Density and density measurement

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Density (“true density”) is the relation of mass and volume. As mass is independent of external conditions, such as buoyancy in air or gravity, it corresponds to weight in vacuo. The symbol of density is the Greek letter rho ρ:

$$\rho = {mass \over volume} \left[ {g \over cm^{3}} \right] or \left[ {kg \over m^{3}} \right]$$

Equation 1: The true density ρ in kg/m³ or g/cm³ of a liquid is defined as its mass m divided by its volume V. The mass m corresponds to the weight in vacuum and is independent of external conditions such as buoyancy in air or gravity.

• See Density units for more information
• Read more about the relationship between mass and density

A multi-layered cocktail (Figure 1): Fluids of higher density such as juices or syrup will sink; they are heavier and have less buoyancy. Fluids of lesser density such as alcohol or water have more buoyancy, they swim on top. 

A material’s volume and state changes with temperature. Temperature therefore has an important influence on the density. Consequently, an accurate density measurement requires accurate temperature determination and good temperature stability^[1].

An excellent example of the temperature dependence of density is the thermometer. With increasing temperature, the volume of alcohol inside the thermometer expands and rises. Same mass but more volume means less density.

Temperature is a vital factor for precise density measurement, in which you need precise temperature control or algorithms for compensation. A temperature difference of 0.1 °C might result in a density error of up to 0.0001 g/cm³.^[2]

The local air pressure and altitude also have an impact on the density of liquids and modern density meters compensate for these influencing factors so they do not affect the results.

Depending on how you measure density, the viscosity of the substance may affect the result. When measuring with a digital density meter, for example, the oscillation frequency measured not only depends on the density of the filled sample but also on its viscosity. Due to the oscillation of the tube (see here for more information on the technology used), shear forces (a sort of friction) occur between the fluid and the tube wall and result in damping. Damping increases with increasing viscosity of the sample and that results in a density over-reading (the density value shown is too high^[3]). Modern density meters compensate this effect and automatically perform a viscosity correction using a special technique in which two different oscillation modes are applied.^[4] 

Water is a unique liquid and reaches the density maximum at a temperature of 3.98 °C. Starting at 3.98 °C upwards, the volume of water increases and it becomes less dense. The same applies when water is cooled, just the other way round.^[5] This anomaly causes lakes to freeze from the top down and water that is colder than 4 °C to freeze and swim on top.

A physical property like density is investigated for several reasons and is therefore reported in several units. The most frequent density unit is kilogram per cubic meter (kg/m³), used in petrochemistry, for example. In other industries, density is reported in gram per cubic centimeter (g/cm³). The conversion factor in this case is 1000 (1 g/cm³ = 1000 kg/m³). Density might also be reported in kilogram per liter (kg/L) or gram per liter (g/L) or converted into a concentration value (to represent, for example, API numbers, °Brix, °Plato).

True density ρ is often confused with apparent density ρ_app. The apparent density of a sample is weight in air per volume:

$$\rho _{app} = {W \over V}$$

Equation 2: The apparent density ρ_app of a sample is defined as the weight in air W divided by the sample’s volume V.

The values of apparent density and true density are different, even if their units are identical. The true density of air at 20 °C as measured in a density meter is 0.0012 g/cm³ whereas the apparent density of air at 20 °C is 0.0000 g/cm³ – air on a balance does not give a reading.

Apparent density can be calculated from true density considering the buoyancy of the sample in air and the weight and density of a reference weight in steel or brass. Nowadays, steel is defined as the material of choice for the weights. Earlier, brass was used.

$$\rho _{app} = {\rho _{true}- \rho _{air} \over 1- \cfrac{\rho _{air}}{\rho _{steel \hspace{1mm} or \hspace{1mm} brass}}}$$

Equation 3: Converting the true density of a sample ρ_true into apparent density ρ_app. This conversion takes the true density of air (ρ_air≈ 0.0012 g/cm³) and true density of brass (ρ_brass = 8.4 g/cm³) or steel (ρ_steel = 8.0 g/cm³) into account.

When do we use the apparent density? When determining filling volumes with a balance from a density result, the apparent density is the required value, for example.

Another vital unit for reporting density is Specific Gravity (SG) as found in several industries. Specific gravity is the measured density of a sample divided by the density of water at a certain temperature. For this reason, it is also called relative density (RD) because it is related to the density of water.

$$SG= {\rho _{true} \over \rho _{W}}$$

Equation 4: Specific Gravity (SG) is defined as sample density ρ divided by the density of water ρ_W at a specific temperature.

