Partial differentiation of air density in mass metrology


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 33 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 33 - 36 

 
 

PARTIAL DIFFERENTIATION OF AIR DENSITY IN MASS METROLOGY 
 

M. L. Win1, T. Sanponpute2, B. Suktat3 

 
1  National Institute of Metrology (Myanmar), Yangon, Myanmar, marlarwin99@gmail.com  

2  National Institute of Metrology (Thailand), Pathumthani, Thailand, tassanai@nimt.or.th  
3  Consultant, Pathumthani, Thailand, bunjobs@hotmail.com    

 

Abstract: 
There are four major uncertainty components to 

be considered when performing mass comparisons. 

They are uncertainties of weighing process, 

reference weight used, air buoyancy, and mass 

comparator. The systematic effect of air buoyancy 

can be greatly reduced if the air density and the 

densities of the test and reference weights are 

known. This paper will emphasis on the uncertainty 

due to air buoyancy correction only. To calculate 

the uncertainty of air density correction, partial 

derivatives of temperature, barometric pressure and 

humidity must be performed. In this paper, two 

methods for partial differentiation of air density 

components are discussed. 

Keywords: air buoyancy; air density; partial 

differentiation of air density; August-Roche-

Magnus approximation equation 

1. INTRODUCTION 

In Mass Metrology, there are many important 

influence quantities have to be taken into account. 

These include thermal stabilisation time, magnetic 

susceptibility, density of standard and test weights, 

environmental conditions, weighing instrument, 

weighing cycles etc. According to OIML R111-1[1, 

equation C.6.5-1], the combined standard 

uncertainty for mass comparison process is 

calculated using equation (1). 

𝑢c(𝑚ct) = √𝑢w
2 (∆𝑚c̅̅ ̅̅ ̅̅ ) + 𝑢

2(𝑚cr) + 𝑢b
2 + 𝑢ba

2  (1) 

where: 

𝑢c(𝑚ct) = combined standard uncertainty of the 
conventional mass of the test weight 

𝑢w(Δ𝑚c̅̅ ̅̅ ̅̅ ) = uncertainty of the weighing process 
𝑢(𝑚cr) = uncertainty of the reference weight  
𝑢b = uncertainty of the air buoyancy 

correction  

𝑢ba
 

= uncertainty of the balance 

 

2. THE SYSTEMATIC EFFECTS OF AIR 
DENSITY 

When highly accurate mass comparisons are 

performed under the laboratory conditions, buoyant 

force that acts upon weights, depending on their 

volume and the air density, can cause significant 

systematic error to the measurement results. The 

systematic effects of air buoyancy can be corrected 

if the air density is known. The air density is a 

function of various influence quantities such as 

temperature, barometric pressure and humidity. To 

correct for the effect of air density, OIML R111-1 

provides the following equation [1, Eq. 10.2-1]: 

𝑚ct = 𝑚cr(1 + 𝐶) + ∆𝑚c̅̅ ̅̅  (2) 

with [1, Eq. 10.2-2]:  

𝐶 = (𝜌a −  𝜌0) [
𝜌t − 𝜌r

𝜌r𝜌t
] (3) 

where:  

𝑚ct = conventional mass of the test weight 
𝑚cr     = conventional mass of the reference weight 
∆𝑚c̅̅ ̅̅  = average weighing difference observed 

between test and reference weight 

𝐶 = correction factor for air buoyancy 
𝜌a = density of moist air 
𝜌0 = density of air as a reference value equal to 

1.2 kg·m-3 
𝜌t = density of the test weight 
𝜌r = density of the reference weight 
 

The uncertainty of air buoyancy correction is 

calculated according to OIML R111-1, as follows 

[1, equation C.6.3-1]: 

𝑢b
2 = [𝑚cr

𝜌r − 𝜌t
𝜌r𝜌t

𝑢(𝜌a)]
2

+ [𝑚cr(𝜌a − 𝜌0)]
2

𝑢2(𝜌t)

𝜌t
4

 

+ 𝑚cr
2 (𝜌a − 𝜌0)[(𝜌a − 𝜌0) − 2(𝜌al − 𝜌0)]

