Simultaneous determination of mass and volume of a set of weights in group weighing


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 17 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 17 - 22 

 
 

SIMULTANEOUS DETERMINATION OF MASS AND VOLUME OF A SET 

OF WEIGHTS IN GROUP WEIGHING 
 

Ch. Wuethrich1, K. Marti2 

 
 1 Federal Institute of Metrology METAS, Switzerland, christian.wuethrich@metas.ch  
 2 Federal Institute of Metrology METAS, Switzerland, kilian.marti@metas.ch 

 

Abstract: 

This work introduces a new technique for the 

determination of mass and volume of a set of weights 

based on closed series (group weighing). A 

traditional closed series is repeated at two different 

air densities. A least squares set of equations, 

involving two Lagrange multipliers, is used to 

determine the mass and the volume of each weight 

simultaneously with a traceability on the mass and 

the volume of the reference weight. 

Keywords: double weighing; closed series; group 

weighing; volume determination 

1. INTRODUCTION 

The determination of mass in air is always 

strongly affected by air buoyancy. Traditionally, the 

volume of weights is determined by hydrostatic 

weighing and used for the correction of air buoyancy. 

Recently, several authors used double weighing in air 

of different density in order to determine the mass 

and the volume of an unknown artefact [1, 2]. The 

determination of the sub-multiples and the multiples 

of the kilogram still requires, in order to have an 

independent definition of the mass at NMI level, the 

volume determination through hydrostatic weighing 

[3]. We propose a technique of closed series in which 

the volume is an unknown of the system and is 

determined simultaneously with the mass by a set of 

double weighings in air of different densities. 

2. THEORETICAL FOUNDATIONS 

2.1. Traditional closed series 
A closed series is traditionally defined by a 

weighing scheme (design matrix) 𝑿 which is given as 
a combination of 0, -1 or +1 depending on whether or 

not the weight is involved in the comparison and 

whether the weight is considered to be a positive or a 

negative contribution to the weighing equation. The 

matrix 𝑿  has p columns, corresponding to the 
number of weights involved in the process and n rows, 

representing the number of weighing operations. It is 

possible to determine the mass values of the weights 

if p = n but it is better, in order to verify the 

consistency of the measurements, to have 

redundancy with n > p. This way, the determination 

of the mass values is realized by calculating the least 

squares of the overdetermined system of equation by 

applying the Lagrange multiplier method described 

by Kochsiek et al. [4]. This technique is widely 

spread in the determination of the conventional value 

of the mass. 

2.2. Closed series with known volumes and air 
buoyancy correction 

In order to determine the true mass, the matrix 𝑿 
is converted to �̂� which incorporates a correction for 
the air buoyancy of each weight at each air density. 

We define the elements of �̂� in the following way: 

�̂�𝑖,𝑗 = 𝑥𝑖,𝑗 (1−
𝜌𝑖
𝐷𝑗
)  (1) 

where 𝜌𝑖 is the air density during weighing i and 𝐷𝑗 

is the material density of weight j [5]. 

2.3. Closed series with unknown masses and 
unknown volumes 

In our innovative technique, the volumes of the 

weights are considered as unknowns of the system. 

We write the equation incorporating the buoyancy 

correction given by the unknown volumes. We repeat 

the weighing scheme 𝑿 and use it at two different 
nominal air densities. Each individual air density is 

defined by its value 𝜌1 to 𝜌2𝑛. We define the new 
matrix 𝒁 with 2p columns and 2n rows.  

In matrix form we can write 

(

𝑧1,1 ⋯ 𝑧1,𝑝
⋮ ⋱ ⋮
𝑧𝑛,1 ⋯ 𝑧𝑛,𝑝

) = (

𝑧𝑛+1,1 ⋯ 𝑧𝑛+1,𝑝
⋮ ⋱ ⋮

𝑧2𝑛,1 ⋯ 𝑧2𝑛,𝑝
) = 𝑋 (2) 

and 

(

𝑧1,𝑝+1 ⋯ 𝑧1,2𝑝
⋮ ⋱ ⋮

𝑧2𝑛,𝑝+1 ⋯ 𝑧2𝑛,2𝑝
) = ⋯ 

… = −(
𝜌1 ⋯ 0
⋮ ⋱ ⋮
0 ⋯ 𝜌2𝑛

) (

𝑧1,1 ⋯ 𝑧1,𝑝
⋮ ⋱ ⋮

𝑧2𝑛,1 ⋯ 𝑧2𝑛,𝑝
) 