SG_20/4 means sample density at 20 °C divided by water density at 4 °C. SG_20/20 means sample density at 20 °C divided by water density at 20 °C. Comparing the results of the same sample SG_20/20 and SG_20/4 shows a difference because the water density is different at 4 °C and 20 °C.

The apparent specific gravity SG_app (sometimes referred to as apparent relative density D_app) is dimensionless, which means it has no unit. It is calculated by dividing the apparent density of a sample ρ_app by the apparent density of pure water ρ_{app, water} at defined temperatures.

$$ {D_{app}{20\over20}} = SG_{app}{20\over20}={{{\rho _{app}}at20^{\circ}C}\over{{\rho _{app, water}}at20^{\circ}C}} $$

Equation 5: Apparent specific gravity D_appT/T (or SG_appT/T) is related to defined temperatures of the apparent density of the sample (ρ_app) and pure water (ρ_{app, water}).

Depending on the measurement – and also on the industry – different units are used. For example: 

Density measurement has a long history – it actually dates back 2200 years to the times of King Hieron II of Syracuse. In a nutshell, King Hieron II had a crown made of pure gold. The goldsmith was accused of having added some silver to the crown. Archimedes, a great mathematician, physicist, engineer, inventor and astronomer, was asked to prove – or disprove – that Hieron’s crown consisted of nothing but gold, without destroying it.

Archimedes went home to think about the problem. Exhausted from all the pondering over this difficult task, he took a bath. (Okay, this part of the story is probably made up but read on to see what he found.) When he sank into a very full bathtub, it started to overflow. He realized that the overflowing water stood in direct relation to the immersion of his body volume into the water. We all know what he did next: he ran across town – naked – yelling: ‘Eureka!’ [I found it!].

In order to verify the gold content of King Hieron’s crown, he took the crown and a bar of pure gold of the exact same weight as the crown. He immersed both objects in a water container. The crown displaced more water than the gold bar, the water went up higher – so the crown obviously had more volume. It had the same mass, but more volume – that means lower average density. The crown was less dense than pure gold, so the gold had to have been mixed with another metal. This experiment helped Archimedes discover the relation of mass to volume.^[6][7] Hydrometers and the hydrostatic balance are both based on the Archimedes principle.

For more information on the relationship between mass and density, see here.

In the 4th century Hypatia of Alexandria, a philosopher, mathematician, teacher, and first female contributor to mathematics in ancient times, developed the hydrometer (which was then called a hydroscope) for measuring the density of liquids.^[8][9]

Weighing precious metals in air and then in water was a common practice among jewelers in Europe when Galileo Galilei, the famous Italian mathematician, astronomer, and physicist, described an instrument in the 16th century that is still used for high-precision density measurements today: The hydrostatic balance. Again, an object is immersed in the fluid, but here the object is attached to a highly sensitive balance, and the density values are read from the movement of the counterweight.^[10]

In 1967 the first-ever digital density meter, DMA 02 C, was presented at the Achema exhibition in Frankfurt by Anton Paar, a high-tech company based in Graz, Austria. DMA 02 C caused a sensation: For the first time it was possible to measure the density of a sample digitally – and that with a precision of 10-6 g/cm³ with a very small volume of a few milliliters. 

Not all density meters are created equal. Density meters differ – sometimes significantly -- depending on the accuracy class, manufacturer, and the characteristics of the samples to be measured.

For an overview of Anton Paar’s benchtop density meters, see here.

You can also measure density in the field with a portable density meter, even in hazardous environments with risk of explosions. See here.

For an overview of commercially available density meters and their features and benefits, see here.

The following describes the parts of a benchtop density meter which all such devices have in common. For density measurement “in the field” there are smaller, lighter, and portable density meters available which are also intrinsically safe.

For a comparison of the different oscillating U-tube measuring methods, see here.

The sensor is the heart of the digital density meter. Typically, the sensors are mostly straight or U-shaped tubes. They can be made of glass, e.g. borosilicate glass 3.3, metals or metal alloys, or plastics depending on the application and resistance towards the sample and cleaning agents.^[22] This page gives details about the 3 different-shaped U-tubes available.