𝑢2(𝜌r)

𝜌r
4

 

(4) 

where: 

𝑢b = uncertainty of the air buoyancy correction 
𝑚cr = conventional mass of the reference weight 
𝜌r = density of the reference weight 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 34 

𝜌t = density of the test weight 
𝑢(𝜌a) = uncertainty of density of moist air 
𝜌a = density of moist air 
𝜌0 = density of the air as a reference value 

equal to 1.2 kg·m-3 
𝑢(𝜌t) = uncertainty of density of the test weight 
𝜌al = air density during the (previous) 

calibration of the reference weight  

𝑢(𝜌r) = uncertainty of density of reference weight 

3. UNCERTAINTY OF AIR DENSITY 

For air density 𝜌a , OIML R111 provides an 
approximation formula [1, equation E.3-1] given as 

equation (5). 

𝜌a =  
0.34848 𝑝 − 0.009 ℎ𝑟 × exp (0.061 𝑡)

273.15 + 𝑡
 (5) 

where: 

𝜌a = air density, in kg·m
-3 

𝑝 = air pressure, in mbar or hPa 
ℎ𝑟 = relative humidity, in % 
𝑡 = air temperature, in C 

Equation (5) has a relative uncertainty of 2 × 10-4 

in the ranges 900 hPa < 𝑝  < 1100 hPa, 
10 °C < 𝑡 < 30 °C and ℎ𝑟 < 80 %. Under reference 
conditions of 𝑝  = 1013.25 hPa, 𝑡  = 20 °C, and 
ℎ𝑟  = 50 %, equation (5) gives an air density of 
1.199 294 kg·m-3. 

The variance of air density is obtained from 

equation (6) [1, equation C.6.3-3]. 

𝑢2(𝜌a) = 𝑢F
2 + [

𝜕𝜌a
𝜕𝑝

𝑢𝑝]
2

+  [
𝜕𝜌a
𝜕𝑡

𝑢𝑡 ]
2

+ [
𝜕𝜌a
𝜕ℎ𝑟

𝑢ℎ𝑟 ]
2

 

(6) 

Under the same reference conditions, the 

following numerical values for sensitivity 

coefficients are given: 

• 𝑢F = [uncertainty of the formula used] (for 
CIPM formula 𝑢F = 10

–4 𝜌a ) 

• 
𝜕𝜌a

𝜕𝑝
=  10−5𝜌a Pa

−1 

• 
𝜕𝜌a

𝜕𝑡
=  −3.4 × 10−3𝜌a K

−1 

• 
𝜕𝜌a

𝜕ℎ𝑟
= −10−2𝜌a 

However, if we calculate the three sensitivity 

coefficients by partial differentiation of 

equation (5), we obtain: 

𝜕𝜌a
𝜕𝑝

=
0.34848

273.15 + 𝑡
 𝜌a 

(7) 

=
0.34848

293.15
 𝜌a = 10

−5 𝜌a Pa
−1 

𝜕𝜌a
𝜕ℎ𝑟

=
−0.009 × exp (0.061𝑡)

273.15 + 𝑡
𝜌a 

(8) 

=
−0.009 × exp (0.061 × 20)

273.15 + 20
𝜌a = −10

−2 𝜌a 

𝜕𝜌a
𝜕𝑡

=  
− 0.34848 𝑝

(273.15 + 𝑡)2
− 

(9) 

0.009ℎ𝑟 [(273.15 + 𝑡)
𝑑 𝑒 0.061𝑡

𝑑𝑡
 − 𝑒0.061𝑡

 𝑑(273.15+𝑡)

𝑑𝑡
]

(273.15 + 𝑡)2
 𝜌a 

=  
− 0.34848 𝑝

(273.15 + 𝑡)2
− 

0.009(50)𝑒 0.061𝑡[293.15 × 0.061 − 1]

(273.15 + 𝑡)2
 𝜌a 

=  
− 0.34848 × 1013.15

(273.15 + 20)2
− 

0.009(50)𝑒 0.061×20[293.15 × 0.061 − 1]

(273.15 + 20)2
 𝜌a 

=  −4.4 × 10−3𝜌a K
−1 

From the above calculations, we found that 

sensitivity coefficient 
𝜕𝜌a

𝜕𝑝
 and 

𝜕𝜌a

𝜕ℎ𝑟
 values are 

exactly the same as OIML R111 values, but not the 

value of  
 𝜕𝜌a

𝜕𝑡
. Therefore, we try to consider how to 

obtain the same 
 𝜕𝜌a

𝜕𝑡
 value as provided by OIML 

R111. 