(3) 

http://www.imeko.org/
mailto:christian.wuethrich@metas.ch
mailto:kilian.marti@metas.ch


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 18 

The unknown values of mass mi and volume vi of 

the weight i are expressed as 

�⃗� ∶= (𝑚1 …𝑚p𝑣1 …𝑣p)
𝑇
 (4) 

where m1 and v1 are the mass and volume of the 

reference weight. The weighing process is given by 

𝑍 ∙ �⃗� = �⃗� + 𝑒 (5) 

where �⃗� = (𝑦1 …𝑦2𝑛)
𝑇 is the weighing differences 

in the individual comparisons and 𝑒 = (𝑒1 …𝑒2𝑛)
𝑇 is 

the error associated with the weighing. The normal 

equation is given by  

𝑍𝑇𝑍 ∙ �⃗� = 𝑍𝑇�⃗� + 𝑍𝑇𝑒 (6) 

To solve this redundant system of equations, we 

apply the Lagrange multiplier method with two 

supplementary equations based on the multipliers 𝜆1 

and 𝜆2 to fix both the mass and the volume of the 

reference weight. We add two columns and two rows 

of zeros and ones to 𝒁𝑻𝒁 and 𝒁𝑻. The matrices and 

vectors become 

𝑍𝑇𝑍 = ⋯

=

(

 
 
 
 
 
 
 
 
 

𝜔1,1 … … … 𝜔1,𝑝 … … … 𝜔1,2𝑝 1 0

⋮ ⋮ ⋮ 0 0
⋮ ⋱ ⋮ ⋱ ⋮ ⋮ ⋮
⋮ ⋮ ⋮ ⋮ ⋮
𝜔𝑝,1 … … … 𝜔𝑝,𝑝 … … … 𝜔𝑝,2𝑝 0 0

⋮ ⋮ ⋮ 0 1
⋮ ⋱ ⋮ ⋱ ⋮ 0 0
⋮ ⋮ ⋮ ⋮ ⋮

𝜔2𝑝,1 … … … 𝜔2𝑝,𝑝 … … … 𝜔2𝑝,2𝑝 0 0

1 0 … … 0 0 0 … 0 0 0
0 0 … … 0 1 0 … 0 0 0)

 
 
 
 
 
 
 
 
 

 
(7) 

 

𝑍𝑇 =

(

 
 
 
 

𝜑1,1 … 𝜑1,𝑛 … 𝜑1,2𝑛 0 0

⋮ ⋱ ⋮ ⋱ ⋮ ⋮ ⋮
𝜑𝑝,1 … 𝜑𝑝,𝑛 … 𝜑𝑝,2𝑛 0 0

⋮ ⋱ ⋮ ⋱ ⋮ ⋮ ⋮
𝜑2𝑝,1 … 𝜑2𝑝,𝑛 … 𝜑2𝑝,2𝑛 0 0

0 … 0 … 0 1 0
0 … 0 … 0 0 1)

 
 
 
 

 (8) 

 

�⃗� = (𝑚1 …𝑚p𝑣1 …𝑣p 𝜆1 𝜆2)
𝑇
 (9) 

 

�⃗� = (𝑦1 …𝑦2𝑛 𝑚𝑟 𝑣𝑟)
𝑇 (10) 

The solution matrix is given by 

𝐿𝑆 ∶= (𝑍
𝑇𝑍)−1𝑍𝑇. (11) 

Finally, the estimated values for the mass and the volume 

are  

〈𝛾〉 = 𝐿𝑆�⃗�. (12) 

2.4. Monte Carlo Simulation 
We used a Monte Carlo simulation to estimate the 

measurement uncertainties of the mass and the 

volume of the weights. For this reason, we vary the 

input quantities which are the measured weighing 

difference from the balance, the air density, and the 

mass and the volume of the reference weight. 