A countermass is linked to the measuring tube to reduce parasitic resonances (“external oscillations”) from other components, e.g. electronic parts. It is linked to the housing of the density meter by elastic supports and acts like a mechanical filter for external oscillations. The countermass has a resonance frequency that lies far below the frequencies used for density measurement. The countermass also ensures that the nodal points of the tube are constantly in position. The sample volume is set by the nodal points and therefore only the mass changes depending on the filled fluid while the volume remains stable.^[23]

In case of sensors made of glass a built-in reference oscillator eliminates not only long-term drifts due to the aging effects of the material but also temperature changes that influence the elasticity. A reference oscillator therefore means that only one single adjustment is used to cover the whole temperature range and provides the possibility to perform temperature scans of a sample.^[23] This is of great benefit if you want to measure the density at different temperatures without having to perform a density adjustment at each temperature. For users who measure at just one temperature or have time to perform a density adjustments at different temperatures, if required, there are density meters without a reference oscillator which also achieve excellent results at up to 4-digit accuracy.

Temperature regulation of the cell is typically performed with Peltier elements, which have now displaced water baths. This advantageous technology has now replaced the use of water baths. Peltier elements make use of the Peltier effect, a heat flux due to electric current: One side of a Peltier element heats up while the other side cools down depending on the direction of the current flow^[24] Peltier elements therefore allow both effective heating and cooling of the measuring cell and additionally provide precise and fast temperature regulation. 

Good Density Measurement™requires care and attention in five basic areas:

Consistently following a few simple guidelines will help you on your way to accurate and reproducible density results.

Since 1967 Anton Paar GmbH has specialized in providing highly accurate and reliable density meters for research and industry. This brochure sums up our experience and insights into measurement practice gained in over forty years. Follow these guidelines and you will be well on your way to accurate and reproducible density results.

View Good Density Measurement™ Guide

Calibrate your density meter on a regular basis in order to secure accurate results 

Various certified reference materials are available from different providers, so the right characteristics have to be considered first. The following points should be kept in mind in order to select the best suitable reference material:

• The reference density of the CRM should be as close to the regular sample densities as possible. Ideally, CRMs slightly above and below the regular sample densities should be chosen.
• The CRM’s composition should be comparable to the regular sample composition; this means that the nature of the sample should be similar to the CRM. This includes the viscosity or vapor pressures, but also whether it’s organic, aqueous, or alcoholic, etc.
• Reference method: We recommend choosing a density standard whose reference value was determined by means of hydrostatic weighing, with only a small measurement uncertainty. Generally speaking, the CRM should be specified 3x better than the instrument.
• Check the certificate for ISO 17034 or ISO 17025 to ensure traceability. Preferably choose ISO 17034-certified reference material, as the stated uncertainty is guaranteed until the end of expiry based on regular stability measurements.

Recalling the above-discussed attributes of density standards, care also needs to be taken when handling these liquids. This already starts with storage conditions and adherence to the date of expiration stated on the certificate. Immediately after opening the sealed CRM container, the liquid has to be injected into the density meter to start the density calibration. Therefore, the right syringe has to be chosen according to chemical compatibility with the CRM.

Calibration of a density meter by an officially accredited lab is the only way to be sure that measurements are traceable back to national standards, such as the International System of Units (SI), globally comparable, and true. See our article on ISO 17025 calibration as well. 

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8. cs-exhibitions.uni-klu.ac.at/index.php (last visited 11.12.2017)
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15. learn.kegerator.com/refractometers-vs-hydrometers/ (last visited 11.12.2017)
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17. H. Fehlauer and H. Wolf Density reference liquids certified by the Physikalisch-Techische Bundesanstalt Meas.Sci.Technol. 17 (2006) 2588–2592
18. www.britannica.com/science/Archimedes-principle (last visited 12.12.2017)
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21. W. Wagner and R. Kleinrahm Merologia 41 (2004) S. 24–39: Densimeters for very accurate density measurements of fluids over large ranges of temperature, pressure, and density
22. EN ISO 15212-1: 1999 Oscillation-type density meters – Part 1: Laboratory instruments
23. Fritz et al. Applications of densiometry, ulrsonic seed measurements, and ultra low shear viscosimetry to aqueaos fluids. J. of Phys.Chem. B Vol 104, No 15, 3463–3470, 2000
24. www.researchgate.net/publication/262179901_Peltier_Effect_in_Semiconductors (last visited 26.03.2018)
25. AT 516420 (B1)
26. Also called digital hydrometers