According to the August-Roche-Magnus 

approximation equation [4, equation 4], 

𝑅𝐻 = 100 ×  
𝑒 

17.625 𝑇𝐷

243.04 + 𝑇𝐷

𝑒 
17.625 𝑡

243.04 + 𝑡

 (10) 

The changes in temperature 𝑡 will also cause the 
relative humidity 𝑅𝐻  to change, if the dew point 
temperature 𝑇𝐷  is kept constant. In this case, we 
need to consider not only the changes in temperature 

but also the correlation between temperature and 

relative humidity according to August-Roche-

Magnus approximation equation. In this equation, 

there are three variables, namely temperature 𝑡 , 
relative humidity 𝑅𝐻  and dew point temperature 
𝑇𝐷. 

The dew point temperature can be calculated by 

using August-Roche-Magnus approximation 

equation [4, equation 5] as follows: 

𝑇𝐷   = 243.04
ln (

𝑅𝐻

100
) + (

17.625 𝑡

243.04+𝑡
)

(17.625 − ln (
𝑅𝐻

100
) −

17.625 𝑡

(243.04+𝑡)
)
 (11) 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 35 

Substituting in values of 𝑅𝐻 = 50 %, 𝑡 = 20 °C 
and 𝑝 = 101 325 Pa, we get: 

𝑇𝐷 = 9.261 ℃ (12) 

If we keep 𝑇𝐷  constant at 9.261 °C while the 
temperature changes from; 𝑡 = 20 °C to 𝑡 = 21 °C, 
from equation (11), we get: 

For 𝑡 = 20 C: 

𝑅𝐻 = 100 ×  
𝑒

17.625×9.261

243.04+9.261

𝑒
17.625×20

243.04+20

= 50 % (13) 

For 𝑡 = 21 C: 

𝑅𝐻 = 100 ×  
𝑒

17.625×9.261

243.04+9.261

𝑒
17.625×21

243.04+21

= 47 % (14) 

From the above examples, the change of 

temperature from 20 C to 21 C can cause the 

relative humidity to change from 50 %RH to 

47 %RH. Thus the air density also changes to: 

𝜌a1 =  
0.34848 × 1013.25 − 0.009(47)e0.061×21

273.15 + 21
 
(15) 

= 1.195 221 kg ∙ m−3 

And thus, from equations (5) and (15): 

𝜕𝜌𝑎
𝜕𝑡

=  
𝜌𝑎1 − 𝜌0

𝜌0
=  

1.195221 − 1.199294

1.199294
 

(16) 

=  −3.4 × 10−3𝜌𝑎 K
−1 

This value of 
𝜕𝜌𝑎

𝜕𝑡
 is now exactly the same as the 

value provided in OIML R111. Therefore, we need 

to consider the correlation between temperature and 

humidity when we perform partial differentiation of 

temperature against the air density. 

Note: In the August-Roche-Magnus 

approximation equation, there are three variables, 

namely temperature 𝑡 , relative humidity 𝑅𝐻  and 
dew point temperature 𝑇𝐷. For our purposes, dew 
point temperature 𝑇𝐷 is considered constant while 
the temperature 𝑡  is changing. In this case any 
changes in temperature will also cause the relative 

humidity to change in the opposite direction. 

Otherwise, we would not get the same value of  
 𝜕𝜌𝑎

𝜕𝑡
 

as that provided in OIML R111. 