Furthermore, we assumed the input quantities to be 

normally distributed 

𝑋~𝒩(𝜇,𝜎2). (13) 

For the mean value, µ, and the standard deviation, 

σ, we use the average value, �̅� , and the standard 
deviation, s, from the measurements. We assume no 

correlation between mass and volume of the 1 kg 

reference mass. After each variation, we re-calculate 

the mass and volume of the weights according to Eq. 

(12). Each re-calculation can be regarded as a random 

experiment, which is either real or, as in this case, 

numerically performed by a computer simulation. 

The number of re-calculations (iterations) is usually 

N = 10 000. The values for the mass and the volume 

of the weights as well as their standard uncertainties 

are given by the mean and standard deviation of the 

simulated values 

�̅�𝑗 =
1

𝑁𝐼𝑡𝑒𝑟
∑ 𝛾𝑗,𝑝

𝑁𝑖𝑡𝑒𝑟

𝑝=1

 (14) 

 

𝑠𝛾𝑗 =
√

1

𝑁𝐼𝑡𝑒𝑟 − 1
∑(𝛾𝑗,𝑝 − �̅�𝑗)

𝑁𝑖𝑡𝑒𝑟

𝑝=1

 (15) 

where j represents the weight and p is the index of the 

iteration. 

3. MEASUREMENT PROCEDURE 

We determined the mass and the volume of a set 

of OIML weights from 10 g to 5 kg through double 

weighing measurements in three different mass 

comparators (see section 4). The first double 

weighing was carried out at about 950 hPa 

(ρair = 1.13 kg/m
3), the second at about 750 hPa 

(ρair = 0.90 kg/m
3). The volumes of the weights were 

additionally measured by hydrostatic weighing in our 

volume comparators (see section 4). The weighing 

schemes to cover the selected range of weights are as 

follows. 
 

(

 
 
 
 
 
 
 
 

−1 0 0 0 0 1 0 0 0 0
−1 1 0 0 0 0 1 0 0 0
0 −1 0 0 0 0 1 0 0 0
0 −1 0 1 1 0 0 0 1 0
0 −1 1 0 0 0 0 1 0 1
0 0 0 −1 0 0 0 0 1 0
0 0 0 −1 0 0 0 1 0 0
0 0 −1 0 0 0 0 1 0 0
0 0 0 −1 1 0 0 0 0 1
0 0 −1 0 1 0 0 0 0 1
0 0 0 0 −1 0 0 0 0 1)

 
 
 
 
 
 
 
 

(

 
 
 
 
 
 
 
 

𝑚100g
𝑚50g
𝑚20∗g
𝑚20g
𝑚10g
𝑚100𝑔
′

𝑚50𝑔
′

𝑚20∗𝑔
′

𝑚20𝑔
′

𝑚10𝑔
′
)

 
 
 
 
 
 
 
 

 (16) 

 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 19 

(

 
 
 
 
 
 

−1 1 0 0 0 1 0 0 0
0 −1 0 0 0 1 0 0 0
0 −1 0 1 1 0 0 1 0
0 −1 1 0 0 0 1 0 1
0 0 0 −1 0 0 0 1 0
0 0 0 −1 0 0 1 0 0
0 0 1 −1 0 0 0 0 0
0 0 0 −1 1 0 0 0 1
0 0 0 0 −1 0 0 0 1)

 
 
 
 
 
 

(

 
 
 
 
 
 
 

𝑚1𝑘g
𝑚500g
𝑚200∗g
𝑚200g
𝑚100g
𝑚500g
′

𝑚200∗g
′

𝑚200g
′

𝑚100g
′

)

 
 
 
 
 
 
 

 (17) 

 

(

 
 
 
 
 
 
 

0 0 0 0 −1 0 0 0 0 1
1 1 1 0 −1 0 0 0 1 0
0 0 0 −1 0 0 0 0 1 0
1 0 1 −1 0 0 0 1 0 0
0 1 0 −1 0 1 1 0 0 0
0 0 −1 0 0 0 0 1 0 0
0 0 −1 0 0 0 1 0 0 0
0 1 −1 0 0 0 0 0 0 0
1 0 −1 0 0 1 0 0 0 0
−1 0 0 0 0 1 0 0 0 0)

 
 
 
 
 
 
 

(

 
 
 
 
 
 
 
 
 

𝑚1𝑘g
𝑚2∗kg
𝑚2kg
𝑚5kg
𝑚10kg
𝑚1kg
′

𝑚2∗kg
′

𝑚2kg
′

𝑚5kg
′

𝑚10kg
′

)

 
 
 
 
 
 
 
 
 

 (18) 

Auxiliary weights are denoted by m'. They are 

necessary to stack individual weights on each other 

so that their sum corresponds to the nominal mass of 

the reference weight.  

