# Calculating Air Density

## Why is Air Density Important?

Any object suspended in a fluid is subject to two opposing vertical forces, gravity is pulling it downwards, whilst fluid buoyancy is pushing it upwards.

If the object is suspended from a mass balance the net result of these forces determines the reading on the scale. People are aware of Archimedes’ principle from an early age and recognise that an object appears lighter when it is in water than in air, but very few are aware that air also imparts a buoyancy force on objects due to the same effect.

This means that whenever an object is weighing in air the reading on the balance will indicate a value less than if it were weighed in vacuum (the hypothetical reading in vacuum would be equivalent to the object’s mass). Therefore for precise weighing it is necessary to correct for this buoyancy force that is proportional to the volume of the object being weighed and the density of the air. The volume of the object is generally obtained by hydrostatic weighing while the measurement of air density and the subsequent application of buoyancy corrections are considered here.

### Air Buoyancy Corrections

Once air density has been calculated, it is possible to calculate the air buoyancy effect on an object if its volume is known. In the mass and weighing community a definition of conventional mass has been established for many years. It was introduced to minimise the effect of changes in buoyancy in the mass calibration of objects. This definition assumes the object being considered is manufactured from steel with a density of exactly 8000 kg/m³, and is being weighed in air of density 1.2 kg/m³. Virtually every mass calibration certificate produced utilises this convention. Before looking any further at air buoyancy corrections, it is necessary to consider the formal definitions of mass and conventional mass

### The Application of Buoyancy Corrections

The table below shows the magnitude of the buoyancy correction when comparing weights of stainless steel with those of another material in air of standard density (1.2 kg/m³) on a true mass basis.

 Material Compared with Stainless Steel Buoyancy Correction (ppm) Platinum Iridium 94 Tungsten 88 Brass 8 Stainless Steel 7.5* Cast Iron 24 Aluminium 294 Silicon 365 Water 875

* This is the result of comparing two types of stainless steel, with densities 7.8 and 8.2 g/cm³

The table shows that even when comparing weights of nominally the same material (such as stainless steel) attention must be paid to buoyancy effects when the best uncertainty is required. When comparing weights of dissimilar materials the effect of air buoyancy becomes more significant and must be applied even for routine calibrations when true mass values are being measured.

When comparing weights in air the corrections become smaller, being equal to the differences of the corrections listed in Table 1. Working on a conventional mass basis, the buoyancy corrections become smaller still, since they depend on the difference in air density from a standard value of 1.2 kg/m³.

The OIML recommendations R 33 use a range for air density of 1.1 to 1.3 kg/m³ (i.e. approximately ± 10% of standard air density), meaning the corrections are about one tenth of the true mass corrections. This, together with the limits specified by OIML R 33 for the density of weights of Classes E1 to M3, means that the maximum correction for any weight is one quarter of its tolerance. This is generally not significant for weights of Class F1 and below (although allowance should be made for the uncertainty contribution of the unapplied correction): but for Class E1 and E2 weights, buoyancy corrections must be applied to achieve the necessary uncertainty values.

## Mass

### True Mass

The mass of a body relates to the amount of material of which it consists. In terms of the calibration of weight, it is referred to as true mass in order to differentiate it from conventional mass, which is generally used to specify the value of weights (see below). A simple way to conceive the concept is to think of it as the amount an object would weigh if it were to be weighed in vacuum (i.e. with no air buoyancy).

The international prototype of the kilogram, from which the mass scale throughout the world is realised, is defined as a true mass of exactly 1 kilogram. All high accuracy comparisons should be performed on a true mass basis (including class E1 calibrations), although values are usually converted to conventional mass when quoted on a certificate.

When measuring the density of an artefact by hydrostatic weighing, the true mass of the artefact and any weights used in air (corrected for air buoyancy) should be used.

### Conventional Mass

This is the value normally quoted on a certificate and is the conventional value of the result of a weighing in air, in accordance with International Recommendation OIML R 33.

For a weight at 20°C, the conventional mass is the mass of a reference weight of density 8000 kg/m³, which it balances in air of a density 1.2 kg/m³. The equations needed to convert between the two types of mass are given below:

$$M_c = M \times (1 + ((\frac{1}{8000} - \frac{1}{p}) \times 1.2))$$

$$M = M_c \times (1 + ((\frac{1}{p} - \frac{1}{8000}) \times 1.2))$$

where:

• M is the (true) Mass value
• Mc is the conventional mass value
• ρ is the density of the object in kg/m³
• 8000 is the conventional material density in kg/m³
• 1.2 is the conventional air density in kg/m³

## References

1. Davis R. S., Equation for the Determination of the Density of Moist Air (1981/91), Metrologia, 1992, 29, 67-70.
2. Giacomo P., Equation for the Determination of the Density of Moist Air (1981), Metrologia, 1982, 18, 33-40.
3. Wexler A., Vapour Pressure Formulation for Water in the Range 0 to 100°C. A revision. J. Res. Nat. Bur. Stand. 1976, 80A, 775-785.
4. Greenspan L., Functional Equations for the Enhancement Factors for CO2-free Moist Air. J. Res. Nat. Bur. Stand. 1976, 80A, 41-44.
5. Good Practice Guidance Note, NPL, Buoyancy Correction and Air Density Measurement.