4. DISCUSSION 

Relative humidity is the ratio, usually expressed 

in %, of the partial pressure of water vapour to the 

saturation (equilibrium) vapour pressure of water at 

a given temperature.  

Relative humidity depends on the temperature 

and pressure of the system of interest. As the air’s 

temperature increases, it can hold more water 

molecules, decreasing its relative humidity. When 

temperatures drop, relative humidity increases. This 

is because colder air does not require as much 

moisture to become saturated as warmer air. 100 % 

relative humidity of the air occurs when the air 

temperature is the same as the dew point value. 

Table 1 shows the relationship between 

temperature and relative humidity. The slope and 

correlation between these two variables are shown 

in Figure 1. In this figure, it shows the correlation 

coefficient close to -1.0. 

Correlation coefficient is used to measure the 

strength of the relationship between two variables. 

The range of values for correlation coefficient 

is -1.0 to 1.0. In other words, the values cannot 

exceed 1.0 nor be less than -1.0. 

Table 1: Relationship between temperature and 

relative humidity 

Temperature 

/ °C 

Relative Humidity 

/ %RH 

16 64.3 

17 60.3 

18 56.6 

19 53.2 

20 50.0 

21 47.0 

22 44.2 

23 41.6 

24 39.2 

 

 

Figure 1: Correlation between temperature and 

relative humidity 

5. SUMMARY 

In mass comparison uncertainty evaluation, we 

need to consider all contributions that are of 

significant. Among them, one of the important 

uncertainty sources is air buoyancy correction. For 

air buoyancy correction, air density is the important 

parameter. Air density depends on the 

environmental conditions such as temperature, 

barometric pressure and humidity. The uncertainty 

of air density 𝑢(𝜌a) is calculated from the standard 
uncertainty of air pressure 𝑢(𝑝), temperature 𝑢(𝑡), 
and relative humidity 𝑢(𝑟ℎ) , together with their 

y = -3,128x + 113,28
R² = 0,9938

30

40

50

60

70

80

15 16 17 18 19 20 21 22 23 24 25

RH
/ %

Temperature / °C 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 36 

sensitivity coefficients, the partial derivatives of 
𝜕𝜌a

𝜕𝑝
, 

𝜕𝜌a

𝜕𝑡
, and 

𝜕𝜌a

𝜕ℎ𝑟
. However, when we use the 

ordinary partial differentiation, the derived value of 
𝜕𝜌a

𝜕𝑡
 is not same as the value given in OIML R111. 

So, we need to consider the correlation between 

temperature and humidity when we perform partial 

differentiation of temperature against the air 

density. 

We use August-Roche-Magnus approximation 

equation to determine the correlation between 

temperature and relative humidity. Finally, we can 

obtain the same value for 
𝜕𝜌a

𝜕𝑡
 as given by 

OIML R111. 

6. ACKNOWLEDGEMENT 

The authors would like to thank PTB 

(Physikalisch-Technische Bundesanstalt) and its 

international cooperation project “Strengthening the 

Quality Infrastructure in Myanmar”, which supports 

and enables us to make this publication possible. 

7. REFERENCES 

[1] OIML R111 – Weights of classes E1, E2, F1, F2, M1, 
M2, M3 Metrological and Technical Requirement, 

2004. 

[2] A. Picard, R. S. Davis, M. Gläser, K. Fujii, 
“Revised formula for the density of moist air 

(CIPM2007)”, Metrologia, vol. 45, 2008, pp. 149-

155. DOI:  

http://dx.doi.org/10.1088/0026-1394/45/2/004  

[3] Guide to the Expression of Uncertainty in 
Measurement, ISO, 1995. 

[4] Thomas K. Thiis, Ingunn Burud, Andreas Flø, 
Dimitrios Kraniotis, Stergiani Charisi, Petter 

Stefansson, Monitoring and Simulation of Diurnal 

Surface Conditions of a Wooden Façade, Procedia 

Environmental Sciences 38, 217, pp. 331-339.  

DOI: https://doi.org/10.1016/j.proenv.2017.03.088  

 

 

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https://doi.org/10.1016/j.proenv.2017.03.088