4. INSTRUMENT SETUP 

We describe the technical specifications of our 

instruments for the double weighing measurements 

and the hydrostatic measurements. 

4.1. Mass comparators with constant air density 
To perform the double weighing measurements in 

different air densities, we used different mass 

comparators from Mettler Toledo: an AT106, a Mone, 

and an AT10005 (Table 1). All comparators are 
enclosed in an air-tight chamber in which the air 

density can be changed by varying the air pressure 

(Figure 1). 

Table 1: Mass comparators with constant air density in the 

mass laboratory in METAS. Readability, repeatability, 

maximum load and weighing range of each comparator are 

shown 

 AT106 Mone AT10005 

Read. 1.0 µg 0.1 µg 10 µg 

Repeat. 1.5 µg 0.5 µg 20 µg 

Max load 111 g 1 001.5 g 10 011 g 

Range 5g – 100 g 100 g – 1 kg 1 kg – 10 kg 

 

To change the air pressure we use a membrane 

pump (813.3, KNF Neuberger). The pressure, 

temperature and humidity of the air inside the 

chamber are recorded during the measurements. The 

pressure is measured by a Druck DPI 141 Precision 

Pressure Indicator with a range of (800 - 1150) hPa 

and a resolution of 0.01 hPa; and a Fluke RPM4 with 

a range of (0 - 280) MPa and a resolution of 1 ppm. 

The temperature is determined with an accuracy of 

0.01° C by means of 10 kΩ normal resistors (Fluke), 

precision thermistors (YSI Inc.), and a digital 

multimeter (Keithley 2010). The humidity is 

measured by a hygrometer (Hygrolab 2, Rotronic AG) 

with a resolution of 1 %RH between (40 – 60) %RH. 

4.2. Volume comparators  

For the hydrostatic weighing, we used two 

different comparators from Mettler-Toledo: an 

AT1005 for the volume comparator VK1, and a 

PR10003 for the volume comparator VK10 (Table 2). 

The suspension is connected to the comparator and 

carries both the test weight and the reference weight 

(VK1) or the test weight only (VK10). The 

suspension is immersed in pure water (ELGA 

Labwater, PURELAB). By means of the well-known 

water density, the volume of the test weight is 

determined. Details can be found elsewhere [6]. 

Table 2: Volume comparators for hydrostatic weighing in 

the mass laboratory in METAS 

 AT1005 PR10003 

Read. 10 µg 1 mg 

Repeat. 20 µg 2 mg 

Max load 1009 g 10100 g 

Range ≤ 1 kg ≤ 10 kg 

5. VALIDATION WITH A SET OF WEIGHTS 

We measured a set of OIML weights ranging from 

10 g to 5 kg in double weighing measurements 

according to the weighing schemes in section 3. We 

calculated the masses and the volumes according to 

the methods presented in sections 2.3 and 2.4. 

The volumes of the weights obtained from the 

double weighing measurements and from the 

hydrostatic measurements are summarised in Table 3 

with the expanded measurement uncertainties in 

brackets. We observed that the measurement 

uncertainty of the volume depends on the nominal 

mass of the weight. Below 1 kg the uncertainty of the 

volume is smaller in the double weighing than in the 

hydrostatic weighing; above 1 kg the reverse is true.  

