (12) for a pressure equal to 101 325 Pa and a temperature equal to 100 °C
as a function of density with views of the roots corresponding to the densities of gas phase
(0.597 61 kg/m3) and the liquid phase (958.349 01 kg/m3) shown as red dots.

6.2. Basic functions

6.2.1. Calculation using the IAPWS-95 formula

This option is available by clicking on the "IAPWS-95" caption located in the banner at the left of the screen (Fig. 1). The input data are the temperature and pressure. The density is displayed by clicking the button "calculate". Fig. 6. - Calculation of water density with IAPWS-95 formula.

Regarding the uncertainty associated with the value of density, IAPWS do not provides any formula but a state diagram of water with several areas in which expanded uncertainties are indicated as percentages. To obtain the uncertainty with DensiCal, just click on "state diagram" on the banner at the left of the screen, and raise the percentage below the red dot and apply it to the value of density (Fig. 7). Fig. 7. - Uncertainty associated to a calculated density
with IAPWS-95 formula. In this example, the expanded uncertainty
is: 998,207 15 × 0,000 1 / 100 # 0,0010 kg/m3.

When the pressure and temperature values are at the limit of the melting curve or saturation curve, a warning message appear. In the case of the melting curve, the densities corresponding to the vapor and liquid phases are provided. The melting and vaporization curves are calculated by approximate models. To take into account the approximations, the alert is triggered on a range of temperature around the melting and saturation curves. The values of these temperature ranges can be modified in the tab "Miscellaneous" of the "settings" window.

6.2.2. Calculation with the CIPM formula

This option is available by clicking on "CIPM" in the banner at the left of the screen (Fig. 8). This formula applies for a pressure value equal to 101 325 Pa. The remaining input data to enter is the temperature. By clicking the button "calculate" density is displayed. Fig. 8. - Calculation of density of water with the CIPM formula.

A warning is displayed when pressure and temperature values are at the limit of the melting curve. The melting curve is calculated by an approximate model. To take into account the approximations, the warning is triggered on a range of values of temperature around the melting curve. The temperature range can be changed in the tab "Miscellaneous" of the settings form.

6.2.3. Modelling curves of melting and saturation

This option is available by clicking on the button "models" and displays the form shown in Figure 9. It can calculate:

• pressure sauration at a given temperature from the model described in ;
• saturation temperature at a given pressure from the model described in ;
• melting pressure at a given temperature from the model described in ;
• melting temperature at a given pressure from the modelling described in . Fig. 9. - Modelling of melting and saturation curves.

6.3. Validation

6.3.1. Density

The density calculated using DensiCal was compared to 210 density values calculated by NIST . These values correspond to a temperature range from 0 °C to 1 000 °C and a pressure range from 0.01 MPa (0.1 bar) to 20 MPa (200 bar) shown in Figure 10. With NIST resolutions, there was no difference between the values of densities calculated by DensiCal and these provided by NIST. The values are provided in Table 1. Fig. 10. - Comparisons between DensiCal
and reference data from NIST. Each red
cross represents a comparison.

Table 1
Comparison between the densities calculated
by DensiCal and calculated by NIST.
The NIST data are taken from Table 3 of .

6.3.2. Saturation temperature

The aim of DensiCal is to calculate densities at the best level of uncertainty. The saturation curve is simply calculated to determine the phase. For this reason an approximate model was used to calculate this curve. The validation consist in determining the appropriate warning thresold. The saturation temperature calculated using DensiCal was compared with 19 values calculated by NIST . These values correspond to a range of pressures from 22.064 Pa to 611.657 MPa, i.e. covering the whole saturation curve. The results are provided in Table 2. With the NIST resolutions, the maximum deviation is 0.008 °C. Consequently, an alert threshold of fusion equal to 0.01 °C seems appropriate.

Table 2
Comparison between saturation temperatures calculated
by DensiCal and calculated by NIST. NIST data are
taken from Table 2 .