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Calculating Water Density
The Importance of Water
The density of pure water is important as a reference quantity, both historically and in the present day. The first definition of the kilogram in 1791 was that it was equal to the mass of one litre of pure water at the temperature at which it was at maximum density, thought then to be 4°C: This is equal to 999.975 modern grams, not too large an error.
Today, water is used as a reference in a variety of density measurement techniques and also in the measurement of volume standards. Despite its popularity, great care must be taken in the use of water as a standard, due to variations in density produced by several effects.
Standard Mean Ocean Water (SMOW)
One of the problems with using water as a standard is its variability in composition. When water was first used as a standard, the existence of different isotopes of oxygen and hydrogen was not known: it was assumed that it was made up of the most common isotope of oxygen (16O) and hydrogen (1H).
We now know that some water molecules contain other isotopes of oxygen (17O and 18O) and deuterium (2H). To overcome this variability, a definition of the isotopic composition of standard water was formulated. This standard water is known as Standard Mean Ocean Water (SMOW).
Tap water (at least in the UK), is substantially derived from rain water which is slightly lighter than SMOW. Once purified it tends to be lighter than SMOW, with a typical density about 0.003 kg/m3 less than SMOW at the same temperature.
Calculating Water Density from 0°C to 40°C
Measurements from two independent National Measurement Institutes, the National Research Laboratory of Metrology (Japan) [2] and the National Measurement Laboratory (Australia) [3], have contributed towards the latest table for the density of standard mean ocean water (SMOW) at an ambient pressure of 101325 Pascals (=1013.25 mbar or 1 standard atmosphere).
These laboratories have used spheres made from either fused quartz or Zerodur, an ultra low expansion glass, as artefacts. Sphere volume was measured using laser interferometry and their masses were measured by weighing in air. Sphere density was calculated from the mass and the volume.
The spheres were then suspended in pure water and weighed. This allowed the density of the water sample to be measured with an uncertainty of less than 1 part per million (3). The Japanese used three spheres to measure the density of eighteen different samples of water at 16°C, whilst the Australians used a hollow sphere to measure the density of nine samples at temperature between 1°C and 40°C. Both laboratories then corrected their densities to a standard atmosphere, and corrected for isotopic ratios.
This work has led to the following formula for the density of SMOW at temperature t°C (in the range 0°C to 40°C):

where
a1 = -3.983035±0.00067°C
a2 = 301.797
a3 = 522528.9
a4 = 69.34881
a5 = 999.974950±0.00084 kg/m3
Note that the maximum density of SMOW is a5 and this occurs at temperature a1. In practice, many people use a value of a5 = 999.972 kg/m3, to take account of the fact that they are using de-ionised or distilled tap water rather than SMOW (see Isotopic abundance correction
The uncertainty in this formula is temperature dependent and is given by:

The density of SMOW at 20°C is 998.206 7 ±0.000 83 kg/m3 when calculated using the formula quoted above.
The table below gives the density of air free pure water (SMOW) at standard atmospheric pressure (1013.25 mbar) using this equation.
The uncertainties in the quoted densities range from 0.000 83 kg/m3 at 0°C to 0.000 88 kg/m3 at 40°C
Calculating Water Density from 40°C to 85°C
The recommended water density equation is only valid up to 40°C. For temperatures above this, it is suggested that a formula developed at the National Research Laboratory of Metrology (Japan) [4] is used.
This work involved varying the temperature of a water sample in a sealed system. A mercury filled capillary tube offered the only potential path for expansion. As the water expanded, mercury was pushed out of the capillary into a weighing bottle whose change in mass was measured.
The density ρ at temperature t°C is given by:

This more complex formula gives the density of SMOW as 992.2167 kg/m3 at 40°C, while the formula in the previous section produces an answer of 992.2152 kg/m3. The uncertainty in this formula is quoted as 1.1 parts per million, although this is probably an optimistic estimate.
Applying Corrections to the Calculated Water Density
The formulae for the calculation of water density are for SMOW. There are several corrections that may need to be applied to this calculation, depending on the required uncertainty. These corrections are insignificant in many applications, but are included here for completeness.
Correcting for Disolved Air
The density of water varies with the amount of air dissolved in it. The standard water density equations assume air-free pure water, with corrections being recommended to take account of water becoming aerated.
At 20°C fully aerated water has a density lower than de-aerated water by just under 2.5 parts per million. Clearly such corrections are only necessary for the most demanding of work and can be ignored in most applications.
It is difficult to maintain water in a de-gassed state in practice, so many users chose to fully aerate their water prior use by vigorous agitation.
Over the temperature range 0°C to 25°C, the correction to the density given by the water density equation (in kgm-3) to take account of the full aeration of water at temperature t°C is given by:

Above this temperature, aeration of water is not a serious problem and most measurements at elevated temperatures do not demand the highest accuracy density uncertainties.
Example
What is the density of air saturated water at 12°C?
The density of SMOW at 12°C is 999.500 kg/m3.
Δρ the density correction is (–4.612 + (0.106 x 12))/1000 = –0.003 34 kg/m3.
The air saturated density is therefore 999.500 – 0.0033 = 999.497 kg/m3.
Correcting for Air Pressure
The standard table for water density is based on a barometric pressure of 101325 Pascal (1 atmosphere). For the very highest accuracy work it is sometimes necessary to make a correction for changes in air pressure.
The corrected density ρ(P), at pressure P in Pascal, is given by the following formula at temperature t between 0°C and 40°C:

where
k0 = 5.074 x 10-10
k1 = -3.26 x 10-12
k2 = 4.16 x 10-14
ρ = uncorrected density (in kg/m3)
Example
At temperature of 20°C and a pressure of 101 325 Pa (1013.25 mbar), the density of SMOW from the standard equation is 998.2067 kg/m3. The barometric pressure is 955 00 Pa (955.0 mbar), what is the change in the water density?
ρ(P) = 998.2067 x [1+(50.74•10-11-(0.326•10-11x20)+(0.000146x202))x(95500-101325)]
= 998.2067 – 998.2041
= 0.0026kg/m3
This change of about 3 parts per million for such a large change in ambient pressure illustrates the fact that this correction is only important for the highest accuracy work.
Correcting for Isotopic Composition
The measurement of isotopic composition is extremely specialised and such data is generally not available for water in use. Fortunately the corrections for changes in this parameter are small, so are rarely applied in day-to-day work.
When the isotopic composition of a sample is known, the following equation is be used to correct the density:
ΡS = ΡSMOW + (0.2333δ18 + 0.0166δd)
where
δ18 = [R18/R18(SMOW)]-1 = [R18/1993x10-6]-1
δ18 = [Rd/Rd(SMOW)]-1 = [R18/156x10-6]-1
where R18 is the ratio of 18O to 16O and Rd is the ratio of deuterium to hydrogen
Preparation of Water
The density of water whose preparation is undertaken using the best procedures is known to about three parts in a million. Such preparation involves the multiple distillation of de-ionised water and its collection in clean glassware.
However, it is not always possible, or necessary, to produce water in this manner, due to factors such as the availability of distilling apparatus, or the time and expense of producing large volumes of distilled water (for example in the calibration of large volume measures, or the hydrostatic weighing of large weights).
A simple low-cost alternative to distilled water is de-ionised water. Perhaps the most straightforward method of producing de-ionised water is to use an in-line de-ionisation cylinder to purify tap water. This requires some maintenance, as the cylinder’s age causing a loss in de-ionising performance. A simple check on the performance of the de-ionising system is to measure the conductivity of the output water on a daily basis (conductivity meters are built into some commercial systems).
In some applications it is more convenient and cost effective to use tap water, due to the extremely large volumes of water required. In such cases it is advisable to use a density meter to check the density of the water prior to each use, as there is a significant variation in the density of mains water, both with time and location. The solubility of air in water decreases quickly as the temperature increases. Tap water is often almost saturated with air at about 12°C. If this water is now allowed to warm to room temperature, say 20°C, air can come out of solution, and coat surfaces with micro air bubbles.
Care should be taken when carrying out multiple distillations, since it is possible for isotopic fractionation to occur. Tests have shown that carrying out two additional distillations can cause the water density to decrease by 0.005 kg/m3. This is due to the fact that water molecules made up of different isotopes have different boiling points, and also different densities. Heavy water (deuterium oxide 2H2O) boils at 101.395°C compared to 99.974°C for “tap” water, and has a density of 1105.3 kg/m3 at 20°C.
Microbes
If a container of pure water is left open to laboratory air for even a short while, the appearance of microbes is almost inevitable, often members of the Pseudomonas family. These seem to grow more quickly in deionised water rather than distilled water.
At low concentrations, these cause an miniscule change to water density, but they do tend to cling to artefacts and suspension wires, so altering the apparent density of the artefact, and appearing to cause irregular changes in surface tension effects. At higher concentrations they both change water density and become a health hazard.
If an object, such as a weight or sphere, which is coated in a film of Pseudomonas is allowed to dry, it can be difficult to remove the dried
microbes without rubbing the surface hard with wet cleaning tissues, obviously undesirable practise with a density artefact or a weight.
We therefore recommend that whenever an object is removed from a water bath, it should be flushed with a strong jet of distilled water before being allowed to dry.
Most microbes are killed if the biocide 2-Bromo-2-nitro-1,3-propanediol (commonly called Bronopol) is added to the water at a concentration of about 50 milligrams per litre. This causes an increase of water density 0.022 kg/m3
Solid Bronopol is toxic by inhalation, in contact with skin, and if swallowed. However, at a concentration of 50 parts per million, it is believed to present no health hazard.
Liquid Head Correction
When an object is suspended in a liquid, such as water, the liquid pressure around the object is higher than the atmospheric pressure at the surface due to the weight of liquid this increase in pressure called here dp is given by the following equation:
dp = h(ρL - ρa)g
where
dp is the pressure in the liquid in Pa (1Pa – 0.01mbar)
h is the depth of immersion in m
ρl is the density of the liquid in kg/m3
ρa is the density of air in kg/m3 (typically 1.2)
g is the acceleration due to gravity in m/s2 (standard value is 9.80665)
Liquid density depends on the pressure, so the liquid head correction dρ must be added to the atmospheric pressure at the surface before calculating the density of the liquid.
Example
A sphere is suspended in a bath of pure water. The centre of the sphere is 1.27 m below the water surface. The water temperature is 22°C, and the atmospheric pressure is 1013 mbar.
What is the water pressure and water density around the sphere.
1013 mbar is 101 300 Pa.
The density of water at 22 and 1013 mbar is given in section 19.2.1 as 997.773 kg/m3
The head pressure Dp is therefore 1.27 x (997.773 – 1.2) x 9.806 65 which equals 124 12 Pa or 124.12 mbar.
The water around the sphere is therefore at a pressure of (1013 + 124.12) = 1137.12 mbar.
At a temperature of 22°C, this gives a density of 997.778 kg/m3.
References
1. Bignell N, Metrologia, 1983, 19, 57-59
2. Masui R, Fuji K and Takenaka M and, Metrologia, 1996, 32, 333-362
3. Patterson J.B and Morris E.C, Metrologia, 1994, 31, 277-288
4. Takenaka M and Masui R, Metrologia, 1990, 27, 165-171
5. Errata, Metrologia, 1991, 28, 107
6. Tenakla M., Gerard G., Davis R., et al, Metrologia, 2001, 38, 301-309