The relative measurement uncertainty of the 

volume depends on the nominal mass as well (Figure 

2). For the double weighing method the correlation 

between the relative measurement uncertainty and 

the nominal mass is negative below 1 kg (-0.60) and 

negative above 1 kg (-0.35). For the hydrostatic 

method the correlation is negative below 1 kg (-0.45) 

and negative above 1 kg (-0.31). Note that weights 

smaller than 1 kg were measured on VK1 whereas 

weights larger than 1 kg were measured on VK10, 

which explains the step in the relative measurement 

uncertainty above 1 kg. 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 20 

Table 3: Calculated volumes and expanded measurement 

uncertainties (k = 2) obtained from the double weighing 

and the hydrostatic weighing measurements 

Nominal Volume / cm³ 

Mass Double weighing Water 

10 g 1.2488(3) 1.2491(7) 

20 g 2.5005(4) 2.5010(8) 

20 g* 2.5004(3) 2.5004(2) 

50 g 6.2444(6) 6.2450(6) 

100 g 12.4889(5) 12.4903(8) 

200 g 24.9838(5) 24.9833(6) 

200 g* 24.9835(6) 24.9832(11) 

500 g 62.4473(11) 62.4460(14) 

1 kg 124.8739(20) 124.8739(20) 

2 kg 249.8089(207) 249.8062(96) 

2 kg* 249.8389(417) 249.8013(96) 

5 kg 624.2566(615) 624.2107(106) 

 

 

Figure 2: Expanded relative uncertainty of the volume of 

the weights obtained from double weighing and 

hydrostatic weighing measurements 

For the hydrostatic weighing, the true (or 

conventional) mass of a weight is required to 

calculate the weight's volume. Therefore, we had 

determined the mass of each weight on our 

comparators beforehand in a direct comparison to a 

reference weight. The results of the true mass 

deviation from the nominal mass value are 

summarised in Table 4. Additionally, in order to 

check the goodness of the equation matrices at the 

two pressures, the true mass deviations from the 

nominal mass values have also been calculated, from 

the first (950 hPa) and from the second (750 hPa) 

double weighing measurements and are shown in 

Table 4 as well. The expanded measurement 

uncertainties are given in brackets and were 

calculated by using equation (15) on the basis of a 

Monte Carlo simulation. All measurement 

uncertainties refer to the uncertainty of mass and 

volume of the 1 kg reference weight (15-0-1kg), 

which were U(m) = 17 µg (k = 2) and U(V) = 2 mm³ 

(k = 2). Note that these uncertainties were established 

before the introduction of the new definition of the 

kilogram. 

Table 4: Calculated true mass deviations from nominal 

mass values and expanded measurement uncertainties 

(k = 2) obtained from the 1st and 2nd double weighing 

(DbW) and from the hydrostatic weighing 

Nominal 

Mass 

Mass deviation from nominal mass 

/ mg 

 1st DbW 2nd DbW previous 

hydrostatic 

weighing 

10 g 0.015(1) 0.015(1) 0.014(3) 

20 g 0.016(1) 0.016(1) 0.004(3) 

20 g* 0.040(1) 0.040(1) 0.032(2) 

50 g 0.011(1) 0.011(1) -0.004(4) 

100 g 0.052(2) 0.052(2) 0.031(2) 

200 g 0.072(3) 0.072(3) 0.049(12) 

200 g* 0.040(4) 0.040(3) 0.023(12) 

500 g 0.088(9) 0.088(9) 0.045(38) 

1 kg 0.117(17) 0.117(17) 0.048(13) 

2 kg 0.591(43) 0.587(42) 0.435(600) 

2 kg* -0.145(58) -0.148(57) -0.340(600) 

5 kg 1.960(110) 1.952(108) 1.568(1500) 

 

6

8
10

2

4

6

8
100

2

4

6

8
1000

U
(V

)/
V

 (
p

p
m

)

1
5

-0
-1

0
g

1
5

-0
-2

0
g

1
5

-1
-2

0
g

1
5

-0
-5

0
g

1
5

-0
-1

0
0

g

1
5

-0
-2

0
0

g

1
5

-1
-2

0
0

g

1
5

-0
-5

0
0

g

1
5

-0
-1

k
g

1
5

-0
-2

k
g

1
5

-1
-2

k
g

1
5

-0
-5

k
g

 Air
 Water

Figure 1: Mass comparators in air-tight enclosures in METAS 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 21 

On the basis of the Monte Carlo simulation we 

analysed the correlation between the estimated values 

of the mass and the volume. For this reason, we 

calculated the standardised form (z-score),  

𝑍 = (𝑋 − 𝜇)/𝜎, of the simulated values, X, of the mass 

and the volume, respectively. 

