Anyone reading a calibration certificate for a weight does not see the word mass, but conventional mass. That is not meaningless jargon. It is an international convention that ensures that weights of different materials, measured in different laboratories and in varying weather, can be compared with each other.
The convention revolves around one physical phenomenon: the buoyancy of the air. In this article we explain what it is, how large the effect is and when it makes a difference in practice. The latter is mainly the case in the fine classes, such as the calibration of F1 and F2 weights.
What a scale actually measures
A scale does not measure mass but a force. That force is the gravitational force on the object, reduced by the buoyancy of the air that the object displaces. The principle is the same as for an object in water, except that air is more than eight hundred times lighter and the effect is therefore much smaller.
Small is not the same as negligible. An object with a large volume displaces a lot of air and therefore appears lighter than a compact object with the same mass. A litre of water with a mass of exactly 1 kg reads about 1.05 g less on a balance adjusted with steel weights. For weights, which are intended precisely for accurate measurements, such an effect therefore counts.
What is conventional mass?
To make this problem manageable, an international convention has been agreed and laid down in OIML R111. The conventional mass of a weight is the mass of an imaginary reference weight that exactly balances it in air, under these fixed conditions:
- the reference weight has a density of 8,000 kg/m³;
- the air has a density of 1.2 kg/m³;
- the temperature is 20 °C.
Conventional mass is therefore not the same as the true mass. For a stainless steel weight, with a density close to 8,000 kg/m³, the difference is very small. For materials with a different density it is larger. The advantage of the convention is that you do not have to think about it in daily use: a balance that has been adjusted with weights works in the same convention.
Air buoyancy in figures
A simple worked example makes it concrete. The buoyancy is equal to the weight of the displaced air. A 1 kg weight with a density of 8,000 kg/m³ has a volume of 125 cm³. Air with a density of 1.2 kg/m³ weighs 1.2 mg per cm³. The displaced air therefore has a mass of 125 × 1.2 = 150 mg.
| Material (1 kg) | Density, indicative | Volume | Displaced air at 1.2 kg/m³ |
|---|---|---|---|
| Reference according to R111 | 8,000 kg/m³ | 125.0 cm³ | 150 mg |
| Brass | 8,400 kg/m³ | 119.0 cm³ | approx. 143 mg |
| Cast iron | 7,200 kg/m³ | 138.9 cm³ | approx. 167 mg |
A cast iron weight with a true mass of exactly 1 kg therefore displaces about 17 mg more air than the reference. Its conventional mass is consequently about 17 mg lower than its true mass. The densities in the table are indicative; the actual density differs per alloy.
Why the actual air density matters
As long as the air weighs exactly 1.2 kg/m³, there is no problem: the conventional mass is then exactly what the comparison shows. But the air density changes with the weather and the temperature. At 20 °C, 1,013 hPa and 50 % relative humidity it is around 1.20 kg/m³. A high-pressure area or a cooler room gives a higher value, a low-pressure area or a warmer room a lower one. A 1 % change in air pressure gives roughly a 1 % change in air density.
When a laboratory compares two weights with different densities, that deviation affects the result. Suppose the air density is 1.15 kg/m³ instead of 1.2 kg/m³, a difference of 0.05 kg/m³. The volume difference between a 1 kg cast iron weight and the reference is 13.9 cm³. Without correction, an error of 13.9 cm³ × 0.05 mg/cm³ then arises, about 0.7 mg. For a stainless steel weight with a density of 7,950 kg/m³ the volume difference is only 0.8 cm³ and the error about 0.04 mg.
When does it become relevant?
Whether those fractions of milligrams matter depends on the class. Compare them with the maximum permissible error (mpe) for a 1 kg weight:
- M1: mpe 50 mg. An error of 0.7 mg is negligible.
- F1: mpe 5 mg, measurement uncertainty at most 1.7 mg. The effect is noticeable and must be included in the uncertainty budget.
- E1: mpe 0.5 mg, measurement uncertainty at most 0.17 mg. Here even the small difference between two stainless steel weights counts.
This is also why OIML R111 sets limits on the density of weights for each class. The finer the class, the closer the density must be to 8,000 kg/m³, so that a normal fluctuation in air density costs only a small part of the permissible error. Partly for that reason, and because of its porous surface, cast iron is only used in the M classes. More about materials in Cast iron or stainless steel weights. An overview of all classes can be found in OIML R111 weight classes.
How a laboratory corrects
A calibration laboratory compares a weight with a reference weight of the same nominal value on a mass comparator. To take air buoyancy into account, three steps are needed.
- Determine the air density. During the weighing, temperature, air pressure and relative humidity are measured. The actual air density is calculated from these using the formula of the CIPM, the International Committee for Weights and Measures.
- Know the volumes. The volume of the reference weight is known. For the weight to be calibrated, in the F and M classes the density of the material is usually assumed, with an uncertainty appropriate to the class.
- Apply the correction. The correction is approximately the difference between the actual air density and 1.2 kg/m³, multiplied by the volume difference between the two weights. The uncertainties of air density and volume are carried into the uncertainty budget.
It also helps to let weights acclimatise and to measure in a room with a stable temperature. Temperature differences cause air currents that disturb the measurement, independently of buoyancy. How the uncertainty eventually ends up on the certificate is explained in Measurement uncertainty on the calibration certificate of weights.
What does this mean for you?
As a user you do not have to calculate air density yourself. The certificate states the conventional mass, and that is exactly the value you need to check or calibrate a scale. Two points of attention remain. Use weights of a suitable material for fine balances, and let weights reach ambient temperature before you measure. If you work with an analytical balance and very small sample quantities, these effects also play a role in your own measurements; see Minimum weight on an analytical balance.
How MASSLAB helps you
MASSLAB calibrates weights in classes F1 to M3 in a climate-controlled room at 20 ±0.5 °C. Our own R111 software records temperature, air pressure and humidity during every weighing, calculates the air density and corrects for it. The result is the conventional mass with a complete uncertainty budget. For E1 and E2 weights we advise you on the appropriate calibration route. More about the working method for precision weights under calibrating F1 and F2 weights.