𝑍𝑚𝑎𝑠𝑠,𝑗(𝑝) =
𝑚𝑗(𝑝)− �̅�𝑗

𝑠𝑚,𝑗
 

𝑍𝑣𝑜𝑙𝑢𝑚𝑒,𝑗(𝑝) =
𝑉𝑗(𝑝) −�̅�𝑗

𝑠𝑉,𝑗
 

(19) 

Figure 3 shows the standardised variables of the 

mass and the volume calculated by a Monte Carlo 

simulation for the weights from 10 g to 5 kg. We 

observed that the correlation between the estimated 

values for mass and volume depends on the nominal 

mass of the weight. Hence, we calculated the 

correlation coefficient  

𝐶𝑜𝑟(𝑥,𝑦) =
∑(𝑥 − �̅�)(𝑦 − �̅�)

√∑(𝑥 − �̅�)2(𝑦 − �̅�)2
 (20) 

between 𝑍𝑚𝑎𝑠𝑠,𝑗  and 𝑍𝑣𝑜𝑙𝑢𝑚𝑒,𝑗 . The correlation 

coefficient is the smallest for the 1 kg weight and 

increases as the nominal mass decreases. Above 1 kg, 

the correlation coefficient is significantly larger and 

seems to be independent of the nominal mass (Table 

5). 

Table 5: Correlation between the error on mass and density 

for the different values of mass 

Nominal 

Mass 

𝑪𝒐𝒓(𝒙,𝒚) Nominal 
Mass 

𝑪𝒐𝒓(𝒙,𝒚) 

10 g 0.83 200 g* 0.07 

20 g 0.74 500 g 0.01 

20 g* 0.66 1 kg 0.00 

50 g 0.52 2 kg 0.54 

100 g 0.09 2 kg* 0.78 

200 g 0.04 5 kg 0.58 

 

To compare the calculated volumes from the 

double weighing and from the hydrostatic weighing 

measurements with each other, we calculated the 

degree of equivalence (DoE) according to Cox [7]. 

The DoE is in good agreement between the two 

methods in the range from 10 g to 5 kg, except for the 

100 g weight (Figure 4). 

 

Figure 4: Degree of equivalence of the calculated volumes 

of the weights between the two measurement methods 

double weighing and hydrostatic weighing. Error bars 

represent expanded uncertainty k = 2. Top: 10 g to 1 kg on 

VK1; Bottom: 2 kg to 5 kg on VK10 

6. DISCUSSION 

The double weighing method offers several 

advantages over hydrostatic weighing: the 

maintenance is less complex, it is less time 

consuming and is easier to perform the measurements, 

-0.002

-0.001

0.000

0.001

0.002

D
e

g
re

e
 o

f 
e

q
u

iv
a

le
n

c
e

 (
c
m

³)

1
5

-0
-1

0
g

1
5

-0
-2

0
g

1
5

-1
-2

0
g

1
5

-0
-5

0
g

1
5

-0
-1

0
0
g

1
5

-0
-2

0
0

g

1
5

-1
-2

0
0
g

1
5

-0
-5

0
0

g

1
5

-0
-1

k
g

VK1

 Air
 Water

-0.08

-0.04

0.00

0.04

0.08

D
e

g
re

e
 o

f 
e

q
u

iv
a

le
n

c
e

 (
c
m

³)

1
5

-0
-2

k
g

1
5

-1
-2

k
g

1
5

-0
-5

k
g

VK10 Air
 Water

Figure 3: Standardised variables of the calculated values for the mass and the volume of the test weights. The values 

come from a Monte-Carlo simulation with N = 10 000 iterations 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 22 

the calculations are simpler, and it does not 

contaminate the surface of the weights. The achieved 

measurement uncertainties are similar to those of the 

hydrostatic weighing, and even better for small 

weights. The discrepancy in the calculated volume of 

the 100 g weight between the double weighing and 

the hydrostatic weighing remains unclear but may be 

related to an error made at the time of hydrostatic 

weighing. The volume value obtained from the 

double weighing measurements was confirmed by 

repeating the double weighing measurements on the 

MonePlus comparator, which is similar to the Mone 

but can also be used in vacuum. The difference 

between the two measured volumes for the 100 g 

weight was only 10 ppm and was thus well within the 

measurement uncertainty (not shown in Figure 4).  

The pairs of weights with the same nominal mass, 

i.e. 20 g, 200 g and 2 kg, do not have the same 

measurement uncertainties on their respective 

volumes. The reason for this is the following: the 

dispersion of the measured weighing values in the 

hydrostatic weighing of 15-0-20g and 15-1-200g is 

larger than the dispersion of the measured weighing 

values of their counterparts 15-1-20g and 15-0-200g, 

respectively. The difference between the volume 

uncertainties of 15-0-2kg and 15-1-2kg in the double 

weighing measurements is more difficult to find out 

because these uncertainties originate from a closed 

series of measurements (group weighing). 

On the basis of the minimum and maximum limits 

for density of class E1 weights [8], we calculated the 

equivalent values for volume (Table 6). 

Table 6: Minimum and maximum limits for density and 

volume of class E1 weights according to OIML R 111-1 

Nominal 

Mass 

Density 

/ kg·m-³ 

Volume 

/ cm³ 

 min max min max 

10 g 7740 8280 1.208 1.292 

20 g 7840 8170 2.448 2.551 

20 g* 7840 8170 2.448 2.551 

50 g 7920 8080 6.188 6.313 

100 g 

7934 8067 

12.396 12.604 

200 g 24.792 25.208 

200 g* 24.792 25.208 

500 g 61.981 63.020 

1 kg 123.962 126.040 

2 kg 247.924 252.080 

2 kg* 247.924 252.080 

5 kg 619.809 630.199 

 

The results in Table 3 show that the volumes of 

our weights, determined by double weighing and 

hydrostatic weighing, are all within the minimum and 

maximum limits given by OIML R 111-1. 

7. SUMMARY 

We are able to achieve the full dissemination of 

multiples and submultiples of a set of weights by 

using group weighings in two different air densities. 

The measurements are traceable to the mass and to 

the volume of the reference weight. The technique is 

promising as there is no contamination from a liquid, 

and it enables measuring the full range of mass values 

covered by the constant air density comparators of a 

laboratory.  This technique is well suited for the 

dissemination of the mass and the volume based on 

the new definition of the kilogram using a silicon 

sphere whose mass and volume are known with high 

precision. 

8. REFERENCES 

[1] M. T. Clarkson, R. S. Davis, C. M. Sutton, J. Coarasa, 
“Determination of volumes of mass standards by 

weighings in air”, Metrologia, vol. 38, no. 1, pp. 17-

24, 2001. 

[2] A. Malengo, W. Bich, “Simultaneous determination 
of mass and volume of a standard by weighings in 

air”, Metrologia, vol. 49, no. 3, pp. 289-293, 2012. 

[3] K. H. Chang, Y. J. Lee, “Hydrostatic weighing at 
KRISS”, Metrologia, vol. 41, no. 2, pp. S95–S99, 

March 2004. 

[4] M. Kochsiek, M. Gläser, Massebestimmung. VCH 
Verlagsgesellschaft mbH, ISBN 3-527-29352-3, 

1997. 

[5] K. Marti, “Dissemination of the kilogram at METAS: 
Extended Method of Lagrange Multipliers for Air 

Buoyancy Correction”, in Proc. of IMEKO 23rd TC3 

Int. Conf., Helsinki, Finland, 2017. 

[6] Y. Su, K. Marti, Ch. Wuethrich, “The determination 
of the volume of weights in the range of 1 g – 5 kg: a 

comparison of hydrostatic weighing and double 

weighing in air using the Monte Carlo Simulation” 

ACTA IMEKO vol. 9, no. 1, pp. 61-68, March 2020. 

[7] M. G. Cox, “The evaluation of key comparison data”, 
Metrologia, vol. 39, pp. 589-595, 2002. 

[8] OIML R 111-1, “Weights of classes E1, E2, F1, F2, M1, 
M1–2, M2, M2–3 and M3 - Part 1: Metrological and 

technical requirements”, OIML TC 9/SC 3, 2004. 

 
 

http://www.imeko.org/